Elementary filters

We saw in Chapter 7 how to predict the frequency and phase response of delay networks. The art of filter design lies in finding a delay network whose transfer function (which controls the frequency and phase response) has a desired shape. We will develop an approach to building such delay networks out of the two types of comb filters developed in Chapter 7: recirculating and non-recirculating. Here we will be interested in the special case where the delay is only one sample in length. In this situation, the frequency responses shown in Figures 7.6 and 7.10 no longer look like combs; the second peak recedes all the way to the sample rate, $2\pi$ radians, when $d=1$. Since only frequencies between 0 and the Nyquist frequency ($\pi$ radians) are audible, in effect there is only one peak when $d=1$.

In the comb filters shown in Chapter 7, the peaks are situated at DC (zero frequency), but we will often wish to place them at other, nonzero frequencies. This is done using delay networks--comb filters--with complex-valued gains.