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HREF="node8.html">Measures of Amplitude</A>
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HREF="node10.html">Controlling Amplitude</A>
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HREF="node19.html">Constant amplitude scaler</A>
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HREF="node67.html">Encapsulation in Pd</A>
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HREF="node69.html">ADSR envelope generator</A>
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HREF="node70.html">Transfer functions for amplitude control</A>
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HREF="node71.html">Additive synthesis: Risset's bell</A>
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HREF="node72.html">Additive synthesis: spectral envelope control</A>
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HREF="node73.html">Polyphonic synthesis: sampler</A>
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HREF="node76.html">Taxonomy of spectra</A>
|
HREF="node76.html">Taxonomy of spectra</A>
|
||||||
<LI><A NAME="tex2html287"
|
<LI><A ID="tex2html287"
|
||||||
HREF="node77.html">Multiplying audio signals</A>
|
HREF="node77.html">Multiplying audio signals</A>
|
||||||
<LI><A NAME="tex2html288"
|
<LI><A ID="tex2html288"
|
||||||
HREF="node78.html">Waveshaping</A>
|
HREF="node78.html">Waveshaping</A>
|
||||||
<LI><A NAME="tex2html289"
|
<LI><A ID="tex2html289"
|
||||||
HREF="node79.html">Frequency and phase modulation</A>
|
HREF="node79.html">Frequency and phase modulation</A>
|
||||||
<LI><A NAME="tex2html290"
|
<LI><A ID="tex2html290"
|
||||||
HREF="node80.html">Examples</A>
|
HREF="node80.html">Examples</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html291"
|
<LI><A ID="tex2html291"
|
||||||
HREF="node81.html">Ring modulation and spectra</A>
|
HREF="node81.html">Ring modulation and spectra</A>
|
||||||
<LI><A NAME="tex2html292"
|
<LI><A ID="tex2html292"
|
||||||
HREF="node82.html">Octave divider and formant adder</A>
|
HREF="node82.html">Octave divider and formant adder</A>
|
||||||
<LI><A NAME="tex2html293"
|
<LI><A ID="tex2html293"
|
||||||
HREF="node83.html">Waveshaping and difference tones</A>
|
HREF="node83.html">Waveshaping and difference tones</A>
|
||||||
<LI><A NAME="tex2html294"
|
<LI><A ID="tex2html294"
|
||||||
HREF="node84.html">Waveshaping using Chebychev polynomials</A>
|
HREF="node84.html">Waveshaping using Chebychev polynomials</A>
|
||||||
<LI><A NAME="tex2html295"
|
<LI><A ID="tex2html295"
|
||||||
HREF="node85.html">Waveshaping using an exponential function</A>
|
HREF="node85.html">Waveshaping using an exponential function</A>
|
||||||
<LI><A NAME="tex2html296"
|
<LI><A ID="tex2html296"
|
||||||
HREF="node86.html">Sinusoidal waveshaping: evenness and oddness</A>
|
HREF="node86.html">Sinusoidal waveshaping: evenness and oddness</A>
|
||||||
<LI><A NAME="tex2html297"
|
<LI><A ID="tex2html297"
|
||||||
HREF="node87.html">Phase modulation and FM</A>
|
HREF="node87.html">Phase modulation and FM</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html298"
|
<LI><A ID="tex2html298"
|
||||||
HREF="node88.html">Exercises</A>
|
HREF="node88.html">Exercises</A>
|
||||||
</UL>
|
</UL>
|
||||||
<BR>
|
<BR>
|
||||||
<LI><A NAME="tex2html299"
|
<LI><A ID="tex2html299"
|
||||||
HREF="node89.html">Designer spectra</A>
|
HREF="node89.html">Designer spectra</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html300"
|
<LI><A ID="tex2html300"
|
||||||
HREF="node90.html">Carrier/modulator model</A>
|
HREF="node90.html">Carrier/modulator model</A>
|
||||||
<LI><A NAME="tex2html301"
|
<LI><A ID="tex2html301"
|
||||||
HREF="node91.html">Pulse trains</A>
|
HREF="node91.html">Pulse trains</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html302"
|
<LI><A ID="tex2html302"
|
||||||
HREF="node92.html">Pulse trains via waveshaping</A>
|
HREF="node92.html">Pulse trains via waveshaping</A>
|
||||||
<LI><A NAME="tex2html303"
|
<LI><A ID="tex2html303"
|
||||||
HREF="node93.html">Pulse trains via wavetable stretching</A>
|
HREF="node93.html">Pulse trains via wavetable stretching</A>
|
||||||
<LI><A NAME="tex2html304"
|
<LI><A ID="tex2html304"
|
||||||
HREF="node94.html">Resulting spectra</A>
|
HREF="node94.html">Resulting spectra</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html305"
|
<LI><A ID="tex2html305"
|
||||||
HREF="node95.html">Movable ring modulation</A>
|
HREF="node95.html">Movable ring modulation</A>
|
||||||
<LI><A NAME="tex2html306"
|
<LI><A ID="tex2html306"
|
||||||
HREF="node96.html">Phase-aligned formant (PAF) generator</A>
|
HREF="node96.html">Phase-aligned formant (PAF) generator</A>
|
||||||
<LI><A NAME="tex2html307"
|
<LI><A ID="tex2html307"
|
||||||
HREF="node97.html">Examples</A>
|
HREF="node97.html">Examples</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html308"
|
<LI><A ID="tex2html308"
|
||||||
HREF="node98.html">Wavetable pulse train</A>
|
HREF="node98.html">Wavetable pulse train</A>
|
||||||
<LI><A NAME="tex2html309"
|
<LI><A ID="tex2html309"
|
||||||
HREF="node99.html">Simple formant generator</A>
|
HREF="node99.html">Simple formant generator</A>
|
||||||
<LI><A NAME="tex2html310"
|
<LI><A ID="tex2html310"
|
||||||
HREF="node100.html">Two-cosine carrier signal</A>
|
HREF="node100.html">Two-cosine carrier signal</A>
|
||||||
<LI><A NAME="tex2html311"
|
<LI><A ID="tex2html311"
|
||||||
HREF="node101.html">The PAF generator</A>
|
HREF="node101.html">The PAF generator</A>
|
||||||
<LI><A NAME="tex2html312"
|
<LI><A ID="tex2html312"
|
||||||
HREF="node102.html">Stretched wavetables</A>
|
HREF="node102.html">Stretched wavetables</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html313"
|
<LI><A ID="tex2html313"
|
||||||
HREF="node103.html">Exercises</A>
|
HREF="node103.html">Exercises</A>
|
||||||
</UL>
|
</UL>
|
||||||
<BR>
|
<BR>
|
||||||
<LI><A NAME="tex2html314"
|
<LI><A ID="tex2html314"
|
||||||
HREF="node104.html">Time shifts and delays</A>
|
HREF="node104.html">Time shifts and delays</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html315"
|
<LI><A ID="tex2html315"
|
||||||
HREF="node105.html">Complex numbers</A>
|
HREF="node105.html">Complex numbers</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html316"
|
<LI><A ID="tex2html316"
|
||||||
HREF="node106.html">Complex sinusoids</A>
|
HREF="node106.html">Complex sinusoids</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html317"
|
<LI><A ID="tex2html317"
|
||||||
HREF="node107.html">Time shifts and phase changes</A>
|
HREF="node107.html">Time shifts and phase changes</A>
|
||||||
<LI><A NAME="tex2html318"
|
<LI><A ID="tex2html318"
|
||||||
HREF="node108.html">Delay networks</A>
|
HREF="node108.html">Delay networks</A>
|
||||||
<LI><A NAME="tex2html319"
|
<LI><A ID="tex2html319"
|
||||||
HREF="node109.html">Recirculating delay networks</A>
|
HREF="node109.html">Recirculating delay networks</A>
|
||||||
<LI><A NAME="tex2html320"
|
<LI><A ID="tex2html320"
|
||||||
HREF="node110.html">Power conservation and complex delay networks</A>
|
HREF="node110.html">Power conservation and complex delay networks</A>
|
||||||
<LI><A NAME="tex2html321"
|
<LI><A ID="tex2html321"
|
||||||
HREF="node111.html">Artificial reverberation</A>
|
HREF="node111.html">Artificial reverberation</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html322"
|
<LI><A ID="tex2html322"
|
||||||
HREF="node112.html">Controlling reverberators</A>
|
HREF="node112.html">Controlling reverberators</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html323"
|
<LI><A ID="tex2html323"
|
||||||
HREF="node113.html">Variable and fractional shifts</A>
|
HREF="node113.html">Variable and fractional shifts</A>
|
||||||
<LI><A NAME="tex2html324"
|
<LI><A ID="tex2html324"
|
||||||
HREF="node114.html">Fidelity of interpolating delay lines</A>
|
HREF="node114.html">Fidelity of interpolating delay lines</A>
|
||||||
<LI><A NAME="tex2html325"
|
<LI><A ID="tex2html325"
|
||||||
HREF="node115.html">Pitch shifting</A>
|
HREF="node115.html">Pitch shifting</A>
|
||||||
<LI><A NAME="tex2html326"
|
<LI><A ID="tex2html326"
|
||||||
HREF="node116.html">Examples</A>
|
HREF="node116.html">Examples</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html327"
|
<LI><A ID="tex2html327"
|
||||||
HREF="node117.html">Fixed, noninterpolating delay line</A>
|
HREF="node117.html">Fixed, noninterpolating delay line</A>
|
||||||
<LI><A NAME="tex2html328"
|
<LI><A ID="tex2html328"
|
||||||
HREF="node118.html">Recirculating comb filter</A>
|
HREF="node118.html">Recirculating comb filter</A>
|
||||||
<LI><A NAME="tex2html329"
|
<LI><A ID="tex2html329"
|
||||||
HREF="node119.html">Variable delay line</A>
|
HREF="node119.html">Variable delay line</A>
|
||||||
<LI><A NAME="tex2html330"
|
<LI><A ID="tex2html330"
|
||||||
HREF="node120.html">Order of execution and lower limits on delay times</A>
|
HREF="node120.html">Order of execution and lower limits on delay times</A>
|
||||||
<LI><A NAME="tex2html331"
|
<LI><A ID="tex2html331"
|
||||||
HREF="node121.html">Order of execution in non-recirculating delay lines</A>
|
HREF="node121.html">Order of execution in non-recirculating delay lines</A>
|
||||||
<LI><A NAME="tex2html332"
|
<LI><A ID="tex2html332"
|
||||||
HREF="node122.html">Non-recirculating comb filter as octave doubler</A>
|
HREF="node122.html">Non-recirculating comb filter as octave doubler</A>
|
||||||
<LI><A NAME="tex2html333"
|
<LI><A ID="tex2html333"
|
||||||
HREF="node123.html">Time-varying complex comb filter: shakers</A>
|
HREF="node123.html">Time-varying complex comb filter: shakers</A>
|
||||||
<LI><A NAME="tex2html334"
|
<LI><A ID="tex2html334"
|
||||||
HREF="node124.html">Reverberator</A>
|
HREF="node124.html">Reverberator</A>
|
||||||
<LI><A NAME="tex2html335"
|
<LI><A ID="tex2html335"
|
||||||
HREF="node125.html">Pitch shifter</A>
|
HREF="node125.html">Pitch shifter</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html336"
|
<LI><A ID="tex2html336"
|
||||||
HREF="node126.html">Exercises</A>
|
HREF="node126.html">Exercises</A>
|
||||||
</UL>
|
</UL>
|
||||||
<BR>
|
<BR>
|
||||||
<LI><A NAME="tex2html337"
|
<LI><A ID="tex2html337"
|
||||||
HREF="node127.html">Filters</A>
|
HREF="node127.html">Filters</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html338"
|
<LI><A ID="tex2html338"
|
||||||
HREF="node128.html">Taxonomy of filters</A>
|
HREF="node128.html">Taxonomy of filters</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html339"
|
<LI><A ID="tex2html339"
|
||||||
HREF="node129.html">Low-pass and high-pass filters</A>
|
HREF="node129.html">Low-pass and high-pass filters</A>
|
||||||
<LI><A NAME="tex2html340"
|
<LI><A ID="tex2html340"
|
||||||
HREF="node130.html">Band-pass and stop-band filters</A>
|
HREF="node130.html">Band-pass and stop-band filters</A>
|
||||||
<LI><A NAME="tex2html341"
|
<LI><A ID="tex2html341"
|
||||||
HREF="node131.html">Equalizing filters</A>
|
HREF="node131.html">Equalizing filters</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html342"
|
<LI><A ID="tex2html342"
|
||||||
HREF="node132.html">Elementary filters</A>
|
HREF="node132.html">Elementary filters</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html343"
|
<LI><A ID="tex2html343"
|
||||||
HREF="node133.html">Elementary non-recirculating filter</A>
|
HREF="node133.html">Elementary non-recirculating filter</A>
|
||||||
<LI><A NAME="tex2html344"
|
<LI><A ID="tex2html344"
|
||||||
HREF="node134.html">Non-recirculating filter, second form</A>
|
HREF="node134.html">Non-recirculating filter, second form</A>
|
||||||
<LI><A NAME="tex2html345"
|
<LI><A ID="tex2html345"
|
||||||
HREF="node135.html">Elementary recirculating filter</A>
|
HREF="node135.html">Elementary recirculating filter</A>
|
||||||
<LI><A NAME="tex2html346"
|
<LI><A ID="tex2html346"
|
||||||
HREF="node136.html">Compound filters</A>
|
HREF="node136.html">Compound filters</A>
|
||||||
<LI><A NAME="tex2html347"
|
<LI><A ID="tex2html347"
|
||||||
HREF="node137.html">Real outputs from complex filters</A>
|
HREF="node137.html">Real outputs from complex filters</A>
|
||||||
<LI><A NAME="tex2html348"
|
<LI><A ID="tex2html348"
|
||||||
HREF="node138.html">Two recirculating filters for the price of one</A>
|
HREF="node138.html">Two recirculating filters for the price of one</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html349"
|
<LI><A ID="tex2html349"
|
||||||
HREF="node139.html">Designing filters</A>
|
HREF="node139.html">Designing filters</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html350"
|
<LI><A ID="tex2html350"
|
||||||
HREF="node140.html">One-pole low-pass filter</A>
|
HREF="node140.html">One-pole low-pass filter</A>
|
||||||
<LI><A NAME="tex2html351"
|
<LI><A ID="tex2html351"
|
||||||
HREF="node141.html">One-pole, one-zero high-pass filter</A>
|
HREF="node141.html">One-pole, one-zero high-pass filter</A>
|
||||||
<LI><A NAME="tex2html352"
|
<LI><A ID="tex2html352"
|
||||||
HREF="node142.html">Shelving filter</A>
|
HREF="node142.html">Shelving filter</A>
|
||||||
<LI><A NAME="tex2html353"
|
<LI><A ID="tex2html353"
|
||||||
HREF="node143.html">Band-pass filter</A>
|
HREF="node143.html">Band-pass filter</A>
|
||||||
<LI><A NAME="tex2html354"
|
<LI><A ID="tex2html354"
|
||||||
HREF="node144.html">Peaking and stop-band filter</A>
|
HREF="node144.html">Peaking and stop-band filter</A>
|
||||||
<LI><A NAME="tex2html355"
|
<LI><A ID="tex2html355"
|
||||||
HREF="node145.html">Butterworth filters</A>
|
HREF="node145.html">Butterworth filters</A>
|
||||||
<LI><A NAME="tex2html356"
|
<LI><A ID="tex2html356"
|
||||||
HREF="node146.html">Stretching the unit circle with rational functions</A>
|
HREF="node146.html">Stretching the unit circle with rational functions</A>
|
||||||
<LI><A NAME="tex2html357"
|
<LI><A ID="tex2html357"
|
||||||
HREF="node147.html">Butterworth band-pass filter</A>
|
HREF="node147.html">Butterworth band-pass filter</A>
|
||||||
<LI><A NAME="tex2html358"
|
<LI><A ID="tex2html358"
|
||||||
HREF="node148.html">Time-varying coefficients</A>
|
HREF="node148.html">Time-varying coefficients</A>
|
||||||
<LI><A NAME="tex2html359"
|
<LI><A ID="tex2html359"
|
||||||
HREF="node149.html">Impulse responses of recirculating filters</A>
|
HREF="node149.html">Impulse responses of recirculating filters</A>
|
||||||
<LI><A NAME="tex2html360"
|
<LI><A ID="tex2html360"
|
||||||
HREF="node150.html">All-pass filters</A>
|
HREF="node150.html">All-pass filters</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html361"
|
<LI><A ID="tex2html361"
|
||||||
HREF="node151.html">Applications</A>
|
HREF="node151.html">Applications</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html362"
|
<LI><A ID="tex2html362"
|
||||||
HREF="node152.html">Subtractive synthesis</A>
|
HREF="node152.html">Subtractive synthesis</A>
|
||||||
<LI><A NAME="tex2html363"
|
<LI><A ID="tex2html363"
|
||||||
HREF="node153.html">Envelope following</A>
|
HREF="node153.html">Envelope following</A>
|
||||||
<LI><A NAME="tex2html364"
|
<LI><A ID="tex2html364"
|
||||||
HREF="node154.html">Single Sideband Modulation</A>
|
HREF="node154.html">Single Sideband Modulation</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html365"
|
<LI><A ID="tex2html365"
|
||||||
HREF="node155.html">Examples</A>
|
HREF="node155.html">Examples</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html366"
|
<LI><A ID="tex2html366"
|
||||||
HREF="node156.html">Prefabricated low-, high-, and band-pass filters</A>
|
HREF="node156.html">Prefabricated low-, high-, and band-pass filters</A>
|
||||||
<LI><A NAME="tex2html367"
|
<LI><A ID="tex2html367"
|
||||||
HREF="node157.html">Prefabricated time-varying band-pass filter</A>
|
HREF="node157.html">Prefabricated time-varying band-pass filter</A>
|
||||||
<LI><A NAME="tex2html368"
|
<LI><A ID="tex2html368"
|
||||||
HREF="node158.html">Envelope followers</A>
|
HREF="node158.html">Envelope followers</A>
|
||||||
<LI><A NAME="tex2html369"
|
<LI><A ID="tex2html369"
|
||||||
HREF="node159.html">Single sideband modulation</A>
|
HREF="node159.html">Single sideband modulation</A>
|
||||||
<LI><A NAME="tex2html370"
|
<LI><A ID="tex2html370"
|
||||||
HREF="node160.html">Using elementary filters directly: shelving and peaking</A>
|
HREF="node160.html">Using elementary filters directly: shelving and peaking</A>
|
||||||
<LI><A NAME="tex2html371"
|
<LI><A ID="tex2html371"
|
||||||
HREF="node161.html">Making and using all-pass filters</A>
|
HREF="node161.html">Making and using all-pass filters</A>
|
||||||
</UL>
|
</UL>
|
||||||
<LI><A NAME="tex2html372"
|
<LI><A ID="tex2html372"
|
||||||
HREF="node162.html">Exercises</A>
|
HREF="node162.html">Exercises</A>
|
||||||
</UL>
|
</UL>
|
||||||
<BR>
|
<BR>
|
||||||
<LI><A NAME="tex2html373"
|
<LI><A ID="tex2html373"
|
||||||
HREF="node163.html">Fourier analysis and resynthesis</A>
|
HREF="node163.html">Fourier analysis and resynthesis</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html374"
|
<LI><A ID="tex2html374"
|
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<H1><A NAME="SECTION00530000000000000000">
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Controlling Amplitude</A>
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Controlling Amplitude</A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.4:</STRONG>
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HREF="node99.html">Simple formant generator</A>
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HREF="node99.html">Simple formant generator</A>
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001053000000000000000">
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<H2><A ID="SECTION001053000000000000000">
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||||||
Two-cosine carrier signal</A>
|
Two-cosine carrier signal</A>
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</H2>
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</H2>
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@@ -86,7 +86,7 @@ and <IMG
|
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<P>
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||||||
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||||||
<DIV ALIGN="CENTER"><A NAME="fig06.16"></A><A NAME="6994"></A>
|
<DIV ALIGN="CENTER"><A ID="fig06.16"></A><A ID="6994"></A>
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.17:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.17:</STRONG>
|
||||||
Cross-fading between sinusoids to make movable
|
Cross-fading between sinusoids to make movable
|
||||||
@@ -169,36 +169,36 @@ never avoid getting phase cancellations where they overlap.
|
|||||||
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HREF="node99.html">Simple formant generator</A>
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HREF="node99.html">Simple formant generator</A>
|
||||||
<B> <A NAME="tex2html2039"
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<B> <A ID="tex2html2039"
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<ADDRESS>
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+24
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@@ -30,48 +30,48 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node97.html">Examples</A>
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HREF="node100.html">Two-cosine carrier signal</A>
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HREF="node100.html">Two-cosine carrier signal</A>
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html2055"
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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||||||
<H2><A NAME="SECTION001054000000000000000">
|
<H2><A ID="SECTION001054000000000000000">
|
||||||
The PAF generator</A>
|
The PAF generator</A>
|
||||||
</H2>
|
</H2>
|
||||||
|
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<P>
|
<P>
|
||||||
|
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||||||
<DIV ALIGN="CENTER"><A NAME="fig06.17"></A><A NAME="7005"></A>
|
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|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.18:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.18:</STRONG>
|
||||||
The phase-aligned formant (PAF) synthesis algorithm.</CAPTION>
|
The phase-aligned formant (PAF) synthesis algorithm.</CAPTION>
|
||||||
@@ -126,7 +126,7 @@ control object is needed:
|
|||||||
|
|
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<P>
|
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|
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||||||
<DIV ALIGN="CENTER"><A NAME="fig06.18"></A><A NAME="7047"></A>
|
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.19:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.19:</STRONG>
|
||||||
Filling in the wavetable for Figure <A HREF="#fig06.17">6.18</A>.</CAPTION>
|
Filling in the wavetable for Figure <A HREF="#fig06.17">6.18</A>.</CAPTION>
|
||||||
@@ -145,7 +145,7 @@ Filling in the wavetable for Figure <A HREF="#fig06.17">6.18</A>.</CAPTION>
|
|||||||
WIDTH="57" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="57" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img623.png"
|
SRC="img623.png"
|
||||||
ALT="\fbox{ $\mathrm{until}$\ }"> :
|
ALT="\fbox{ $\mathrm{until}$\ }"> :
|
||||||
<A NAME="7054"></A>When the left, "start" inlet is banged, output sequential bangs (with no
|
<A ID="7054"></A>When the left, "start" inlet is banged, output sequential bangs (with no
|
||||||
elapsed time between them) iteratively, until the right, "stop" inlet is
|
elapsed time between them) iteratively, until the right, "stop" inlet is
|
||||||
banged. The stopping "bang" message must originate somehow from the
|
banged. The stopping "bang" message must originate somehow from the
|
||||||
<TT>until</TT> object's outlet; otherwise, the outlet will send "bang" messages
|
<TT>until</TT> object's outlet; otherwise, the outlet will send "bang" messages
|
||||||
@@ -231,36 +231,36 @@ the fundamental dropping, not rising, in amplitude as the string decays.
|
|||||||
<P>
|
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<HR>
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<HR>
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HREF="node100.html">Two-cosine carrier signal</A>
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|
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+11
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@@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION001055000000000000000">
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Stretched wavetables</A>
|
Stretched wavetables</A>
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</H2>
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</H2>
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+11
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@@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION001060000000000000000">
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Exercises</A>
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Exercises</A>
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+46
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HREF="node4.html">Contents</A></B>
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<H1><A NAME="SECTION001100000000000000000"></A>
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Time shifts and delays
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Time shifts and delays
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@@ -83,7 +83,7 @@ But now suppose you played it at 5:00 and 5:00:01 on the same day (on two
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different playback systems, since the music lasts much longer than one second).
