diff --git a/book.html b/book.html index 04d6d50..d6580ae 100644 --- a/book.html +++ b/book.html @@ -27,7 +27,7 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next @@ -35,20 +35,20 @@ original version by: Nikos Drakos, CBLU, University of Leeds SRC="up_g.png"> previous - contents - index
- Next: Next: #1 -   Contents -   Index

@@ -60,503 +60,503 @@ original version by: Nikos Drakos, CBLU, University of Leeds --> - + diff --git a/node10.html b/node10.html index 55d636f..0e7c81d 100644 --- a/node10.html +++ b/node10.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Frequency - Up: Up: Sinusoids, amplitude and frequency - Previous: Previous: Units of Amplitude -   Contents -   Index

-

+

Controlling Amplitude

@@ -82,7 +82,7 @@ nominal amplitude -
+
Figure 1.4: The relationship between "MIDI" pitch and frequency in cycles per @@ -166,36 +166,36 @@ considered later in Chapter 5.

- next - up - previous - contents - index
- Next: Next: Frequency - Up: Up: Sinusoids, amplitude and frequency - Previous: Previous: Units of Amplitude -   Contents -   Index
diff --git a/node100.html b/node100.html index 1a3181e..85e6d2e 100644 --- a/node100.html +++ b/node100.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: The PAF generator - Up: Up: Examples - Previous: Previous: Simple formant generator -   Contents -   Index

-

+

Two-cosine carrier signal

@@ -86,7 +86,7 @@ and -
+
Figure 6.17: Cross-fading between sinusoids to make movable @@ -169,36 +169,36 @@ never avoid getting phase cancellations where they overlap.

- next - up - previous - contents - index
- Next: Next: The PAF generator - Up: Up: Examples - Previous: Previous: Simple formant generator -   Contents -   Index
diff --git a/node101.html b/node101.html index f2161b0..cbd7806 100644 --- a/node101.html +++ b/node101.html @@ -30,48 +30,48 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Stretched wavetables - Up: Up: Examples - Previous: Previous: Two-cosine carrier signal -   Contents -   Index

-

+

The PAF generator

-

+
@@ -126,7 +126,7 @@ control object is needed:

-

+
Figure 6.18: The phase-aligned formant (PAF) synthesis algorithm.
@@ -145,7 +145,7 @@ Filling in the wavetable for Figure 6.18. WIDTH="57" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img623.png" ALT="\fbox{ $\mathrm{until}$\ }"> : -When the left, "start" inlet is banged, output sequential bangs (with no +When the left, "start" inlet is banged, output sequential bangs (with no elapsed time between them) iteratively, until the right, "stop" inlet is banged. The stopping "bang" message must originate somehow from the until 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.

- next - up - previous - contents - index
- Next: Next: Stretched wavetables - Up: Up: Examples - Previous: Previous: Two-cosine carrier signal -   Contents -   Index
diff --git a/node102.html b/node102.html index e0ca216..80d8df1 100644 --- a/node102.html +++ b/node102.html @@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Exercises - Up: Up: Examples - Previous: Previous: The PAF generator -   Contents -   Index

-

+

Stretched wavetables

diff --git a/node103.html b/node103.html index 3a1c0f1..af64f8e 100644 --- a/node103.html +++ b/node103.html @@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Time shifts and delays - Up: Up: Designer spectra - Previous: Previous: Stretched wavetables -   Contents -   Index

-

+

Exercises

diff --git a/node104.html b/node104.html index a2159f7..3158cb3 100644 --- a/node104.html +++ b/node104.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Complex numbers - Up: Up: book - Previous: Previous: Exercises -   Contents -   Index

-

- +

+
Time shifts and delays

@@ -83,7 +83,7 @@ But now suppose you played it at 5:00 and 5:00:01 on the same day (on two different playback systems, since the music lasts much longer than one second). Now the sound is much different. The difference, whatever it is, clearly resides in neither of the two individual sounds, but rather in the -interference +interference between the two. This interference can be perceived in at least four different ways:
@@ -138,96 +138,96 @@ section of this chapter we will develop the additional background needed.


-Subsections +Subsections
- next - up - previous - contents - index
- Next: Next: Complex numbers - Up: Up: book - Previous: Previous: Exercises -   Contents -   Index
diff --git a/node105.html b/node105.html index 11baaa9..1f58f65 100644 --- a/node105.html +++ b/node105.html @@ -30,50 +30,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Complex sinusoids - Up: Up: Time shifts and delays - Previous: Previous: Time shifts and delays -   Contents -   Index

-

- +

+
Complex numbers

Complex -numbers are written as: +numbers are written as:

-Subsections +Subsections
- next - up - previous - contents - index
- Next: Next: Complex sinusoids - Up: Up: Time shifts and delays - Previous: Previous: Time shifts and delays -   Contents -   Index
diff --git a/node106.html b/node106.html index 964e1fe..da8f4a0 100644 --- a/node106.html +++ b/node106.html @@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Time shifts and phase - Up: Up: Complex numbers - Previous: Previous: Complex numbers -   Contents -   Index

-

+

Complex sinusoids

@@ -104,7 +104,7 @@ arguments and magnitudes.

-

+

Figure 6.19: Filling in the wavetable for Figure 6.18.
Figure 7.2: The powers of a complex number $A$ is called the - + complex amplitude.

@@ -426,36 +426,36 @@ end of whatever we're doing.

- next - up - previous - contents - index
- Next: Next: Time shifts and phase - Up: Up: Complex numbers - Previous: Previous: Complex numbers -   Contents -   Index
diff --git a/node107.html b/node107.html index ab7d695..c6cc827 100644 --- a/node107.html +++ b/node107.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Delay networks - Up: Up: Time shifts and delays - Previous: Previous: Complex sinusoids -   Contents -   Index

-

- +

+
Time shifts and phase changes

@@ -152,7 +152,7 @@ further property that, if you time shift a sinusoid of frequency time invariance, +time invariance, makes it easy to analyze the effects of time shifts--and linear combinations of them--by considering separately what the operations do on individual sinusoids. @@ -234,36 +234,36 @@ since the sinusoid advances
- next - up - previous - contents - index
- Next: Next: Delay networks - Up: Up: Time shifts and delays - Previous: Previous: Complex sinusoids -   Contents -   Index
diff --git a/node108.html b/node108.html index ad63a9e..9e97df6 100644 --- a/node108.html +++ b/node108.html @@ -30,50 +30,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Recirculating delay networks - Up: Up: Time shifts and delays - Previous: Previous: Time shifts and phase -   Contents -   Index

-

- +

+
Delay networks

-

+
Figure 7.3: 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 $d$ samples corresponds to a - + delay of $R$ is the sample rate. Figure 7.3 shows one example of a -linear delay network: +linear delay network: an assembly of delay units, possibly with amplitude 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 @@ -117,13 +117,13 @@ so that the gains and delay times do not change with time.

