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
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
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
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., .
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
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.
so that the amplitude response is
For , the feedback-comb amplitude response reduces to
Note that produces resonant peaks at
The filtered-feedback comb filter (FFBCF) uses filtered feedback instead of just a feedback gain.
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 . 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 . (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).
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:
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
Comb filters can be changed slowly over time to produce the following digital audio ``effects'', among others: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)
Tapped Delay Line (TDL)