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<H1><A ID="SECTION001030000000000000000"></A>
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<A ID="sect6.carrier"></A>
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<BR>
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Movable ring modulation
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</H1>
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<P>
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We turn now to the carrier signal, seeking ways to make it more controllable.
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We would particularly like to be able to slide the spectral energy continuously
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up and
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down in frequency. Simply ramping the frequency of the carrier oscillator
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will not accomplish this, since the spectra won't be harmonic except when the
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carrier is an integer multiple of the fundamental frequency.
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<P>
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In the stretched wavetable approach we can accomplish this simply by sampling
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a sinusoid and transposing it to the desired "pitch". The transposed pitch
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isn't heard as a periodicity since the wavetable itself is read periodically at
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the fundamental frequency. Instead, the sinusoid is transposed as a spectral
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envelope.
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<P>
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Figure <A HREF="#fig06.07">6.7</A> shows a carrier signal produced in this way, tuned to
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produce a formant centered at 1.5 times the fundamental frequency. The signal
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has no outright discontinuity at the phase wraparound frequency, but it does
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have a discontinuity in slope, which, if not removed by applying a suitable
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modulation signal, would have very audible high-frequency components.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.07"></A><A ID="6894"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.7:</STRONG>
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Waveform for a wavetable-based carrier signal tuned to 1.5 times
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the fundamental. Two periods are shown.</CAPTION>
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<TR><TD><IMG
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WIDTH="353" HEIGHT="109" BORDER="0"
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SRC="img585.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.07.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Using this idea we can make a complete description of how to use the block
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diagram of Figure <A HREF="node90.html#fig06.03">6.3</A> to produce a desired formant. The wavetable
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lookup on the left hand side would hold a sinusoid (placed symmetrically so
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that the phase is zero at the center of the wavetable). The right-hand-side
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wavetable would hold a Hann or other appropriate window function.
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If we desire the fundamental frequency to be <IMG
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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 $">, the
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formant center frequency to be <IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img464.png"
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ALT="$\omega_c$">, and the bandwidth to be <IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img586.png"
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ALT="$\omega_b$">,
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we set the "stretch" parameter to the <I>center frequency quotient</I>
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defined as <!-- MATH
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${\omega_c}/\omega$
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-->
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<IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img587.png"
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ALT="${\omega_c}/\omega$">, and the index of
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modulation to the <I>bandwidth quotient</I>, <!-- MATH
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${\omega_b}/\omega$
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-->
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<IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img588.png"
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ALT="${\omega_b}/\omega$">.
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<P>
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The output signal is simply a sample of a cosine wave at the
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desired center frequency, repeated at the (unrelated in general) desired
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period, and windowed to take out the discontinuities at period boundaries.
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<P>
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Although we aren't able to derive this result yet (we will need Fourier
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analysis), it will turn out that, in the main lobe of the formant, the phases
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are all zero at the center of the waveform (i.e., the components are all
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cosines if we consider the phase to be zero at the center of
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the waveform). This means we may
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superpose any number of these formants to build a more complex spectrum and the
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amplitudes of the partials will combine by addition. (The sidelobes don't
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behave so well: they are alternately of opposite sign and will produce
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cancellation patterns; but we can often just shrug them off as a small,
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uncontrollable, residual signal.)
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<P>
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This method leads to an interesting generalization, which is to take a sequence
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of recorded wavetables, align all their component phases to those of
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cosines, and use them in place of the cosine function as the carrier signal.
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The phase alignment is necessary to allow coherent cross-fading between
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samples so that the spectral envelope can change smoothly. If, for example, we
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use successive snippets of a vocal sample as input, we get a strikingly
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effective vocoder; see Section <A HREF="node177.html#sect9-bash">9.6</A>.