In general there are two ways of thinking about delay networks. We can think in the -time domain, +time domain, in which we draw waveforms as functions of time (or of the index $n$), and consider delays as time shifts. Alternatively we may think in the -frequency domain, +frequency domain, 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 wrought by the network, as a function of the frequency. We'll now look at the @@ -131,7 +131,7 @@ delay network of Figure 7.3 in each of the two ways in t

-

+
Figure 7.4: The time domain view of the delay network of Figure 7.3. @@ -155,7 +155,7 @@ delayed copy of itself.

A frequently used test function is an -impulse, +impulse, 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 any other digital audio signal--because any signal -

+
Figure 7.5: Analysis, in the complex plane, of the frequency-domain behavior of @@ -483,19 +483,19 @@ The quantity $\vert H\vert$ is called the -gain +gain of the delay network at the angular frequency $\omega $, and is graphed in Figure 7.6. The frequency-dependent gain of a delay network (that is, the gain as a function of frequency) is called the network's -frequency response. +frequency response.

Since the network has greater gain at some frequencies than at others, it may be considered as a -filter +filter 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 $\omega $, this kind of delay network is called a (non-recirculating) -comb filter. +comb filter.

-

+
Figure 7.6: Gain of the delay network of Figure 7.3, shown as a function @@ -596,36 +596,36 @@ over the entire range of possible delay times.

- next - up - previous - contents - index
- Next: Next: Recirculating delay networks - Up: Up: Time shifts and delays - Previous: Previous: Time shifts and phase -   Contents -   Index
diff --git a/node109.html b/node109.html index 381c06e..5093ece 100644 --- a/node109.html +++ b/node109.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Power conservation and complex - Up: Up: Time shifts and delays - Previous: Previous: Delay networks -   Contents -   Index

-

- +

+
Recirculating delay networks

@@ -80,7 +80,7 @@ the network to engender yet others.

The simplest example of a recirculating network is the - + recirculating comb filter whose block diagram is shown in Figure 7.7. As with the earlier, simple comb filter, the input signal is sent down a delay line whose @@ -97,7 +97,7 @@ multiplied by a number -

+
Figure 7.7: Block diagram for a recirculating comb filter. Here $g$. In general, a delay network's output given an impulse as input is called the network's -impulse response. +impulse response.

-

+
Figure 7.8: 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 network whose output eventually falls toward zero after its input terminates is called -stable; +stable; one whose output grows without bound is called unstable.

@@ -420,7 +420,7 @@ frequency response -

+
Figure 7.9: Diagram in the complex plane for approximating the output gain

-

+
Figure 7.10: Frequency response of the recirculating comb filter with 8.

- next - up - previous - contents - index
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diff --git a/node11.html b/node11.html index 1285885..f4f6b46 100644 --- a/node11.html +++ b/node11.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Synthesizing a sinusoid - Up: Up: Sinusoids, amplitude and frequency - Previous: Previous: Controlling Amplitude -   Contents -   Index

-

+

Frequency

@@ -78,7 +78,7 @@ the musical interval between them.

The Western musical scale divides the -octave (the musical interval associated with a ratio of 2:1) into +octave (the musical interval associated with a ratio of 2:1) into twelve equal sub-intervals, each of which therefore corresponds to a ratio of - next - up - previous - contents - index
- Next: Next: Synthesizing a sinusoid - Up: Up: Sinusoids, amplitude and frequency - Previous: Previous: Controlling Amplitude -   Contents -   Index

diff --git a/node110.html b/node110.html index edaa5a9..54d0b5f 100644 --- a/node110.html +++ b/node110.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Artificial reverberation - Up: Up: Time shifts and delays - Previous: Previous: Recirculating delay networks -   Contents -   Index

-

+

Power conservation and complex delay networks

@@ -90,7 +90,7 @@ the gain, suitably defined, is exactly one.

-

+
Figure 7.11: First fundamental building block for unitary delay networks: @@ -163,7 +163,7 @@ where unitary. To start with, we can put any number of delays in parallel, as +unitary. To start with, we can put any number of delays in parallel, as shown in Figure 7.11. Whatever the total power of the inputs, 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.

-

+
Figure 7.12: Second fundamental building block for unitary delay networks: @@ -329,7 +329,7 @@ s = \sin(\theta)

for an -angle of rotation angle of rotation $\theta$. @@ -423,7 +423,7 @@ unit magnitude and argument If we perform a rotation on a pair of signals and then invert one (but not the other) of them, the result is a - + reflection. 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 @@ -486,7 +486,7 @@ a) because each signal need only be multiplied by the one quantity -
+
Figure 7.13: Details about rotation (and reflection) matrix operations: (a) @@ -535,7 +535,7 @@ recirculating networks that still enjoy flat frequency responses.

-

+
Figure 7.14: Flat frequency response in recirculating networks: (a) in general, @@ -655,7 +655,7 @@ let the transformation $W$ output, the result is the well-known -all-pass filter. +all-pass filter. With some juggling, and letting @@ -670,36 +670,36 @@ of which we will visit later in this book.

- next - up - previous - contents - index
- Next: Next: Artificial reverberation - Up: Up: Time shifts and delays - Previous: Previous: Recirculating delay networks -   Contents -   Index
diff --git a/node111.html b/node111.html index a4e13f0..4ba7904 100644 --- a/node111.html +++ b/node111.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Controlling reverberators - Up: Up: Time shifts and delays - Previous: Previous: Power conservation and complex -   Contents -   Index

-

+

Artificial reverberation

@@ -101,13 +101,13 @@ is not designed correctly. To make an artificial reverberator using a delay network, we must fill two competing demands simultaneously. First, the delay lines must be long enough to prevent -coloration in the output as a result of comb filtering. +coloration in the output as a result of comb filtering. (Even if we move beyond the simple comb filter of Section 7.4, the frequency response will tend to have peaks and valleys whose spacing varies inversely with total delay time.) On the other hand, we should not hear individual echoes; the -echo density should ideally be at least one thousand per second. +echo density should ideally be at least one thousand per second.

In pursuit of these aims, we assemble some number of delay lines and @@ -138,7 +138,7 @@ and 80 milliseconds. The figure shows three such stages.

-

+
Figure 7.15: Reverberator design using power-preserving transformations and @@ -273,45 +273,45 @@ and lengthy tuning by trial, error, and critical listening.


