# Acoustic Modeling with Digital Delay

*Delay effects*, such as *phasing*, *flanging*,
chorus, and artificial
reverberation,
as well as digital waveguide models (Chapter 6), are built
using delay lines, simple digital filter sections, and sometimes
nonlinear elements and modulation. We will focus on these elements in
the simpler context of delay effects before using them for sound
synthesis.

## Delay Lines

The *delay line* is an elementary functional
unit which models *acoustic propagation delay*. It is a
fundamental building block of both delay-effects processors and
digital-waveguide synthesis models. The function of a delay line is
to introduce a time delay between its input and output, as shown in
Fig.2.1.

Let the input signal be denoted , and let the delay-line length be samples. Then the output signal is specified by the relation

where for .

Before the digital era, delay lines were expensive and imprecise in ``analog'' form. For example, ``spring reverberators'' (common in guitar amplifiers) use metal springs as analog delay lines; while adequate for that purpose, they are highly dispersive and prone to noise pick-up. Large delays require prohibitively long springs or coils in analog implementations. In the digital domain, on the other hand, delay by samples is trivially implemented, and non-integer delays can be implemented using interpolation techniques, as discussed later in §4.1.

### A Software Delay Line

In software, a delay line is often implemented using a *circular
buffer*. Let `D` denote an array of length . Then we can
implement the -sample delay line in the `C` programming
language as shown in Fig.2.2.

/* delayline.c */ static double D[M]; // initialized to zero static long ptr=0; // read-write offset double delayline(double x) { double y = D[ptr]; // read operation D[ptr++] = x; // write operation if (ptr >= M) { ptr -= M; } // wrap ptr // ptr %= M; // modulo-operator syntax return y; } |

Delay lines of this type are typically used in *digital
reverberators* and other acoustic simulators involving *fixed*
propagation delays. Later, in Chapter 5, we will consider
*time-varying* delay lengths.

## Acoustic Wave Propagation Simulation

Delay lines can be used to simulate *acoustic wave propagation*.
We start with the simplest case of a pure *traveling wave*,
followed by the more general case of *spherical waves*. We then
look at the details of a simple *acoustic echo* simulation using
a delay line to model the difference in time-of-arrival between the
direct and reflected signals.

### Traveling Waves

In acoustic wave propagation, pure delays can be used to simulate
*traveling waves*. A traveling wave is any kind of wave which
propagates in a single direction with negligible change in shape. An
important class of traveling waves is ``plane waves'' in air which
create ``standing waves'' in rectangular enclosures such as
``shoebox'' shaped concert halls. Also, far away from any acoustic
source (where ``far'' is defined as ``many wavelengths''), the direct
sound emanating from any source can be well approximated as a plane
wave, and thus as a traveling wave.

Another case in which plane waves dominate is the *cylindrical
bore*, such as the bore of a clarinet or the straight tube segments of
a trumpet. Additionally, the *vocal tract* is generally
simulated using plane waves, though in this instance there is a
higher degree of approximation error.

*Transverse* and *longitudinal* waves in a vibrating string, such as on a
guitar, are also nearly perfect traveling waves, and they can be
simulated to a very high degree of perceptual accuracy by
approximating them as ideal, while implementing slight losses and
dispersion once per period (*i.e.*, at one particular point along the
``virtual string'').

In a conical bore, we find sections of *spherical waves* taking
the place of plane waves. However, they still ``travel'' like plane
waves, and we can still use a delay line to simulate their
propagation. The same applies to spherical waves created by a ``point
source.'' Spherical waves will be considered on
page .

### Damped Traveling Waves

The delay line shown in Fig.2.1 on page can be used
to simulate any traveling wave, where the traveling wave must
propagate in one direction with a fixed waveshape. If a traveling
wave *attenuates* as it propagates, with the same attenuation
factor at each frequency, the attenuation can be simulated by a simple
*scaling* of the delay line output (or input), as shown in
Fig.2.3. This is perhaps the simplest example of
the important principle of *lumping distributed losses* at
discrete points. That is, it is not necessary to implement a small
attenuation for each time-step of wave propagation; the same
result is obtained at the delay-line output if propagation is
``lossless'' within the delay line, and the total cumulative
attenuation is applied at the output. The input-output
simulation is exact, while the signal samples inside the delay line
are simulated with a slight gain error. If the internal signals are
needed later, they can be tapped out using correcting gains. For
example, the signal half way along the delay line can be tapped using
a coefficient of in order to make it an exact second output.
In summary, computational efficiency can often be greatly increased at
no cost to accuracy by lumping losses only at the outputs and points
of interaction with other simulations.

Modeling traveling-wave attenuation by a scale factor is only exact
physically when all frequency components decay at the same rate. For
accurate acoustic modeling, it is usually necessary to replace the
constant scale factor by a *digital filter* which
implements *frequency-dependent attenuation*, as depicted in
Fig.2.4. In principle, a linear time-invariant (LTI) filter
can provide an independent attenuation factor at each frequency.
Section 2.3 addresses this case in more detail.
Frequency-dependent damping substitution will be used in artificial
reverberation design in §3.7.4.