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different playback systems, since the music lasts much longer than one second).
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Now the sound is much different. The difference, whatever it is,
|
Now the sound is much different. The difference, whatever it is,
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clearly resides in neither of the two individual sounds, but rather in the
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clearly resides in neither of the two individual sounds, but rather in the
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<A NAME="7751"></A><I>interference</I>
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<A ID="7751"></A><I>interference</I>
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between the two. This interference can be perceived in at least four different
|
between the two. This interference can be perceived in at least four different
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ways:
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ways:
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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HREF="node105.html">Complex numbers</A>
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<LI><A NAME="tex2html2097"
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|
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|
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HREF="node109.html">Recirculating delay networks</A>
|
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<LI><A NAME="tex2html2101"
|
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|
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HREF="node111.html">Artificial reverberation</A>
|
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|
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|
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||||||
<LI><A NAME="tex2html2107"
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|
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<LI><A NAME="tex2html2109"
|
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<LI><A NAME="tex2html2113"
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HREF="node122.html">Non-recirculating comb filter as octave doubler</A>
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HREF="node123.html">Time-varying complex comb filter: shakers</A>
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HREF="node124.html">Reverberator</A>
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HREF="node125.html">Pitch shifter</A>
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<LI><A NAME="tex2html2117"
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HREF="node126.html">Exercises</A>
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+32
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@@ -30,50 +30,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<B> <A NAME="tex2html2127"
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<H1><A NAME="SECTION001110000000000000000"></A>
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<H1><A ID="SECTION001110000000000000000"></A>
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||||||
<A NAME="sect7.complex"></A>
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<A ID="sect7.complex"></A>
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<BR>
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<BR>
|
||||||
Complex numbers
|
Complex numbers
|
||||||
</H1>
|
</H1>
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
Complex
|
Complex
|
||||||
<A NAME="7761"></A>numbers are written as:
|
<A ID="7761"></A>numbers are written as:
|
||||||
<BR><P></P>
|
<BR><P></P>
|
||||||
<DIV ALIGN="CENTER">
|
<DIV ALIGN="CENTER">
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||||||
<!-- MATH
|
<!-- MATH
|
||||||
@@ -122,9 +122,9 @@ line) to graph it, as shown in Figure <A HREF="#fig07.01">7.1</A>. The quantiti
|
|||||||
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img21.png"
|
SRC="img21.png"
|
||||||
ALT="$b$"> are called the
|
ALT="$b$"> are called the
|
||||||
<A NAME="7764"></A><I>real</I>
|
<A ID="7764"></A><I>real</I>
|
||||||
and
|
and
|
||||||
<A NAME="7766"></A><I>imaginary parts</I> of <IMG
|
<A ID="7766"></A><I>imaginary parts</I> of <IMG
|
||||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img20.png"
|
SRC="img20.png"
|
||||||
ALT="$Z$">, written as:
|
ALT="$Z$">, written as:
|
||||||
@@ -165,7 +165,7 @@ b = \mathrm{im}(Z)
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.01"></A><A NAME="7772"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.01"></A><A ID="7772"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.1:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.1:</STRONG>
|
||||||
A number, <IMG
|
A number, <IMG
|
||||||
@@ -191,7 +191,7 @@ If <IMG
|
|||||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img20.png"
|
SRC="img20.png"
|
||||||
ALT="$Z$"> is a complex number, its
|
ALT="$Z$"> is a complex number, its
|
||||||
<A NAME="7775"></A><A NAME="7776"></A><I>magnitude</I> (or <I>absolute value</I>),
|
<A ID="7775"></A><A ID="7776"></A><I>magnitude</I> (or <I>absolute value</I>),
|
||||||
written as <IMG
|
written as <IMG
|
||||||
WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img629.png"
|
SRC="img629.png"
|
||||||
@@ -218,7 +218,7 @@ point <IMG
|
|||||||
<BR CLEAR="ALL">
|
<BR CLEAR="ALL">
|
||||||
<P></P>
|
<P></P>
|
||||||
and its
|
and its
|
||||||
<A NAME="7781"></A><I>argument</I>,
|
<A ID="7781"></A><I>argument</I>,
|
||||||
written as <IMG
|
written as <IMG
|
||||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img632.png"
|
SRC="img632.png"
|
||||||
@@ -306,8 +306,8 @@ A complex number may be written in terms of its real and imaginary parts
|
|||||||
WIDTH="76" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="76" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img637.png"
|
SRC="img637.png"
|
||||||
ALT="$Z=a+bi$"> (this is called
|
ALT="$Z=a+bi$"> (this is called
|
||||||
<A NAME="7785"></A><I>rectangular form</I>), or alternatively in
|
<A ID="7785"></A><I>rectangular form</I>), or alternatively in
|
||||||
<A NAME="7787"></A><I>polar form</I>,
|
<A ID="7787"></A><I>polar form</I>,
|
||||||
in terms of <IMG
|
in terms of <IMG
|
||||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img467.png"
|
SRC="img467.png"
|
||||||
@@ -511,45 +511,45 @@ imaginary parts.
|
|||||||
<P>
|
<P>
|
||||||
<BR><HR>
|
<BR><HR>
|
||||||
<!--Table of Child-Links-->
|
<!--Table of Child-Links-->
|
||||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||||
|
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html2132"
|
<LI><A ID="tex2html2132"
|
||||||
HREF="node106.html">Complex sinusoids</A>
|
HREF="node106.html">Complex sinusoids</A>
|
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<ADDRESS>
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+23
-23
@@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001111000000000000000">
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<H2><A ID="SECTION001111000000000000000">
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Complex sinusoids</A>
|
Complex sinusoids</A>
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</H2>
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</H2>
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||||||
|
|
||||||
@@ -104,7 +104,7 @@ arguments and magnitudes.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.02"></A><A NAME="7829"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.02"></A><A ID="7829"></A>
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||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.2:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.2:</STRONG>
|
||||||
The powers of a complex number <IMG
|
The powers of a complex number <IMG
|
||||||
@@ -360,7 +360,7 @@ sinusoid of this form, <IMG
|
|||||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img23.png"
|
SRC="img23.png"
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ALT="$A$"> is called the
|
ALT="$A$"> is called the
|
||||||
<A NAME="7846"></A>
|
<A ID="7846"></A>
|
||||||
<I>complex amplitude</I>.
|
<I>complex amplitude</I>.
|
||||||
|
|
||||||
<P>
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<P>
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||||||
@@ -426,36 +426,36 @@ end of whatever we're doing.
|
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+23
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@@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION001120000000000000000"></A>
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<A NAME="sect7.phase"></A>
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<BR>
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<BR>
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Time shifts and phase changes
|
Time shifts and phase changes
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</H1>
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@@ -152,7 +152,7 @@ further property that, if you time shift a sinusoid of frequency <IMG
|
|||||||
result is another sinusoid of the same frequency; time shifting never
|
result is another sinusoid of the same frequency; time shifting never
|
||||||
introduces frequencies that weren't present in the signal before it was
|
introduces frequencies that weren't present in the signal before it was
|
||||||
shifted. This property, called
|
shifted. This property, called
|
||||||
<A NAME="7855"></A><I>time invariance</I>,
|
<A ID="7855"></A><I>time invariance</I>,
|
||||||
makes it easy to analyze the effects of time shifts--and linear combinations
|
makes it easy to analyze the effects of time shifts--and linear combinations
|
||||||
of them--by considering separately what the operations do on individual
|
of them--by considering separately what the operations do on individual
|
||||||
sinusoids.
|
sinusoids.
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@@ -234,36 +234,36 @@ since the sinusoid advances <IMG
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+35
-35
@@ -30,50 +30,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION001130000000000000000"></A>
|
<H1><A ID="SECTION001130000000000000000"></A>
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||||||
<A NAME="sect7.network"></A>
|
<A ID="sect7.network"></A>
|
||||||
<BR>
|
<BR>
|
||||||
Delay networks
|
Delay networks
|
||||||
</H1>
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</H1>
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||||||
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<P>
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||||||
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||||||
<DIV ALIGN="CENTER"><A NAME="fig07.03"></A><A NAME="7867"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.03"></A><A ID="7867"></A>
|
||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.3:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.3:</STRONG>
|
||||||
A delay network. Here we add the incoming signal to a delayed
|
A delay network. Here we add the incoming signal to a delayed
|
||||||
@@ -95,7 +95,7 @@ successive moments in time, then time shifting the signal by <IMG
|
|||||||
SRC="img28.png"
|
SRC="img28.png"
|
||||||
ALT="$d$"> samples
|
ALT="$d$"> samples
|
||||||
corresponds to a
|
corresponds to a
|
||||||
<A NAME="7870"></A>
|
<A ID="7870"></A>
|
||||||
<I>delay</I>
|
<I>delay</I>
|
||||||
of <IMG
|
of <IMG
|
||||||
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||||
@@ -105,7 +105,7 @@ of <IMG
|
|||||||
SRC="img36.png"
|
SRC="img36.png"
|
||||||
ALT="$R$"> is the sample rate.
|
ALT="$R$"> is the sample rate.
|
||||||
Figure <A HREF="#fig07.03">7.3</A> shows one example of a
|
Figure <A HREF="#fig07.03">7.3</A> shows one example of a
|
||||||
<A NAME="7873"></A><I>linear delay network</I>:
|
<A ID="7873"></A><I>linear delay network</I>:
|
||||||
an assembly of delay units, possibly with amplitude
|
an assembly of delay units, possibly with amplitude
|
||||||
scaling operations, combined using addition and subtraction. The output
|
scaling operations, combined using addition and subtraction. The output
|
||||||
is a linear function of the input, in the sense that adding two signals at the
|
is a linear function of the input, in the sense that adding two signals at the
|
||||||
@@ -117,13 +117,13 @@ so that the gains and delay times do not change with time.
|
|||||||
<P>
|
<P>
|
||||||
In general there are two ways of thinking about delay networks. We can think
|
In general there are two ways of thinking about delay networks. We can think
|
||||||
in the
|
in the
|
||||||
<A NAME="7875"></A><I>time domain</I>,
|
<A ID="7875"></A><I>time domain</I>,
|
||||||
in which we draw waveforms as functions of time (or of the index <IMG
|
in which we draw waveforms as functions of time (or of the index <IMG
|
||||||
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img75.png"
|
SRC="img75.png"
|
||||||
ALT="$n$">), and
|
ALT="$n$">), and
|
||||||
consider delays as time shifts. Alternatively we may think in the
|
consider delays as time shifts. Alternatively we may think in the
|
||||||
<A NAME="7877"></A><I>frequency domain</I>,
|
<A ID="7877"></A><I>frequency domain</I>,
|
||||||
in which we dose the input with a complex sinusoid (so that its output is a
|
in which we dose the input with a complex sinusoid (so that its output is a
|
||||||
sinusoid at the same frequency) and report the amplitude and/or phase change
|
sinusoid at the same frequency) and report the amplitude and/or phase change
|
||||||
wrought by the network, as a function of the frequency. We'll now look at the
|
wrought by the network, as a function of the frequency. We'll now look at the
|
||||||
@@ -131,7 +131,7 @@ delay network of Figure <A HREF="#fig07.03">7.3</A> in each of the two ways in t
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.04"></A><A NAME="8340"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.04"></A><A ID="8340"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.4:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.4:</STRONG>
|
||||||
The time domain view of the delay network of Figure <A HREF="#fig07.03">7.3</A>.
|
The time domain view of the delay network of Figure <A HREF="#fig07.03">7.3</A>.
|
||||||
@@ -155,7 +155,7 @@ delayed copy of itself.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
A frequently used test function is an
|
A frequently used test function is an
|
||||||
<A NAME="7886"></A><I>impulse</I>,
|
<A ID="7886"></A><I>impulse</I>,
|
||||||
which is a pulse lasting only one sample. The utility of this is that, if we
|
which is a pulse lasting only one sample. The utility of this is that, if we
|
||||||
know the output of the network for an impulse, we can find the output for
|
know the output of the network for an impulse, we can find the output for
|
||||||
any other digital audio signal--because any signal <IMG
|
any other digital audio signal--because any signal <IMG
|
||||||
@@ -271,7 +271,7 @@ combined additively.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.05"></A><A NAME="8344"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.05"></A><A ID="8344"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.5:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.5:</STRONG>
|
||||||
Analysis, in the complex plane, of the frequency-domain behavior of
|
Analysis, in the complex plane, of the frequency-domain behavior of
|
||||||
@@ -483,19 +483,19 @@ The quantity <IMG
|
|||||||
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img30.png"
|
SRC="img30.png"
|
||||||
ALT="$\vert H\vert$"> is called the
|
ALT="$\vert H\vert$"> is called the
|
||||||
<A NAME="7917"></A><I>gain</I>
|
<A ID="7917"></A><I>gain</I>
|
||||||
of the delay network at the angular frequency <IMG
|
of the delay network at the angular frequency <IMG
|
||||||
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img27.png"
|
SRC="img27.png"
|
||||||
ALT="$\omega $">, and is graphed in
|
ALT="$\omega $">, and is graphed in
|
||||||
Figure <A HREF="#fig07.06">7.6</A>. The frequency-dependent gain of a delay network (that
|
Figure <A HREF="#fig07.06">7.6</A>. The frequency-dependent gain of a delay network (that
|
||||||
is, the gain as a function of frequency) is called the network's
|
is, the gain as a function of frequency) is called the network's
|
||||||
<A NAME="7920"></A><I>frequency response</I>.
|
<A ID="7920"></A><I>frequency response</I>.
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
Since the network has greater gain at some frequencies
|
Since the network has greater gain at some frequencies
|
||||||
than at others, it may be considered as a
|
than at others, it may be considered as a
|
||||||
<A NAME="7922"></A><I>filter</I>
|
<A ID="7922"></A><I>filter</I>
|
||||||
that can be used to separate certain components of a sound from others.
|
that can be used to separate certain components of a sound from others.
|
||||||
Because of the shape of this particular gain expression as a function of
|
Because of the shape of this particular gain expression as a function of
|
||||||
<IMG
|
<IMG
|
||||||
@@ -503,11 +503,11 @@ Because of the shape of this particular gain expression as a function of
|
|||||||
SRC="img27.png"
|
SRC="img27.png"
|
||||||
ALT="$\omega $">, this kind of delay network
|
ALT="$\omega $">, this kind of delay network
|
||||||
is called a (non-recirculating)
|
is called a (non-recirculating)
|
||||||
<A NAME="7924"></A><I>comb filter</I>.
|
<A ID="7924"></A><I>comb filter</I>.
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.06"></A><A NAME="8359"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.06"></A><A ID="8359"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.6:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.6:</STRONG>
|
||||||
Gain of the delay network of Figure <A HREF="#fig07.03">7.3</A>, shown as a function
|
Gain of the delay network of Figure <A HREF="#fig07.03">7.3</A>, shown as a function
|
||||||
@@ -596,36 +596,36 @@ over the entire range of possible delay times.
|
|||||||
<P>
|
<P>
|
||||||
<HR>
|
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|
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+29
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@@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION001140000000000000000"></A>
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<H1><A ID="SECTION001140000000000000000"></A>
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<A NAME="sect7.recirculatingcomb"></A>
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<A ID="sect7.recirculatingcomb"></A>
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<BR>
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<BR>
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Recirculating delay networks
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Recirculating delay networks
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</H1>
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</H1>
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@@ -80,7 +80,7 @@ the network to engender yet others.
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<P>
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<P>
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The simplest example of a recirculating network is the
|
The simplest example of a recirculating network is the
|
||||||
<A NAME="7935"></A>
|
<A ID="7935"></A>
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<I>recirculating comb filter</I>
|
<I>recirculating comb filter</I>
|
||||||
whose block diagram is shown in Figure <A HREF="#fig07.07">7.7</A>. As with the
|
whose block diagram is shown in Figure <A HREF="#fig07.07">7.7</A>. As with the
|
||||||
earlier, simple comb filter, the input signal is sent down a delay line whose
|
earlier, simple comb filter, the input signal is sent down a delay line whose
|
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@@ -97,7 +97,7 @@ multiplied by a number <IMG
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<DIV ALIGN="CENTER"><A NAME="fig07.07"></A><A NAME="7940"></A>
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<DIV ALIGN="CENTER"><A ID="fig07.07"></A><A ID="7940"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.7:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.7:</STRONG>
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||||||
Block diagram for a recirculating comb filter. Here <IMG
|
Block diagram for a recirculating comb filter. Here <IMG
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@@ -128,11 +128,11 @@ gain <IMG
|
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SRC="img29.png"
|
SRC="img29.png"
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ALT="$g$">. In general, a delay network's output given an impulse as input is
|
ALT="$g$">. In general, a delay network's output given an impulse as input is
|
||||||
called the network's
|
called the network's
|
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<A NAME="7944"></A><I>impulse response</I>.
|
<A ID="7944"></A><I>impulse response</I>.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig07.08"></A><A NAME="7948"></A>
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<DIV ALIGN="CENTER"><A ID="fig07.08"></A><A ID="7948"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.8:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.8:</STRONG>
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||||||
Time-domain analysis of the recirculating comb filter, using
|
Time-domain analysis of the recirculating comb filter, using
|
||||||
@@ -154,7 +154,7 @@ a larger magnitude than the previous one. Instead of falling exponentially
|
|||||||
as they do in the figure, they would grow exponentially. A recirculating
|
as they do in the figure, they would grow exponentially. A recirculating
|
||||||
network whose output eventually falls toward zero after its input terminates
|
network whose output eventually falls toward zero after its input terminates
|
||||||
is called
|
is called
|
||||||
<A NAME="7951"></A><I>stable</I>;
|
<A ID="7951"></A><I>stable</I>;
|
||||||
one whose output grows without bound is called <I>unstable</I>.
|
one whose output grows without bound is called <I>unstable</I>.
|
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<P>
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@@ -420,7 +420,7 @@ frequency response <IMG
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<DIV ALIGN="CENTER"><A NAME="fig07.09"></A><A NAME="7976"></A>
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||||||
<TABLE>
|
<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.9:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.9:</STRONG>
|
||||||
Diagram in the complex plane for approximating the output gain <IMG
|
Diagram in the complex plane for approximating the output gain <IMG
|
||||||
@@ -449,7 +449,7 @@ is chosen to give a gain about 3 dB below the peak.</CAPTION>
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<DIV ALIGN="CENTER"><A NAME="fig07.10"></A><A NAME="7981"></A>
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.10:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.10:</STRONG>
|
||||||
Frequency response of the recirculating comb filter with <IMG
|
Frequency response of the recirculating comb filter with <IMG
|
||||||
@@ -671,36 +671,36 @@ filters in Chapter <A HREF="node127.html#chapter-filter">8</A>.