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<P>
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Another technique for making carrier signals that can be slid continuously
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up and down in frequency while maintaining a fundamental frequency
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is simply to cross-fade between harmonics. The carrier signal is then:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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c(\phi) = c(\omega n) = p \cos( k \omega n) + q \cos( (k+1) \omega n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="313" HEIGHT="28" BORDER="0"
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SRC="img589.png"
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ALT="\begin{displaymath}
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c(\phi) = c(\omega n) = p \cos( k \omega n) + q \cos( (k+1) \omega n)
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
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WIDTH="67" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img590.png"
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ALT="$p + q = 1$"> and <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$"> is an integer, all three chosen so that
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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(k + q) * {\omega} = {\omega_c}
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\end{displaymath}
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-->
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<IMG
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WIDTH="109" HEIGHT="28" BORDER="0"
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SRC="img591.png"
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ALT="\begin{displaymath}
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(k + q) * {\omega} = {\omega_c}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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so that the spectral center of mass of the two cosines is placed at
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<IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img464.png"
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ALT="$\omega_c$">. (Note that we make the amplitudes of the two cosines add to one
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instead of setting the total power to one; we do this because the modulator
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will operate phase-coherently on them.) To accomplish this we simply
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set <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$"> and <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$"> to be the integer and fractional part, respectively,
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of the center frequency quotient <!-- MATH
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$\omega_c/\omega$
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-->
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<IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img593.png"
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ALT="$\omega_c/\omega$">.
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<P>
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The simplest way of making a control interface for this synthesis technique
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would be to use
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ramps to update <IMG
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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 $"> and <IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img464.png"
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ALT="$\omega_c$">, and then to compute <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$"> and <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">
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as audio signals from the ramped, smoothly varying <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> and <IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img464.png"
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ALT="$\omega_c$">.
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Oddly enough, despite the fact that <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">, <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$">, and <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$"> are discontinuous
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functions of <!-- MATH
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$\omega_c/\omega$
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-->
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<IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img593.png"
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ALT="$\omega_c/\omega$">, the carrier <IMG
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WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img594.png"
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ALT="$c(\phi)$"> turns out to vary
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continuously with <!-- MATH
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$\omega_c/\omega$
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-->
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<IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img593.png"
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ALT="$\omega_c/\omega$">, and so if the desired center frequency
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<IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img464.png"
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ALT="$\omega_c$"> is ramped from value to value the result is a continuous sweep in
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center frequency. However, more work is needed if discontinuous changes
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in center frequency are needed. This is not an unreasonable thing to wish
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for, being analogous to changing the frequency of an oscillator discontinuously.
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<P>
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There turns out to be a good way to accomodate this.
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The trick to updating <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$"> and <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$"> is to note that <IMG
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WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img595.png"
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ALT="$c(\phi) = 1$"> whenever <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\phi$">
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is a multiple of <IMG
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WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="$2\pi $">, regardless of the choice of <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">, <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$">, and <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$"> as long
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as <IMG
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WIDTH="67" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img590.png"
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ALT="$p + q = 1$">. Hence, we may make discontinuous changes in <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">, <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$">, and <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$"> once
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per period (right when the phase is a multiple of <IMG
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WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="$2\pi $">), without making
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discontinuities in the carrier signal.
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<P>
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In the specific case of FM, if we wish we can now go back and modify the
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original formulation to:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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p \cos ( n {\omega_2} t + r \cos ({\omega_1} t)) +
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\end{displaymath}
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-->
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<IMG
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WIDTH="176" HEIGHT="28" BORDER="0"
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SRC="img596.png"
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ALT="\begin{displaymath}
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p \cos ( n {\omega_2} t + r \cos ({\omega_1} t)) +
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P> <BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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+ q \cos ( (n+1) {\omega_2} t + r \cos ({\omega_1} t))
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\end{displaymath}
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-->
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<IMG
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WIDTH="212" HEIGHT="28" BORDER="0"
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SRC="img597.png"
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ALT="\begin{displaymath}
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+ q \cos ( (n+1) {\omega_2} t + r \cos ({\omega_1} t))
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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This allows us to add glissandi (which are heard as dipthongs) to Chowning's
|
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original phase-modulation-based vocal synthesis technique.
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|
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<P>
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<B> Up:</B> <A ID="tex2html1962"
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HREF="node89.html">Designer spectra</A>
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<B> Previous:</B> <A ID="tex2html1956"
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HREF="node94.html">Resulting spectra</A>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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