### Dispersive Traveling Waves

In many acoustic systems, such as *piano strings*
(§9.4.1,§C.6), wave propagation is also
significantly *dispersive*. A wave-propagation medium is said to
be dispersive if the speed of wave propagation is not the same at all
frequencies. As a result, a propagating wave shape will ``disperse''
(change shape) as its various frequency components travel at different
speeds. Dispersive propagation in one direction can be simulated
using a delay line in series with a *nonlinear phase* filter, as
indicated in Fig.2.5. If there is no damping, the filter
must be *all-pass* [449], *i.e.*,
for
all
.

### Converting Propagation Distance to Delay Length

We may regard the delay-line memory itself as the fixed ``air'' which propagates sound samples at a fixed speed ( meters per second at degrees Celsius and 1 atmosphere). The input signal can be associated with a sound source, and the output signal (see Fig.2.1 on page ) can be associated with the listening point. If the listening point is meters away from the source, then the delay line length needs to be

### Spherical Waves from a Point Source

Acoustic theory tells us that a *point source* produces a
*spherical wave* in an ideal isotropic (uniform) medium such as air.
Furthermore, the sound from any radiating surface can be computed as
the sum of spherical wave contributions from each point on the surface
(including any relevant reflections). The *Huygens-Fresnel principle*
explains wave propagation itself as the superposition of spherical
waves generated at each point along a wavefront (see, *e.g.*,
[349, p. 175]). Thus, all linear acoustic wave propagation
can be seen as a superposition of spherical traveling waves.

To a good first approximation, wave energy is *conserved* as it
propagates through the air. In a spherical pressure wave of radius
, the energy of the wavefront is spread out over the spherical
surface area . Therefore, the energy per unit area of an
expanding spherical pressure wave decreases as . This is
called *spherical spreading loss*. It is also an example of an
*inverse square law* which is found repeatedly in the physics of
conserved quantities in three-dimensional space. Since energy is
proportional to amplitude squared, an inverse square law for energy
translates to a decay law for amplitude.

The sound-pressure amplitude of a traveling wave is proportional to the square-root of its energy per unit area. Therefore, in a spherical traveling wave, acoustic amplitude is proportional to , where is the radius of the sphere. In terms of Cartesian coordinates, the amplitude at the point due to a point source located at is given by

*i.e.*, where ), and denotes the distance from the point to :

In summary, every point of a radiating sound source emits spherical traveling waves in all directions which decay as , where is the distance from the source. The amplitude-decay by can be considered a consequence of energy conservation for propagating waves. (The energy spreads out over the surface of an expanding sphere.) We often visualize such waves as ``rays'' emanating from the source, and we can simulate them as a delay line along with a scaling coefficient (see Fig.2.7). In contrast, since plane waves propagate with no decay at all, each ``ray'' can be considered lossless, and the simulation involves only a delay line with no scale factor, as shown in Fig.2.1 on page .

### Reflection of Spherical or Plane Waves

When a spreading spherical wave reaches a wall or other obstacle, it
is either reflected or scattered. A wavefront is *reflected* when
it impinges on a surface which is flat over at least a few wavelengths
in each direction.^{3.1} Reflected
wavefronts can be easily mapped using *ray tracing*, *i.e.*, the
reflected ray leaves at an angle to the surface equal to the angle of
incidence (``law of reflection''). Wavefront reflection is also
called *specular reflection*, especially when considering light
waves.

A wave is *scattered* when it encounters a surface which has
variations on the scale of the spatial wavelength. A scattering
reflection is also called a *diffuse reflection*. As a special
case, objects smaller than a wavelength yield a diffuse reflection
which approaches a spherical wave as the object approaches zero
volume. More generally, each point of a scatterer can be seen as
emitting a new spherically spreading wavefront in response to the
incoming wave--a decomposition known as Huygen's principle, as
mentioned in the previous section. The same process happens in
reflection, but the hemispheres emitted by each point of the flat
reflecting surface combine to form a more organized wavefront which is
the same as the incident wave but traveling in a new direction.

The distinction between specular and diffuse reflections is dependent on frequency. Since sound travels approximately 1 foot per millisecond, a cube 1 foot on each side will ``specularly reflect'' directed ``beams'' of sound energy above KHz, and will ``diffuse'' or scatter sound energy below KHz. A good concert hall, for example, will have plenty of diffusion. As a general rule, reverberation should be diffuse in order to avoid ``standing waves'' (isolated energetic modes). In other words, in reverberation, we wish to spread the sound energy uniformly in both time and space, and we do not want any specific spatial or temporal patterns in the reverberation.

### An Acoustic Echo Simulator

An acoustic *echo* is one of the simplest acoustic modeling
problems. Echoes occur when a sound arrives via more than one
acoustic propagation path, as shown in Fig.2.8. We may hear a
discrete echo, for example, if we clap our hands standing in front of
a large flat wall outdoors, such as the side of a building. To be
perceived as an echo, however, the reflection must arrive well after
the direct signal (or previous echo).

A common cause of echoes is ``multipath'' wave propagation, as diagrammed in Fig.2.8. The acoustic source is denoted by `S', the listener by `L', and they are at the same height meters from a reflecting surface. The direct path is meters long, while the length of the single reflection is meters. These quantities are of course related by the Pythagorean theorem:

Figure 2.9 illustrates an echo simulator for the case of a direct
signal and single echo, as shown in Fig.2.8. It is common
practice to pull out and discard any *common delay* which affects
all signals equally, since such a delay does not affect timbre; thus,
the direct signal delay is not implemented at all in Fig.2.9.
Similarly, it is not necessary to implement the *attenuation* of
the direct signal due to propagation, since it is the *relative
amplitude* of the direct signal and its echoes which affect timbre.

