Feedforward Comb Filters

The feedforward comb filter is normally implemented as shown in Fig. 1.17, in which the direct signal ``feeds forward'' around the delay line and sums (scaled) with the delay-line output.

Figure 1.17: The feedforward comb filter.

The ``difference equation'' for the feedforward comb filter is

$$y(n) = b_0 x(n) + b_M x(n-M). \protect$$ (2.2)

We see that the feedforward comb filter is a particular type of FIR filter. It is also a type of TDL in which the output is formed as a linear combination of the delay line's input and output.

Note that the feedforward comb filter can implement the echo simulator of Fig. 1.8 by setting $b_0=1$ and $b_M=g$. Thus, the feedforward comb filter 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'' $b_0 x(n)$ plus an attenuated, delayed signal $b_M x(n-M)$, where the attenuation is due to ``air absorption'' and/or spherical spreading losses, and the delay can be ascribed to acoustic propagation over the distance $cMT$ meters, where $T$ is the sampling period in seconds, and $c$ is sound speed in meters per second. In cases where the simulated propagation delay needs to be more accurate than the nearest integer number of samples $M$, some kind of delay-line interpolation needs to be used (which we address in §3.2). Similarly, when air absorption needs to be simulated more accurately, the constant attenuation factor $b_M$ can be replaced by a linear, time-invariant filter $G(z)$ giving a different attenuation at every frequency. Due to the physics of air absorption, $G(z)$ is generally lowpass in character [327, p. 560], [44,298].