Gain Range for Feedforward Comb Filter

$$y(n) = b_0 \,x(n) +b_M \,x(n-M)$$

For a sinewave input, with $b_0,b_M>0$ :

• Gain is maximum ($b_0+b_M$ ) when a whole number of periods fits in $M$ samples:
  $$\omega_k T = k\frac{2\pi}{M}, \quad k=0,1,2,\dots$$
  (the DFT basis frequencies for length $M$ DFTs)

• Gain is minimum ( $\left\vert b_0-b_M\right\vert$ ) when an odd number of half-periods fits in $M$ samples:
  $$\omega_k T = (2k+1)\frac{\pi}{M}, \quad k=0,1,2,\dots$$