From the geometry in Fig.2.8, we see that the delay-line length in Fig.2.9 should be

### Program for Acoustic Echo Simulation

The following main program (Fig.2.10) simulates a simple
acoustic echo using the `delayline` function in
Fig.2.2. It reads a sound file and writes a sound file
containing a single, discrete echo at the specified delay. For
simplicity, utilities from the free Synthesis Tool Kit (STK) (Version
`4.2.x`) are used for sound input/output [86].^{3.2}

/* Acoustic echo simulator, main C++ program. Compatible with STK version 4.2.1. Usage: main inputsoundfile Writes main.wav as output soundfile */ #include "FileWvIn.h" /* STK soundfile input support */ #include "FileWvOut.h" /* STK soundfile output support */ static const int M = 20000; /* echo delay in samples */ static const int g = 0.8; /* relative gain factor */ #include "delayline.c" /* defined previously */ int main(int argc, char *argv[]) { long i; Stk::setSampleRate(FileRead(argv[1]).fileRate()); FileWvIn input(argv[1]); /* read input soundfile */ FileWvOut output("main"); /* creates main.wav */ long nsamps = input.getSize(); for (i=0;i<nsamps+M;i++) { StkFloat insamp = input.tick(); output.tick(insamp + g * delayline(insamp)); } } |

In summary, a delay line simulates the *time delay* associated
with wave propagation in a particular direction. Attenuation (*e.g.*,
by ) associated with ray propagation can of course be simulated
by multiplying the delay-line output by some constant .

## Lossy Acoustic Propagation

Attenuation of waves by spherical spreading, as described in
§2.2.5 above, is not the only source of amplitude decay
in a traveling wave. In air, there is always significant additional
loss caused by *air absorption*. Air absorption varies with
frequency, with high frequencies usually being more attenuated than
low frequencies, as discussed in §B.7.15. Wave
propagation in *vibrating strings* undergoes an analogous
absorption loss, as does the propagation of nearly every other kind of
wave in the physical world. To simulate such propagation losses, we
can use a delay line in series with a nondispersive filter, as
illustrated in §2.2.2 above. In practice, the desired attenuation
at each frequency becomes the desired magnitude frequency-response of
the filter in Fig.2.4, and filter-design software
(typically matlab) is used to compute the filter coefficients to
approximate the desired frequency response in some optimal way. The
phase response may be linear, minimum, or left unconstrained when
damping-filter dispersion is not considered harmful. There is
typically a frequency-dependent weighting on the approximation error
corresponding to audio perceptual importance (*e.g.*, the weighting
is a simple example that increases accuracy at low frequencies).
Some filter-design methods are summarized in §8.6.

### Exponentially Decaying Traveling Waves

Let
denote the decay factor associated with
propagation of a plane wave over distance at frequency
rad/sec. For an ideal plane wave, there is no ``spreading
loss'' (attenuation by ). Under uniform conditions, the
amount of attenuation (in dB) is proportional to the distance
traveled; in other words, the attenuation factors for two successive
segments of a propagation path are *multiplicative*:

*exponential*function of distance .

^{3.3}

*Frequency-independent air
absorption* is easily modeled in an acoustic simulation by making
the substitution

### Frequency-Dependent Air-Absorption Filtering

More generally, *frequency-dependent* air
absorption can be modeled using the substitution

*filtering per sample*in the propagation medium. Since air absorption cannot amplify a wave at any frequency, we have . A lossy delay line for plane-wave simulation is thus described by

For spherical waves, the loss due to spherical spreading is of the form

### Dispersive Traveling Waves

In addition to frequency-dependent attenuation, LTI filters can
provide a *frequency-dependent delay*. This can be used to
simulate *dispersive wave
propagation*, as
introduced in §2.2.3.

### Summary

Up to now, we have been concerned with the simulation of
*traveling waves* in *linear, time-invariant (LTI) media*.
The main example considered was wave propagation in air, but waves on
vibrating strings behave analogously. We saw that the point-to-point
propagation of a traveling plane wave in an LTI medium can be
simulated simply using only a *delay line* and an *LTI
filter*. The delay line simulates propagation delay, while the filter
further simulates (1) an independent attenuation factor at each
frequency by means of its amplitude response (*e.g.*, to simulate air
absorption), and (2) a frequency-dependent propagation speed using its
phase response (to simulate dispersion). If there is additionally
spherical spreading loss, the amplitude may be further attenuated by
, where is the distance from the source. For more details
about the acoustics of plane waves and spherical waves, see, *e.g.*,
[318,349]. Appendix B contains a bit more about
elementary acoustics,

So far we have considered only traveling waves going in one direction.
The next simplest case is 1D acoustic systems, such as vibrating
strings and acoustic tubes, in which traveling waves may propagate in
*two* directions. Such systems are simulated using a pair of
delay lines called a *digital waveguide*.

## Digital Waveguides

A (lossless) *digital waveguide* is defined as a
*bidirectional delay line* at some wave impedance
[430,433].
Figure 2.11 illustrates one digital waveguide.