-Subsections +Subsections
- next - up - previous - contents - index
- Next: Next: Controlling reverberators - Up: Up: Time shifts and delays - Previous: Previous: Power conservation and complex -   Contents -   Index
diff --git a/node112.html b/node112.html index 9edc652..f7c1d54 100644 --- a/node112.html +++ b/node112.html @@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Variable and fractional shifts - Up: Up: Artificial reverberation - Previous: Previous: Artificial reverberation -   Contents -   Index

-

+

Controlling reverberators

@@ -88,7 +88,7 @@ moments of stable pitch.

-

+
Figure 7.16: Controlling a reverberator to capture sounds selectively: (a) @@ -135,36 +135,36 @@ decay.

- next - up - previous - contents - index
- Next: Next: Variable and fractional shifts - Up: Up: Artificial reverberation - Previous: Previous: Artificial reverberation -   Contents -   Index
diff --git a/node113.html b/node113.html index 17735ca..9fa4a90 100644 --- a/node113.html +++ b/node113.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Fidelity of interpolating delay - Up: Up: Time shifts and delays - Previous: Previous: Controlling reverberators -   Contents -   Index

-

- +

+
Variable and fractional shifts

@@ -181,7 +181,7 @@ result from the changes.

-

+
Figure 7.17: A variable length delay line, whose output is the input from some @@ -350,7 +350,7 @@ t[n] = y[n] - y[n-1] = 1 - (d[n] - d[n-1])

-If If $d[n]$ does not change with This is called the -Doppler effect, and it occurs in nature as well. +Doppler effect, and it occurs in nature as well. The air that sound travels through can sometimes be thought of as a delay line. Changing the length of the delay line corresponds to moving the listener toward or away from a stationary sound source; the Doppler effect @@ -391,36 +391,36 @@ diagonal region.

- next - up - previous - contents - index
- Next: Next: Fidelity of interpolating delay - Up: Up: Time shifts and delays - Previous: Previous: Controlling reverberators -   Contents -   Index
diff --git a/node114.html b/node114.html index d152c1e..e670a2d 100644 --- a/node114.html +++ b/node114.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Pitch shifting - Up: Up: Time shifts and delays - Previous: Previous: Variable and fractional shifts -   Contents -   Index

-

+

Fidelity of interpolating delay lines

@@ -200,7 +200,7 @@ itself, however, the gain is zero.

-

+
Figure 7.18: Gain of a four-point interpolating delay line with a delay halfway @@ -221,36 +221,36 @@ the way up to 20000 Hertz.

- next - up - previous - contents - index
- Next: Next: Pitch shifting - Up: Up: Time shifts and delays - Previous: Previous: Variable and fractional shifts -   Contents -   Index
diff --git a/node115.html b/node115.html index 9fbe9fb..3ebbfc0 100644 --- a/node115.html +++ b/node115.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Examples - Up: Up: Time shifts and delays - Previous: Previous: Fidelity of interpolating delay -   Contents -   Index

-

- +

+
Pitch shifting

@@ -87,7 +87,7 @@ in the strip.

-

+
Figure 7.19: Vibrato using a variable delay line. Since the pitch shift @@ -161,7 +161,7 @@ This ranges in value between -
+
Figure 7.20: Piecewise linear delay functions to maintain a constant transposition @@ -187,7 +187,7 @@ to avoid discontinuities.

-

+
Figure 7.21: 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 $s$ is sometimes called the -window size. It corresponds roughly to the sample length in a +window size. It corresponds roughly to the sample length in a looping sampler (Section 2.2).

@@ -330,7 +330,7 @@ Figure 7.22.

-

+
Figure 7.22: The pitch shifter's delay reading pattern using two delay lines, @@ -373,7 +373,7 @@ f = {{(t - 1) R} \over s}

-The window size The window size $s$ should be chosen small enough, if possible, so that the @@ -445,36 +445,36 @@ artifacts such as difference tones.

- next - up - previous - contents - index
- Next: Next: Examples - Up: Up: Time shifts and delays - Previous: Previous: Fidelity of interpolating delay -   Contents -   Index
diff --git a/node116.html b/node116.html index e06585e..8ef6a66 100644 --- a/node116.html +++ b/node116.html @@ -30,68 +30,68 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Fixed, noninterpolating delay line - Up: Up: Time shifts and delays - Previous: Previous: Pitch shifting -   Contents -   Index

-

+

Examples


-Subsections +Subsections diff --git a/node117.html b/node117.html index 25a4d37..37756b2 100644 --- a/node117.html +++ b/node117.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Recirculating comb filter - Up: Up: Examples - Previous: Previous: Examples -   Contents -   Index

-

+

Fixed, noninterpolating delay line

@@ -75,7 +75,7 @@ an input signal. Two new objects are needed:

-

+
WIDTH="94" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img819.png" ALT="\fbox{ \texttt{delwrite\~}}">: -define and write to a delay line. The first creation argument gives the name of the +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 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. @@ -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" SRC="img821.png" ALT="\fbox{ \texttt{delread\~}}">: -read from (or "tap") a delay line. The first creation argument gives the name +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 delwrite~ object; this is how Pd knows which delwrite~ to associate with the delread~ object). The @@ -130,36 +130,36 @@ possible when the delay time changes.

- next - up - previous - contents - index
- Next: Next: Recirculating comb filter - Up: Up: Examples - Previous: Previous: Examples -   Contents -   Index
diff --git a/node118.html b/node118.html index f93df04..a393bc5 100644 --- a/node118.html +++ b/node118.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Variable delay line - Up: Up: Examples - Previous: Previous: Fixed, noninterpolating delay line -   Contents -   Index

-

+

Recirculating comb filter

@@ -81,7 +81,7 @@ of Section 7.4.

-

+

Figure 7.23: Example patch G01.delay.pd, showing a noninterpolating delay with @@ -95,7 +95,7 @@ a delay time controlled in milliseconds.
diff --git a/node119.html b/node119.html index a1848bc..ac34ef7 100644 --- a/node119.html +++ b/node119.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Order of execution and - Up: Up: Examples - Previous: Previous: Recirculating comb filter -   Contents -   Index

-

+

Variable delay line

@@ -76,7 +76,7 @@ new object is introduced here:

-

+

Figure 7.24: Recirculating delay (still noninterpolating).
@@ -95,7 +95,7 @@ The flanger: an interpolating, variable delay line. WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img825.png" ALT="\fbox{ \texttt{vd\~}}">: -Read from a delay line, with a time-varying delay time. As with the +Read from a delay line, with a time-varying delay time. As with the delread~ object, this reads from a delay line whose name is specified as a creation argument. Instead of using a second argument and/or control messages to specify the delay time, for the vd~ object the delay in milliseconds is specified by an incoming audio signal. @@ -136,36 +136,36 @@ characteristic way.