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<B> <A ID="tex2html704"
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<H1><A NAME="SECTION00540000000000000000">
|
<H1><A ID="SECTION00540000000000000000">
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Frequency</A>
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Frequency</A>
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</H1>
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</H1>
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@@ -78,7 +78,7 @@ the musical interval between them.
|
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|
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<P>
|
<P>
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||||||
The Western musical scale divides the
|
The Western musical scale divides the
|
||||||
<A NAME="1095"></A><I>octave</I> (the musical interval associated with a ratio of 2:1) into
|
<A ID="1095"></A><I>octave</I> (the musical interval associated with a ratio of 2:1) into
|
||||||
twelve equal sub-intervals, each of which therefore corresponds to a ratio
|
twelve equal sub-intervals, each of which therefore corresponds to a ratio
|
||||||
of <!-- MATH
|
of <!-- MATH
|
||||||
${2 ^ {1/{12}}}$
|
${2 ^ {1/{12}}}$
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||||||
@@ -87,12 +87,12 @@ of <!-- MATH
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SRC="img110.png"
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||||||
ALT="${2 ^ {1/{12}}}$">. For historical reasons this sub-interval is called a
|
ALT="${2 ^ {1/{12}}}$">. For historical reasons this sub-interval is called a
|
||||||
<A NAME="1098"></A><I>half-step</I>.
|
<A ID="1098"></A><I>half-step</I>.
|
||||||
A convenient logarithmic scale for pitch is simply to
|
A convenient logarithmic scale for pitch is simply to
|
||||||
count the number of half-steps from a reference pitch--allowing fractions to
|
count the number of half-steps from a reference pitch--allowing fractions to
|
||||||
permit us to specify pitches which don't fall on a note of the Western scale.
|
permit us to specify pitches which don't fall on a note of the Western scale.
|
||||||
The most commonly used logarithmic pitch scale is
|
The most commonly used logarithmic pitch scale is
|
||||||
<A NAME="1100"></A>"MIDI pitch", in which the pitch 69 is assigned to a frequency of 440 cycles
|
<A ID="1100"></A>"MIDI pitch", in which the pitch 69 is assigned to a frequency of 440 cycles
|
||||||
per second--the A above middle C. To convert between a MIDI pitch <IMG
|
per second--the A above middle C. To convert between a MIDI pitch <IMG
|
||||||
WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img111.png"
|
SRC="img111.png"
|
||||||
@@ -102,10 +102,10 @@ in cycles per second <IMG
|
|||||||
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img112.png"
|
SRC="img112.png"
|
||||||
ALT="$f$">, apply the
|
ALT="$f$">, apply the
|
||||||
<A NAME="1101"></A>Pitch/Frequency Conversion formulas:
|
<A ID="1101"></A>Pitch/Frequency Conversion formulas:
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
<A NAME="eq-pitchmidi"></A>
|
<A ID="eq-pitchmidi"></A>
|
||||||
<BR><P></P>
|
<BR><P></P>
|
||||||
<DIV ALIGN="CENTER">
|
<DIV ALIGN="CENTER">
|
||||||
<!-- MATH
|
<!-- MATH
|
||||||
@@ -162,7 +162,7 @@ description of MIDI, its capabilities and limitations, see
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
A half-step comes to a ratio of about 1.059 to 1, or about a six percent
|
A half-step comes to a ratio of about 1.059 to 1, or about a six percent
|
||||||
increase in frequency. Half-steps are further divided into <A NAME="1107"></A><I>cents</I>, each cent being one hundredth of a half-step. As a rule of
|
increase in frequency. Half-steps are further divided into <A ID="1107"></A><I>cents</I>, each cent being one hundredth of a half-step. As a rule of
|
||||||
thumb, it might take about three cents to make a discernible change in the
|
thumb, it might take about three cents to make a discernible change in the
|
||||||
pitch of a musical tone. At middle C this comes to a difference of about 1/2
|
pitch of a musical tone. At middle C this comes to a difference of about 1/2
|
||||||
cycle per second. A graph of frequency as a function of MIDI pitch, over a
|
cycle per second. A graph of frequency as a function of MIDI pitch, over a
|
||||||
@@ -171,36 +171,36 @@ two-octave range, is shown in Figure <A HREF="node10.html#fig01.04">1.4</A>.
|
|||||||
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+29
-29
@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node109.html">Recirculating delay networks</A>
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HREF="node109.html">Recirculating delay networks</A>
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<B> <A NAME="tex2html2196"
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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|
||||||
<H1><A NAME="SECTION001150000000000000000">
|
<H1><A ID="SECTION001150000000000000000">
|
||||||
Power conservation and complex delay networks</A>
|
Power conservation and complex delay networks</A>
|
||||||
</H1>
|
</H1>
|
||||||
|
|
||||||
@@ -90,7 +90,7 @@ the gain, suitably defined, is exactly one.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.11"></A><A NAME="8003"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.11"></A><A ID="8003"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.11:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.11:</STRONG>
|
||||||
First fundamental building block for unitary delay networks:
|
First fundamental building block for unitary delay networks:
|
||||||
@@ -163,7 +163,7 @@ where <IMG
|
|||||||
It turns out that a wide range of interesting delay networks has the property
|
It turns out that a wide range of interesting delay networks has the property
|
||||||
that the total power output equals the total power input;
|
that the total power output equals the total power input;
|
||||||
they are called
|
they are called
|
||||||
<A NAME="8013"></A><I>unitary</I>. To start with, we can put any number of delays in parallel, as
|
<A ID="8013"></A><I>unitary</I>. To start with, we can put any number of delays in parallel, as
|
||||||
shown in Figure <A HREF="#fig07.11">7.11</A>. Whatever the total power of the inputs,
|
shown in Figure <A HREF="#fig07.11">7.11</A>. Whatever the total power of the inputs,
|
||||||
the total power of the outputs has to equal it.
|
the total power of the outputs has to equal it.
|
||||||
|
|
||||||
@@ -233,7 +233,7 @@ of a collection of signals must must be preserved by rotation.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.12"></A><A NAME="8024"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.12"></A><A ID="8024"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.12:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.12:</STRONG>
|
||||||
Second fundamental building block for unitary delay networks:
|
Second fundamental building block for unitary delay networks:
|
||||||
@@ -329,7 +329,7 @@ s = \sin(\theta)
|
|||||||
<BR CLEAR="ALL">
|
<BR CLEAR="ALL">
|
||||||
<P></P>
|
<P></P>
|
||||||
for an
|
for an
|
||||||
<A NAME="8036"></A><I>angle of rotation</I> <IMG
|
<A ID="8036"></A><I>angle of rotation</I> <IMG
|
||||||
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img634.png"
|
SRC="img634.png"
|
||||||
ALT="$\theta$">.
|
ALT="$\theta$">.
|
||||||
@@ -423,7 +423,7 @@ unit magnitude and argument <IMG
|
|||||||
<P>
|
<P>
|
||||||
If we perform a rotation on a pair of signals and then invert one (but not the
|
If we perform a rotation on a pair of signals and then invert one (but not the
|
||||||
other) of them, the result is a
|
other) of them, the result is a
|
||||||
<A NAME="8050"></A>
|
<A ID="8050"></A>
|
||||||
<I>reflection</I>.
|
<I>reflection</I>.
|
||||||
This also preserves total signal power, since we can invert any or all of a
|
This also preserves total signal power, since we can invert any or all of a
|
||||||
collection of signals without changing the total power. In two dimensions, a
|
collection of signals without changing the total power. In two dimensions, a
|
||||||
@@ -486,7 +486,7 @@ a) because each signal need only be multiplied by the one quantity <IMG
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.13"></A><A NAME="8389"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.13"></A><A ID="8389"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.13:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.13:</STRONG>
|
||||||
Details about rotation (and reflection) matrix operations: (a)
|
Details about rotation (and reflection) matrix operations: (a)
|
||||||
@@ -535,7 +535,7 @@ recirculating networks that still enjoy flat frequency responses.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.14"></A><A NAME="8069"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.14"></A><A ID="8069"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.14:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.14:</STRONG>
|
||||||
Flat frequency response in recirculating networks: (a) in general,
|
Flat frequency response in recirculating networks: (a) in general,
|
||||||
@@ -655,7 +655,7 @@ let the transformation <IMG
|
|||||||
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img31.png"
|
SRC="img31.png"
|
||||||
ALT="$W$"> output, the result is the well-known
|
ALT="$W$"> output, the result is the well-known
|
||||||
<A NAME="8086"></A><A NAME="8087"></A><I>all-pass filter</I>.
|
<A ID="8086"></A><A ID="8087"></A><I>all-pass filter</I>.
|
||||||
With some juggling, and letting <!-- MATH
|
With some juggling, and letting <!-- MATH
|
||||||
$c = \cos(\theta)$
|
$c = \cos(\theta)$
|
||||||
-->
|
-->
|
||||||
@@ -670,36 +670,36 @@ of which we will visit later in this book.
|
|||||||
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|
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HREF="node109.html">Recirculating delay networks</A>
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<H1><A NAME="SECTION001160000000000000000">
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<H1><A ID="SECTION001160000000000000000">
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Artificial reverberation</A>
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Artificial reverberation</A>
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</H1>
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</H1>
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@@ -101,13 +101,13 @@ is not designed correctly.
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To make an artificial reverberator using a delay network, we must
|
To make an artificial reverberator using a delay network, we must
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fill two competing demands simultaneously. First, the delay lines must
|
fill two competing demands simultaneously. First, the delay lines must
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be long enough to prevent
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be long enough to prevent
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<A NAME="8091"></A><I>coloration</I> in the output as a result of comb filtering.
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<A ID="8091"></A><I>coloration</I> in the output as a result of comb filtering.
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(Even if we move beyond the simple comb filter
|
(Even if we move beyond the simple comb filter
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of Section <A HREF="node109.html#sect7.recirculatingcomb">7.4</A>, the frequency response will tend to
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of Section <A HREF="node109.html#sect7.recirculatingcomb">7.4</A>, the frequency response will tend to
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have peaks and
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have peaks and
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valleys whose spacing varies inversely with total delay time.) On the other
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valleys whose spacing varies inversely with total delay time.) On the other
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hand, we should not hear individual echoes; the
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hand, we should not hear individual echoes; the
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<A NAME="8094"></A><I>echo density</I> should ideally be at least one thousand per second.
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<A ID="8094"></A><I>echo density</I> should ideally be at least one thousand per second.
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<P>
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In pursuit of these aims, we assemble some number of delay lines and
|
In pursuit of these aims, we assemble some number of delay lines and
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@@ -138,7 +138,7 @@ and 80 milliseconds. The figure shows three such stages.
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<DIV ALIGN="CENTER"><A NAME="fig07.15"></A><A NAME="8099"></A>
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<DIV ALIGN="CENTER"><A ID="fig07.15"></A><A ID="8099"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.15:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.15:</STRONG>
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Reverberator design using power-preserving transformations and
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Reverberator design using power-preserving transformations and
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@@ -273,45 +273,45 @@ and lengthy tuning by trial, error, and critical listening.
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<P>
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<P>
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<BR><HR>
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<BR><HR>
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<!--Table of Child-Links-->
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html2215"
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HREF="node112.html">Controlling reverberators</A>
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+22
-22
@@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<B> <A NAME="tex2html2223"
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<B> <A ID="tex2html2223"
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HREF="node4.html">Contents</A></B>
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<H2><A NAME="SECTION001161000000000000000">
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<H2><A ID="SECTION001161000000000000000">
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Controlling reverberators</A>
|
Controlling reverberators</A>
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</H2>
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</H2>
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@@ -88,7 +88,7 @@ moments of stable pitch.
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<DIV ALIGN="CENTER"><A NAME="fig07.16"></A><A NAME="8114"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.16:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.16:</STRONG>
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Controlling a reverberator to capture sounds selectively: (a)
|
Controlling a reverberator to capture sounds selectively: (a)
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@@ -135,36 +135,36 @@ decay.
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+25
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Variable and fractional shifts
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Variable and fractional shifts
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.17:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.17:</STRONG>
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A variable length delay line, whose output is the input from some
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A variable length delay line, whose output is the input from some
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@@ -350,7 +350,7 @@ t[n] = y[n] - y[n-1] = 1 - (d[n] - d[n-1])
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<A NAME="eq.momentarydel"></A>If <IMG
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This is called the
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This is called the
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<A NAME="8132"></A><I>Doppler effect</I>, and it occurs in nature as well.
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<A ID="8132"></A><I>Doppler effect</I>, and it occurs in nature as well.
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The air that sound travels through can sometimes be thought of as a delay
|
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||||||
line. Changing the length of the delay line corresponds to moving the
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line. Changing the length of the delay line corresponds to moving the
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listener toward or away from a stationary sound source; the Doppler effect
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listener toward or away from a stationary sound source; the Doppler effect
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@@ -391,36 +391,36 @@ diagonal region.
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+22
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<H1><A NAME="SECTION001180000000000000000">
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<H1><A ID="SECTION001180000000000000000">
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Fidelity of interpolating delay lines</A>
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Fidelity of interpolating delay lines</A>
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</H1>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.18:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.18:</STRONG>
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Gain of a four-point interpolating delay line with a delay halfway
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Gain of a four-point interpolating delay line with a delay halfway
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@@ -221,36 +221,36 @@ the way up to 20000 Hertz.
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<H1><A NAME="SECTION001190000000000000000"></A>
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Pitch shifting
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|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.19"></A><A NAME="8151"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.19"></A><A ID="8151"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.19:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.19:</STRONG>
|
||||||
Vibrato using a variable delay line. Since the pitch shift
|
Vibrato using a variable delay line. Since the pitch shift
|
||||||
@@ -161,7 +161,7 @@ This ranges in value between <IMG
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.20"></A><A NAME="8159"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.20"></A><A ID="8159"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.20:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.20:</STRONG>
|
||||||
Piecewise linear delay functions to maintain a constant transposition
|
Piecewise linear delay functions to maintain a constant transposition
|
||||||
@@ -187,7 +187,7 @@ to avoid discontinuities.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.21"></A><A NAME="8165"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.21"></A><A ID="8165"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.21:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.21:</STRONG>
|
||||||
Using a variable delay line as a pitch shifter. The sawtooth wave
|
Using a variable delay line as a pitch shifter. The sawtooth wave
|
||||||
@@ -230,7 +230,7 @@ pitch shifter since it is essentially free. The quantity <IMG
|
|||||||
SRC="img208.png"
|
SRC="img208.png"
|
||||||
ALT="$s$"> is sometimes
|
ALT="$s$"> is sometimes
|
||||||
called the
|
called the
|
||||||
<A NAME="8171"></A><I>window size</I>. It corresponds roughly to the sample length in a
|
<A ID="8171"></A><I>window size</I>. It corresponds roughly to the sample length in a
|
||||||
looping sampler (Section <A HREF="node28.html#sect2.sampling">2.2</A>).
|
looping sampler (Section <A HREF="node28.html#sect2.sampling">2.2</A>).
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
@@ -330,7 +330,7 @@ Figure <A HREF="#fig07.22">7.22</A>.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.22"></A><A NAME="8183"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.22"></A><A ID="8183"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.22:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.22:</STRONG>
|
||||||
The pitch shifter's delay reading pattern using two delay lines,
|
The pitch shifter's delay reading pattern using two delay lines,
|
||||||
@@ -373,7 +373,7 @@ f = {{(t - 1) R} \over s}
|
|||||||
</DIV>
|
</DIV>
|
||||||
<BR CLEAR="ALL">
|
<BR CLEAR="ALL">
|
||||||
<P></P>
|
<P></P>
|
||||||
<A NAME="sect7.psformula"></A>The window size <IMG
|
<A ID="sect7.psformula"></A>The window size <IMG
|
||||||
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||||
SRC="img208.png"
|
SRC="img208.png"
|
||||||
ALT="$s$"> should be chosen small enough, if possible, so that the
|
ALT="$s$"> should be chosen small enough, if possible, so that the
|
||||||
@@ -445,36 +445,36 @@ artifacts such as difference tones.
|
|||||||
<P>
|
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|
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HREF="node114.html">Fidelity of interpolating delay</A>
|
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<ADDRESS>
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|||||||
+21
-21
@@ -30,68 +30,68 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node104.html">Time shifts and delays</A>
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<B> Previous:</B> <A ID="tex2html2271"
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HREF="node115.html">Pitch shifting</A>
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HREF="node115.html">Pitch shifting</A>
|
||||||
<B> <A NAME="tex2html2279"
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<B> <A ID="tex2html2279"
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html2281"
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<B> <A ID="tex2html2281"
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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<!--End of Navigation Panel-->
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|
|
||||||
<H1><A NAME="SECTION0011100000000000000000">
|
<H1><A ID="SECTION0011100000000000000000">
|
||||||
Examples</A>
|
Examples</A>
|
||||||
</H1>
|
</H1>
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
<BR><HR>
|
<BR><HR>
|
||||||
<!--Table of Child-Links-->
|
<!--Table of Child-Links-->
|
||||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||||
|
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html2284"
|
<LI><A ID="tex2html2284"
|
||||||
HREF="node117.html">Fixed, noninterpolating delay line</A>
|
HREF="node117.html">Fixed, noninterpolating delay line</A>
|
||||||
<LI><A NAME="tex2html2285"
|
<LI><A ID="tex2html2285"
|
||||||
HREF="node118.html">Recirculating comb filter</A>
|
HREF="node118.html">Recirculating comb filter</A>
|
||||||
<LI><A NAME="tex2html2286"
|
<LI><A ID="tex2html2286"
|
||||||
HREF="node119.html">Variable delay line</A>
|
HREF="node119.html">Variable delay line</A>
|
||||||
<LI><A NAME="tex2html2287"
|
<LI><A ID="tex2html2287"
|
||||||
HREF="node120.html">Order of execution and lower limits on delay times</A>
|
HREF="node120.html">Order of execution and lower limits on delay times</A>
|
||||||
<LI><A NAME="tex2html2288"
|
<LI><A ID="tex2html2288"
|
||||||
HREF="node121.html">Order of execution in non-recirculating delay lines</A>
|
HREF="node121.html">Order of execution in non-recirculating delay lines</A>
|
||||||
<LI><A NAME="tex2html2289"
|
<LI><A ID="tex2html2289"
|
||||||
HREF="node122.html">Non-recirculating comb filter as octave doubler</A>
|
HREF="node122.html">Non-recirculating comb filter as octave doubler</A>
|
||||||
<LI><A NAME="tex2html2290"
|
<LI><A ID="tex2html2290"
|
||||||
HREF="node123.html">Time-varying complex comb filter: shakers</A>
|
HREF="node123.html">Time-varying complex comb filter: shakers</A>
|
||||||
<LI><A NAME="tex2html2291"
|
<LI><A ID="tex2html2291"
|
||||||
HREF="node124.html">Reverberator</A>
|
HREF="node124.html">Reverberator</A>
|
||||||
<LI><A NAME="tex2html2292"
|
<LI><A ID="tex2html2292"
|
||||||
HREF="node125.html">Pitch shifter</A>
|
HREF="node125.html">Pitch shifter</A>
|
||||||
</UL>
|
</UL>
|
||||||
<!--End of Table of Child-Links-->
|
<!--End of Table of Child-Links-->
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||||||
|
|||||||
+24
-24
@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
|
|||||||
|
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HREF="node118.html">Recirculating comb filter</A>
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HREF="node118.html">Recirculating comb filter</A>
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<B> Up:</B> <A NAME="tex2html2300"
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<B> Up:</B> <A ID="tex2html2300"
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HREF="node116.html">Examples</A>
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HREF="node116.html">Examples</A>
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||||||
<B> Previous:</B> <A NAME="tex2html2294"
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<B> Previous:</B> <A ID="tex2html2294"
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HREF="node116.html">Examples</A>
|
HREF="node116.html">Examples</A>
|
||||||
<B> <A NAME="tex2html2302"
|
<B> <A ID="tex2html2302"
|
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HREF="node4.html">Contents</A></B>
|
HREF="node4.html">Contents</A></B>
|
||||||
<B> <A NAME="tex2html2304"
|
<B> <A ID="tex2html2304"
|
||||||
HREF="node201.html">Index</A></B>
|
HREF="node201.html">Index</A></B>
|
||||||
<BR>
|
<BR>
|
||||||
<BR>
|
<BR>
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<!--End of Navigation Panel-->
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||||||
|
|
||||||
<H2><A NAME="SECTION0011101000000000000000">
|
<H2><A ID="SECTION0011101000000000000000">
|
||||||
Fixed, noninterpolating delay line</A>
|
Fixed, noninterpolating delay line</A>
|
||||||
</H2>
|
</H2>
|
||||||
|
|
||||||
@@ -75,7 +75,7 @@ an input signal. Two new objects are needed:
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.23"></A><A NAME="8194"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.23"></A><A ID="8194"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.23:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.23:</STRONG>
|
||||||
Example patch G01.delay.pd, showing a noninterpolating delay with
|
Example patch G01.delay.pd, showing a noninterpolating delay with
|
||||||
@@ -95,7 +95,7 @@ a delay time controlled in milliseconds.</CAPTION>
|
|||||||
WIDTH="94" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="94" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img819.png"
|
SRC="img819.png"
|
||||||
ALT="\fbox{ \texttt{delwrite\~}}">:
|
ALT="\fbox{ \texttt{delwrite\~}}">:
|
||||||
<A NAME="8424"></A>define and write to a delay line. The first creation argument gives the name of the
|
<A ID="8424"></A>define and write to a delay line. The first creation argument gives the name of the
|
||||||
delay line (and two delay lines may not share the same name). The
|
delay line (and two delay lines may not share the same name). The
|
||||||
second creation argument is the length of the delay line in milliseconds. The
|
second creation argument is the length of the delay line in milliseconds. The
|
||||||
inlet takes an audio signal and writes it continuously into the delay line.
|
inlet takes an audio signal and writes it continuously into the delay line.
|
||||||
@@ -108,7 +108,7 @@ inlet takes an audio signal and writes it continuously into the delay line.
|
|||||||
WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img821.png"
|
SRC="img821.png"
|
||||||
ALT="\fbox{ \texttt{delread\~}}">:
|
ALT="\fbox{ \texttt{delread\~}}">:
|
||||||
<A NAME="8425"></A>read from (or "tap") a delay line. The first creation argument gives the name
|
<A ID="8425"></A>read from (or "tap") a delay line. The first creation argument gives the name
|
||||||
of the delay line (which should agree with the name of the corresponding
|
of the delay line (which should agree with the name of the corresponding
|
||||||
<TT>delwrite~</TT> object; this is how Pd knows which <TT>delwrite~</TT> to
|
<TT>delwrite~</TT> object; this is how Pd knows which <TT>delwrite~</TT> to
|
||||||
associate with the <TT>delread~</TT> object). The
|
associate with the <TT>delread~</TT> object). The
|
||||||
@@ -130,36 +130,36 @@ possible when the delay time changes.
|
|||||||
<P>
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+12
-12
@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node117.html">Fixed, noninterpolating delay line</A>
|
HREF="node117.html">Fixed, noninterpolating delay line</A>
|
||||||
<B> <A NAME="tex2html2316"
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<B> <A ID="tex2html2316"
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HREF="node4.html">Contents</A></B>
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||||||
|
|
||||||
<H2><A NAME="SECTION0011102000000000000000">
|
<H2><A ID="SECTION0011102000000000000000">
|
||||||
Recirculating comb filter</A>
|
Recirculating comb filter</A>
|
||||||
</H2>
|
</H2>
|
||||||
|
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||||||
@@ -81,7 +81,7 @@ of Section <A HREF="node109.html#sect7.recirculatingcomb">7.4</A>.
|
|||||||
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||||||
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|
<P>
|
||||||
|
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||||||
<DIV ALIGN="CENTER"><A NAME="fig07.24"></A><A NAME="8208"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.24"></A><A ID="8208"></A>
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.24:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.24:</STRONG>
|
||||||
Recirculating delay (still noninterpolating).</CAPTION>
|
Recirculating delay (still noninterpolating).</CAPTION>
|
||||||
|
|||||||
+23
-23
@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<B> Next:</B> <A NAME="tex2html2334"
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HREF="node120.html">Order of execution and</A>
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HREF="node120.html">Order of execution and</A>
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<B> Up:</B> <A NAME="tex2html2328"
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<B> Up:</B> <A ID="tex2html2328"
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HREF="node118.html">Recirculating comb filter</A>
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HREF="node118.html">Recirculating comb filter</A>
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<B> <A ID="tex2html2330"
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html2332"
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<B> <A ID="tex2html2332"
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION0011103000000000000000">
|
<H2><A ID="SECTION0011103000000000000000">
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Variable delay line</A>
|
Variable delay line</A>
|
||||||
</H2>
|
</H2>
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||||||
|
|
||||||
@@ -76,7 +76,7 @@ new object is introduced here:
|
|||||||
|
|
||||||
<P>
|
<P>
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||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig07.25"></A><A NAME="8215"></A>
|
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.25:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.25:</STRONG>
|
||||||
The flanger: an interpolating, variable delay line.</CAPTION>
|
The flanger: an interpolating, variable delay line.</CAPTION>
|
||||||
@@ -95,7 +95,7 @@ The flanger: an interpolating, variable delay line.</CAPTION>
|
|||||||
WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img825.png"
|
SRC="img825.png"
|
||||||
ALT="\fbox{ \texttt{vd\~}}">:
|
ALT="\fbox{ \texttt{vd\~}}">:
|
||||||
<A NAME="8426"></A>Read from a delay line, with a time-varying delay time. As with the
|
<A ID="8426"></A>Read from a delay line, with a time-varying delay time. As with the
|
||||||
<TT>delread~</TT> object, this reads from a delay line whose name is specified
|
<TT>delread~</TT> object, this reads from a delay line whose name is specified
|
||||||
as a creation argument. Instead of using a second argument and/or
|
as a creation argument. Instead of using a second argument and/or
|
||||||
control messages to specify the delay time, for the <TT>vd~</TT> object the delay in milliseconds is specified by an incoming audio signal.
