Comb Filters
Comb filters are basic building blocks for digital audio effects. Theacoustic echo simulation in Fig.2.9 is one instance of a combfilter. This section presents the two basic comb-filter types,feedforward and feedback, and gives a frequency-responseanalysis.
Feedforward Comb Filters
The feedforward comb filter is shown in Fig.2.23. Thedirect signal ``feeds forward'' around the delay line. The outputis a linear combination of the direct and delayed signal.
 |
The ``difference equation'' [449] for the feedforward comb filter is
 | (3.2) |
We see that the feedforward comb filter is a particular type of FIRfilter. It is also a special case of a TDL.
Note that the feedforward comb filter can implement the echo simulatorof Fig.2.9 by setting 
and 
. Thus, it is is acomputational physical model of a single discrete echo. Thisis one of the simplest examples of acoustic modeling using signalprocessing elements. The feedforward comb filter models thesuperposition of a ``direct signal'' 
plus an attenuated,delayed signal 
, where the attenuation (by 
) isdue to ``air absorption'' and/or spherical spreading losses, and thedelay 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 moreaccurate than the nearest integer number of samples 
, some kind ofdelay-line interpolation needs to be used (the subject of§4.1). Similarly, when air absorption needs to besimulated more accurately, the constant attenuation factor 
canbe replaced by a linear, time-invariant filter 
giving adifferent attenuation at every frequency. Due to the physics of airabsorption, is generally lowpass in character [349, p. 560], [47,318].
Feedback Comb Filters
The feedback comb filter uses feedback instead of afeedforward signal, as shown in Fig.2.24 (drawn in ``direct form 2''[449]).
 |
A difference equation describing the feedback comb filter can bewritten in ``direct form 1'' [449] as3.9
The feedback comb filter is a special case of an Infinite ImpulseResponse (IIR) (``recursive'') digital filter, since there isfeedback from the delayed output to the input [449].The feedback comb filter can be regarded as a computational physicalmodel of a series of echoes, exponentially decaying anduniformly spaced in time. For example, the special case
is a computational model of an ideal plane wave bouncing back andforth between two parallel walls; in such a model, 
represents thetotal round-trip attenuation (two wall-to-wall traversals, includingtwo 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 anever-ending, growing series of echoes.
Sometimes the output signal is taken from the end of the delay line insteadof the beginning, in which case the difference equation becomes
This choice of output merely delays the output signal by samples.
Feedforward Comb Filter Amplitude Response
Comb filters get their name from the ``comb-like'' appearance of theiramplitude response (gain versus frequency), as shown inFigures 2.25, 2.26, and 2.27.For a review of frequency-domain analysisof digital filters, see, e.g., [449].
 |
(diagrammed in Fig.
'').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 inflangers, these nulls are moved slowly over time bymodulating the delay length . Doing this smoothly requiresinterpolated delay lines (see Chapter 4 andChapter 5).
FeedbackComb Filter Amplitude Response
Figure 2.26 shows a family of feedback-comb-filteramplitude responses, obtained using a selection of feedbackcoefficients.
 |
(Fig.
) with Figure 2.27 shows a similar family obtained usingnegated feedback coefficients; the opposite sign of the feedbackexchanges the peaks and valleys in the amplitude response.
 |
, 
, and 
.a) Linear amplitude scale. b) As introduced in §2.6.2 above, a class of feedback combfilters 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 feedbacksign-inverted.
For 
, the feedback-comb amplitude responsereduces to
and for 
to
which exactly inverts the amplitude response of the feedforwardcomb filter with gain (Eq.(2.4)).
Note that 
produces resonant peaks at
while for 
, the peaks occur midway between these values.
Filtered-Feedback Comb Filters
The filtered-feedback comb filter (FFBCF) uses filteredfeedback instead of just a feedback gain.
Denoting the feedback-filter transfer function by 
, thetransfer function of the filtered-feedback comb filter can be writtenas
Note that when is a causal filter, the FFBCF can beconsidered mathematically a special case of the general allpoletransfer function in which the first 
denominator coefficientsare constrained to be zero:
It is this ``sparseness'' of the filter coefficients that makes theFFBCF more computationally efficient than other, more general-purpose,IIR filter structures.
In §2.6.2 above, we mentioned the physical interpretationof a feedback-comb-filter as simulating a plane-wave bouncing back andforth between two walls. Inserting a lowpass filter in the feedbackloop further simulates frequency dependent losses incurredduring a propagation round-trip, as naturally occurs in real rooms.
The main physical sources of plane-wave attenuation are airabsorption (§B.7.15) and the coefficient ofabsorption at each wall [349]. Additional ``losses'' forplane waves in real rooms occur due to scattering. (The planewave hits something other than a wall and reflects off in manydifferent directions.) A particular scatterer used in concert hallsis textured wall surfaces. In ray-tracing simulations,reflections from such walls are typically modeled as having aspecular and diffuse component. Generally speaking,wavelengths that are large compared with the ``grain size'' of thewall texture reflect specularly (with some attenuation due to any wallmotion), while wavelengths on the order of or smaller than the texturegrain size are scattered in various directions, contributing to thediffuse component of reflection.
The filtered-feedback comb filter has many applications in computermusic. It was evidently first suggested for artificial reverberationby Schroeder [412, p. 223], and first implemented by Moorer[314]. (Reverberation applicationsare discussed further in §3.6.) In the physicalinterpretation [428,207] of the Karplus-Strongalgorithm [236,233], the FFBCF can be regarded as atransfer-function physical-model of a vibrating string. In digitalwaveguide modeling of string and wind instruments, FFBCFs aretypically derived routinely as a computationally optimized equivalentforms based on some initial waveguide model developed in terms ofbidirectional 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 aparallel combination of three feedforward comb filters, each as inFig.2.23. To see this, we simply add the three comb-filter transferfunctions 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 onlyrequires 
elements.
Equivalence of Series Combs to TDLs
It is also straightforward to show that a series combination offeedforward comb filters produces a sparsely tapped delay line aswell. Considering the case of two sections, we have
which yields
Thus, the TDL of Fig.2.19 is equivalent also to the seriescombination of two feedforward comb filters. Note that thesame TDL structure results irrespective of the series ordering of thecomponent comb filters.
Time Varying Comb Filters
Comb filters can be changed slowly over time to produce the followingdigital audio ``effects'', among others:
Since all of these effects involve modulating delay length overtime, and since time-varying delay lines typically requireinterpolation, these applications will be discussed afterChapter 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 applicationis artificial reverberation, addressed inChapter 3.
Next Section:
Feedback Delay Networks (FDN)
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Tapped Delay Line (TDL)