As before, each delay line contains a sampled acoustic traveling wave.
However, since we now have a *bidirectional* delay line, we have
*two* traveling waves, one to the ``left'' and one to the
``right'', say. It has been known since 1747 [100] that
the vibration of an ideal string
can be described as the sum of two traveling waves going in opposite
directions. (See Appendix C for a mathematical derivation of this
important fact.) Thus, while a single delay line can model an
acoustic plane wave, a *bidirectional* delay line (a digital
waveguide) can model any one-dimensional linear acoustic system such
as a violin string, clarinet bore, flute pipe, trumpet-valve pipe, or
the like. Of course, in real acoustic strings and bores, the 1D
waveguides exhibit some loss and
dispersion^{3.4} so that we will need some *filtering* in
the waveguide to obtain an accurate physical model of such systems.
The *wave impedance* (derived in Chapter 6) is
needed for connecting digital waveguides to other physical simulations
(such as another digital waveguide or finite-difference model).

### Physical Outputs

Physical variables (force, pressure, velocity, ...) are obtained by
*summing* traveling-wave components, as shown in
Fig.2.12, and more elaborated in
Fig.2.13.

It is important to understand that the two traveling waves in a
digital waveguide are now *components* of a more general acoustic
vibration. The physical wave vibration is obtained by *summing*
the left- and right-going traveling waves. A traveling wave by itself
in one of the delay lines is no longer regarded as ``physical'' unless
the signal in the opposite-going delay line is zero. Traveling waves
are efficient for simulation, but they are not easily estimated from
real-world measurements [476], except when the
traveling-wave component in one direction can be arranged to be zero.

Note that traveling-wave components are not necessarily *unique*.
For example, we can add a constant to the right-going wave and
subtract the same constant from the left-going wave without altering
the (physical) sum [263]. However, as
derived in Appendix C (§C.3.6), 1D traveling-wave components
are uniquely specified by *two* linearly independent physical
variables along the waveguide, such as position and velocity
(vibrating strings) or pressure and velocity (acoustic tubes).

### Physical Inputs

A digital waveguide *input signal* corresponds to a
*disturbance* of the 1D propagation medium. For example, a
vibrating string is *plucked* or *bowed* by such an external
disturbance. The result of the disturbance is wave propagation to the
left and right of the input point. By physical symmetry, the
amplitude of the left- and right-going propagating disturbances will
normally be equal.^{3.5} If
the disturbance *superimposes* with the waves already passing
through at that point (an idealized case), then it is purely an
*additive input*, as shown in Fig.2.14.

Note that the superimposing input of
Fig.2.14 is the graph-theoretic
*transpose* of the ideal output shown in
Fig.2.13. In other words, the
superimposing input injects by means of two *transposed taps*.
Transposed taps are discussed further in §2.5.2 below.

In practical reality, physical driving inputs do not merely
superimpose with the current state of the driven system. Instead,
there is normally some amount of *interaction* with the current
system state (when it is nonzero), as discussed further in the next
section. Note that there are similarly no ideal outputs as depicted
in Fig.2.13. Real physical ouputs must
present some kind of *load* on the system (energy must be
extracted). Superimposing inputs and non-loading outputs are ideals
that are often approximated in real-world systems. Of course, in the
virtual world, they are no problem at all--in fact, they are usually
easier to implement, and more efficient.

### Interacting Physical Input

Figure 2.15 shows the general case of an input signal that interacts with the state of the system at one point along the waveguide. Since the interaction is physical, it only depends on the ``incoming state'' (traveling-wave components) and the driving input signal.

A less general but commonly encountered case is shown in Fig.2.16. This case requires the ``outgoing disturbance'' to be distributed equally to the left and right, and it sums with the incoming waves to produce the outgoing waves.

Figure 2.17 shows a further reduction in
generality--also commonly encountered--in which the interaction
depends only on the *amplitude* of the simulated physical
variable (such as string velocity or displacement). The incoming
amplitude is formed as the sum of the incoming traveling-wave
components. We will encounter examples of this nature in later
chapters (such as Chapter 9). It provides realistic models
of physical excitations such as a guitar plectra, violin bows, and
woodwind reeds.

If an output signal is desired at this precise point, it can be computed as the incoming amplitude plus twice the outgoing disturbance signal (equivalent to summing the inputs of the two outgoing delay lines).

Note that the above examples all involve waveguide excitation at a single spatial point. While this can give a sufficiently good approximation to physical reality in many applications, one should also consider excitations that are spread out over multiple spatial samples (even just two).

We will develop the topic of digital waveguide modeling more
systematically in Chapter 6 and Appendix C, among other
places in this book. This section is intended only as a high-level
preview and overview. For the next several chapters, we will restrict
attention to normal signal processing structures in which signals may
have physical units (such as acoustic pressure), and delay lines hold
sampled acoustic waves propagating in one direction, but successive
processing blocks do not ``load each other down'' or connect
``bidirectionally'' (as every truly physical interaction must, by
Newton's third law^{3.6}).
Thus, when one processing block feeds a signal to a next block, an
``ideal output'' drives an ``ideal input''. This is typical in
digital signal processing: Loading effects and return
waves^{3.7} are
neglected.^{3.8}

## Tapped Delay Line (TDL)

A *tapped delay line* (TDL) is a delay line with at least one
``tap''. A delay-line *tap* extracts a signal output from
somewhere within the delay line, optionally scales it, and usually
sums with other taps for form an output signal. A tap may be
*interpolating* or *non-interpolating*. A non-interpolating
tap extracts the signal at some fixed integer delay relative to the
input. Thus, a tap implements a shorter delay line within a larger
one, as shown in
Fig.2.18.