- next - up - previous - contents - index
- Next: Next: Order of execution and - Up: Up: Examples - Previous: Previous: Recirculating comb filter -   Contents -   Index
diff --git a/node12.html b/node12.html index 2f366d1..2b41ee2 100644 --- a/node12.html +++ b/node12.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Superposing Signals - Up: Up: Sinusoids, amplitude and frequency - Previous: Previous: Frequency -   Contents -   Index

-

- +

+
Synthesizing a sinusoid

@@ -75,10 +75,10 @@ Synthesizing a sinusoid In most widely used audio synthesis and processing packages (Csound, Max/MSP, and Pd, for instance), the audio operations are specified as networks of -unit generators[unit generators[Mat69] which pass audio signals among themselves. The user of the software package specifies the network, sometimes called a -patch, +patch, 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 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 introduce two unit generators. First we need a pure sinusoid which is made with an -oscillator. Figure 1.5 (part a) shows a pictorial +oscillator. Figure 1.5 (part a) shows a pictorial representation of 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.

-

+
Figure 7.25: The flanger: an interpolating, variable delay line.
Figure 1.5: Block diagrams for (a) a sinusoidal oscillator; (b) controlling the @@ -131,7 +131,7 @@ negative in value.

-

+
Figure 1.6: Two amplitude functions (parts a, c), and (parts b, d), the result of @@ -183,7 +183,7 @@ Suitable amplitude control functions $y[n]$ may be made using an -envelope generator. +envelope generator. Figure 1.7 shows a network in which an envelope generator is used to control the amplitude of an oscillator. Envelope generators vary widely in design, but we will focus on the simplest @@ -252,7 +252,7 @@ Envelope generators are described in more detail in Section +
@@ -266,36 +266,36 @@ Using an envelope generator to control amplitude.

- next - up - previous - contents - index
- Next: Next: Superposing Signals - Up: Up: Sinusoids, amplitude and frequency - Previous: Previous: Frequency -   Contents -   Index
diff --git a/node120.html b/node120.html index 344ea87..82a0020 100644 --- a/node120.html +++ b/node120.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Order of execution in - Up: Up: Examples - Previous: Previous: Variable delay line -   Contents -   Index

-

+

Order of execution and lower limits on delay times

@@ -94,7 +94,7 @@ delay operations might, for example).

-

+

Figure 1.7: Using an envelope generator to control amplitude.
Figure 7.26: Order of execution of tilde objects in Pd: (a), an acyclic network. @@ -249,7 +249,7 @@ a new object:

-

+
WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img834.png" ALT="\fbox{ \texttt{switch\~}}">: -Set the local block size of the patch window the object sits in. Block sizes +Set the local block size of the patch window the object sits in. Block sizes are normally powers of two. The switch~ object, in addition, can be used to turn audio computation within the window on and off, using control messages. 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.

Putting a pulse (or other excitation signal) into a recirculating comb filter to make a pitch is sometimes called -Karplus-Strong synthesis, +Karplus-Strong synthesis, having been described in a paper by them [KS83], although the idea 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,

- next - up - previous - contents - index
- Next: Next: Order of execution in - Up: Up: Examples - Previous: Previous: Variable delay line -   Contents -   Index
diff --git a/node121.html b/node121.html index 2e1cb95..cad3112 100644 --- a/node121.html +++ b/node121.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Non-recirculating comb filter as - Up: Up: Examples - Previous: Previous: Order of execution and -   Contents -   Index

-

+

Order of execution in non-recirculating delay lines

@@ -84,7 +84,7 @@ G05.execution.order.pd (Figure 7.28).

-

+

Figure 7.27: A patch using block size control to lower the loop delay below @@ -277,7 +277,7 @@ with a block~ object and a recirculating delay network.
Figure 7.28: Using subpatches to ensure that delay lines are written before they @@ -125,36 +125,36 @@ delays below the 64 sample block size.

- next - up - previous - contents - index
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diff --git a/node122.html b/node122.html index 2d68389..674bb33 100644 --- a/node122.html +++ b/node122.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Time-varying complex comb filter: - Up: Up: Examples - Previous: Previous: Order of execution in -   Contents -   Index

-

+

Non-recirculating comb filter as octave doubler

@@ -83,7 +83,7 @@ would have got by using speed change to do the transposition.

-

+
Figure 7.29: An "octave doubler" uses pitch information (obtained using @@ -165,36 +165,36 @@ up-shifting for best results.)

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diff --git a/node123.html b/node123.html index f9a0310..1abb413 100644 --- a/node123.html +++ b/node123.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Reverberator - Up: Up: Examples - Previous: Previous: Non-recirculating comb filter as -   Contents -   Index

-

+

Time-varying complex comb filter: shakers

@@ -79,7 +79,7 @@ fourth "tap" is the original, un-delayed signal.

-

+
Figure 7.30: A "shaker", a four-tap comb filter with randomly varying gains @@ -135,36 +135,36 @@ subject of Chapter 8).

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diff --git a/node124.html b/node124.html index ebb3dd4..6943939 100644 --- a/node124.html +++ b/node124.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Pitch shifter - Up: Up: Examples - Previous: Previous: Time-varying complex comb filter: -   Contents -   Index

-

+

Reverberator

@@ -81,7 +81,7 @@ rotations of -
+
@@ -110,7 +110,7 @@ reinserted into them, so the reverberation lasts perpetually.

-

+
Figure 7.31: An artificial reverberator.
@@ -141,36 +141,36 @@ sonic qualities described as "presence", "warmth", "clarity", and so on.

- next - up - previous - contents - index
- Next: Next: Pitch shifter - Up: Up: Examples - Previous: Previous: Time-varying complex comb filter: -   Contents -   Index
diff --git a/node125.html b/node125.html index 16cf2be..d9dea51 100644 --- a/node125.html +++ b/node125.html @@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Exercises - Up: Up: Examples - Previous: Previous: Reverberator -   Contents -   Index

-

+

Pitch shifter

@@ -77,7 +77,7 @@ minimum plus a window size (the "window" control.)

-

+

Figure 7.32: The echo generator used in the reverberator.
Figure 7.33: A pitch shifter using two variable taps into a delay @@ -141,36 +141,36 @@ envelopes and summed.

