# 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; } |

*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

*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 :

### 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).

*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)); } } |

*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

### 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.

*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.*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.

^{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

`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

*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].

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.

*negated*feedback coefficients; the opposite sign of the feedback exchanges the peaks and valleys in the amplitude response.

*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

### 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

*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:

*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

*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

### 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:*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

*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

*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

## 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 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*

### 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].

*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

*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

### 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}

*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.*,

*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