Simplified Feedback Comb Filter

Special case: $b_0=1$ , $-a_M = g \; \Rightarrow$

$$\begin{eqnarray*} y(n) &=& x(n) + g\, y(n-M)\\ [10pt] H(z) &=& \frac{1}{1-g\,z^{-M}} \end{eqnarray*}$$

• Impulse response is a series of echoes, exponentially decaying and uniformly spaced in time:
  $$\begin{eqnarray*} H(z) &=& \frac{1}{1-g\,z^{-M}} \;=\;1 + g\,z^{-M} + g^2\,z^{-2M} + \cdots\\ [10pt] &\;\longleftrightarrow\;& \delta(n) + g\,\delta(n-M) + g^2\,\delta(n-2M) + \cdots\\ [10pt] &=& [1, \underbrace{0, \ldots, 0}_{M-1},g,\underbrace{0, \ldots, 0}_{M-1},g^2,0,\ldots\,] \end{eqnarray*}$$
  

• Models a plane wave between parallel walls
• Models wave propagation on a guitar string
• $g$ = round-trip gain coefficient:
  • two wall-to-wall traversals (two wall reflections)
  • two string traversals (two endpoint reflections)