The z-transform of a finite impulse response (FIR) filter is defined as follows:
where z ≡ ej2πf, bk is the set of filter coefficients, and N is the order of the FIR filter. In Equation A, zk represents the roots of the polynomial H(z). H(zk) = 0 for all zk, so zk represents the zeroes of the filter H(z). The number of zeroes in a filter must equal the filter order N.
Similarly, the z-transform of an infinite impulse response (IIR) filter is defined by the following equation:
where zk and pfk represent the roots of the numerator polynomial and denominator polynomial, respectively. pk represents the poles of an IIR filter H(z). IIR filters have poles and zeroes, and FIR filters have only zeroes.
From a mathematical point of view, the pole-zero plot and frequency response provide the same information. Based on the frequency response, you can obtain a pole-zero plot. Conversely, from the pole-zero plot, you can compute the frequency response.
The following figure illustrates a pole-zero plot for a particular IIR filter. The half-circle corresponds to |z| = 1, or the unit circle. The small circles along the half-circle represent zeroes. Each × represents a pole.
The pole-zero plot and frequency response characterize digital filters from the following perspectives: