Feedback Comb Filters

• A feedback comb filter is illustrated in Fig. 12.

  Figure 12: A feedback comb filter block diagram.

  

  The difference equation for this filter is given by:

  $$y[n] = b_{0}x[n] - a_{M}y[n-M].$$
  For stability, the coefficient $\vert a_{M}\vert \leq 1$.

• The transfer function of the feedback comb filter is
  $$H(z) = \frac{b_{0}}{1 + a_{M} z^{-M}},$$
  from which the amplitude response is found as
  $$G(\omega) \ensuremath{\stackrel{\Delta}{=}}\left\vert H(e^{j \ome... ...+ a_{M} e^{-j \omega M}}\right\vert, \hspace{0.2in} -\pi \leq \omega \leq \pi.$$

• The magnitude response of a feedback comb filter, calculated with the Matlab script combs.m, is shown in Fig. 13 for $M = 5$, $b_{0} = 1$, and $a_{M}$ = -0.1, -0.5, and -0.9.

  Figure 13: Magnitude response of a feedback comb filter with $M = 5$, $b_{0} = 1$, and $a_{M}$ = -0.1, -0.5, and -0.9.