## 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.

**Next Section:**

Feedback Delay Networks (FDN)

**Previous Section:**

Tapped Delay Line (TDL)