Feedback Comb Filters

The feedback comb filter uses feedback instead of a feedforward signal, as shown in Fig.2.24 (drawn in ``direct form 2'' [452]).

Figure 2.24: The feedback comb filter.

A difference equation describing the feedback comb filter can be written in ``direct form 1'' [452] as^3.9

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

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 [452]. 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 becomes

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

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