|
control messages to specify the delay time, for the <TT>vd~</TT> object the delay in milliseconds is specified by an incoming audio signal.
|
||||||
@@ -136,36 +136,36 @@ characteristic way.
|
|||||||
<P>
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HREF="node120.html">Order of execution and</A>
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HREF="node120.html">Order of execution and</A>
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HREF="node118.html">Recirculating comb filter</A>
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HREF="node118.html">Recirculating comb filter</A>
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+29
-29
@@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node11.html">Frequency</A>
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HREF="node11.html">Frequency</A>
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<B> <A NAME="tex2html718"
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<B> <A ID="tex2html718"
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html720"
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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<H1><A NAME="SECTION00550000000000000000"></A>
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<H1><A ID="SECTION00550000000000000000"></A>
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||||||
<A NAME="sect1.synth"></A>
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<A ID="sect1.synth"></A>
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||||||
<BR>
|
<BR>
|
||||||
Synthesizing a sinusoid
|
Synthesizing a sinusoid
|
||||||
</H1>
|
</H1>
|
||||||
@@ -75,10 +75,10 @@ Synthesizing a sinusoid
|
|||||||
In most widely used audio synthesis and processing packages (Csound, Max/MSP,
|
In most widely used audio synthesis and processing packages (Csound, Max/MSP,
|
||||||
and Pd, for instance), the audio operations are specified as
|
and Pd, for instance), the audio operations are specified as
|
||||||
networks of
|
networks of
|
||||||
<A NAME="1112"></A><I>unit generators</I>[<A
|
<A ID="1112"></A><I>unit generators</I>[<A
|
||||||
HREF="node202.html#r-mathews69">Mat69</A>] which pass audio signals among themselves. The user of
|
HREF="node202.html#r-mathews69">Mat69</A>] which pass audio signals among themselves. The user of
|
||||||
the software package specifies the network, sometimes called a
|
the software package specifies the network, sometimes called a
|
||||||
<A NAME="1115"></A><I>patch</I>,
|
<A ID="1115"></A><I>patch</I>,
|
||||||
which essentially corresponds to the synthesis algorithm to be used, and then
|
which essentially corresponds to the synthesis algorithm to be used, and then
|
||||||
worries about how to control the various unit generators in time. In this
|
worries about how to control the various unit generators in time. In this
|
||||||
section, we'll use abstract block diagrams to describe patches, but in the
|
section, we'll use abstract block diagrams to describe patches, but in the
|
||||||
@@ -91,14 +91,14 @@ details.
|
|||||||
To show how to produce a sinusoid with time-varying amplitude we'll need to
|
To show how to produce a sinusoid with time-varying amplitude we'll need to
|
||||||
introduce two unit generators. First we need a pure sinusoid which is
|
introduce two unit generators. First we need a pure sinusoid which is
|
||||||
made with an
|
made with an
|
||||||
<A NAME="1118"></A><I>oscillator</I>. Figure <A HREF="#fig01.05">1.5</A> (part a) shows a pictorial
|
<A ID="1118"></A><I>oscillator</I>. Figure <A HREF="#fig01.05">1.5</A> (part a) shows a pictorial
|
||||||
representation of
|
representation of
|
||||||
a sinusoidal oscillator as an icon. The input is a frequency (in cycles per second), and
|
a sinusoidal oscillator as an icon. The input is a frequency (in cycles per second), and
|
||||||
the output is a sinusoid of peak amplitude one.
|
the output is a sinusoid of peak amplitude one.
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig01.05"></A><A NAME="1123"></A>
|
<DIV ALIGN="CENTER"><A ID="fig01.05"></A><A ID="1123"></A>
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||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.5:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.5:</STRONG>
|
||||||
Block diagrams for (a) a sinusoidal oscillator; (b) controlling the
|
Block diagrams for (a) a sinusoidal oscillator; (b) controlling the
|
||||||
@@ -131,7 +131,7 @@ negative in value.
|
|||||||
|
|
||||||
<P>
|
<P>
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||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig01.06"></A><A NAME="1351"></A>
|
<DIV ALIGN="CENTER"><A ID="fig01.06"></A><A ID="1351"></A>
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||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.6:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.6:</STRONG>
|
||||||
Two amplitude functions (parts a, c), and (parts b, d), the result of
|
Two amplitude functions (parts a, c), and (parts b, d), the result of
|
||||||
@@ -183,7 +183,7 @@ Suitable amplitude control functions <IMG
|
|||||||
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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||||||
SRC="img2.png"
|
SRC="img2.png"
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||||||
ALT="$y[n]$"> may be made using an
|
ALT="$y[n]$"> may be made using an
|
||||||
<A NAME="1134"></A><I>envelope generator</I>.
|
<A ID="1134"></A><I>envelope generator</I>.
|
||||||
Figure <A HREF="#fig01.07">1.7</A> shows a network in which an envelope generator is used to
|
Figure <A HREF="#fig01.07">1.7</A> shows a network in which an envelope generator is used to
|
||||||
control the amplitude of an oscillator. Envelope generators vary widely in
|
control the amplitude of an oscillator. Envelope generators vary widely in
|
||||||
design, but we will focus on the simplest
|
design, but we will focus on the simplest
|
||||||
@@ -252,7 +252,7 @@ Envelope generators are described in more detail in Section <A HREF="node59.html
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig01.07"></A><A NAME="1148"></A>
|
<DIV ALIGN="CENTER"><A ID="fig01.07"></A><A ID="1148"></A>
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||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.7:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.7:</STRONG>
|
||||||
Using an envelope generator to control amplitude.</CAPTION>
|
Using an envelope generator to control amplitude.</CAPTION>
|
||||||
@@ -266,36 +266,36 @@ Using an envelope generator to control amplitude.</CAPTION>
|
|||||||
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+25
-25
@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node121.html">Order of execution in</A>
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HREF="node119.html">Variable delay line</A>
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HREF="node119.html">Variable delay line</A>
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<B> <A NAME="tex2html2344"
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<B> <A ID="tex2html2344"
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html2346"
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<B> <A ID="tex2html2346"
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION0011104000000000000000">
|
<H2><A ID="SECTION0011104000000000000000">
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Order of execution and lower limits on delay times</A>
|
Order of execution and lower limits on delay times</A>
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</H2>
|
</H2>
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||||||
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@@ -94,7 +94,7 @@ delay operations might, for example).
|
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|
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||||||
<P>
|
<P>
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||||||
|
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||||||
<DIV ALIGN="CENTER"><A NAME="fig07.26"></A><A NAME="8233"></A>
|
<DIV ALIGN="CENTER"><A ID="fig07.26"></A><A ID="8233"></A>
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.26:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.26:</STRONG>
|
||||||
Order of execution of tilde objects in Pd: (a), an acyclic network.
|
Order of execution of tilde objects in Pd: (a), an acyclic network.
|
||||||
@@ -249,7 +249,7 @@ a new object:
|
|||||||
|
|
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<P>
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<P>
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|
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||||||
<DIV ALIGN="CENTER"><A NAME="fig07.27"></A><A NAME="8407"></A>
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<DIV ALIGN="CENTER"><A ID="fig07.27"></A><A ID="8407"></A>
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||||||
<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.27:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.27:</STRONG>
|
||||||
A patch using block size control to lower the loop delay below
|
A patch using block size control to lower the loop delay below
|
||||||
@@ -277,7 +277,7 @@ with a <TT>block~</TT> object and a recirculating delay network.</CAPTION>
|
|||||||
WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img834.png"
|
SRC="img834.png"
|
||||||
ALT="\fbox{ \texttt{switch\~}}">:
|
ALT="\fbox{ \texttt{switch\~}}">:
|
||||||
<A NAME="8427"></A><A NAME="8428"></A>Set the local block size of the patch window the object sits in. Block sizes
|
<A ID="8427"></A><A ID="8428"></A>Set the local block size of the patch window the object sits in. Block sizes
|
||||||
are normally powers of two. The <TT>switch~</TT> object, in addition, can be used to
|
are normally powers of two. The <TT>switch~</TT> object, in addition, can be used to
|
||||||
turn audio computation within the window on and off, using control messages.
|
turn audio computation within the window on and off, using control messages.
|
||||||
Additional creation arguments can set the local sample rate and specify
|
Additional creation arguments can set the local sample rate and specify
|
||||||
@@ -298,7 +298,7 @@ minimum achievable delay is one sample instead of the default 64.
|
|||||||
<P>
|
<P>
|
||||||
Putting a pulse (or other excitation signal) into a recirculating comb filter
|
Putting a pulse (or other excitation signal) into a recirculating comb filter
|
||||||
to make a pitch is sometimes called
|
to make a pitch is sometimes called
|
||||||
<A NAME="8260"></A><I>Karplus-Strong synthesis</I>,
|
<A ID="8260"></A><I>Karplus-Strong synthesis</I>,
|
||||||
having been described in a paper by them [<A
|
having been described in a paper by them [<A
|
||||||
HREF="node202.html#r-karplus83">KS83</A>], although the idea
|
HREF="node202.html#r-karplus83">KS83</A>], although the idea
|
||||||
seems to be older. It shows up for example in Paul Lansky's 1979 piece,
|
seems to be older. It shows up for example in Paul Lansky's 1979 piece,
|
||||||
@@ -307,36 +307,36 @@ seems to be older. It shows up for example in Paul Lansky's 1979 piece,
|
|||||||
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HREF="node120.html">Order of execution and</A>
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<H2><A NAME="SECTION0011105000000000000000">
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Order of execution in non-recirculating delay lines</A>
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Order of execution in non-recirculating delay lines</A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.28:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.28:</STRONG>
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Using subpatches to ensure that delay lines are written before they
|
Using subpatches to ensure that delay lines are written before they
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@@ -125,36 +125,36 @@ delays below the 64 sample block size.
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HREF="node4.html">Contents</A></B>
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+22
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HREF="node121.html">Order of execution in</A>
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION0011106000000000000000">
|
<H2><A ID="SECTION0011106000000000000000">
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||||||
Non-recirculating comb filter as octave doubler</A>
|
Non-recirculating comb filter as octave doubler</A>
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||||||
</H2>
|
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||||||
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||||||
@@ -83,7 +83,7 @@ would have got by using speed change to do the transposition.
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<DIV ALIGN="CENTER"><A NAME="fig07.29"></A><A NAME="8278"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.29:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.29:</STRONG>
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An "octave doubler" uses pitch information (obtained using
|
An "octave doubler" uses pitch information (obtained using
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@@ -165,36 +165,36 @@ up-shifting for best results.)
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HREF="node124.html">Reverberator</A>
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HREF="node122.html">Non-recirculating comb filter as</A>
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HREF="node4.html">Contents</A></B>
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<H2><A NAME="SECTION0011107000000000000000">
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Time-varying complex comb filter: shakers</A>
|
Time-varying complex comb filter: shakers</A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.30:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.30:</STRONG>
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A "shaker", a four-tap comb filter with randomly varying gains
|
A "shaker", a four-tap comb filter with randomly varying gains
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@@ -135,36 +135,36 @@ subject of Chapter 8).
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HREF="node122.html">Non-recirculating comb filter as</A>
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HREF="node4.html">Contents</A></B>
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+23
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@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node123.html">Time-varying complex comb filter:</A>
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HREF="node123.html">Time-varying complex comb filter:</A>
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<H2><A NAME="SECTION0011108000000000000000">
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Reverberator</A>
|
Reverberator</A>
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</H2>
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<DIV ALIGN="CENTER"><A NAME="fig07.31"></A><A NAME="8301"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.31:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.31:</STRONG>
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An artificial reverberator.</CAPTION>
|
An artificial reverberator.</CAPTION>
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@@ -110,7 +110,7 @@ reinserted into them, so the reverberation lasts perpetually.
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<DIV ALIGN="CENTER"><A NAME="fig07.32"></A><A NAME="8307"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.32:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.32:</STRONG>
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The echo generator used in the reverberator.</CAPTION>
|
The echo generator used in the reverberator.</CAPTION>
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@@ -141,36 +141,36 @@ sonic qualities described as "presence", "warmth", "clarity", and so on.
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||||||
<H2><A NAME="SECTION0011109000000000000000">
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Pitch shifter</A>
|
Pitch shifter</A>
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||||||
</H2>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.33:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.33:</STRONG>
|
||||||
A pitch shifter using two variable taps into a delay
|
A pitch shifter using two variable taps into a delay
|
||||||
@@ -141,36 +141,36 @@ envelopes and summed.
|
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Exercises</A>
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HREF="node104.html">Time shifts and delays</A>
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+61
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HREF="book.html">book</A>
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Filters
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Filters
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@@ -79,11 +79,11 @@ variably depending on frequency. When the delay times used are very short,
|
|||||||
the most important properties of a delay network become its frequency and
|
the most important properties of a delay network become its frequency and
|
||||||
phase response. A delay network that is designed specifically for its
|
phase response. A delay network that is designed specifically for its
|
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frequency or phase response is called a
|
frequency or phase response is called a
|
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<A NAME="10048"></A><I>filter</I>.
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<A ID="10048"></A><I>filter</I>.
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<DIV ALIGN="CENTER"><A NAME="fig08.01"></A><A NAME="10052"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.1:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.1:</STRONG>
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||||||
Representations of a filter: (a) in a block diagram; (b) a graph of its
|
Representations of a filter: (a) in a block diagram; (b) a graph of its
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||||||
@@ -116,7 +116,7 @@ depends on <IMG
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ALT="$H(\omega)$">, which is called the
|
ALT="$H(\omega)$">, which is called the
|
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<A NAME="10057"></A><I>transfer function</I>
|
<A ID="10057"></A><I>transfer function</I>
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of the filter.
|
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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HREF="node128.html">Taxonomy of filters</A>
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HREF="node129.html">Low-pass and high-pass filters</A>
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<LI><A NAME="tex2html2445"
|
<LI><A ID="tex2html2445"
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HREF="node130.html">Band-pass and stop-band filters</A>
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|
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<LI><A NAME="tex2html2448"
|
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HREF="node133.html">Elementary non-recirculating filter</A>
|
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|
<LI><A ID="tex2html2449"
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HREF="node134.html">Non-recirculating filter, second form</A>
|
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|
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HREF="node135.html">Elementary recirculating filter</A>
|
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|
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|
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HREF="node137.html">Real outputs from complex filters</A>
|
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|
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HREF="node138.html">Two recirculating filters for the price of one</A>
|
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|
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<LI><A NAME="tex2html2454"
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HREF="node139.html">Designing filters</A>
|
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|
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|
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<LI><A NAME="tex2html2455"
|
<LI><A ID="tex2html2455"
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HREF="node140.html">One-pole low-pass filter</A>
|
HREF="node140.html">One-pole low-pass filter</A>
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<LI><A NAME="tex2html2456"
|
<LI><A ID="tex2html2456"
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HREF="node141.html">One-pole, one-zero high-pass filter</A>
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<LI><A NAME="tex2html2457"
|
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HREF="node142.html">Shelving filter</A>
|
HREF="node142.html">Shelving filter</A>
|
||||||
<LI><A NAME="tex2html2458"
|
<LI><A ID="tex2html2458"
|
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HREF="node143.html">Band-pass filter</A>
|
HREF="node143.html">Band-pass filter</A>
|
||||||
<LI><A NAME="tex2html2459"
|
<LI><A ID="tex2html2459"
|
||||||
HREF="node144.html">Peaking and stop-band filter</A>
|
HREF="node144.html">Peaking and stop-band filter</A>
|
||||||
<LI><A NAME="tex2html2460"
|
<LI><A ID="tex2html2460"
|
||||||
HREF="node145.html">Butterworth filters</A>
|
HREF="node145.html">Butterworth filters</A>
|
||||||
<LI><A NAME="tex2html2461"
|
<LI><A ID="tex2html2461"
|
||||||
HREF="node146.html">Stretching the unit circle with rational functions</A>
|
HREF="node146.html">Stretching the unit circle with rational functions</A>
|
||||||
<LI><A NAME="tex2html2462"
|
<LI><A ID="tex2html2462"
|
||||||
HREF="node147.html">Butterworth band-pass filter</A>
|
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|
||||||
<LI><A NAME="tex2html2463"
|
<LI><A ID="tex2html2463"
|
||||||
HREF="node148.html">Time-varying coefficients</A>
|
HREF="node148.html">Time-varying coefficients</A>
|
||||||
<LI><A NAME="tex2html2464"
|
<LI><A ID="tex2html2464"
|
||||||
HREF="node149.html">Impulse responses of recirculating filters</A>
|
HREF="node149.html">Impulse responses of recirculating filters</A>
|
||||||
<LI><A NAME="tex2html2465"
|
<LI><A ID="tex2html2465"
|
||||||
HREF="node150.html">All-pass filters</A>
|
HREF="node150.html">All-pass filters</A>
|
||||||
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|
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|
||||||
<BR>
|
<BR>
|
||||||
<LI><A NAME="tex2html2466"
|
<LI><A ID="tex2html2466"
|
||||||
HREF="node151.html">Applications</A>
|
HREF="node151.html">Applications</A>
|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html2467"
|
<LI><A ID="tex2html2467"
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||||||
HREF="node152.html">Subtractive synthesis</A>
|
HREF="node152.html">Subtractive synthesis</A>
|
||||||
<LI><A NAME="tex2html2468"
|
<LI><A ID="tex2html2468"
|
||||||
HREF="node153.html">Envelope following</A>
|
HREF="node153.html">Envelope following</A>
|
||||||
<LI><A NAME="tex2html2469"
|
<LI><A ID="tex2html2469"
|
||||||
HREF="node154.html">Single Sideband Modulation</A>
|
HREF="node154.html">Single Sideband Modulation</A>
|
||||||
</UL>
|
</UL>
|
||||||
<BR>
|
<BR>
|
||||||
<LI><A NAME="tex2html2470"
|
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|
||||||
HREF="node155.html">Examples</A>
|
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|
||||||
<UL>
|
<UL>
|
||||||
<LI><A NAME="tex2html2471"
|
<LI><A ID="tex2html2471"
|
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HREF="node156.html">Prefabricated low-, high-, and band-pass filters</A>
|
HREF="node156.html">Prefabricated low-, high-, and band-pass filters</A>
|
||||||
<LI><A NAME="tex2html2472"
|
<LI><A ID="tex2html2472"
|
||||||
HREF="node157.html">Prefabricated time-varying band-pass filter</A>
|
HREF="node157.html">Prefabricated time-varying band-pass filter</A>
|
||||||
<LI><A NAME="tex2html2473"
|
<LI><A ID="tex2html2473"
|
||||||
HREF="node158.html">Envelope followers</A>
|
HREF="node158.html">Envelope followers</A>
|
||||||
<LI><A NAME="tex2html2474"
|
<LI><A ID="tex2html2474"
|
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HREF="node159.html">Single sideband modulation</A>
|
HREF="node159.html">Single sideband modulation</A>
|
||||||
<LI><A NAME="tex2html2475"
|
<LI><A ID="tex2html2475"
|
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HREF="node160.html">Using elementary filters directly: shelving and peaking</A>
|
HREF="node160.html">Using elementary filters directly: shelving and peaking</A>
|
||||||
<LI><A NAME="tex2html2476"
|
<LI><A ID="tex2html2476"
|
||||||
HREF="node161.html">Making and using all-pass filters</A>
|
HREF="node161.html">Making and using all-pass filters</A>
|
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</UL>
|
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|
||||||
<BR>
|
<BR>
|
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<LI><A NAME="tex2html2477"
|
<LI><A ID="tex2html2477"
|
||||||
HREF="node162.html">Exercises</A>
|
HREF="node162.html">Exercises</A>
|
||||||
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|
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HREF="node128.html">Taxonomy of filters</A>
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<B> <A NAME="tex2html2438"
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<B> <A ID="tex2html2438"
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HREF="node4.html">Contents</A></B>
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<ADDRESS>
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|||||||
+15
-15
@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node129.html">Low-pass and high-pass filters</A>
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HREF="node129.html">Low-pass and high-pass filters</A>
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<B> Up:</B> <A NAME="tex2html2485"
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<B> Up:</B> <A ID="tex2html2485"
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HREF="node127.html">Filters</A>
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HREF="node127.html">Filters</A>
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<B> Previous:</B> <A NAME="tex2html2479"
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<B> Previous:</B> <A ID="tex2html2479"
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HREF="node127.html">Filters</A>
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HREF="node127.html">Filters</A>
|
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<B> <A NAME="tex2html2487"
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<B> <A ID="tex2html2487"
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html2489"
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<B> <A ID="tex2html2489"
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HREF="node201.html">Index</A></B>
|
HREF="node201.html">Index</A></B>
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<H1><A NAME="SECTION001210000000000000000">
|
<H1><A ID="SECTION001210000000000000000">
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Taxonomy of filters</A>
|
Taxonomy of filters</A>
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</H1>
|
</H1>
|
||||||
|
|
||||||
@@ -80,14 +80,14 @@ applications of filters in electronic music.
|
|||||||
<P>
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<P>
|
||||||
<BR><HR>
|
<BR><HR>
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||||||
<!--Table of Child-Links-->
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html2492"
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HREF="node129.html">Low-pass and high-pass filters</A>
|
HREF="node129.html">Low-pass and high-pass filters</A>
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<LI><A NAME="tex2html2493"
|
<LI><A ID="tex2html2493"
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HREF="node130.html">Band-pass and stop-band filters</A>
|
HREF="node130.html">Band-pass and stop-band filters</A>
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<LI><A NAME="tex2html2494"
|
<LI><A ID="tex2html2494"
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HREF="node131.html">Equalizing filters</A>
|
HREF="node131.html">Equalizing filters</A>
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+29
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HREF="node130.html">Band-pass and stop-band filters</A>
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<B> Up:</B> <A NAME="tex2html2502"
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HREF="node128.html">Taxonomy of filters</A>
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html2506"
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<B> <A ID="tex2html2506"
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HREF="node201.html">Index</A></B>
|
HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001211000000000000000">
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Low-pass and high-pass filters</A>
|
Low-pass and high-pass filters</A>
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</H2>
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</H2>
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|
||||||
@@ -73,13 +73,13 @@ Low-pass and high-pass filters</A>
|
|||||||
By far the most frequent purpose for using a filter is extracting either
|
By far the most frequent purpose for using a filter is extracting either
|
||||||
the low-frequency or the high-frequency portion of an audio signal, attenuating
|
the low-frequency or the high-frequency portion of an audio signal, attenuating
|
||||||
the rest. This is accomplished using a
|
the rest. This is accomplished using a
|
||||||
<A NAME="10062"></A><A NAME="10063"></A><I>low-pass</I> or
|
<A ID="10062"></A><A ID="10063"></A><I>low-pass</I> or
|
||||||
<A NAME="10065"></A><A NAME="10066"></A><I>high-pass</I>
|
<A ID="10065"></A><A ID="10066"></A><I>high-pass</I>
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filter.
|
filter.
|
||||||
|
|
||||||
<P>
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<P>
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||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig08.02"></A><A NAME="10070"></A>
|
<DIV ALIGN="CENTER"><A ID="fig08.02"></A><A ID="10070"></A>
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||||||
<TABLE>
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<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.2:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.2:</STRONG>
|
||||||
Terminology for describing the frequency response of low-pass and
|
Terminology for describing the frequency response of low-pass and
|
||||||
@@ -104,7 +104,7 @@ and computation time we put into it, the closer we can get.
|
|||||||
Figure <A HREF="#fig08.02">8.2</A> shows the frequency response of a low-pass
|
Figure <A HREF="#fig08.02">8.2</A> shows the frequency response of a low-pass
|
||||||
filter. Frequency is divided into three bands, labeled on the
|
filter. Frequency is divided into three bands, labeled on the
|
||||||
horizontal axis. The
|
horizontal axis. The
|
||||||
<A NAME="10074"></A><I>passband</I>
|
<A ID="10074"></A><I>passband</I>
|
||||||
is the region (frequency band) where the filter should pass its input through
|
is the region (frequency band) where the filter should pass its input through
|
||||||
to its output with unit gain.
|
to its output with unit gain.
|
||||||
For a low-pass filter (as shown), the passband reaches from a frequency of
|
For a low-pass filter (as shown), the passband reaches from a frequency of
|
||||||
@@ -113,18 +113,18 @@ would appear on the right-hand side of the graph and would extend from the
|
|||||||
frequency limit up to the highest frequency possible. Any
|
frequency limit up to the highest frequency possible. Any
|
||||||
realizable filter's passband will be only approximately flat;
|
realizable filter's passband will be only approximately flat;
|
||||||
the deviation from flatness is called the
|
the deviation from flatness is called the
|
||||||
<A NAME="10076"></A><I>ripple</I>,
|
<A ID="10076"></A><I>ripple</I>,
|
||||||
and is often specified by giving the ratio between the highest and lowest gain
|
and is often specified by giving the ratio between the highest and lowest gain
|
||||||
in the passband, expressed in decibels. The ideal low-pass or high-pass filter
|
in the passband, expressed in decibels. The ideal low-pass or high-pass filter
|
||||||
would have a ripple of 0 dB.
|
would have a ripple of 0 dB.