Tapped delay lines efficiently simulate *multiple echoes* from
the same source signal. As a result, they are extensively used in the
field of *artificial reverberation*.

### Example Tapped Delay Line

An example of a TDL with two internal taps is shown in Fig.2.19. The total delay line length is samples, and the internal taps are located at delays of and samples, respectively. The output signal is a linear combination of the input signal , the delay-line output , and the two tap signals and .

The difference equation of the TDL in Fig.2.19 is, by inspection,

### Transposed Tapped Delay Line

In many applications, the *transpose* of a tapped delay line is
desired, as shown in Fig.2.20, which is the transpose of the tapped
delay line shown in Fig.2.19. A transposed TDL is obtained from a
normal TDL by formal *transposition* of the system diagram. The
transposition operation is also called *flow-graph reversal*
[333, pp. 153-155]. A flow-graph is transposed
by reversing all signal paths, which necessitates signal branchpoints
becoming sums, and sums becoming branchpoints. For single-input,
single-output systems, the transfer function is the same, but the
input and output are interchanged. This ``flow-graph reversal
theorem'' derives from *Mason's gain formula* for signal flow
graphs.
Transposition is used to convert direct-forms I and II of a digital
filter to direct-forms III and IV, respectively
[333].

### TDL for Parallel Processing

When multiplies and additions can be performed in parallel, the
computational complexity of a tapped delay line is
multiplies and
additions, where is the number of
taps. This computational complexity is achieved by arranging the
additions into a *binary tree*, as shown in Fig.2.21 for the
case .

### General Causal FIR Filters

The most general case--a TDL having a tap after *every* delay
element--is the general causal *Finite Impulse Response (FIR)*
filter, shown in Fig.2.22. It is restricted to be *causal*
because the output may not depend on *``future''* inputs
, , etc. The FIR filter is also called a
*transversal filter*. FIR filters are described in greater
detail in [449].

The *difference equation* for the th-order FIR filter in Fig.2.22
is, by inspection,

*transfer function*is

The STK class for implementing arbitrary direct-form FIR filters is
called Fir. (There is also a class for IIR filters named Iir.)
In Matlab and Octave, the built-in function `filter` is
normally used.

## Comb Filters

Comb filters are basic building blocks for digital audio effects. The
acoustic echo simulation in Fig.2.9 is one instance of a comb
filter. This section presents the two basic comb-filter types,
*feedforward* and *feedback*, and gives a frequency-response
analysis.

### Feedforward Comb Filters

The *feedforward comb filter* is shown in Fig.2.23. The
direct signal ``feeds forward'' around the delay line. The output
is a linear combination of the direct and delayed signal.

The ``difference equation'' [449] for the feedforward comb filter is

We see that the feedforward comb filter is a particular type of FIR filter. It is also a special case of a TDL.

Note that the feedforward comb filter can implement the echo simulator
of Fig.2.9 by setting and . Thus, it is is a
*computational physical model* of a single discrete echo. This
is one of the simplest examples of acoustic modeling using signal
processing elements. The feedforward comb filter models the
superposition of a ``direct signal'' plus an attenuated,
delayed signal
, where the attenuation (by ) is
due to ``air absorption'' and/or spherical spreading losses, and the
delay is due to acoustic propagation over the distance meters,
where is the sampling period in seconds, and is sound speed.
In cases where the simulated propagation delay needs to be more
accurate than the nearest integer number of samples , some kind of
*delay-line interpolation* needs to be used (the subject of
§4.1). Similarly, when air absorption needs to be
simulated more accurately, the constant attenuation factor can
be replaced by a linear, time-invariant filter giving a
different attenuation at every frequency. Due to the physics of air
absorption, is generally lowpass in character [349, p. 560], [47,318].

### Feedback Comb Filters

The *feedback comb filter* uses feedback instead of a
feedforward signal, as shown in Fig.2.24 (drawn in ``direct form 2''
[449]).

A difference equation describing the feedback comb filter can be
written in ``direct form 1'' [449] as^{3.9}

*feedback*from the delayed output to the input [449]. The feedback comb filter can be regarded as a computational physical model of a

*series*of echoes, exponentially decaying and uniformly spaced in time. For example, the special case

For *stability*, the feedback coefficient must be less than
in magnitude, *i.e.*,
. Otherwise, if
,
each echo will be louder than the previous echo, producing a
never-ending, growing series of echoes.

Sometimes the output signal is taken from the end of the delay line instead of the beginning, in which case the difference equation becomes

### Feedforward Comb Filter Amplitude Response

Comb filters get their name from the ``comb-like'' appearance of their
amplitude response (gain versus frequency), as shown in
Figures 2.25, 2.26, and 2.27.
For a review of frequency-domain analysis
of digital filters, see, *e.g.*, [449].

The transfer function of the feedforward comb filter Eq.(2.2) is

so that the amplitude response (gain versus frequency) is

This is plotted in Fig.2.25 for , , and , , and . When , we get the simplified result

*nulls*, which are points (frequencies) of zero gain in the amplitude response. Note that in

*flangers*, these nulls are

*moved*slowly over time by modulating the delay length . Doing this smoothly requires interpolated delay lines (see Chapter 4 and Chapter 5).

### Feedback Comb Filter Amplitude Response

Figure 2.26 shows a family of *feedback*-comb-filter
amplitude responses, obtained using a selection of feedback
coefficients.

