Feedback Comb Filter

$$\begin{eqnarray*} -a_M &=& \hbox{Feedback coefficient (need $\vert a_M\vert<1$\ for stability)}\\ M &=& \hbox{Delay-line length in samples} \end{eqnarray*}$$

Direct-Form-II Difference Equation (see above figure)

$$\begin{eqnarray*} v(n) &=& x(n) - a_M y(n-M)\\ y(n) &=& b_0 v(n) \end{eqnarray*}$$

Direct-Form-I Difference Equation (commute gain $b_0$ to input)

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

Transfer Function

$$H(z) = \frac{b_0}{1 + a_M z^{-M}}$$

Frequency Response

$$H(e^{j\omega T}) = \frac{b_0}{1 + a_M e^{-jM\omega T}}$$