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
The
|
The
|
||||||
<A NAME="10078"></A><I>stopband</I>
|
<A ID="10078"></A><I>stopband</I>
|
||||||
of a low-pass or high-pass filter is the frequency
|
of a low-pass or high-pass filter is the frequency
|
||||||
band over which the filter is intended not to transmit its input.
|
band over which the filter is intended not to transmit its input.
|
||||||
The
|
The
|
||||||
<A NAME="10080"></A><I>stopband attenuation</I>
|
<A ID="10080"></A><I>stopband attenuation</I>
|
||||||
is the difference, in decibels, between the lowest gain in the passband
|
is the difference, in decibels, between the lowest gain in the passband
|
||||||
and the highest gain in the stopband. Ideally this would
|
and the highest gain in the stopband. Ideally this would
|
||||||
be infinite; the higher the better.
|
be infinite; the higher the better.
|
||||||
@@ -134,42 +134,42 @@ Finally, a realizable filter, whose frequency response is always a
|
|||||||
continuous function of frequency, must have a frequency
|
continuous function of frequency, must have a frequency
|
||||||
band over which the gain drops from the passband gain to the stopband
|
band over which the gain drops from the passband gain to the stopband
|
||||||
gain; this is called the
|
gain; this is called the
|
||||||
<A NAME="10082"></A><I>transition band</I>.
|
<A ID="10082"></A><I>transition band</I>.
|
||||||
The thinner this band can be made, the more nearly ideal the filter.
|
The thinner this band can be made, the more nearly ideal the filter.
|
||||||
|
|
||||||
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|
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|
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+24
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@@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION00560000000000000000"></A>
|
<H1><A ID="SECTION00560000000000000000"></A>
|
||||||
<A NAME="sect1.combine"></A>
|
<A ID="sect1.combine"></A>
|
||||||
<BR>
|
<BR>
|
||||||
Superposing Signals
|
Superposing Signals
|
||||||
</H1>
|
</H1>
|
||||||
@@ -153,7 +153,7 @@ If we fix a window from <IMG
|
|||||||
SRC="img128.png"
|
SRC="img128.png"
|
||||||
ALT="$N+M-1$"> as usual, we can write out the
|
ALT="$N+M-1$"> as usual, we can write out the
|
||||||
mean power of the sum of two signals:
|
mean power of the sum of two signals:
|
||||||
<A NAME="eq-meanpowersum"></A>
|
<A ID="eq-meanpowersum"></A>
|
||||||
<BR><P></P>
|
<BR><P></P>
|
||||||
<DIV ALIGN="CENTER">
|
<DIV ALIGN="CENTER">
|
||||||
<!-- MATH
|
<!-- MATH
|
||||||
@@ -174,7 +174,7 @@ P \{x[n] + y[n]\} = P \{x[n]\} + P \{y[n]\}
|
|||||||
<BR CLEAR="ALL">
|
<BR CLEAR="ALL">
|
||||||
<P></P>
|
<P></P>
|
||||||
where we have introduced the
|
where we have introduced the
|
||||||
<A NAME="1161"></A><I>covariance</I> of two signals:
|
<A ID="1161"></A><I>covariance</I> of two signals:
|
||||||
<BR><P></P>
|
<BR><P></P>
|
||||||
<DIV ALIGN="CENTER">
|
<DIV ALIGN="CENTER">
|
||||||
<!-- MATH
|
<!-- MATH
|
||||||
@@ -276,36 +276,36 @@ amplitude <IMG
|
|||||||
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|
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+27
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@@ -30,50 +30,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H2><A NAME="SECTION001212000000000000000"></A>
|
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|
||||||
<A NAME="sect8.bandpassdef"></A>
|
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|
||||||
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|
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|
||||||
Band-pass and stop-band filters
|
Band-pass and stop-band filters
|
||||||
</H2>
|
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|
||||||
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|
||||||
<P>
|
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|
||||||
A
|
A
|
||||||
<A NAME="10086"></A><A NAME="10087"></A><I>band-pass filter</I>
|
<A ID="10086"></A><A ID="10087"></A><I>band-pass filter</I>
|
||||||
admits frequencies within a given band, rejecting frequencies below it and
|
admits frequencies within a given band, rejecting frequencies below it and
|
||||||
above it. Figure <A HREF="#fig08.03">8.3</A> shows the frequency response of a band-pass
|
above it. Figure <A HREF="#fig08.03">8.3</A> shows the frequency response of a band-pass
|
||||||
filter, with the key parameters labelled. A stop-band filter
|
filter, with the key parameters labelled. A stop-band filter
|
||||||
@@ -82,7 +82,7 @@ frequencies outside it.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig08.03"></A><A NAME="10092"></A>
|
<DIV ALIGN="CENTER"><A ID="fig08.03"></A><A ID="10092"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.3:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.3:</STRONG>
|
||||||
Terminology for describing the frequency response of band-pass and
|
Terminology for describing the frequency response of band-pass and
|
||||||
@@ -99,9 +99,9 @@ contiguous stopband surrounded by two passbands.</CAPTION>
|
|||||||
<P>
|
<P>
|
||||||
In practice, a simpler language is often used for describing bandpass filters,
|
In practice, a simpler language is often used for describing bandpass filters,
|
||||||
as shown in Figure <A HREF="#fig08.04">8.4</A>. Here there are only two parameters: a
|
as shown in Figure <A HREF="#fig08.04">8.4</A>. Here there are only two parameters: a
|
||||||
<A NAME="10096"></A><I>center frequency</I>
|
<A ID="10096"></A><I>center frequency</I>
|
||||||
and a
|
and a
|
||||||
<A NAME="10098"></A><I>bandwidth</I>.
|
<A ID="10098"></A><I>bandwidth</I>.
|
||||||
The passband is considered to be the region where the filter has at least half
|
The passband is considered to be the region where the filter has at least half
|
||||||
the power gain as at the peak (i.e., the gain is within 3 decibels of its
|
the power gain as at the peak (i.e., the gain is within 3 decibels of its
|
||||||
maximum). The bandwidth is the width, in frequency units, of the passband.
|
maximum). The bandwidth is the width, in frequency units, of the passband.
|
||||||
@@ -110,7 +110,7 @@ midpoint of the passband.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig08.04"></A><A NAME="10102"></A>
|
<DIV ALIGN="CENTER"><A ID="fig08.04"></A><A ID="10102"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.4:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.4:</STRONG>
|
||||||
A simplified view of a band-pass filter, showing bandwidth and
|
A simplified view of a band-pass filter, showing bandwidth and
|
||||||
@@ -125,36 +125,36 @@ center frequency.</CAPTION>
|
|||||||
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|
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+27
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<H2><A NAME="SECTION001213000000000000000"></A>
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<H2><A ID="SECTION001213000000000000000"></A>
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||||||
<A NAME="sect8.equalizing"></A>
|
<A ID="sect8.equalizing"></A>
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||||||
<BR>
|
<BR>
|
||||||
Equalizing filters
|
Equalizing filters
|
||||||
</H2>
|
</H2>
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
In some applications, such as
|
In some applications, such as
|
||||||
<A NAME="10107"></A><I>equalization</I>,
|
<A ID="10107"></A><I>equalization</I>,
|
||||||
the goal isn't to pass signals of certain frequencies while stopping others
|
the goal isn't to pass signals of certain frequencies while stopping others
|
||||||
altogether, but to make controllable adjustments, boosting or attenuating
|
altogether, but to make controllable adjustments, boosting or attenuating
|
||||||
a signal, over a frequency range, by a desired gain. Two filter types
|
a signal, over a frequency range, by a desired gain. Two filter types
|
||||||
are useful for this. First, a
|
are useful for this. First, a
|
||||||
<A NAME="10109"></A><A NAME="10110"></A><I>shelving filter</I>
|
<A ID="10109"></A><A ID="10110"></A><I>shelving filter</I>
|
||||||
(Figure <A HREF="#fig08.05">8.5</A>) is used for selectively boosting or reducing either the
|
(Figure <A HREF="#fig08.05">8.5</A>) is used for selectively boosting or reducing either the
|
||||||
low or high end of the frequency range. Below a selectable crossover frequency,
|
low or high end of the frequency range. Below a selectable crossover frequency,
|
||||||
the filter tends toward a low-frequency gain, and above it it tends toward a
|
the filter tends toward a low-frequency gain, and above it it tends toward a
|
||||||
@@ -86,7 +86,7 @@ and high-frequency gain can all be adjusted independently.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig08.05"></A><A NAME="10115"></A>
|
<DIV ALIGN="CENTER"><A ID="fig08.05"></A><A ID="10115"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.5:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.5:</STRONG>
|
||||||
A shelving filter, showing low and high frequency gain, and
|
A shelving filter, showing low and high frequency gain, and
|
||||||
@@ -100,7 +100,7 @@ crossover frequency.</CAPTION>
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
Second, a
|
Second, a
|
||||||
<A NAME="10118"></A><A NAME="10119"></A><I>peaking filter</I>
|
<A ID="10118"></A><A ID="10119"></A><I>peaking filter</I>
|
||||||
(Figure <A HREF="#fig08.06">8.6</A>) is capable of boosting or attenuating signals within
|
(Figure <A HREF="#fig08.06">8.6</A>) is capable of boosting or attenuating signals within
|
||||||
a range of frequencies. The center frequency and bandwidth (which together
|
a range of frequencies. The center frequency and bandwidth (which together
|
||||||
control the range of frequencies affected), and the in-band and out-of-band
|
control the range of frequencies affected), and the in-band and out-of-band
|
||||||
@@ -108,7 +108,7 @@ gains are separately adjustible.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig08.06"></A><A NAME="10124"></A>
|
<DIV ALIGN="CENTER"><A ID="fig08.06"></A><A ID="10124"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.6:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.6:</STRONG>
|
||||||
A peaking filter, with controllable center frequency, bandwidth,
|
A peaking filter, with controllable center frequency, bandwidth,
|
||||||
@@ -128,36 +128,36 @@ adjust bands in between.
|
|||||||
<P>
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+28
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@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A NAME="SECTION001220000000000000000">
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Elementary filters</A>
|
Elementary filters</A>
|
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</H1>
|
</H1>
|
||||||
|
|
||||||
@@ -103,55 +103,55 @@ filters--with complex-valued gains.
|
|||||||
<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html2549"
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<LI><A ID="tex2html2549"
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HREF="node133.html">Elementary non-recirculating filter</A>
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HREF="node133.html">Elementary non-recirculating filter</A>
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<LI><A NAME="tex2html2550"
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<LI><A ID="tex2html2550"
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HREF="node134.html">Non-recirculating filter, second form</A>
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HREF="node134.html">Non-recirculating filter, second form</A>
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<LI><A NAME="tex2html2551"
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<LI><A ID="tex2html2551"
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HREF="node135.html">Elementary recirculating filter</A>
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HREF="node135.html">Elementary recirculating filter</A>
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<LI><A NAME="tex2html2552"
|
<LI><A ID="tex2html2552"
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HREF="node136.html">Compound filters</A>
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HREF="node136.html">Compound filters</A>
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<LI><A NAME="tex2html2553"
|
<LI><A ID="tex2html2553"
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HREF="node137.html">Real outputs from complex filters</A>
|
HREF="node137.html">Real outputs from complex filters</A>
|
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<LI><A NAME="tex2html2554"
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<LI><A ID="tex2html2554"
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HREF="node138.html">Two recirculating filters for the price of one</A>
|
HREF="node138.html">Two recirculating filters for the price of one</A>
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</UL>
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</UL>
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HREF="node127.html">Filters</A>
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+26
-26
@@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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HREF="node4.html">Contents</A></B>
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<H2><A NAME="SECTION001221000000000000000"></A>
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<H2><A ID="SECTION001221000000000000000"></A>
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<A NAME="sect8.nonrecirculating"></A>
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<A ID="sect8.nonrecirculating"></A>
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Elementary non-recirculating filter
|
Elementary non-recirculating filter
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</H2>
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@@ -75,7 +75,7 @@ Elementary non-recirculating filter
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|||||||
The non-recirculating comb
|
The non-recirculating comb
|
||||||
filter may be generalized to yield the design shown in Figure <A HREF="#fig08.07">8.7</A>.
|
filter may be generalized to yield the design shown in Figure <A HREF="#fig08.07">8.7</A>.
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This is the
|
This is the
|
||||||
<A NAME="10133"></A><A NAME="10134"></A><I>elementary non-recirculating filter</I>,
|
<A ID="10133"></A><A ID="10134"></A><I>elementary non-recirculating filter</I>,
|
||||||
of the first form. Its single, complex-valued parameter <IMG
|
of the first form. Its single, complex-valued parameter <IMG
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WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.png"
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@@ -84,7 +84,7 @@ complex gain of the delayed signal subtracted from the original one.
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<DIV ALIGN="CENTER"><A NAME="fig08.07"></A><A NAME="10680"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.7:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.7:</STRONG>
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||||||
A delay network with a single-sample delay and a complex
|
A delay network with a single-sample delay and a complex
|
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@@ -250,7 +250,7 @@ plane, which is equal to the distance from <IMG
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<DIV ALIGN="CENTER"><A NAME="fig08.08"></A><A NAME="10688"></A>
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<DIV ALIGN="CENTER"><A ID="fig08.08"></A><A ID="10688"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.8:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.8:</STRONG>
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||||||
Diagram for calculating the frequency response of the
|
Diagram for calculating the frequency response of the
|
||||||
@@ -300,7 +300,7 @@ function for three different values of <IMG
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<DIV ALIGN="CENTER"><A NAME="fig08.09"></A><A NAME="10689"></A>
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<DIV ALIGN="CENTER"><A ID="fig08.09"></A><A ID="10689"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.9:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.9:</STRONG>
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||||||
Frequency response of the elementary non-recirculating filter
|
Frequency response of the elementary non-recirculating filter
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||||||
@@ -322,36 +322,36 @@ same argument (-2 radians), but with varying absolute value (magnitude) <IMG
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+24
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<B> <A NAME="tex2html2578"
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001222000000000000000"></A>
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<H2><A ID="SECTION001222000000000000000"></A>
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<A NAME="sect8.secondform"></A>
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<A ID="sect8.secondform"></A>
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<BR>
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<BR>
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Non-recirculating filter, second form
|
Non-recirculating filter, second form
|
||||||
</H2>
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</H2>
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||||||
@@ -80,7 +80,7 @@ Instead of multiplying the delay output by <IMG
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SRC="img42.png"
|
SRC="img42.png"
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ALT="$Q$"> we multiply the direct signal
|
ALT="$Q$"> we multiply the direct signal
|
||||||
by its
|
by its
|
||||||
<A NAME="10166"></A><I>complex conjugate</I>
|
<A ID="10166"></A><I>complex conjugate</I>
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||||||
<IMG
|
<IMG
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WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img872.png"
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SRC="img872.png"
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@@ -180,7 +180,7 @@ occasionally lead us to prefer the second form.
|
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|
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<P>
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|
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<DIV ALIGN="CENTER"><A NAME="fig08.10"></A><A NAME="10180"></A>
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.10:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.10:</STRONG>
|
||||||
The elementary non-recirculating filter, second form.</CAPTION>
|
The elementary non-recirculating filter, second form.</CAPTION>
|
||||||
@@ -194,36 +194,36 @@ The elementary non-recirculating filter, second form.</CAPTION>
|
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<H2><A NAME="SECTION001223000000000000000"></A>
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<H2><A ID="SECTION001223000000000000000"></A>
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<A NAME="sect8.recirculating"></A>
|
<A ID="sect8.recirculating"></A>
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||||||
<BR>
|
<BR>
|
||||||
Elementary recirculating filter
|
Elementary recirculating filter
|
||||||
</H2>
|
</H2>
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||||||
<P>
|
<P>
|
||||||
The
|
The
|
||||||
<A NAME="10185"></A><A NAME="10186"></A><I>elementary recirculating filter</I> is the recirculating comb filter of
|
<A ID="10185"></A><A ID="10186"></A><I>elementary recirculating filter</I> is the recirculating comb filter of
|
||||||
Figure <A HREF="node109.html#fig07.07">7.7</A> with a
|
Figure <A HREF="node109.html#fig07.07">7.7</A> with a
|
||||||
complex-valued feedback gain <IMG
|
complex-valued feedback gain <IMG
|
||||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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@@ -133,7 +133,7 @@ output grows exponentially as the delayed sample recirculates.
|
|||||||
|
|
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<P>
|
<P>
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||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig08.11"></A><A NAME="10197"></A>
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<DIV ALIGN="CENTER"><A ID="fig08.11"></A><A ID="10197"></A>
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.11:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.11:</STRONG>
|
||||||
The elementary recirculating filter: (a) block diagram; (b)
|
The elementary recirculating filter: (a) block diagram; (b)
|
||||||
@@ -158,36 +158,36 @@ working out the impulse response of the combined network).
|
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<H2><A NAME="SECTION001224000000000000000">
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Compound filters</A>
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Compound filters</A>
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</H2>
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||||||
|
|
||||||
<P>
|
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|
||||||
We can use the recirculating and non-recirculating filters developed here to
|
We can use the recirculating and non-recirculating filters developed here to
|
||||||
create a
|
create a
|
||||||
<A NAME="10201"></A><A NAME="10202"></A><I>compound filter</I> by putting several elementary ones in series. If the parameters
|
<A ID="10201"></A><A ID="10202"></A><I>compound filter</I> by putting several elementary ones in series. If the parameters
|
||||||
of the non-recirculating ones (of the first type) are <!-- MATH
|
of the non-recirculating ones (of the first type) are <!-- MATH
|
||||||
${Q_1}, \ldots, {Q_j}$
|
${Q_1}, \ldots, {Q_j}$
|
||||||
-->
|
-->
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+21
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@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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Real outputs from complex filters</A>
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Real outputs from complex filters</A>
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+21
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<H2><A NAME="SECTION001226000000000000000">
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Two recirculating filters for the price of one</A>
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Two recirculating filters for the price of one</A>
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HREF="node140.html">One-pole low-pass filter</A>
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<H1><A NAME="SECTION001230000000000000000">
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Designing filters</A>
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from the field of complex analysis. A plot showing the poles and zeroes
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associated with a filter is unimaginatively called a <A NAME="10312"></A><I>pole-zero plot</I>.
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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HREF="node145.html">Butterworth filters</A>
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<LI><A NAME="tex2html2657"
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HREF="node146.html">Stretching the unit circle with rational functions</A>
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HREF="node147.html">Butterworth band-pass filter</A>
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HREF="node149.html">Impulse responses of recirculating filters</A>
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HREF="node150.html">All-pass filters</A>
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+30
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HREF="node13.html">Superposing Signals</A>
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<H1><A NAME="SECTION00570000000000000000"></A>
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<A NAME="sect1.fourier"></A>
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Periodic Signals
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Periodic Signals
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@@ -105,7 +105,7 @@ the smallest <IMG
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ALT="$\tau$"> if any at which a signal repeats is called the signal's
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<A NAME="1175"></A><I>period</I>.
|
<A ID="1175"></A><I>period</I>.
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In discussing periods of digital audio signals, we quickly run into the
|
In discussing periods of digital audio signals, we quickly run into the
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difficulty of describing signals whose "period" isn't an integer, so that the
|
difficulty of describing signals whose "period" isn't an integer, so that the
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equation above doesn't make sense. For now we'll effectively
|
equation above doesn't make sense. For now we'll effectively
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@@ -149,7 +149,7 @@ with frequencies <!-- MATH
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SRC="img138.png"
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SRC="img138.png"
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ALT="$2 \pi / \omega$"> samples.
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ALT="$2 \pi / \omega$"> samples.
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Such a sum is called a <I>Fourier Series</I>:
|
Such a sum is called a <I>Fourier Series</I>:
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<A NAME="eq-fourierseries"></A>
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<!-- MATH
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<!-- MATH
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@@ -184,12 +184,12 @@ multiples of <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">. They are called the
|
ALT="$\omega $">. They are called the
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<A NAME="1187"></A><I>harmonics</I>
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<A ID="1187"></A><I>harmonics</I>
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of <IMG
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of <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">, which in turn is called the
|
ALT="$\omega $">, which in turn is called the
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<A NAME="1189"></A><I>fundamental</I>. In terms of pitch, the harmonics
|
<A ID="1189"></A><I>fundamental</I>. In terms of pitch, the harmonics
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<!-- MATH
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<!-- MATH
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$\omega, 2 \omega, \ldots$
|
$\omega, 2 \omega, \ldots$
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-->
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-->
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@@ -205,7 +205,7 @@ pitches of the same scale; the third and sixth miss only by 2 cents and the
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fifth misses by 14.
|
fifth misses by 14.
