Feedforward Comb Filter Amplitude Response

Comb filters get their name from the ``comb-like'' appearance of their amplitude response (gain versus frequency), as shown in Figures 2.25, 2.26, and 2.27. For a review of frequency-domain analysis of digital filters, see, e.g., [452].

Figure: Amplitude responses of the feed forward comb-filter $y(n) = x(n) + g x(n-M)$ (diagrammed in Fig.2.23) with $M=5$ and $g=0.1$ , $0.5$ , and $0.9$ . a) Linear amplitude scale. b) Decibel scale. The frequency axis goes from 0 to the sampling rate (instead of only half the sampling rate, which is more typical for real filters) in order to display the fact that the number of notches is exactly $M=5$ (as opposed to ``$2.5$ '').

The transfer function of the feedforward comb filter Eq.(2.2) is

$$H(z) = b_0+b_M\,z^{-M},$$ (3.3)

so that the amplitude response (gain versus frequency) is

$$G(\omega) \isdef \left\vert H(\ejo)\right\vert = \left\vert b_0 + b_M e^{-j\omega M}\right\vert, \quad -\pi \leq \omega \leq \pi. \protect$$ (3.4)

This is plotted in Fig.2.25 for $M=5$ , $b_0=1$ , and $b_M=0.1$ , $0.5$ , and $0.9$ . When $b_0=b_M=1$ , we get the simplified result

$$G(\omega) = \left\vert 1 + e^{-j\omega M}\right\vert = \left\vert e^{-j\omega M/2}\right\vert\left\vert e^{j\omega M/2} + e^{-j\omega M/2}\right\vert = 2\left\vert\cos\left(\omega\frac{M}{2}\right)\right\vert.$$

In this case, we obtain $M$ nulls, which are points (frequencies) of zero gain in the amplitude response. Note that in flangers, these nulls are moved slowly over time by modulating the delay length $M$ . Doing this smoothly requires interpolated delay lines (see Chapter 4 and Chapter 5).