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diff --git a/node126.html b/node126.html index 17810a6..fd5d670 100644 --- a/node126.html +++ b/node126.html @@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Filters - Up: Up: Time shifts and delays - Previous: Previous: Pitch shifter -   Contents -   Index

-

+

Exercises

@@ -174,36 +174,36 @@ function of angular frequency
- next - up - previous - contents - index
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diff --git a/node127.html b/node127.html index 7fecd59..5dc54b7 100644 --- a/node127.html +++ b/node127.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Taxonomy of filters - Up: Up: book - Previous: Previous: Exercises -   Contents -   Index

-

- +

+
Filters

@@ -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 phase response. A delay network that is designed specifically for its frequency or phase response is called a -filter. +filter.

-

+
Figure 8.1: Representations of a filter: (a) in a block diagram; (b) a graph of its @@ -116,7 +116,7 @@ depends on $H(\omega)$, which is called the -transfer function +transfer function of the filter.

@@ -168,128 +168,128 @@ when its parameters change quickly with time.


-Subsections +Subsections
- next - up - previous - contents - index
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diff --git a/node128.html b/node128.html index 9de581d..4d946fb 100644 --- a/node128.html +++ b/node128.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Low-pass and high-pass filters - Up: Up: Filters - Previous: Previous: Filters -   Contents -   Index

-

+

Taxonomy of filters

@@ -80,14 +80,14 @@ applications of filters in electronic music.


-Subsections +Subsections diff --git a/node129.html b/node129.html index b4dd09f..2a9800a 100644 --- a/node129.html +++ b/node129.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Band-pass and stop-band filters - Up: Up: Taxonomy of filters - Previous: Previous: Taxonomy of filters -   Contents -   Index

-

+

Low-pass and high-pass filters

@@ -73,13 +73,13 @@ Low-pass and high-pass filters 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 rest. This is accomplished using a -low-pass or -high-pass +low-pass or +high-pass filter.

-

+
Figure 8.2: 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 8.2 shows the frequency response of a low-pass filter. Frequency is divided into three bands, labeled on the horizontal axis. The -passband +passband is the region (frequency band) where the filter should pass its input through to its output with unit gain. 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 realizable filter's passband will be only approximately flat; the deviation from flatness is called the -ripple, +ripple, 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 would have a ripple of 0 dB.

The -stopband +stopband of a low-pass or high-pass filter is the frequency band over which the filter is intended not to transmit its input. The -stopband attenuation +stopband attenuation is the difference, in decibels, between the lowest gain in the passband and the highest gain in the stopband. Ideally this would 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 band over which the gain drops from the passband gain to the stopband gain; this is called the -transition band. +transition band. The thinner this band can be made, the more nearly ideal the filter.

- next - up - previous - contents - index
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diff --git a/node13.html b/node13.html index 8720bd8..4b0fe4f 100644 --- a/node13.html +++ b/node13.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Periodic Signals - Up: Up: Sinusoids, amplitude and frequency - Previous: Previous: Synthesizing a sinusoid -   Contents -   Index

-

- +

+
Superposing Signals

@@ -153,7 +153,7 @@ If we fix a window from $N+M-1$ as usual, we can write out the mean power of the sum of two signals: - +

- next - up - previous - contents - index
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diff --git a/node130.html b/node130.html index 72a32fd..4c95a60 100644 --- a/node130.html +++ b/node130.html @@ -30,50 +30,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Equalizing filters - Up: Up: Taxonomy of filters - Previous: Previous: Low-pass and high-pass filters -   Contents -   Index

-

- +

+
Band-pass and stop-band filters

A -band-pass filter +band-pass filter admits frequencies within a given band, rejecting frequencies below it and above it. Figure 8.3 shows the frequency response of a band-pass filter, with the key parameters labelled. A stop-band filter @@ -82,7 +82,7 @@ frequencies outside it.

-

+

In practice, a simpler language is often used for describing bandpass filters, as shown in Figure 8.4. Here there are only two parameters: a -center frequency +center frequency and a -bandwidth. +bandwidth. 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 maximum). The bandwidth is the width, in frequency units, of the passband. @@ -110,7 +110,7 @@ midpoint of the passband.

-

+
Figure 8.3: Terminology for describing the frequency response of band-pass and @@ -99,9 +99,9 @@ contiguous stopband surrounded by two passbands.

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diff --git a/node131.html b/node131.html index 1de5525..78cbbee 100644 --- a/node131.html +++ b/node131.html @@ -29,55 +29,55 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Elementary filters - Up: Up: Taxonomy of filters - Previous: Previous: Band-pass and stop-band filters -   Contents -   Index

-

- +

+
Equalizing filters

In some applications, such as -equalization, +equalization, the goal isn't to pass signals of certain frequencies while stopping others altogether, but to make controllable adjustments, boosting or attenuating a signal, over a frequency range, by a desired gain. Two filter types are useful for this. First, a -shelving filter +shelving filter (Figure 8.5) is used for selectively boosting or reducing either the 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 @@ -86,7 +86,7 @@ and high-frequency gain can all be adjusted independently.

-

+
Figure 8.4: A simplified view of a band-pass filter, showing bandwidth and @@ -125,36 +125,36 @@ center frequency.

Second, a -peaking filter +peaking filter (Figure 8.6) is capable of boosting or attenuating signals within 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 @@ -108,7 +108,7 @@ gains are separately adjustible.

-

+
Figure 8.5: A shelving filter, showing low and high frequency gain, and @@ -100,7 +100,7 @@ crossover frequency.
Figure 8.6: A peaking filter, with controllable center frequency, bandwidth, @@ -128,36 +128,36 @@ adjust bands in between.

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diff --git a/node132.html b/node132.html index 4788dac..e470daa 100644 --- a/node132.html +++ b/node132.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Elementary non-recirculating filter - Up: Up: Filters - Previous: Previous: Equalizing filters -   Contents -   Index

-

+

Elementary filters

@@ -103,55 +103,55 @@ filters--with complex-valued gains.