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<P>
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<P>
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<A NAME="r-intervals"></A>Put another way, the frequency ratio 3:2 (a perfect fifth in Western
|
<A ID="r-intervals"></A>Put another way, the frequency ratio 3:2 (a perfect fifth in Western
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terminology) is almost exactly
|
terminology) is almost exactly
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seven half-steps, 4:3 (a perfect fourth) is just as near to five half-steps,
|
seven half-steps, 4:3 (a perfect fourth) is just as near to five half-steps,
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and the ratios 5:4 and 6:5 (perfect major and minor thirds) are fairly close to
|
and the ratios 5:4 and 6:5 (perfect major and minor thirds) are fairly close to
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@@ -221,7 +221,7 @@ by the three component sinusoids.
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<DIV ALIGN="CENTER"><A NAME="fig01.08"></A><A NAME="1196"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.8:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.8:</STRONG>
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A Fourier series, showing three sinusoids and their sum. The
|
A Fourier series, showing three sinusoids and their sum. The
|
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@@ -242,7 +242,7 @@ block diagram for doing this.
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<DIV ALIGN="CENTER"><A NAME="fig01.09"></A><A NAME="1202"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.9:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.9:</STRONG>
|
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Using many oscillators to synthesize a waveform with desired harmonic
|
Using many oscillators to synthesize a waveform with desired harmonic
|
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@@ -256,43 +256,43 @@ amplitudes.</CAPTION>
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This is an example of
|
This is an example of
|
||||||
<A NAME="1205"></A><I>additive synthesis</I>; more generally the term can be applied to networks
|
<A ID="1205"></A><I>additive synthesis</I>; more generally the term can be applied to networks
|
||||||
in which the frequencies of the oscillators are independently controllable.
|
in which the frequencies of the oscillators are independently controllable.
|
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The early days of computer music rang with the sound of additive synthesis.
|
The early days of computer music rang with the sound of additive synthesis.
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<H2><A NAME="SECTION001231000000000000000"></A>
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<A NAME="sect08.low-pass"></A>
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One-pole low-pass filter
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One-pole low-pass filter
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.12:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.12:</STRONG>
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One-pole low-pass filter: (a) pole-zero diagram; (b)
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One-pole low-pass filter: (a) pole-zero diagram; (b)
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@@ -164,36 +164,36 @@ cycles per second.
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<H2><A NAME="SECTION001232000000000000000"></A>
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<A NAME="sect08.highpass"></A>
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One-pole, one-zero high-pass filter
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One-pole, one-zero high-pass filter
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This is inaudible, but, since it specifies electrical power that is sent
|
This is inaudible, but, since it specifies electrical power that is sent
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to your speakers, its presence reduces the level of loudness you can
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to your speakers, its presence reduces the level of loudness you can
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reach without distortion. Another name for a constant signal component is
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reach without distortion. Another name for a constant signal component is
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<A NAME="10330"></A>"DC", meaning "direct current".
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<A ID="10330"></A>"DC", meaning "direct current".
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An easy and practical way to remove the zero-frequency component from an audio
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An easy and practical way to remove the zero-frequency component from an audio
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@@ -135,7 +135,7 @@ in the previous example we plotted it all the way up to the sample rate,
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.13:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.13:</STRONG>
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One-pole, one-zero high-pass filter: (a) pole-zero diagram; (b)
|
One-pole, one-zero high-pass filter: (a) pole-zero diagram; (b)
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@@ -150,36 +150,36 @@ frequency response (from zero to Nyquist frequency).</CAPTION>
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<H2><A NAME="SECTION001233000000000000000"></A>
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<A NAME="sect8.shelving"></A>
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Shelving filter
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Shelving filter
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</H2>
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@@ -96,7 +96,7 @@ diagrammed in Figure <A HREF="#fig08.14">8.14</A>.
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.14:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.14:</STRONG>
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One-pole, one-zero shelving filter: (a) pole-zero diagram; (b)
|
One-pole, one-zero shelving filter: (a) pole-zero diagram; (b)
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Band-pass filter
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.15:</STRONG>
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Two-pole band-pass filter: (a) pole-zero diagram; (b)
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Peaking and stop-band filter
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A peaking filter: (a) pole-zero diagram; (b)
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A peaking filter: (a) pole-zero diagram; (b)
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Butterworth filters</A>
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Butterworth filters</A>
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@@ -86,7 +86,7 @@ whose two bands are flatter and separated by a narrower transition region.
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<P>
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A procedure borrowed from the analog filtering world transforms real,
|
A procedure borrowed from the analog filtering world transforms real,
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one-pole, one-zero filters to corresponding
|
one-pole, one-zero filters to corresponding
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<A NAME="10375"></A>
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<A ID="10375"></A>
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<I>Butterworth filters</I>,
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<I>Butterworth filters</I>,
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which have narrower transition regions. This procedure is described clearly
|
which have narrower transition regions. This procedure is described clearly
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and elegantly in the last chapter of [<A
|
and elegantly in the last chapter of [<A
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@@ -259,7 +259,7 @@ which, after some algebra, gives an angular frequency equal to
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<DIV ALIGN="CENTER"><A NAME="fig08.17"></A><A NAME="10393"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.17:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.17:</STRONG>
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Replacing a real-valued pole or zero (shown as a solid dot) with an
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Replacing a real-valued pole or zero (shown as a solid dot) with an
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@@ -300,7 +300,7 @@ becomes a shelving filter and then a high-pass one.
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<DIV ALIGN="CENTER"><A NAME="fig08.18"></A><A NAME="10399"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.18:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.18:</STRONG>
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Butterworth low-pass filter with three poles
|
Butterworth low-pass filter with three poles
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@@ -338,36 +338,36 @@ unit gain at the Nyquist frequency, and the others for unit gain at DC. </CAPTIO
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<H2><A NAME="SECTION001237000000000000000">
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Stretching the unit circle with rational functions</A>
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Stretching the unit circle with rational functions</A>
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</H2>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.19:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.19:</STRONG>
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One-pole, one-zero low-pass filter: (a) pole-zero plot; (b)
|
One-pole, one-zero low-pass filter: (a) pole-zero plot; (b)
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@@ -444,36 +444,36 @@ result is a band-pass filter with center frequency <IMG
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<H2><A NAME="SECTION001238000000000000000">
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Butterworth band-pass filter</A>
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Butterworth band-pass filter</A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.20:</STRONG>
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Butterworth band-pass filter: (a) pole-zero diagram; (b)
|
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@@ -422,36 +422,36 @@ low-pass filter used.</CAPTION>
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<H2><A NAME="SECTION001239000000000000000"></A>
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Time-varying coefficients
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Time-varying coefficients
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.21:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.21:</STRONG>
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Normalizing a recirculating elementary filter:
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Normalizing a recirculating elementary filter:
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+24
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<H2><A NAME="SECTION0012310000000000000000">
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Impulse responses of recirculating filters</A>
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Impulse responses of recirculating filters</A>
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ALT="$e$"> is Euler's constant, about
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2.718. The filter can be said to have a
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2.718. The filter can be said to have a
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<A NAME="10483"></A><I>settling time</I> of <IMG
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.22:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.22:</STRONG>
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The impulse response of three elementary recirculating (one-pole)
|
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ALT="$q$"> is the
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<A NAME="10499"></A><I>quality</I> of the filter, defined as the center frequency divided by
|
<A ID="10499"></A><I>quality</I> of the filter, defined as the center frequency divided by
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bandwidth. Resonant filters are often specified in terms of the center
|
bandwidth. Resonant filters are often specified in terms of the center
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frequency and "q" in place of bandwidth.
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frequency and "q" in place of bandwidth.
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About the Software Examples</A>
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<H2><A NAME="SECTION0012311000000000000000">
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.23:</STRONG>
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Phase response of all-pass filters with different pole locations <IMG
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HREF="node127.html">Filters</A>
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Applications</A>
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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HREF="node152.html">Subtractive synthesis</A>
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<LI><A NAME="tex2html2829"
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HREF="node153.html">Envelope following</A>
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HREF="node154.html">Single Sideband Modulation</A>
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<H2><A NAME="SECTION001241000000000000000"></A>
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<A NAME="sect08.subtractive"></A>
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Subtractive synthesis
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Subtractive synthesis
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<A NAME="10517"></A><I>Subtractive synthesis</I> is the technique of using filters to shape the
|
<A ID="10517"></A><I>Subtractive synthesis</I> is the technique of using filters to shape the
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||||||
spectral envelope of a sound, forming another sound, usually preserving
|
spectral envelope of a sound, forming another sound, usually preserving
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||||||
qualities of the original sound such as pitch, roughness, noisiness, or
|
qualities of the original sound such as pitch, roughness, noisiness, or
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graniness. The spectral envelope of the resulting sound is the product of the
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graniness. The spectral envelope of the resulting sound is the product of the
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@@ -82,7 +82,7 @@ filter, and result.
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<DIV ALIGN="CENTER"><A NAME="fig08.24"></A><A NAME="10522"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.24:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.24:</STRONG>
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Subtractive synthesis: (a) spectrum of input sound; (b) filter frequency
|
Subtractive synthesis: (a) spectrum of input sound; (b) filter frequency
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@@ -113,7 +113,7 @@ brass instruments over the life of a note.
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.25:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.25:</STRONG>
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ADSR-controlled subtractive synthesis.</CAPTION>
|
ADSR-controlled subtractive synthesis.</CAPTION>
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@@ -127,36 +127,36 @@ ADSR-controlled subtractive synthesis.</CAPTION>
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<H2><A NAME="SECTION001242000000000000000"></A>
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<A NAME="sect8.envelopefollower"></A>
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Envelope following
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Envelope following
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</H2>
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@@ -81,7 +81,7 @@ power over an interval of time long enough that its variations won't
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|||||||
show up in the power estimate, but short enough that changes in signal level
|
show up in the power estimate, but short enough that changes in signal level
|
||||||
are quickly reported. A computation that provides a
|
are quickly reported. A computation that provides a
|
||||||
time-varying power estimate of a signal is called an
|
time-varying power estimate of a signal is called an
|
||||||
<A NAME="10533"></A><I>envelope follower</I>.
|
<A ID="10533"></A><I>envelope follower</I>.
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||||||
|
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<P>
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<P>
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||||||
The output of a low-pass filter can be viewed as a moving average of its input.
|
The output of a low-pass filter can be viewed as a moving average of its input.
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@@ -133,7 +133,7 @@ the moving average.
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<DIV ALIGN="CENTER"><A NAME="fig08.26"></A><A NAME="10539"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.26:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.26:</STRONG>
|
||||||
Envelope follower. The output is the average power of the input
|
Envelope follower. The output is the average power of the input
|
||||||
@@ -232,7 +232,7 @@ estimates the average power.
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<DIV ALIGN="CENTER"><A NAME="fig08.27"></A><A NAME="10551"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.27:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.27:</STRONG>
|
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Envelope following from the spectral point of view: (a) an
|
Envelope following from the spectral point of view: (a) an
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@@ -260,36 +260,36 @@ getting a quick response and a smooth result.
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HREF="node153.html">Envelope following</A>
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<H2><A NAME="SECTION001243000000000000000"></A>
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Single Sideband Modulation
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Single Sideband Modulation
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HREF="node156.html">Prefabricated low-, high-, and</A>
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HREF="node127.html">Filters</A>
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HREF="node154.html">Single Sideband Modulation</A>
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Examples</A>
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html2885"
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HREF="node156.html">Prefabricated low-, high-, and band-pass filters</A>
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<LI><A NAME="tex2html2886"
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HREF="node157.html">Prefabricated time-varying band-pass filter</A>
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|
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HREF="node160.html">Using elementary filters directly: shelving and peaking</A>
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HREF="node161.html">Making and using all-pass filters</A>
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<H2><A NAME="SECTION001251000000000000000">
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Prefabricated low-, high-, and band-pass filters</A>
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Prefabricated low-, high-, and band-pass filters</A>
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</H2>
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WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1014.png"
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SRC="img1014.png"
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ALT="\fbox{\texttt{lop\~}}">:
|
ALT="\fbox{\texttt{lop\~}}">:
|
||||||
<A NAME="10842"></A>one-pole low-pass filter. The left inlet takes a signal to be filtered, and
|
<A ID="10842"></A>one-pole low-pass filter. The left inlet takes a signal to be filtered, and
|
||||||
the right inlet takes control messages to set the
|
the right inlet takes control messages to set the
|
||||||
cutoff frequency of the filter. The filter is normalized so that the
|
cutoff frequency of the filter. The filter is normalized so that the
|
||||||
gain is one at frequency 0.
|
gain is one at frequency 0.
|
||||||
@@ -97,7 +97,7 @@ gain is one at frequency 0.
|
|||||||
WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img1016.png"
|
SRC="img1016.png"
|
||||||
ALT="\fbox{ \texttt{hip\~}}">:
|
ALT="\fbox{ \texttt{hip\~}}">:
|
||||||
<A NAME="10843"></A>one-pole, one-zero high-pass filter, with the same inputs and outputs as
|
<A ID="10843"></A>one-pole, one-zero high-pass filter, with the same inputs and outputs as
|
||||||
<TT>lop~</TT>, normalized to have a gain of one at the Nyquist frequency.
|
<TT>lop~</TT>, normalized to have a gain of one at the Nyquist frequency.
|
||||||
|
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||||||
<P>
|
<P>
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||||||
@@ -108,7 +108,7 @@ gain is one at frequency 0.
|
|||||||
WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img1017.png"
|
SRC="img1017.png"
|
||||||
ALT="\fbox{ \texttt{bp\~}}">:
|
ALT="\fbox{ \texttt{bp\~}}">:
|
||||||
<A NAME="10844"></A>resonant filter. The middle inlet takes control messages to set the center
|
<A ID="10844"></A>resonant filter. The middle inlet takes control messages to set the center
|
||||||
frequency, and the right inlet to set "q".
|
frequency, and the right inlet to set "q".
|
||||||
|
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||||||
<P>
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<P>
|
||||||
@@ -119,12 +119,12 @@ frequency, and the right inlet to set "q".
|
|||||||
WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img1019.png"
|
SRC="img1019.png"
|
||||||
ALT="\fbox{ \texttt{noise\~}}">:
|
ALT="\fbox{ \texttt{noise\~}}">:
|
||||||
<A NAME="10845"></A>white noise generator. Each sample is an independent
|
<A ID="10845"></A>white noise generator. Each sample is an independent
|
||||||
pseudo-random number, uniformly distributed from -1 to 1.
|
pseudo-random number, uniformly distributed from -1 to 1.
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig08.28"></A><A NAME="10600"></A>
|
<DIV ALIGN="CENTER"><A ID="fig08.28"></A><A ID="10600"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.28:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.28:</STRONG>
|
||||||
Using prefabricated filters in Pd: (a) a low-pass filter, with
|
Using prefabricated filters in Pd: (a) a low-pass filter, with
|
||||||
@@ -146,36 +146,36 @@ frequency) component of a signal.
|
|||||||
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|
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|
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<H2><A NAME="SECTION001252000000000000000">
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<H2><A ID="SECTION001252000000000000000">
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||||||
Prefabricated time-varying band-pass filter</A>
|
Prefabricated time-varying band-pass filter</A>
|
||||||
</H2>
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</H2>
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||||||
|
|
||||||
@@ -82,7 +82,7 @@ using the <TT>vcf~</TT> object, introduced here:
|
|||||||
WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
|
||||||
SRC="img1022.png"
|
SRC="img1022.png"
|
||||||
ALT="\fbox{ \texttt{vcf\~}}">:
|
ALT="\fbox{ \texttt{vcf\~}}">:
|
||||||
<A NAME="10846"></A>a "voltage controlled" band-pass filter,
|
<A ID="10846"></A>a "voltage controlled" band-pass filter,
|
||||||
similar to <TT>bp~</TT>, but with a signal inlet to control center frequency.
|
similar to <TT>bp~</TT>, but with a signal inlet to control center frequency.
|
||||||
Both <TT>bp~</TT> and <TT>vcf~</TT> are one-pole resonant filters as
|
Both <TT>bp~</TT> and <TT>vcf~</TT> are one-pole resonant filters as
|
||||||
developed in Section <A HREF="node143.html#sect8.twopolebandpass">8.3.4</A>; <TT>bp~</TT> outputs only
|
developed in Section <A HREF="node143.html#sect8.twopolebandpass">8.3.4</A>; <TT>bp~</TT> outputs only
|
||||||
@@ -91,7 +91,7 @@ real and imaginary parts separately.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig08.29"></A><A NAME="10612"></A>
|
<DIV ALIGN="CENTER"><A ID="fig08.29"></A><A ID="10612"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.29:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.29:</STRONG>
|
||||||
The <TT>vcf~</TT> band-pass filter, with its center frequency
|
The <TT>vcf~</TT> band-pass filter, with its center frequency
|
||||||
@@ -119,36 +119,36 @@ subtractive synthesis is demonstrated in example H05.filter.floyd.pd.
|
|||||||
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Envelope followers</A>
|
Envelope followers</A>
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||||||
</H2>
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||||||
<A NAME="sect8.heterodyning"></A>
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<A ID="sect8.heterodyning"></A>
|
||||||
<P>
|
<P>
|
||||||
Example H06.envelope.follower.pd shows a simple and self-explanatory realization of the
|
Example H06.envelope.follower.pd shows a simple and self-explanatory realization of the
|
||||||
envelope follower described in Section <A HREF="node153.html#sect8.envelopefollower">8.4.2</A>. An
|
envelope follower described in Section <A HREF="node153.html#sect8.envelopefollower">8.4.2</A>. An
|
||||||
@@ -84,7 +84,7 @@ strength.
|
|||||||
|
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||||||
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||||||
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<DIV ALIGN="CENTER"><A NAME="fig08.30"></A><A NAME="10622"></A>
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||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.30:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.30:</STRONG>
|
||||||
Analyzing the spectrum of a sound: (a) band-pass filtering a sampled
|
Analyzing the spectrum of a sound: (a) band-pass filtering a sampled
|
||||||
@@ -113,36 +113,36 @@ analysis, the subject of Chapter 9.
|
|||||||
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+12
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HREF="node158.html">Envelope followers</A>
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HREF="node158.html">Envelope followers</A>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001254000000000000000">
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Single sideband modulation</A>
|
Single sideband modulation</A>
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</H2>
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<DIV ALIGN="CENTER"><A NAME="fig08.31"></A><A NAME="10628"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.31:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.31:</STRONG>
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Using an all-pass filter network to
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Using an all-pass filter network to
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||||||
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+31
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HREF="node15.html">About the Software Examples</A>
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HREF="node15.html">About the Software Examples</A>
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<B> <A NAME="tex2html776"
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<B> <A ID="tex2html776"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html778"
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<B> <A ID="tex2html778"
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HREF="node201.html">Index</A></B>
|
HREF="node201.html">Index</A></B>
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||||||
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<BR>
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||||||
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|
||||||
<H2><A NAME="SECTION00581000000000000000">
|
<H2><A ID="SECTION00581000000000000000">
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||||||
Quick Introduction to Pd</A>
|
Quick Introduction to Pd</A>
|
||||||
</H2>
|
</H2>
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
Pd documents are called
|
Pd documents are called
|
||||||
<A NAME="1211"></A><I>patches</I>. They correspond roughly to the
|
<A ID="1211"></A><I>patches</I>. They correspond roughly to the
|
||||||
boxes in the abstract block diagrams shown earlier in this chapter, but in
|
boxes in the abstract block diagrams shown earlier in this chapter, but in
|
||||||
detail they
|
detail they
|
||||||
are quite different, because Pd is an implementation
|
are quite different, because Pd is an implementation
|
||||||
@@ -80,7 +80,7 @@ environment, not a specification language.
|
|||||||
<P>
|
<P>
|
||||||
A Pd patch, such as the ones shown in
|
A Pd patch, such as the ones shown in
|
||||||
Figure <A HREF="#fig01.10">1.10</A>, consists of a collection of
|
Figure <A HREF="#fig01.10">1.10</A>, consists of a collection of
|
||||||
<A NAME="1214"></A><I>boxes</I>
|
<A ID="1214"></A><I>boxes</I>
|
||||||
connected in a network.
|
connected in a network.
|
||||||
The border of a box tells you how its text is interpreted and how the box
|
The border of a box tells you how its text is interpreted and how the box
|
||||||
functions. In part (a) of the figure we see three types of boxes. From
|
functions. In part (a) of the figure we see three types of boxes. From
|
||||||
@@ -88,7 +88,7 @@ top to bottom they are:
|
|||||||
|
|
||||||
<UL>
|
<UL>
|
||||||
<LI>a
|
<LI>a
|
||||||
<A NAME="1217"></A><A NAME="1218"></A><I>message box</I>.
|
<A ID="1217"></A><A ID="1218"></A><I>message box</I>.
|
||||||
Message boxes, with a flag-shaped border, interpret the text as a message to
|
Message boxes, with a flag-shaped border, interpret the text as a message to
|
||||||
send whenever the box is
|
send whenever the box is
|
||||||
activated (by an incoming message or with a pointing device). The message in this
|
activated (by an incoming message or with a pointing device). The message in this
|
||||||
@@ -97,7 +97,7 @@ case consists simply of the number "21".
|
|||||||
<P>
|
<P>
|
||||||
</LI>
|
</LI>
|
||||||
<LI>an
|
<LI>an
|
||||||
<A NAME="1220"></A><A NAME="1221"></A><I>object box</I>.
|
<A ID="1220"></A><A ID="1221"></A><I>object box</I>.
|
||||||
Object boxes have a rectangular border; they interpret the text to create
|
Object boxes have a rectangular border; they interpret the text to create
|
||||||
objects
|
objects
|
||||||
when you load a patch. Object boxes may hold hundreds of different
|
when you load a patch. Object boxes may hold hundreds of different
|
||||||
@@ -106,18 +106,18 @@ signal processing modules to be introduced later--depending on the text
|
|||||||
inside. In this example, the box holds an adder. In most Pd patches, the
|
inside. In this example, the box holds an adder. In most Pd patches, the
|
||||||
majority of boxes are of type "object". The first word typed into an object
|
majority of boxes are of type "object". The first word typed into an object
|
||||||
box specifies its
|
box specifies its
|
||||||
<A NAME="1223"></A><I>class</I>,
|
<A ID="1223"></A><I>class</I>,
|
||||||
which in this case is just "+". Any additional (blank-space-separated) words
|
which in this case is just "+". Any additional (blank-space-separated) words
|
||||||
appearing in the box are called
|
appearing in the box are called
|
||||||
<A NAME="1225"></A><A NAME="1226"></A><I>creation arguments</I>,
|
<A ID="1225"></A><A ID="1226"></A><I>creation arguments</I>,
|
||||||
which specify the initial state of the object when it is created.
