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 1.19, 1.20, and 1.21. For a review of frequency-domain analysis of digital filters, see, e.g., [426].

Figure: Amplitude responses of the feed forward comb-filter $H(z) = x(n) + g x(n-M)$ (diagrammed in Fig. 1.17) 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. (1.2) is

$$H(z) = b_0+b_M\,z^{-M},$$ (2.3)

so that the amplitude response (gain versus frequency) is

$$G(\omega) \isdef \left\vert H(e^{j\omega})\right\vert = \left\vert b_0 + b_M e^{-j\omega M}\right\vert, \quad -\pi \leq \omega \leq \pi. \protect$$ (2.4)

This is plotted in Fig. 1.19 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

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

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 3 and Appendix I).