-Subsections +Subsections
- next - up - previous - contents - index
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diff --git a/node133.html b/node133.html index 66ee7c9..22ab48a 100644 --- a/node133.html +++ b/node133.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Non-recirculating filter, second form - Up: Up: Elementary filters - Previous: Previous: Elementary filters -   Contents -   Index

-

- +

+
Elementary non-recirculating filter

@@ -75,7 +75,7 @@ Elementary non-recirculating filter The non-recirculating comb filter may be generalized to yield the design shown in Figure 8.7. This is the -elementary non-recirculating filter, +elementary non-recirculating filter, of the first form. Its single, complex-valued parameter -
+
Figure 8.7: A delay network with a single-sample delay and a complex @@ -250,7 +250,7 @@ plane, which is equal to the distance from -
+
Figure 8.8: Diagram for calculating the frequency response of the @@ -300,7 +300,7 @@ function for three different values of -
+
Figure 8.9: Frequency response of the elementary non-recirculating filter @@ -322,36 +322,36 @@ same argument (-2 radians), but with varying absolute value (magnitude)
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diff --git a/node134.html b/node134.html index d2285be..0272dda 100644 --- a/node134.html +++ b/node134.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Elementary recirculating filter - Up: Up: Elementary filters - Previous: Previous: Elementary non-recirculating filter -   Contents -   Index

-

- +

+
Non-recirculating filter, second form

@@ -80,7 +80,7 @@ Instead of multiplying the delay output by $Q$ we multiply the direct signal by its -complex conjugate +complex conjugate -
+
@@ -194,36 +194,36 @@ The elementary non-recirculating filter, second form.

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diff --git a/node135.html b/node135.html index af12035..03ca536 100644 --- a/node135.html +++ b/node135.html @@ -30,50 +30,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Compound filters - Up: Up: Elementary filters - Previous: Previous: Non-recirculating filter, second form -   Contents -   Index

-

- +

+
Elementary recirculating filter

The -elementary recirculating filter is the recirculating comb filter of +elementary recirculating filter is the recirculating comb filter of Figure 7.7 with a complex-valued feedback gain -

+
Figure 8.10: The elementary non-recirculating filter, second form.
Figure 8.11: 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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diff --git a/node136.html b/node136.html index d93edde..66e2c98 100644 --- a/node136.html +++ b/node136.html @@ -30,49 +30,49 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Real outputs from complex - Up: Up: Elementary filters - Previous: Previous: Elementary recirculating filter -   Contents -   Index

-

+

Compound filters

We can use the recirculating and non-recirculating filters developed here to create a -compound filter by putting several elementary ones in series. If the parameters +compound filter by putting several elementary ones in series. If the parameters of the non-recirculating ones (of the first type) are diff --git a/node137.html b/node137.html index e3c7a35..5d5b1b8 100644 --- a/node137.html +++ b/node137.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Two recirculating filters for - Up: Up: Elementary filters - Previous: Previous: Compound filters -   Contents -   Index

-

+

Real outputs from complex filters

@@ -246,36 +246,36 @@ either real-valued, or else appears in a pair with its complex conjugate.

- next - up - previous - contents - index
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diff --git a/node138.html b/node138.html index e8d16d5..5c7cdc3 100644 --- a/node138.html +++ b/node138.html @@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Designing filters - Up: Up: Elementary filters - Previous: Previous: Real outputs from complex -   Contents -   Index

-

+

Two recirculating filters for the price of one

@@ -297,36 +297,36 @@ explicitly.

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diff --git a/node139.html b/node139.html index c37f6ba..93832f4 100644 --- a/node139.html +++ b/node139.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: One-pole low-pass filter - Up: Up: Filters - Previous: Previous: Two recirculating filters for -   Contents -   Index

-

+

Designing filters

@@ -109,7 +109,7 @@ and each of the $P_i$ with an "x" (a "pole"); their names are borrowed from the field of complex analysis. A plot showing the poles and zeroes -associated with a filter is unimaginatively called a pole-zero plot. +associated with a filter is unimaginatively called a pole-zero plot.

When

-Subsections +Subsections
- next - up - previous - contents - index
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diff --git a/node14.html b/node14.html index d847ec9..3dc9bb4 100644 --- a/node14.html +++ b/node14.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: About the Software Examples - Up: Up: Sinusoids, amplitude and frequency - Previous: Previous: Superposing Signals -   Contents -   Index

-

- +

+
Periodic Signals

@@ -105,7 +105,7 @@ the smallest $\tau$ if any at which a signal repeats is called the signal's -period. +period. In discussing periods of digital audio signals, we quickly run into the difficulty of describing signals whose "period" isn't an integer, so that the equation above doesn't make sense. For now we'll effectively @@ -149,7 +149,7 @@ with frequencies @@ -205,7 +205,7 @@ pitches of the same scale; the third and sixth miss only by 2 cents and the fifth misses by 14.

-Put another way, the frequency ratio 3:2 (a perfect fifth in Western +Put another way, the frequency ratio 3:2 (a perfect fifth in Western terminology) is almost exactly seven half-steps, 4:3 (a perfect fourth) is just as near to five half-steps, and the ratios 5:4 and 6:5 (perfect major and minor thirds) are fairly close to @@ -221,7 +221,7 @@ by the three component sinusoids.

-

+
Figure 1.8: A Fourier series, showing three sinusoids and their sum. The @@ -242,7 +242,7 @@ block diagram for doing this.

-

+

This is an example of -additive synthesis; more generally the term can be applied to networks +additive synthesis; more generally the term can be applied to networks in which the frequencies of the oscillators are independently controllable. The early days of computer music rang with the sound of additive synthesis.

- next - up - previous - contents - index
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diff --git a/node140.html b/node140.html index 6e88d09..e07e456 100644 --- a/node140.html +++ b/node140.html @@ -30,50 +30,50 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: One-pole, one-zero high-pass filter - Up: Up: Designing filters - Previous: Previous: Designing filters -   Contents -   Index

-

- +

+
One-pole low-pass filter

-

+
Figure 1.9: Using many oscillators to synthesize a waveform with desired harmonic @@ -256,43 +256,43 @@ amplitudes.
Figure 8.12: One-pole low-pass filter: (a) pole-zero diagram; (b) @@ -164,36 +164,36 @@ cycles per second.

- next - up - previous - contents - index
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diff --git a/node141.html b/node141.html index 90e01c5..4f39065 100644 --- a/node141.html +++ b/node141.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Shelving filter - Up: Up: Designing filters - Previous: Previous: One-pole low-pass filter -   Contents -   Index

-

- +

+
One-pole, one-zero high-pass filter

@@ -78,7 +78,7 @@ Section 5.3 almost always contain a This is inaudible, but, since it specifies electrical power that is sent to your speakers, its presence reduces the level of loudness you can reach without distortion. Another name for a constant signal component is -"DC", meaning "direct current". +"DC", meaning "direct current".

An easy and practical way to remove the zero-frequency component from an audio @@ -135,7 +135,7 @@ in the previous example we plotted it all the way up to the sample rate,

-

+

- next - up - previous - contents - index
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diff --git a/node142.html b/node142.html index 8bb82f7..3d3de1b 100644 --- a/node142.html +++ b/node142.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Band-pass filter - Up: Up: Designing filters - Previous: Previous: One-pole, one-zero high-pass filter -   Contents -   Index

-

- +

+
Shelving filter

@@ -96,7 +96,7 @@ diagrammed in Figure 8.14.