|
which specify the initial state of the object when it is created.
|
||||||
|
|
||||||
<P>
|
<P>
|
||||||
</LI>
|
</LI>
|
||||||
<LI>a
|
<LI>a
|
||||||
<A NAME="1228"></A><A NAME="1229"></A><I>number box</I>.
|
<A ID="1228"></A><A ID="1229"></A><I>number box</I>.
|
||||||
Number boxes are a particular type of
|
Number boxes are a particular type of
|
||||||
<A NAME="1231"></A><A NAME="1232"></A><I>GUI box</I>. Others include push buttons and
|
<A ID="1231"></A><A ID="1232"></A><I>GUI box</I>. Others include push buttons and
|
||||||
toggle switches; these will come up later in the examples.
|
toggle switches; these will come up later in the examples.
|
||||||
The number box
|
The number box
|
||||||
has a punched-card-shaped border, with a nick out of its top right corner.
|
has a punched-card-shaped border, with a nick out of its top right corner.
|
||||||
@@ -137,7 +137,7 @@ from one box to the next; outputs of boxes are on the bottom and inputs on top.
|
|||||||
|
|
||||||
<P>
|
<P>
|
||||||
|
|
||||||
<DIV ALIGN="CENTER"><A NAME="fig01.10"></A><A NAME="1238"></A>
|
<DIV ALIGN="CENTER"><A ID="fig01.10"></A><A ID="1238"></A>
|
||||||
<TABLE>
|
<TABLE>
|
||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.10:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.10:</STRONG>
|
||||||
(a) three types of boxes in Pd (message, object, and GUI); (b) a simple patch to output a
|
(a) three types of boxes in Pd (message, object, and GUI); (b) a simple patch to output a
|
||||||
@@ -153,7 +153,7 @@ sinusoid.</CAPTION>
|
|||||||
Figure <A HREF="#fig01.10">1.10</A> (part b) shows a Pd patch which
|
Figure <A HREF="#fig01.10">1.10</A> (part b) shows a Pd patch which
|
||||||
makes a sinusoid with controllable frequency and amplitude. The connecting
|
makes a sinusoid with controllable frequency and amplitude. The connecting
|
||||||
patch lines are of two types here; the thin ones are for carrying sporadic
|
patch lines are of two types here; the thin ones are for carrying sporadic
|
||||||
<A NAME="1242"></A><I>messages</I>,
|
<A ID="1242"></A><I>messages</I>,
|
||||||
and the thicker ones (connecting the oscillator, the multiplier, and the output
|
and the thicker ones (connecting the oscillator, the multiplier, and the output
|
||||||
<TT>dac~</TT> object) carry digital audio signals. Since Pd is a real-time
|
<TT>dac~</TT> object) carry digital audio signals. Since Pd is a real-time
|
||||||
program, the audio signals flow in a continuous stream. On the other hand, the
|
program, the audio signals flow in a continuous stream. On the other hand, the
|
||||||
@@ -173,36 +173,36 @@ and "<TT>osc~</TT>".
|
|||||||
<P>
|
<P>
|
||||||
<HR>
|
<HR>
|
||||||
<!--Navigation Panel-->
|
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<H2><A NAME="SECTION001255000000000000000">
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Using elementary filters directly: shelving and peaking</A>
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Using elementary filters directly: shelving and peaking</A>
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<DIV ALIGN="CENTER"><A NAME="fig08.32"></A><A NAME="10641"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.32:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.32:</STRONG>
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Building filters from elementary, raw ones: (a) shelving; (b) peaking.
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Building filters from elementary, raw ones: (a) shelving; (b) peaking.
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@@ -114,7 +114,7 @@ First we introduce the six Pd objects that realize elementary filters:
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WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1030.png"
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ALT="\fbox{ \texttt{rpole\~}}">:
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ALT="\fbox{ \texttt{rpole\~}}">:
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<A NAME="10847"></A><A NAME="10848"></A><A NAME="10849"></A>elementary filters with real-valued coefficients operating on real-valued
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<A ID="10847"></A><A ID="10848"></A><A ID="10849"></A>elementary filters with real-valued coefficients operating on real-valued
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signals. The three implement non-recirculating filters of the first and
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signals. The three implement non-recirculating filters of the first and
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second types, and the recirculating filter. They all have one inlet, at
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second types, and the recirculating filter. They all have one inlet, at
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right, to supply the coefficient that sets the location of the zero or pole.
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right, to supply the coefficient that sets the location of the zero or pole.
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@@ -144,7 +144,7 @@ audio signals. No stability check is performed.
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WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1036.png"
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SRC="img1036.png"
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ALT="\fbox{ \texttt{cpole\~}}">:
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ALT="\fbox{ \texttt{cpole\~}}">:
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<A NAME="10850"></A><A NAME="10851"></A><A NAME="10852"></A>elementary filters with complex-valued coefficients, operating on complex-valued
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<A ID="10850"></A><A ID="10851"></A><A ID="10852"></A>elementary filters with complex-valued coefficients, operating on complex-valued
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||||||
signals, corresponding to the real-valued ones above. Instead of two inlets and
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signals, corresponding to the real-valued ones above. Instead of two inlets and
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one outlet, each of these filters has four inlets (real and imaginary part of
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one outlet, each of these filters has four inlets (real and imaginary part of
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||||||
the signal to filter, and real and imaginary part of the coefficient) and
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the signal to filter, and real and imaginary part of the coefficient) and
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@@ -175,36 +175,36 @@ easy reuse.
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<H2><A NAME="SECTION001256000000000000000">
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<H2><A ID="SECTION001256000000000000000">
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Making and using all-pass filters</A>
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Making and using all-pass filters</A>
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</H2>
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<DIV ALIGN="CENTER"><A NAME="fig08.33"></A><A NAME="10672"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.33:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.33:</STRONG>
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All-pass filters: (a) making an all-pass filter from elementary
|
All-pass filters: (a) making an all-pass filter from elementary
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||||||
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+21
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Exercises</A>
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Exercises</A>
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+34
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HREF="node162.html">Exercises</A>
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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<H1><A NAME="SECTION001300000000000000000"></A>
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<A NAME="chapter-fourier"></A><A NAME="chapter-fft"></A>
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<BR>
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Fourier analysis and resynthesis
|
Fourier analysis and resynthesis
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@@ -78,7 +78,7 @@ amplitude and phase of a sinusoidal component of a signal (Page
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<A HREF="node158.html#sect8.heterodyning"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>). In this chapter we will refine this technique
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SRC="crossref.png"></A>). In this chapter we will refine this technique
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into what is called
|
into what is called
|
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<A NAME="12298"></A><I>Fourier analysis</I>.
|
<A ID="12298"></A><I>Fourier analysis</I>.
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In its simplest form, Fourier analysis takes as input any periodic
|
In its simplest form, Fourier analysis takes as input any periodic
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signal (of period <IMG
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signal (of period <IMG
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@@ -104,63 +104,63 @@ Finally, we will develop some standard applications such as the phase vocoder.
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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HREF="node166.html">Fourier transform as additive synthesis</A>
|
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HREF="node167.html">Properties of Fourier transforms</A>
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|
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|
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HREF="node173.html">Narrow-band companding</A>
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<LI><A NAME="tex2html3009"
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|
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HREF="node179.html">Fourier analysis and resynthesis in Pd</A>
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HREF="node180.html">Narrow-band companding: noise suppression</A>
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HREF="node181.html">Timbre stamp ("vocoder")</A>
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HREF="node182.html">Phase vocoder time bender</A>
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HREF="node183.html">Exercises</A>
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<H1><A NAME="SECTION001310000000000000000">
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Fourier analysis of periodic signals</A>
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Fourier analysis of periodic signals</A>
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<BR CLEAR="ALL">
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<P></P>
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This is such a useful formula that it gets its own notation. The
|
This is such a useful formula that it gets its own notation. The
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<A NAME="12325"></A><I>Fourier transform</I>
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<A ID="12325"></A><I>Fourier transform</I>
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of a signal <IMG
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of a signal <IMG
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HREF="node165.html">Periodicity of the Fourier transform</A>
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HREF="node166.html">Fourier transform as additive synthesis</A>
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+11
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<H2><A NAME="SECTION001311000000000000000">
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Periodicity of the Fourier transform</A>
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Periodicity of the Fourier transform</A>
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<H2><A NAME="SECTION001312000000000000000"></A>
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Fourier transform as additive synthesis
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Fourier transform as additive synthesis
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<H1><A NAME="SECTION001320000000000000000">
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Properties of Fourier transforms</A>
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Properties of Fourier transforms</A>
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</H1>
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@@ -79,14 +79,14 @@ result to any other sinusoid.
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<!--Table of Child-Links-->
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html3075"
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HREF="node168.html">Fourier transform of DC</A>
|
HREF="node168.html">Fourier transform of DC</A>
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<LI><A NAME="tex2html3076"
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HREF="node169.html">Shifts and phase changes</A>
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HREF="node169.html">Shifts and phase changes</A>
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<LI><A NAME="tex2html3077"
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<LI><A ID="tex2html3077"
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HREF="node170.html">Fourier transform of a sinusoid</A>
|
HREF="node170.html">Fourier transform of a sinusoid</A>
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</UL>
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+25
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HREF="node167.html">Properties of Fourier transforms</A>
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HREF="node167.html">Properties of Fourier transforms</A>
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001321000000000000000">
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Fourier transform of DC</A>
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Fourier transform of DC</A>
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</H2>
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@@ -330,7 +330,7 @@ where <IMG
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WIDTH="49" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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WIDTH="49" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1090.png"
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SRC="img1090.png"
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ALT="${D_N}(k)$">, known as the
|
ALT="${D_N}(k)$">, known as the
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<A NAME="12394"></A><I>Dirichlet kernel</I>,
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<A ID="12394"></A><I>Dirichlet kernel</I>,
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is defined as
|
is defined as
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<BR><P></P>
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<BR><P></P>
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@@ -430,7 +430,7 @@ the <IMG
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WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1092.png"
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SRC="img1092.png"
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ALT="$k=0$">. This is surrounded by one-unit-wide
|
ALT="$k=0$">. This is surrounded by one-unit-wide
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<A NAME="12409"></A><I>sidelobes</I>,
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<A ID="12409"></A><I>sidelobes</I>,
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||||||
alternating in sign and gradually decreasing in magnitude as <IMG
|
alternating in sign and gradually decreasing in magnitude as <IMG
|
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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SRC="img58.png"
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@@ -461,7 +461,7 @@ radians over the peak from <IMG
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<DIV ALIGN="CENTER"><A NAME="fig09.01"></A><A NAME="12413"></A>
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<DIV ALIGN="CENTER"><A ID="fig09.01"></A><A ID="12413"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.1:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.1:</STRONG>
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The Fourier transform of a signal consisting of all ones. Here
|
The Fourier transform of a signal consisting of all ones. Here
|
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@@ -480,7 +480,7 @@ page and the imaginary axis pointing away from it.</CAPTION>
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<DIV ALIGN="CENTER"><A NAME="fig09.02"></A><A NAME="12418"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.2:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.2:</STRONG>
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The Dirichlet kernel, for <IMG
|
The Dirichlet kernel, for <IMG
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@@ -497,36 +497,36 @@ The Dirichlet kernel, for <IMG
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+24
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001322000000000000000"></A>
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<A NAME="sect9.shift"></A>
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Shifts and phase changes
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Shifts and phase changes
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</H2>
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<P></P>
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<P></P>
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(The third line is just the second one with the terms summed in a
|
(The third line is just the second one with the terms summed in a
|
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different order). We therefore get the Time Shift Formula for Fourier
|
different order). We therefore get the Time Shift Formula for Fourier
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Transforms: <A NAME="12441"></A>
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Transforms: <A ID="12441"></A>
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<BR><P></P>
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<!-- MATH
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<!-- MATH
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@@ -430,7 +430,7 @@ The Fourier transform is:
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<BR CLEAR="ALL">
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<P></P>
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<P></P>
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We therefore get the Phase Shift Formula for Fourier Transforms:
|
We therefore get the Phase Shift Formula for Fourier Transforms:
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<A NAME="12464"></A>
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<A ID="12464"></A>
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<!-- MATH
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<!-- MATH
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@@ -454,36 +454,36 @@ We therefore get the Phase Shift Formula for Fourier Transforms:
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To run the patches, you must first download, install, and run Pd.
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Instructions for
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doing this appear in Pd's on-line HTML documentation, which you can find at
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Choose the revision that corresponds to the text you're reading (or perhaps
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just the latest one) and download the archive
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<H2><A NAME="SECTION001323000000000000000">
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Fourier transform of a sinusoid</A>
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Fourier transform of a sinusoid</A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.3:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.3:</STRONG>
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Fourier transforms of complex sinusoids, with <IMG
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Fourier transforms of complex sinusoids, with <IMG
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.4:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.4:</STRONG>
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A complex sinusoid with frequency <!-- MATH
|
A complex sinusoid with frequency <!-- MATH
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<B> <A NAME="tex2html3113"
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<H1><A NAME="SECTION001330000000000000000">
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Fourier analysis of non-periodic signals</A>
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Fourier analysis of non-periodic signals</A>
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</H1>
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</H1>
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@@ -110,12 +110,12 @@ Given a signal <IMG
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ALT="$N-1$">,
|
ALT="$N-1$">,
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the technique is to envelope the signal before doing the Fourier analysis.
|
the technique is to envelope the signal before doing the Fourier analysis.
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The envelope shape is known as a
|
The envelope shape is known as a
|
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<A NAME="12499"></A><I>window function</I>.
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<A ID="12499"></A><I>window function</I>.
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Given a window function <IMG
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Given a window function <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1123.png"
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SRC="img1123.png"
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ALT="$w[n]$">, the
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ALT="$w[n]$">, the
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<A NAME="12501"></A><I>windowed Fourier transform</I>
|
<A ID="12501"></A><I>windowed Fourier transform</I>
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is:
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is:
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||||||
<BR><P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<DIV ALIGN="CENTER">
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@@ -136,7 +136,7 @@ is:
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<P></P>
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<P></P>
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Much ink has been spilled over the design of suitable window functions for
|
Much ink has been spilled over the design of suitable window functions for
|
||||||
particular situations, but here we will consider the simplest one, named the
|
particular situations, but here we will consider the simplest one, named the
|
||||||
<A NAME="12504"></A><A NAME="12505"></A><I>Hann</I>
|
<A ID="12504"></A><A ID="12505"></A><I>Hann</I>
|
||||||
window function (the name is sometimes corrupted to "Hanning" in DSP circles).
|
window function (the name is sometimes corrupted to "Hanning" in DSP circles).
|
||||||
The Hann window is:
|
The Hann window is:
|
||||||
<BR><P></P>
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<BR><P></P>
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@@ -302,7 +302,7 @@ Dirichlet kernel components are also shown separately.
|
|||||||
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<P>
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<P>
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||||||
|
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<DIV ALIGN="CENTER"><A NAME="fig09.05"></A><A NAME="12536"></A>
|
<DIV ALIGN="CENTER"><A ID="fig09.05"></A><A ID="12536"></A>
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.5:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.5:</STRONG>
|
||||||
The magnitude M(k) of the Fourier transform of the Hann
|
The magnitude M(k) of the Fourier transform of the Hann
|
||||||
@@ -450,7 +450,7 @@ Reducing the amplitude of the sidelobes reduces this interference.
|
|||||||
|
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<P>
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<P>
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|
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<DIV ALIGN="CENTER"><A NAME="fig09.06"></A><A NAME="12558"></A>
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<DIV ALIGN="CENTER"><A ID="fig09.06"></A><A ID="12558"></A>
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||||||
<TABLE>
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<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.6:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.6:</STRONG>
|
||||||
The Hann-windowed Fourier transform of a signal with two
|
The Hann-windowed Fourier transform of a signal with two
|
||||||
@@ -507,36 +507,36 @@ four-period rule may be relaxed to three or even slightly less.
|
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+27
-27
@@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds
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<B> <A ID="tex2html3141"
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html3143"
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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<H1><A NAME="SECTION001340000000000000000">
|
<H1><A ID="SECTION001340000000000000000">
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Fourier analysis and reconstruction of audio signals</A>
|
Fourier analysis and reconstruction of audio signals</A>
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</H1>
|
</H1>
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@@ -90,7 +90,7 @@ modified transform.
|
|||||||
|
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|
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<DIV ALIGN="CENTER"><A NAME="fig09.07"></A><A NAME="12565"></A>
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<DIV ALIGN="CENTER"><A ID="fig09.07"></A><A ID="12565"></A>
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||||||
<TABLE>
|
<TABLE>
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||||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.7:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.7:</STRONG>
|
||||||
Sliding-window analysis and resynthesis of an audio signal using
|
Sliding-window analysis and resynthesis of an audio signal using
|
||||||
@@ -107,7 +107,7 @@ Fourier transform with a desired frequency response.</CAPTION>
|
|||||||
Figure <A HREF="#fig09.07">9.7</A> shows how to carry out a Fourier analysis, modification,
|
Figure <A HREF="#fig09.07">9.7</A> shows how to carry out a Fourier analysis, modification,
|
||||||
and reconstruction of an audio signal. The first step is to divide the
|
and reconstruction of an audio signal. The first step is to divide the
|
||||||
signal into
|
signal into
|
||||||
<A NAME="12569"></A><I>windows</I>,
|
<A ID="12569"></A><I>windows</I>,
|
||||||
which are segments of the signal, of <IMG
|
which are segments of the signal, of <IMG
|
||||||
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
|
SRC="img3.png"
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@@ -137,7 +137,7 @@ Let <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img25.png"
|
SRC="img25.png"
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ALT="$H$"> denote the
|
ALT="$H$"> denote the
|
||||||
<A NAME="12571"></A><I>hop size</I>,
|
<A ID="12571"></A><I>hop size</I>,
|
||||||
the number of samples each window is advanced past the
|
the number of samples each window is advanced past the
|
||||||
previous window. Then for each <!-- MATH
|
previous window. Then for each <!-- MATH
|
||||||
$m = \ldots, 0, 1, \ldots$
|
$m = \ldots, 0, 1, \ldots$
|
||||||
@@ -310,47 +310,47 @@ to vary smoothly between the input and some transformed version of it.
|
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<P>
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<P>
|
||||||
<BR><HR>
|
<BR><HR>
|
||||||
<!--Table of Child-Links-->
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<!--Table of Child-Links-->
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||||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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||||||
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HREF="node173.html">Narrow-band companding</A>
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<LI><A NAME="tex2html3147"
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<LI><A ID="tex2html3147"
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HREF="node174.html">Timbre stamping (classical vocoder)</A>
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HREF="node174.html">Timbre stamping (classical vocoder)</A>
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<H2><A NAME="SECTION001341000000000000000">
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<H2><A ID="SECTION001341000000000000000">
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Narrow-band companding</A>
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Narrow-band companding</A>
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</H2>
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<DIV ALIGN="CENTER"><A NAME="fig09.08"></A><A NAME="12579"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.8:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.8:</STRONG>
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Block diagram for narrow-band noise suppression by companding.</CAPTION>
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Block diagram for narrow-band noise suppression by companding.</CAPTION>
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@@ -84,12 +84,12 @@ Block diagram for narrow-band noise suppression by companding.</CAPTION>
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<P>
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<P>
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A
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A
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<A NAME="12582"></A><I>compander</I>
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<A ID="12582"></A><I>compander</I>
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||||||
is a tool that amplifies a signal with a variable gain, depending on the
|
is a tool that amplifies a signal with a variable gain, depending on the
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signal's measured amplitude. The term is a contraction of "compressor" and
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signal's measured amplitude. The term is a contraction of "compressor" and
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"expander". A compressor's gain decreases as the input level increases, so
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"expander". A compressor's gain decreases as the input level increases, so
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||||||
that the
|
that the
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<A NAME="12584"></A><I>dynamic range</I>,
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<A ID="12584"></A><I>dynamic range</I>,
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||||||
that is, the overall variation in signal level, is reduced. An expander does
|
that is, the overall variation in signal level, is reduced. An expander does
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||||||
the reverse, increasing the dynamic range. Frequently the gain depends not only
|
the reverse, increasing the dynamic range. Frequently the gain depends not only
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on the immediate signal level but on its history; for instance the rate of change
|
on the immediate signal level but on its history; for instance the rate of change
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@@ -132,7 +132,7 @@ simply a function of the magnitude of the current sample: <IMG
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|||||||
The patch diagrammed in Figure <A HREF="#fig09.08">9.8</A> shows one very useful
|
The patch diagrammed in Figure <A HREF="#fig09.08">9.8</A> shows one very useful
|
||||||
application of companding, called
|
application of companding, called
|
||||||
a
|
a
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<A NAME="12587"></A><I>noise gate</I>. Here the gain <IMG
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<A ID="12587"></A><I>noise gate</I>. Here the gain <IMG
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WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1159.png"
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SRC="img1159.png"
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ALT="$g[m]$"> depends on the channel amplitude
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ALT="$g[m]$"> depends on the channel amplitude
|
||||||
@@ -257,36 +257,36 @@ undistorted.
|
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<H2><A NAME="SECTION001342000000000000000">
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<H2><A ID="SECTION001342000000000000000">
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Timbre stamping (classical vocoder)</A>
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Timbre stamping (classical vocoder)</A>
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</H2>
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</H2>
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<DIV ALIGN="CENTER"><A NAME="fig09.09"></A><A NAME="12598"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.9:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.9:</STRONG>
|
||||||
Block diagram for timbre stamping (AKA "vocoding'').</CAPTION>
|
Block diagram for timbre stamping (AKA "vocoding'').</CAPTION>
|
||||||
@@ -87,11 +87,11 @@ filter capable of making one sound take on the evolving spectral envelope of
|
|||||||
another. This is
|
another. This is
|
||||||
widely known in electronic music circles as
|
widely known in electronic music circles as
|
||||||
a
|
a
|
||||||
<A NAME="12601"></A><I>vocoder</I>,
|
<A ID="12601"></A><I>vocoder</I>,
|
||||||
named, not quite accurately, after the original
|
named, not quite accurately, after the original
|
||||||
Bell Laboratories vocal analysis/synthesis device. The
|
Bell Laboratories vocal analysis/synthesis device. The
|
||||||
technique described here is more accurately called
|
technique described here is more accurately called
|
||||||
<A NAME="12603"></A><I>timbre stamping</I>.
|
<A ID="12603"></A><I>timbre stamping</I>.
|
||||||
Two input signals are used, one to be filtered, and the other to control the
|
Two input signals are used, one to be filtered, and the other to control the
|
||||||
filter via its time-varying spectral envelope. The windowed Fourier
|
filter via its time-varying spectral envelope. The windowed Fourier
|
||||||
transform is used both on the control signal input to estimate its spectral
|
transform is used both on the control signal input to estimate its spectral
|
||||||
@@ -136,36 +136,36 @@ suitable limiting function is applied to the gain before using it.
|
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<H1><A NAME="SECTION001350000000000000000"></A>
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<A NAME="sect9.phase"></A>
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Phase
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Phase
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@@ -82,7 +82,7 @@ time requires analyzing and reconstructing the phase.
