Feedback Comb Filters

The feedback comb filter uses feedback instead of a feedforward signal, as shown in Fig. 1.18 (drawn in ``direct form 2'' [426]).

Figure 1.18: The feedback comb filter.

The difference equation describing the feedback comb filter can be written in ``direct form 1'' [426] as^2.5

$$y(n) = b_0 x(n) - a_M y(n-M).$$

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 [426]. 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

$$y(n) = x(n) + g\, y(n-M)$$

is a computational model of an ideal plane wave bouncing back and forth between two parallel walls; in such a model, $g$ represents the total round-trip attenuation (two wall-to-wall traversals, including two reflections).

For stability, the feedback coefficient $a_M$ must be less than $1$ in magnitude, i.e., $\left\vert a_M\right\vert<1$. Otherwise, if $\left\vert a_M\right\vert>1$, 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 can be written as

$$y(n) = b_M x(n-M) - a_M y(n-M) .$$

This choice of output merely delays the output signal by $M$ samples.