-

+
Figure 8.13: One-pole, one-zero high-pass filter: (a) pole-zero diagram; (b) @@ -150,36 +150,36 @@ frequency response (from zero to Nyquist frequency).
Figure 8.14: One-pole, one-zero shelving filter: (a) pole-zero diagram; (b) @@ -211,36 +211,36 @@ as desired. For example, in the figure,
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diff --git a/node143.html b/node143.html index baebc11..0ccb5a9 100644 --- a/node143.html +++ b/node143.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Peaking and stop-band filter - Up: Up: Designing filters - Previous: Previous: Shelving filter -   Contents -   Index

-

- +

+
Band-pass filter

@@ -146,7 +146,7 @@ The resulting pole-zero plot is as shown in Figure 8.15.

-

+
Figure 8.15: Two-pole band-pass filter: (a) pole-zero diagram; (b) @@ -206,36 +206,36 @@ so that the gain drops to zero at angular frequencies
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diff --git a/node144.html b/node144.html index 87642fd..117fc9c 100644 --- a/node144.html +++ b/node144.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Butterworth filters - Up: Up: Designing filters - Previous: Previous: Band-pass filter -   Contents -   Index

-

- +

+
Peaking and stop-band filter

@@ -126,7 +126,7 @@ to the one-pole, one-zero high-pass filter above.

-

+
Figure 8.16: A peaking filter: (a) pole-zero diagram; (b) diff --git a/node145.html b/node145.html index 5c7e04c..5bd7b8b 100644 --- a/node145.html +++ b/node145.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Stretching the unit circle - Up: Up: Designing filters - Previous: Previous: Peaking and stop-band filter -   Contents -   Index

-

+

Butterworth filters

@@ -86,7 +86,7 @@ whose two bands are flatter and separated by a narrower transition region.

A procedure borrowed from the analog filtering world transforms real, one-pole, one-zero filters to corresponding - + Butterworth filters, which have narrower transition regions. This procedure is described clearly and elegantly in the last chapter of [ -

+
Figure 8.17: Replacing a real-valued pole or zero (shown as a solid dot) with an @@ -300,7 +300,7 @@ becomes a shelving filter and then a high-pass one.

-

+
Figure 8.18: Butterworth low-pass filter with three poles @@ -338,36 +338,36 @@ unit gain at the Nyquist frequency, and the others for unit gain at DC.
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diff --git a/node146.html b/node146.html index 4fcae65..556ed13 100644 --- a/node146.html +++ b/node146.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Butterworth band-pass filter - Up: Up: Designing filters - Previous: Previous: Butterworth filters -   Contents -   Index

-

+

Stretching the unit circle with rational functions

@@ -421,7 +421,7 @@ points of highest gain for the new filter.

-

+
Figure 8.19: One-pole, one-zero low-pass filter: (a) pole-zero plot; (b) @@ -444,36 +444,36 @@ result is a band-pass filter with center frequency
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diff --git a/node147.html b/node147.html index 2c15abf..1547da7 100644 --- a/node147.html +++ b/node147.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Time-varying coefficients - Up: Up: Designing filters - Previous: Previous: Stretching the unit circle -   Contents -   Index

-

+

Butterworth band-pass filter

@@ -402,7 +402,7 @@ are shown in Figure 8.20.

-

+

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diff --git a/node148.html b/node148.html index 692db1d..92cbcc2 100644 --- a/node148.html +++ b/node148.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Impulse responses of recirculating - Up: Up: Designing filters - Previous: Previous: Butterworth band-pass filter -   Contents -   Index

-

- +

+
Time-varying coefficients

@@ -96,7 +96,7 @@ factor of two.

-

+
Figure 8.20: Butterworth band-pass filter: (a) pole-zero diagram; (b) @@ -422,36 +422,36 @@ low-pass filter used.
Figure 8.21: Normalizing a recirculating elementary filter: @@ -132,36 +132,36 @@ is needed.

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diff --git a/node149.html b/node149.html index 996f1c4..9809c45 100644 --- a/node149.html +++ b/node149.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: All-pass filters - Up: Up: Designing filters - Previous: Previous: Time-varying coefficients -   Contents -   Index

-

+

Impulse responses of recirculating filters

@@ -116,7 +116,7 @@ where $e$ is Euler's constant, about 2.718. The filter can be said to have a -settling time of settling time of $n$ samples. In the figure, -
+
Figure 8.22: The impulse response of three elementary recirculating (one-pole) @@ -223,43 +223,43 @@ where $q$ is the -quality of the filter, defined as the center frequency divided by +quality of the filter, defined as the center frequency divided by bandwidth. Resonant filters are often specified in terms of the center frequency and "q" in place of bandwidth.

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diff --git a/node15.html b/node15.html index 30c2db8..a5be097 100644 --- a/node15.html +++ b/node15.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Quick Introduction to Pd - Up: Up: Sinusoids, amplitude and frequency - Previous: Previous: Periodic Signals -   Contents -   Index

-

+

About the Software Examples

@@ -106,47 +106,47 @@ patches from one to the other, but the two aren't truly compatible.


-Subsections +Subsections
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diff --git a/node150.html b/node150.html index 6506cb0..4120172 100644 --- a/node150.html +++ b/node150.html @@ -29,42 +29,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Applications - Up: Up: Designing filters - Previous: Previous: Impulse responses of recirculating -   Contents -   Index

-

+

All-pass filters

@@ -85,7 +85,7 @@ introduces a phase change of $\omega $. Another class of filters, called -all-pass filters, +all-pass filters, can make phase changes which are more interesting functions of -
+
Figure 8.23: Phase response of all-pass filters with different pole locations
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diff --git a/node151.html b/node151.html index dfd50ee..0952917 100644 --- a/node151.html +++ b/node151.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Subtractive synthesis - Up: Up: Filters - Previous: Previous: All-pass filters -   Contents -   Index

-

+

Applications

@@ -79,14 +79,14 @@ musical applications.


-Subsections +Subsections diff --git a/node152.html b/node152.html index c8b96f6..c4e20dd 100644 --- a/node152.html +++ b/node152.html @@ -30,49 +30,49 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Envelope following - Up: Up: Applications - Previous: Previous: Applications -   Contents -   Index

-

- +

+
Subtractive synthesis

-Subtractive synthesis is the technique of using filters to shape the +Subtractive synthesis is the technique of using filters to shape the spectral envelope of a sound, forming another sound, usually preserving qualities of the original sound such as pitch, roughness, noisiness, or graniness. The spectral envelope of the resulting sound is the product of the @@ -82,7 +82,7 @@ filter, and result.