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<DIV ALIGN="CENTER"><A NAME="fig09.10"></A><A NAME="12612"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.10:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.10:</STRONG>
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Phase in windowed Fourier analysis: (a) a complex sinusoid analyzed
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Phase in windowed Fourier analysis: (a) a complex sinusoid analyzed
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@@ -268,7 +268,7 @@ from somewhere else and can't be assumed to be in line with our wishes.
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<DIV ALIGN="CENTER"><A NAME="fig09.11"></A><A NAME="12631"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.11:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.11:</STRONG>
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Propagating phases in resynthesis. Each phase, such as that of
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Propagating phases in resynthesis. Each phase, such as that of
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@@ -286,7 +286,7 @@ input phases.</CAPTION>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.12:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.12:</STRONG>
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Phases of one channel of the analysis windows and two successive
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Phases of one channel of the analysis windows and two successive
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@@ -522,45 +522,45 @@ S[m, k] \; = \;
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<H2><A NAME="SECTION001351000000000000000"></A>
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<A NAME="sect9.phaserelationship"></A>
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<BR>
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Phase relationships between channels
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Phase relationships between channels
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ALT="$S[m, k]$"> above so that whenever there happens to be
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ALT="$S[m, k]$"> above so that whenever there happens to be
|
||||||
a solution to the over-constrained system of equations, the reconstruction
|
a solution to the over-constrained system of equations, the reconstruction
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||||||
algorithm homes in on the solution. This approach is called
|
algorithm homes in on the solution. This approach is called
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<A NAME="12657"></A><I>phase locking</I>
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<A ID="12657"></A><I>phase locking</I>
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[<A
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[<A
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HREF="node202.html#r-puckette95b">Puc95b</A>], and has the virtue of simplicity although more
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HREF="node202.html#r-puckette95b">Puc95b</A>], and has the virtue of simplicity although more
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sophisticated techniques are available [<A
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sophisticated techniques are available [<A
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Phase bashing
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Phase bashing
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.13:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.13:</STRONG>
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Phase-bashing a recorded sound (here, a sinusoid with rising
|
Phase-bashing a recorded sound (here, a sinusoid with rising
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+16
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<H1><A NAME="SECTION001370000000000000000">
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Examples</A>
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Examples</A>
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html3229"
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HREF="node179.html">Fourier analysis and resynthesis in Pd</A>
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<LI><A NAME="tex2html3230"
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HREF="node180.html">Narrow-band companding: noise suppression</A>
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HREF="node181.html">Timbre stamp ("vocoder")</A>
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HREF="node182.html">Phase vocoder time bender</A>
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<H2><A NAME="SECTION001371000000000000000">
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Fourier analysis and resynthesis in Pd</A>
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Fourier analysis and resynthesis in Pd</A>
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</H2>
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@@ -75,7 +75,7 @@ Fourier transform of an audio signal using the <TT>fft~</TT> object:
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<DIV ALIGN="CENTER"><A NAME="fig09.14"></A><A NAME="12858"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
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Fourier analysis in Pd: (a) the <!-- MATH
|
Fourier analysis in Pd: (a) the <!-- MATH
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@@ -103,7 +103,7 @@ Hann windowing function.</CAPTION>
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WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1216.png"
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ALT="\fbox{\texttt{fft\~}}">:
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ALT="\fbox{\texttt{fft\~}}">:
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<A NAME="12898"></A><A NAME="12696"></A>Fast Fourier transform. The two inlets take audio signals representing the real
|
<A ID="12898"></A><A ID="12696"></A>Fast Fourier transform. The two inlets take audio signals representing the real
|
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and imaginary parts of a complex-valued signal. The window size <IMG
|
and imaginary parts of a complex-valued signal. The window size <IMG
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@@ -134,7 +134,7 @@ subwindow which in turn is shown in part (c). New objects are:
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WIDTH="56" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1218.png"
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ALT="\fbox{\texttt{rfft\~}}">:
|
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<A NAME="12899"></A>real Fast Fourier transform. The imaginary part of the input is assumed to
|
<A ID="12899"></A>real Fast Fourier transform. The imaginary part of the input is assumed to
|
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be zero. Only the first <IMG
|
be zero. Only the first <IMG
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WIDTH="60" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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@@ -150,7 +150,7 @@ are determined by symmetry). This takes half the computation time of the
|
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WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1221.png"
|
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ALT="\fbox{\texttt{tabreceive\~}}">:
|
ALT="\fbox{\texttt{tabreceive\~}}">:
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<A NAME="12900"></A>repeatedly outputs the contents of a wavetable. Each
|
<A ID="12900"></A>repeatedly outputs the contents of a wavetable. Each
|
||||||
block of computation outputs the same first <IMG
|
block of computation outputs the same first <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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@@ -169,7 +169,7 @@ graphing.
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<DIV ALIGN="CENTER"><A NAME="fig09.15"></A><A NAME="12707"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.15:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.15:</STRONG>
|
||||||
Fourier analysis and resynthesis, using <TT>block~</TT> to specify an
|
Fourier analysis and resynthesis, using <TT>block~</TT> to specify an
|
||||||
@@ -196,7 +196,7 @@ We need one new object for the inverse Fourier transform:
|
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ALT="\fbox{\texttt{rifft\~}}">:
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<A NAME="12901"></A>real inverse Fast Fourier transform. Using the first <IMG
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<A ID="12901"></A>real inverse Fast Fourier transform. Using the first <IMG
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+19
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<H1><A NAME="SECTION00590000000000000000"></A>
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Examples
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Examples
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html807"
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HREF="node19.html">Constant amplitude scaler</A>
|
HREF="node19.html">Constant amplitude scaler</A>
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<LI><A NAME="tex2html808"
|
<LI><A ID="tex2html808"
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HREF="node20.html">Amplitude control in decibels</A>
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<LI><A ID="tex2html809"
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HREF="node21.html">Smoothed amplitude control with an envelope generator</A>
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HREF="node22.html">Major triad</A>
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HREF="node23.html">Conversion between frequency and pitch</A>
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<LI><A NAME="tex2html812"
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HREF="node24.html">More additive synthesis</A>
|
HREF="node24.html">More additive synthesis</A>
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+24
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<H2><A NAME="SECTION001372000000000000000">
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Narrow-band companding: noise suppression</A>
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Narrow-band companding: noise suppression</A>
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</H2>
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<DIV ALIGN="CENTER"><A NAME="fig09.16"></A><A NAME="12719"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.16:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.16:</STRONG>
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Noise suppression as an example of narrow-band companding: (a)
|
Noise suppression as an example of narrow-band companding: (a)
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@@ -164,7 +164,7 @@ figure. To compute the mask we are using two new new objects:
|
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WIDTH="56" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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ALT="\fbox{\texttt{bang\~}}">:
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<A NAME="12903"></A>send a bang in advance of each block of computation. The bang appears at the
|
<A ID="12903"></A>send a bang in advance of each block of computation. The bang appears at the
|
||||||
logical time of the first sample in each block (the earliest logical time whose
|
logical time of the first sample in each block (the earliest logical time whose
|
||||||
control computation affects that block and not the previous one), following
|
control computation affects that block and not the previous one), following
|
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the scheme shown
|
the scheme shown
|
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@@ -178,7 +178,7 @@ in Figure <A HREF="node42.html#fig03.02">3.2</A>.
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WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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ALT="\fbox{\texttt{tabsend\~}}">:
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<A NAME="12904"></A>the companion object for <TT>tabreceive~</TT>, repeatedly copies its input to
|
<A ID="12904"></A>the companion object for <TT>tabreceive~</TT>, repeatedly copies its input to
|
||||||
the contents of a table, affecting up to the first <IMG
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+22
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HREF="node180.html">Narrow-band companding: noise suppression</A>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001373000000000000000">
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<H2><A ID="SECTION001373000000000000000">
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Timbre stamp ("vocoder")</A>
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Timbre stamp ("vocoder")</A>
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</H2>
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<DIV ALIGN="CENTER"><A NAME="fig09.17"></A><A NAME="12742"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.17:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.17:</STRONG>
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Timbre stamp.</CAPTION>
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Timbre stamp.</CAPTION>
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@@ -132,36 +132,36 @@ unchanged, and values in between give a smooth interpolation between the two.
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SRC="index.png"></A>
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HREF="node182.html">Phase vocoder time bender</A>
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HREF="node178.html">Examples</A>
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HREF="node180.html">Narrow-band companding: noise suppression</A>
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HREF="node4.html">Contents</A></B>
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HREF="node201.html">Index</A></B>
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+24
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SRC="index.png"></A>
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<B> Next:</B> <A NAME="tex2html3286"
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<B> Next:</B> <A ID="tex2html3286"
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HREF="node183.html">Exercises</A>
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<B> Previous:</B> <A NAME="tex2html3276"
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HREF="node181.html">Timbre stamp ("vocoder")</A>
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HREF="node201.html">Index</A></B>
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<H2><A NAME="SECTION001374000000000000000">
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Phase vocoder time bender</A>
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<DIV ALIGN="CENTER"><A NAME="fig09.18"></A><A NAME="12750"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.18:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.18:</STRONG>
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Phase vocoder for time stretching and contraction.</CAPTION>
|
Phase vocoder for time stretching and contraction.</CAPTION>
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@@ -99,7 +99,7 @@ the location in the recorded sound we hear. Two new objects are used:
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WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1241.png"
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SRC="img1241.png"
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ALT="\fbox{\texttt{lrshift\~}}">:
|
ALT="\fbox{\texttt{lrshift\~}}">:
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<A NAME="12905"></A>shift a block left or right (according to its creation argument). If the
|
<A ID="12905"></A>shift a block left or right (according to its creation argument). If the
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argument is positive, each block of the output is the input shifted that number
|
argument is positive, each block of the output is the input shifted that number
|
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of spaces to the right, filling zeros in as needed on the left. A negative
|
of spaces to the right, filling zeros in as needed on the left. A negative
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argument shifts to the left, filling zeros in at the right.
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argument shifts to the left, filling zeros in at the right.
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@@ -112,7 +112,7 @@ argument shifts to the left, filling zeros in at the right.
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WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1243.png"
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SRC="img1243.png"
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ALT="\fbox{\texttt{q8\_rsqrt\~}}">:
|
ALT="\fbox{\texttt{q8\_rsqrt\~}}">:
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<A NAME="12906"></A>quick and approximate reciprocal square root. Outputs the reciprocal of the
|
<A ID="12906"></A>quick and approximate reciprocal square root. Outputs the reciprocal of the
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square root of its input, good to about a part in 256, using much less
|
square root of its input, good to about a part in 256, using much less
|
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computation than a full-precision square root and reciprocal would.
|
computation than a full-precision square root and reciprocal would.
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@@ -213,36 +213,36 @@ can be made using daisy-chained cross-fades.
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SRC="index.png"></A>
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<B> Previous:</B> <A NAME="tex2html3276"
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<B> Previous:</B> <A ID="tex2html3276"
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HREF="node181.html">Timbre stamp ("vocoder")</A>
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HREF="node181.html">Timbre stamp ("vocoder")</A>
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<B> <A NAME="tex2html3282"
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<B> <A ID="tex2html3282"
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html3284"
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<B> <A ID="tex2html3284"
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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<ADDRESS>
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+21
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Exercises</A>
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Exercises</A>
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+42
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HREF="node4.html">Contents</A></B>
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<H1><A NAME="SECTION001400000000000000000"></A>
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Classical waveforms
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Classical waveforms
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@@ -82,12 +82,12 @@ controllable endpoints. This approach is historically at least as important
|
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as the others, and was dominant during the
|
as the others, and was dominant during the
|
||||||
analog synthesizer period, approximately 1965-1985.
|
analog synthesizer period, approximately 1965-1985.
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For lack of a better name, we'll use the term
|
For lack of a better name, we'll use the term
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<A NAME="14228"></A><I>classical waveforms</I>
|
<A ID="14228"></A><I>classical waveforms</I>
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to denote waveforms composed of line segments.
|
to denote waveforms composed of line segments.
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<DIV ALIGN="CENTER"><A NAME="fig10.01"></A><A NAME="14232"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.1:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.1:</STRONG>
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Classical waveforms: (a) the sawtooth, (b) the triangle, and (c)
|
Classical waveforms: (a) the sawtooth, (b) the triangle, and (c)
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@@ -102,7 +102,7 @@ shown as functions of a continuous variable (not sampled).</CAPTION>
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They include the
|
They include the
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<A NAME="14235"></A><A NAME="14236"></A><A NAME="14237"></A><I>sawtooth</I>, <I>triangle</I>, and <I>rectangle</I> waves pictured in Figure
|
<A ID="14235"></A><A ID="14236"></A><A ID="14237"></A><I>sawtooth</I>, <I>triangle</I>, and <I>rectangle</I> waves pictured in Figure
|
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<A HREF="#fig10.01">10.1</A>, among many other possibilities. The salient features of
|
<A HREF="#fig10.01">10.1</A>, among many other possibilities. The salient features of
|
||||||
classical waveforms are either discontinuous jumps (changes in value) or
|
classical waveforms are either discontinuous jumps (changes in value) or
|
||||||
corners (changes in slope). In the figure, the sawtooth and rectangle waves
|
corners (changes in slope). In the figure, the sawtooth and rectangle waves
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@@ -150,87 +150,87 @@ foldover is produced, and careful handling will help us control it.
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<P>
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<BR><HR>
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<LI><A NAME="tex2html3313"
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||||||
HREF="node185.html">Symmetries and Fourier series</A>
|
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||||||
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||||||
<LI><A NAME="tex2html3314"
|
<LI><A ID="tex2html3314"
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||||||
HREF="node186.html">Sawtooth waves and symmetry</A>
|
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||||||
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|
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HREF="node187.html">Dissecting classical waveforms</A>
|
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|
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HREF="node188.html">Fourier series of the elementary waveforms</A>
|
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<LI><A NAME="tex2html3317"
|
<LI><A ID="tex2html3317"
|
||||||
HREF="node189.html">Sawtooth wave</A>
|
HREF="node189.html">Sawtooth wave</A>
|
||||||
<LI><A NAME="tex2html3318"
|
<LI><A ID="tex2html3318"
|
||||||
HREF="node190.html">Parabolic wave</A>
|
HREF="node190.html">Parabolic wave</A>
|
||||||
<LI><A NAME="tex2html3319"
|
<LI><A ID="tex2html3319"
|
||||||
HREF="node191.html">Square and symmetric triangle waves</A>
|
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<LI><A NAME="tex2html3320"
|
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HREF="node192.html">General (non-symmetric) triangle wave</A>
|
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|
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<LI><A NAME="tex2html3321"
|
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HREF="node193.html">Predicting and controlling foldover</A>
|
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|
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||||||
<LI><A NAME="tex2html3322"
|
<LI><A ID="tex2html3322"
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||||||
HREF="node194.html">Over-sampling</A>
|
HREF="node194.html">Over-sampling</A>
|
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<LI><A NAME="tex2html3323"
|
<LI><A ID="tex2html3323"
|
||||||
HREF="node195.html">Sneaky triangle waves</A>
|
HREF="node195.html">Sneaky triangle waves</A>
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<LI><A NAME="tex2html3324"
|
<LI><A ID="tex2html3324"
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||||||
HREF="node196.html">Transition splicing</A>
|
HREF="node196.html">Transition splicing</A>
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|
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HREF="node197.html">Examples</A>
|
HREF="node197.html">Examples</A>
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|
<UL>
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<LI><A NAME="tex2html3326"
|
<LI><A ID="tex2html3326"
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||||||
HREF="node198.html">Combining sawtooth waves</A>
|
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<LI><A NAME="tex2html3327"
|
<LI><A ID="tex2html3327"
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||||||
HREF="node199.html">Strategies for band-limiting sawtooth waves</A>
|
HREF="node199.html">Strategies for band-limiting sawtooth waves</A>
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<LI><A NAME="tex2html3328"
|
<LI><A ID="tex2html3328"
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HREF="node200.html">Exercises</A>
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HREF="node200.html">Exercises</A>
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Symmetries and Fourier series</A>
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Symmetries and Fourier series</A>
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<H2><A NAME="SECTION001411000000000000000">
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Sawtooth waves and symmetry</A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.2:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.2:</STRONG>
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Using a symmetry relation to extract even and odd harmonics from a
|
Using a symmetry relation to extract even and odd harmonics from a
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@@ -97,7 +97,7 @@ waveform with slope zero except at the discontinuities. The discontinuities
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coming from the original sawtooth wave jump in the same direction (negative to
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coming from the original sawtooth wave jump in the same direction (negative to
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positive), but those coming from the shifted one are negated and jump from
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positive to negative. The result is a
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a particular rectangle wave in which the two component segments
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<H1><A NAME="SECTION001420000000000000000">
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<H1><A ID="SECTION001420000000000000000">
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Dissecting classical waveforms</A>
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Dissecting classical waveforms</A>
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</H1>
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</H1>
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@@ -79,7 +79,7 @@ waveform having only jumps (discontinuities in value) but no corners
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amplitudes. We then develop the idea further, showing how to generate waveforms
|
amplitudes. We then develop the idea further, showing how to generate waveforms
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with corners (either in addition to, or instead of, jumps) using
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with corners (either in addition to, or instead of, jumps) using
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another elementary waveform we'll call the
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another elementary waveform we'll call the
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<A NAME="14273"></A><I>parabolic wave</I>.
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<A ID="14273"></A><I>parabolic wave</I>.
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<P>
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<P>
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Suppose first that a waveform of period <IMG
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Suppose first that a waveform of period <IMG
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@@ -121,7 +121,7 @@ a jump from a lower to a higher value.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig10.03"></A><A NAME="14282"></A>
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<DIV ALIGN="CENTER"><A ID="fig10.03"></A><A ID="14282"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.3:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.3:</STRONG>
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Dissecting a waveform: (a) the original waveform with two
|
Dissecting a waveform: (a) the original waveform with two
|
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@@ -242,7 +242,7 @@ this to happen in general the jumps must add to zero:
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<P>
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig10.04"></A><A NAME="14301"></A>
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<DIV ALIGN="CENTER"><A ID="fig10.04"></A><A ID="14301"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.4:</STRONG>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.4:</STRONG>
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The parabolic wave.</CAPTION>
|
The parabolic wave.</CAPTION>
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@@ -328,7 +328,7 @@ An example is shown graphically in Figure <A HREF="#fig10.05">10.5</A>.
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<P>
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig10.05"></A><A NAME="14320"></A>
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<DIV ALIGN="CENTER"><A ID="fig10.05"></A><A ID="14320"></A>
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<TABLE>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.5:</STRONG>
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.5:</STRONG>
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||||||
Decomposing a triangle wave (part a) into two parabolic waves
|
Decomposing a triangle wave (part a) into two parabolic waves
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||||||
@@ -367,36 +367,36 @@ connect with itself.
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<B> <A ID="tex2html3381"
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HREF="node201.html">Index</A></B>
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<H1><A NAME="SECTION001430000000000000000"></A>
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<A NAME="sect10.spectra"></A>
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<A ID="sect10.spectra"></A>
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<BR>
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<BR>
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Fourier series of the elementary waveforms
|
Fourier series of the elementary waveforms
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||||||
</H1>
|
</H1>
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@@ -125,7 +125,7 @@ of a signal <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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SRC="img80.png"
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ALT="$x[n]$"> with its
|
ALT="$x[n]$"> with its
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<A NAME="14332"></A><I>first difference</I>,
|
<A ID="14332"></A><I>first difference</I>,
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||||||
defined as <IMG
|
defined as <IMG
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WIDTH="105" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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WIDTH="105" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1303.png"
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SRC="img1303.png"
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@@ -248,51 +248,51 @@ Now we plug this result in to evaluate:
|
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<P></P>
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<P></P>
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<BR><HR>
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<BR><HR>
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<!--Table of Child-Links-->
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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HREF="node189.html">Sawtooth wave</A>
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<LI><A NAME="tex2html3385"
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HREF="node190.html">Parabolic wave</A>
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HREF="node190.html">Parabolic wave</A>
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HREF="node191.html">Square and symmetric triangle waves</A>
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<LI><A NAME="tex2html3387"
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<LI><A ID="tex2html3387"
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HREF="node192.html">General (non-symmetric) triangle wave</A>
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<B> <A ID="tex2html3397"
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HREF="node4.html">Contents</A></B>
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html3399"
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<B> <A ID="tex2html3399"
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HREF="node201.html">Index</A></B>
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HREF="node201.html">Index</A></B>
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<BR>
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<BR>
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<BR>
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<BR>
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<H2><A NAME="SECTION001431000000000000000">
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<H2><A ID="SECTION001431000000000000000">
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Sawtooth wave</A>
|
Sawtooth wave</A>
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</H2>
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</H2>
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First we apply this to the sawtooth wave <IMG
|
First we apply this to the sawtooth wave <IMG
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@@ -107,7 +107,7 @@ Ignoring the constant offset of <IMG
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WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img1314.png"
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SRC="img1314.png"
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ALT="$-{1 \over N}$">, this gives an
|
ALT="$-{1 \over N}$">, this gives an
|
||||||
<A NAME="14349"></A><I>impulse</I>,
|
<A ID="14349"></A><I>impulse</I>,
|
||||||
zero everywhere except one sample per cycle. The summation in
|
zero everywhere except one sample per cycle. The summation in
|
||||||
the Fourier transform only has one term, and we get:
|
the Fourier transform only has one term, and we get:
|
||||||
<BR><P></P>
|
<BR><P></P>
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@@ -305,36 +305,36 @@ s[n] \approx {1 \over \pi} \left [
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<P>
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<P>
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<HR>
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<B> Next:</B> <A NAME="tex2html3401"
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HREF="node190.html">Parabolic wave</A>
|
HREF="node190.html">Parabolic wave</A>
|
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<B> Up:</B> <A NAME="tex2html3395"
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<B> Up:</B> <A ID="tex2html3395"
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HREF="node188.html">Fourier series of the</A>
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HREF="node188.html">Fourier series of the</A>
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HREF="node188.html">Fourier series of the</A>
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HREF="node4.html">Contents</A></B>
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<ADDRESS>
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<ADDRESS>
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