Figure 2.27 shows a similar family obtained using
*negated* feedback coefficients; the opposite sign of the feedback
exchanges the peaks and valleys in the amplitude response.

As introduced in §2.6.2 above, a class of feedback comb filters can be defined as any difference equation of the form

*z*transform of both sides and solving for , the transfer function of the feedback comb filter is found to be

so that the amplitude response is

*sign-inverted*.

For , the feedback-comb amplitude response reduces to

Note that produces resonant peaks at

### Filtered-Feedback Comb Filters

The *filtered-feedback comb filter* (FFBCF) uses filtered
feedback instead of just a feedback gain.

Denoting the feedback-filter transfer function by , the transfer function of the filtered-feedback comb filter can be written as

In §2.6.2 above, we mentioned the physical interpretation
of a feedback-comb-filter as simulating a plane-wave bouncing back and
forth between two walls. Inserting a lowpass filter in the feedback
loop further simulates frequency dependent *losses* incurred
during a propagation round-trip, as naturally occurs in real rooms.

The main physical sources of plane-wave attenuation are *air
absorption* (§B.7.15) and the *coefficient of
absorption* at each wall [349]. Additional ``losses'' for
plane waves in real rooms occur due to *scattering*. (The plane
wave hits something other than a wall and reflects off in many
different directions.) A particular scatterer used in concert halls
is *textured wall surfaces*. In ray-tracing simulations,
reflections from such walls are typically modeled as having a
*specular* and *diffuse* component. Generally speaking,
wavelengths that are large compared with the ``grain size'' of the
wall texture reflect specularly (with some attenuation due to any wall
motion), while wavelengths on the order of or smaller than the texture
grain size are scattered in various directions, contributing to the
diffuse component of reflection.

The filtered-feedback comb filter has many applications in computer music. It was evidently first suggested for artificial reverberation by Schroeder [412, p. 223], and first implemented by Moorer [314]. (Reverberation applications are discussed further in §3.6.) In the physical interpretation [428,207] of the Karplus-Strong algorithm [236,233], the FFBCF can be regarded as a transfer-function physical-model of a vibrating string. In digital waveguide modeling of string and wind instruments, FFBCFs are typically derived routinely as a computationally optimized equivalent forms based on some initial waveguide model developed in terms of bidirectional delay-lines (``digital waveguides'') (see §6.10.1 for an example).

For *stability*, the amplitude-response of the feedback-filter
must be less than in magnitude at all frequencies, *i.e.*,
.

### Equivalence of Parallel Combs to TDLs

It is easy to show that the TDL of Fig.2.19 is equivalent to a
*parallel combination* of three feedforward comb filters, each as in
Fig.2.23. To see this, we simply add the three comb-filter transfer
functions of Eq.(2.3) and equate coefficients:

which implies

We see that parallel comb filters require *more delay memory*
(
elements) than the corresponding TDL, which only
requires
elements.

### Equivalence of Series Combs to TDLs

It is also straightforward to show that a *series combination* of
feedforward comb filters produces a sparsely tapped delay line as
well. Considering the case of two sections, we have

which yields

*series*combination of

*two*feedforward comb filters. Note that the same TDL structure results irrespective of the series ordering of the component comb filters.

### Time Varying Comb Filters

Comb filters can be changed slowly over time to produce the following digital audio ``effects'', among others:

Since all of these effects involve modulating*delay length*over time, and since time-varying delay lines typically require

*interpolation*, these applications will be discussed after Chapter 5 which covers variable delay lines. For now, we will pursue what can be accomplished using

*fixed*(time-invariant) delay lines. Perhaps the most important application is

*artificial reverberation*, addressed in Chapter 3.

## Feedback Delay Networks (FDN)

The FDN can be seen as a *vector feedback comb filter*,^{3.10}obtained by replacing the delay line with a diagonal delay matrix
(defined in Eq.(2.10) below), and replacing the feedback gain
by the product of a diagonal matrix
times an orthogonal
matrix
, as shown in
Fig.2.28 for . The time-update for this FDN can be written
as

with the outputs given by

(3.7) |

or, in frequency-domain vector notation,

(3.8) | |||

(3.9) |

where

### FDN and State Space Descriptions

When
in Eq.(2.10), the FDN (Fig.2.28)
reduces to a normal *state-space model* (§1.3.7),

The matrix is the

*state transition matrix*. The vector holds the

*state variables*that determine the state of the system at time . The

*order*of a state-space system is equal to the number of state variables,

*i.e.*, the dimensionality of . The input and output signals have been trivially redefined as

to follow normal convention for state-space form.

Thus, an FDN can be viewed as a generalized state-space model for a class of th-order linear systems--``generalized'' in the sense that unit delays are replaced by arbitrary delays. This correspondence is valuable for analysis because tools for state-space analysis are well known and included in many software libraries such as with matlab.

### Single-Input, Single-Output (SISO) FDN

When there is only one input signal , the input vector in Fig.2.28 can be defined as the scalar input times a vector of gains:

Note that when
, this system is capable of realizing
*any* transfer function of the form

*z*transform of the impulse response of the system.

The more general case shown in Fig.2.29 can be handled in one of
two ways: (1) the matrices
can be *augmented*
to order
such that the three delay lines are replaced
by unit-sample delays, or (2) ordinary state-space analysis
may be *generalized* to non-unit delays, yielding

In FDN reverberation applications,
, where
is an orthogonal matrix, for reasons addressed below, and
is a
diagonal matrix of lowpass filters, each having gain bounded by 1. In
certain applications, the subset of orthogonal matrices known as
*circulant matrices* have advantages [385].