-

+
Figure 8.24: Subtractive synthesis: (a) spectrum of input sound; (b) filter frequency @@ -113,7 +113,7 @@ brass instruments over the life of a note.

-

+
@@ -127,36 +127,36 @@ ADSR-controlled subtractive synthesis.

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diff --git a/node153.html b/node153.html index 06daf34..eae2860 100644 --- a/node153.html +++ b/node153.html @@ -30,43 +30,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Single Sideband Modulation - Up: Up: Applications - Previous: Previous: Subtractive synthesis -   Contents -   Index

-

- +

+
Envelope following

@@ -81,7 +81,7 @@ power over an interval of time long enough that its variations won't show up in the power estimate, but short enough that changes in signal level are quickly reported. A computation that provides a time-varying power estimate of a signal is called an -envelope follower. +envelope follower.

The output of a low-pass filter can be viewed as a moving average of its input. @@ -133,7 +133,7 @@ the moving average.

-

+
Figure 8.25: ADSR-controlled subtractive synthesis.
Figure 8.26: Envelope follower. The output is the average power of the input @@ -232,7 +232,7 @@ estimates the average power.

-

+
Figure 8.27: Envelope following from the spectral point of view: (a) an @@ -260,36 +260,36 @@ getting a quick response and a smooth result.

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diff --git a/node154.html b/node154.html index 640793c..072ab8d 100644 --- a/node154.html +++ b/node154.html @@ -29,43 +29,43 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Examples - Up: Up: Applications - Previous: Previous: Envelope following -   Contents -   Index

-

- +

+
Single Sideband Modulation

@@ -362,36 +362,36 @@ only positive frequencies.

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diff --git a/node155.html b/node155.html index bc41a7a..52d2463 100644 --- a/node155.html +++ b/node155.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Prefabricated low-, high-, and - Up: Up: Filters - Previous: Previous: Single Sideband Modulation -   Contents -   Index

-

+

Examples

@@ -78,20 +78,20 @@ that require specially designed filters.


-Subsections +Subsections diff --git a/node156.html b/node156.html index c268825..600127a 100644 --- a/node156.html +++ b/node156.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Prefabricated time-varying band-pass filter - Up: Up: Examples - Previous: Previous: Examples -   Contents -   Index

-

+

Prefabricated low-, high-, and band-pass filters

@@ -84,7 +84,7 @@ We will need four new Pd objects: WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img1014.png" ALT="\fbox{\texttt{lop\~}}">: -one-pole low-pass filter. The left inlet takes a signal to be filtered, and +one-pole low-pass filter. The left inlet takes a signal to be filtered, and the right inlet takes control messages to set the cutoff frequency of the filter. The filter is normalized so that the gain is one at frequency 0. @@ -97,7 +97,7 @@ gain is one at frequency 0. WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img1016.png" ALT="\fbox{ \texttt{hip\~}}">: -one-pole, one-zero high-pass filter, with the same inputs and outputs as +one-pole, one-zero high-pass filter, with the same inputs and outputs as lop~, normalized to have a gain of one at the Nyquist frequency.

@@ -108,7 +108,7 @@ gain is one at frequency 0. WIDTH="44" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img1017.png" ALT="\fbox{ \texttt{bp\~}}">: -resonant filter. The middle inlet takes control messages to set the center +resonant filter. The middle inlet takes control messages to set the center frequency, and the right inlet to set "q".

@@ -119,12 +119,12 @@ frequency, and the right inlet to set "q". WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img1019.png" ALT="\fbox{ \texttt{noise\~}}">: -white noise generator. Each sample is an independent +white noise generator. Each sample is an independent pseudo-random number, uniformly distributed from -1 to 1.

-

+
Figure 8.28: Using prefabricated filters in Pd: (a) a low-pass filter, with @@ -146,36 +146,36 @@ frequency) component of a signal.

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diff --git a/node157.html b/node157.html index 69876ac..5a110be 100644 --- a/node157.html +++ b/node157.html @@ -30,42 +30,42 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Envelope followers - Up: Up: Examples - Previous: Previous: Prefabricated low-, high-, and -   Contents -   Index

-

+

Prefabricated time-varying band-pass filter

@@ -82,7 +82,7 @@ using the vcf~ object, introduced here: WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="img1022.png" ALT="\fbox{ \texttt{vcf\~}}">: -a "voltage controlled" band-pass filter, +a "voltage controlled" band-pass filter, similar to bp~, but with a signal inlet to control center frequency. Both bp~ and vcf~ are one-pole resonant filters as developed in Section 8.3.4; bp~ outputs only @@ -91,7 +91,7 @@ real and imaginary parts separately.

-

+
Figure 8.29: The vcf~ 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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diff --git a/node158.html b/node158.html index 5651a36..a7eeabb 100644 --- a/node158.html +++ b/node158.html @@ -30,45 +30,45 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Single sideband modulation - Up: Up: Examples - Previous: Previous: Prefabricated time-varying band-pass filter -   Contents -   Index

-

+

Envelope followers

- +

Example H06.envelope.follower.pd shows a simple and self-explanatory realization of the envelope follower described in Section 8.4.2. An @@ -84,7 +84,7 @@ strength.

-

+
Figure 8.30: 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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diff --git a/node159.html b/node159.html index f8817f3..1934095 100644 --- a/node159.html +++ b/node159.html @@ -30,48 +30,48 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: Using elementary filters directly: - Up: Up: Examples - Previous: Previous: Envelope followers -   Contents -   Index

-

+

Single sideband modulation

-

+
Figure 8.31: Using an all-pass filter network to diff --git a/node16.html b/node16.html index 45cc118..3a651e2 100644 --- a/node16.html +++ b/node16.html @@ -30,48 +30,48 @@ original version by: Nikos Drakos, CBLU, University of Leeds - next - up - previous - contents - index
- Next: Next: How to find and - Up: Up: About the Software Examples - Previous: Previous: About the Software Examples -   Contents -   Index

-

+

Quick Introduction to Pd

Pd documents are called -patches. They correspond roughly to the +patches. They correspond roughly to the boxes in the abstract block diagrams shown earlier in this chapter, but in detail they are quite different, because Pd is an implementation @@ -80,7 +80,7 @@ environment, not a specification language.

A Pd patch, such as the ones shown in Figure 1.10, consists of a collection of -boxes +boxes connected in a network. 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 @@ -88,7 +88,7 @@ top to bottom they are: