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.

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.
Figure 2.23:
The feedforward comb filter.
 |
The ``
difference equation'' [
449] for the feedforward comb filter is
 |
(3.2) |
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].
The
feedback comb filter uses feedback instead of a
feedforward
signal, as shown in Fig.
2.24 (drawn in ``
direct form 2''
[
449]).
Figure 2.24:
The feedback comb filter.
 |
A
difference equation describing the feedback comb filter can be
written in ``direct form 1'' [
449] as
3.9
The feedback comb filter is a special case of an Infinite
Impulse
Response (IIR) (``recursive'')
digital filter, since there is
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
is a computational model of an ideal
plane wave bouncing back and
forth between two parallel walls; in such a model,

represents the
total round-trip attenuation (two wall-to-wall traversals, including
two reflections).
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
This choice of output merely delays the output signal by

samples.
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].
Figure:
Amplitude responses of the
feed forward comb-filter
(diagrammed in Fig.2.23) with
and
,
, and
.
a) Linear amplitude scale. b) Decibel scale. The frequency axis goes
from 0 to the sampling rate (instead of only half the
sampling rate, which is more typical for real filters) in order to
display the fact that the number of notches is exactly
(as
opposed to ``
'').
![\includegraphics[width=\twidth ]{eps/ffcfar}](http://www.dsprelated.com/josimages_new/pasp/img499.png) |
The
transfer function of the feedforward comb
filter Eq.

(
2.2) is
 |
(3.3) |
so that the amplitude response (gain versus frequency) is
 |
(3.4) |
This is plotted in Fig.
2.25 for

,

, and

,

, and

.
When

, we get the simplified result
In this case, we obtain
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).
Figure
2.26 shows a family of
feedback-
comb-filter
amplitude responses, obtained using a selection of feedback
coefficients.
Figure:
Amplitude response of the feedback
comb-filter
(Fig.2.24 with
and
) with
and
,
, and
. a) Linear
amplitude scale. b) Decibel scale.
![\includegraphics[width=\twidth ]{eps/fbcfar}](http://www.dsprelated.com/josimages_new/pasp/img505.png) |
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.
Figure:
Amplitude response of the phase-inverted feedback comb-filter, i.e., as in Fig.2.26 with negated
,
, and
.
a) Linear amplitude scale. b) Decibel scale.
![\includegraphics[width=\twidth ]{eps/fbcfiar}](http://www.dsprelated.com/josimages_new/pasp/img506.png) |
As introduced in §
2.6.2 above, a class of feedback comb
filters can be defined as any
difference equation of the form
Taking the
z transform of both sides and solving for

,
the
transfer function of the feedback comb filter is found to be
 |
(3.5) |
so that the amplitude response is
This is plotted in Fig.
2.26 for

and

,

, and

. Figure
2.27 shows the same case but with the feedback
sign-inverted.
For

, the feedback-comb amplitude response
reduces to
and for

to
which exactly inverts the amplitude response of the
feedforward
comb filter with gain

(Eq.

(
2.4)).
Note that

produces resonant peaks at
while for

, the peaks occur midway between these values.
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
Note that when

is a
causal filter, the FFBCF can be
considered mathematically a special case of the general allpole
transfer function in which the first

denominator coefficients
are constrained to be zero:
It is this ``sparseness'' of the filter coefficients that makes the
FFBCF more computationally efficient than other, more general-purpose,
IIR filter structures.
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
Thus, the TDL of Fig.
2.19 is equivalent also to the
series
combination of
two feedforward
comb filters. Note that the
same TDL structure results irrespective of the series ordering of the
component
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)