### FDN Stability

Stability of the FDN is assured when some *norm* [451] of
the state vector
decreases over time when the input signal is
zero [220, ``Lyapunov stability theory'']. That is, a
sufficient condition for FDN stability is

for all , where denotes the norm of , and

for all , where denotes the

*norm*, defined by

The *matrix norm* corresponding to any vector norm
may be defined for the matrix
as

*spectral norm*. Thus, Eq.(2.13) can be restated as

where denotes the spectral norm of .

It can be shown [167] that the spectral norm of a matrix
is given by the largest singular value of
(``
''), and that this is equal to the
square-root of the largest eigenvalue of
, where
denotes the matrix transpose of the real matrix
.^{3.11}

Since every *orthogonal matrix*
has spectral norm
1,^{3.12} a wide variety of stable
feedback matrices can be parametrized as

An alternative stability proof may be based on showing that an FDN is a special case of a passive digital waveguide network (derived in §C.15). This analysis reveals that the FDN is lossless if and only if the feedback matrix has unit-modulus eigenvalues and linearly independent eigenvectors.

## Allpass Filters

The *allpass filter* is an important building block for digital
audio signal processing systems. It is called ``allpass'' because all
frequencies are ``passed'' in the same sense as in ``lowpass'',
``highpass'', and ``bandpass'' filters. In other words, the amplitude
response of an allpass filter is 1 at each frequency, while the phase
response (which determines the delay versus frequency) can be arbitrary.

In practice, a filter is often said to be allpass if the amplitude
response is any nonzero constant. However, in this book, the term
``allpass'' refers to *unity gain* at each frequency.

In this section, we will first make an allpass filter by cascading a
feedback comb-filter with a feedforward comb-filter. This structure,
known as the *Schroeder allpass comb filter*, or simply the
*Schroeder allpass* section, is used extensively in the fields of
artificial reverberation and digital audio effects. Next we will look
at creating allpass filters by *nesting* them; allpass filters
are nested by replacing delay elements (which are allpass filters
themselves) with arbitrary allpass filters. Finally, we will consider
the general case, and characterize the set of all single-input,
single-output allpass filters. The general case, including
multi-input, multi-output (MIMO) allpass filters, is treated in
[449, Appendix D].

### Allpass from Two Combs

An *allpass filter* can be defined as any filter having a gain of
at all frequencies (but typically different delays at different
frequencies).

It is well known that the series combination of a feedforward and feedback comb filter (having equal delays) creates an allpass filter when the feedforward coefficient is the negative of the feedback coefficient.

Figure 2.30 shows a combination feedforward/feedback
comb filter structure which shares the same delay line.^{3.13} By inspection of Fig.2.30, the difference
equation is

This can be recognized as a digital filter in direct form II [449]. Thus, the system of Fig.2.30 can be interpreted as the series combination of a feedback comb filter (Fig.2.24) taking to followed by a feedforward comb filter (Fig.2.23) taking to . By the commutativity of LTI systems, we can interchange the order to get

Substituting the right-hand side of the first equation above for in the second equation yields more simply

This can be recognized as direct form I [449], which requires delays instead of ; however, unlike direct-form II, direct-form I cannot suffer from ``internal'' overflow--overflow can happen only at the output.

The coefficient symbols and here have been chosen to
correspond to standard notation for the *transfer function*

An allpass filter is obtained when , or, in the case of real coefficients, when . To see this, let . Then we have

### Nested Allpass Filters

An interesting property of allpass filters is that they can be
*nested* [412,152,153].
That is, if and
denote unity-gain allpass transfer functions, then both
and
are allpass filters. A proof can be
based on the observation that, since
, can
be viewed as a conformal map
[326] which maps the unit circle in the plane to itself;
therefore, the set of all such maps is closed under functional
composition. Nested allpass filters were proposed for use in artificial
reverberation by Schroeder [412, p. 222].

An important class of nested allpass filters is obtained by nesting first-order allpass filters of the form

*two-multiplier lattice filter*section [297]. In the lattice form, it is clear that replacing by just extends the lattice to the right, as shown in Fig.2.32.

The equivalence of nested allpass filters to lattice filters has computational significance since it is well known that the two-multiply lattice sections can be replaced by one-multiply lattice sections [297,314].

In summary, *nested first-order allpass filters are equivalent to
lattice filters made of two-multiply lattice sections*. In
§C.8.4, a one-multiply section is derived which is not
only less expensive to implement in hardware, but it additionally has
a direct interpretation as a physical model.

### More General Allpass Filters

We have so far seen two types of allpass filters:

- The series combination of feedback and feedforward comb-filters is allpass when their delay lines are the same length and their feedback and feedforward coefficents are the same. An example is shown in Fig.2.30.
- Any delay element in an allpass filter can be replaced by an allpass filter to obtain a new (typically higher order) allpass filter. The special case of nested first-order allpass filters yielded the lattice digital filter structure of Fig.2.32.

