Designing filters

The frequency response of a series of elementary recirculating and non-recirculating filters can be estimated graphically by plotting all the coefficients ${Q_1}, \ldots, {Q_j}$ and ${P_1}, \ldots, {P_k}$ on the complex plane and reasoning as in Figure 8.8. The overall frequency response is the product of all the distances from the point $Z$ to each of the $Q_i$, divided by the product of the distances to each of the $P_i$.

One customarily marks each of the $Q_i$ with an ``o" (this is called a ``zero") and each of the $P_i$ with an ``x" (called a ``pole"); their names are borrowed from the field of complex analysis. A plot showing the poles and zeros associated with a filter is unimaginatively called a pole-zero plot.

When $Z$ is close to a zero the frequency response tends to dip, and when it is close to a pole, the frequency response tends to rise. The effect of a pole or a zero is more pronounced, and also more local, if it is close to the unit circle that $Z$ is constrained to lie on. Poles must lie within the unit circle for a stable filter. Zeros may lie on or outside it, but any zero $Q$ outside the unit circle may be replaced by one within it, at the point $1/\overline{Q}$, to give a constant multiple of the same frequency response. Except in special cases we will keep the zeros inside the circle as well as the poles.

In the rest of thie section we will show how to construct several of the filter types most widely used in electronic music. The theory of digital filter design is vast, and we will only give an introduction here. A deeper treatment is available online from Julius Smith at ccrma.stanford.edu. See also [Ste96] for a fuller treatment of filtering theory in the context and language of Digital Signal Processing.