**Definition: **
A linear, time-invariant filter is said to be
*lossless* if it *preserves signal
energy* for every input signal. That is, if the input signal is
, and the output signal is
, we must have

Notice that only stable filters can be lossless since, otherwise,
is generally infinite, even when
is finite. We
further assume all filters are *causal*^{3.14} for
simplicity. It is straightforward to show the following:

It can be shown [449, Appendix C] that stable, linear, time-invariant (LTI) filter transfer function is lossless if and only if

Thus, ``lossless'' and ``unity-gain allpass'' are synonymous. For an allpass filter with gain at each frequency, the energy gain of the filter is for every input signal . Since we can describe such a filter as an allpass times a constant gain, the term ``allpass'' will refer here to the case .

### Example Allpass Filters

- The simplest allpass filter is a unit-modulus gain
- A lossless FIR filter can consist only of a single nonzero tap:
- The transfer function of every finite-order, causal,
lossless IIR digital filter (recursive allpass filter) can be written as

where , , and . The polynomial can be obtained by reversing the order of the coefficients in and conjugating them. (The factor serves to restore negative powers of and hence causality.)

### Gerzon Nested MIMO Allpass

An interesting generalization of the single-input, single-output Schroeder allpass filter (defined in §2.8.1) was proposed by Gerzon [157] for use in artificial reverberation systems.

The starting point can be the first-order allpass of Fig.2.31a on
page , or the allpass made from two comb-filters depicted
in Fig.2.30 on
page .^{3.15}In either case,

- all signal paths are converted from scalars to
*vectors*of dimension , - the delay element (or delay line) is replaced by an arbitrary
*unitary matrix frequency response*.^{3.16}

Let
denote the input vector with components
, and let
denote
the corresponding vector of *z* transforms. Denote the output
vector by
. The resulting vector difference equation becomes,
in the frequency domain (cf. Eq.(2.15))

*paraunitary matrix transfer function*[500], [449, Appendix C].

Note that to avoid implementing
twice,
should
be realized in vector direct-form II, *viz.*,

where denotes the *unit-delay operator* (
).

To avoid a delay-free loop, the paraunitary matrix must include at
least one pure delay in every row, *i.e.*,
where
is paraunitary and causal.

In [157], Gerzon suggested using of the form

*orthogonal*matrix, and

is a diagonal matrix of pure delays, with the lengths chosen to be mutually prime (as suggested by Schroeder [417] for a series combination of Schroeder allpass sections). This structure is very close to the that of typical feedback delay networks (FDN), but unlike FDNs, which are ``vector feedback comb filters,'' the vectorized Schroeder allpass is a true multi-input, multi-output (MIMO)

*allpass*filter.

Gerzon further suggested replacing the feedback and feedforward gains by digital filters having an amplitude response bounded by 1. In principle, this allows the network to be arbitrarily different at each frequency.

Gerzon's vector Schroeder allpass is used in the IRCAM Spatialisateur [218].

## Allpass Digital Waveguide Networks

We now describe the class of multi-input, multi-output (MIMO) allpass
filters which can be made using *closed waveguide networks*. We
will see that feedback delay networks can be obtained as a special
case.

### Signal Scattering

The digital waveguide was introduced in §2.4. A basic fact from
acoustics is that traveling waves only happen in a *uniform
medium*. For a medium to be uniform, its *wave impedance*^{3.17}must be *constant*. When a traveling wave
encounters a *change* in the wave impedance, it will
*reflect*, at least partially. If the reflection is not total,
it will also partially *transmit* into the new impedance. This
is called *scattering* of the traveling wave.

Let denote the constant impedance in some waveguide, such as a stretched steel string or acoustic bore. Then signal scattering is caused by a change in wave impedance from to . We can depict the partial reflection and transmission as shown in Fig.2.33.

The computation of reflection and transmission in both directions, as
shown in Fig.2.33 is called a
*scattering junction*.

As derived in Appendix C, for force or pressure waves, the
*reflection coefficient* is given by

That is, the coefficient of reflection for a traveling pressure wave leaving impedance and entering impedance is given by the

*impedance step over the impedance sum*. The

*reflection coefficient*fully characterizes the scattering junction.

For *velocity* traveling waves, the reflection coefficient is
just the negative of that for force/pressure waves, or (see
Appendix C).

Signal scattering is *lossless*, *i.e.*, wave energy is neither
created nor destroyed. An implication of this is that the
*transmission coefficient*
for a traveling pressure wave leaving impedance and entering
impedance is given by

### Digital Waveguide Networks

A *Digital Waveguide Network* (DWN) consists of any number of
digital waveguides interconnected by scattering junctions. For
example, when two digital waveguides are connected together at their
endpoints, we obtain a two-port scattering junction as shown in
Fig.2.33. When three or more waveguides are
connected at a point, we obtain a *multiport scattering
junction*, as discussed in §C.8. In other words, a digital
waveguide network is formed whenever digital waveguides having
arbitrary wave impedances are interconnected. Since DWNs are
lossless, they provide a systematic means of building a very large
class of MIMO allpass filters.

Consider the following question:

In other words, how do we addUnder what conditions may I feed a signal from one point inside a given allpass filter to some other point (adding them) without altering signal energy at any frequency?

*feedback paths*anywhere and everywhere, thereby maximizing the richness of the recursive feedback structure, while maintaining an overall allpass structure?

The *digital waveguide* approach to allpass design
[430] answers this question by maintaining a *physical
interpretation* for all delay elements in the system. Allpass filters
are made out of *lossless digital waveguides* arranged in
*closed, energy conserving networks*. See Appendix C for further
discussion.

**Next Section:**

Artificial Reverberation

**Previous Section:**

Introduction to Physical Signal Models