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The z-transform of a finite impulse response (FIR) filter is defined as follows:
(A) |
where z ≡ e^{j2πf}, b_{k} is the set of filter coefficients, and N is the order of the FIR filter. In Equation A, z_{k} represents the roots of the polynomial H(z). H(z_{k}) = 0 for all z_{k}, so z_{k} 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:
(B) |
where z_{k} and pf_{k} represent the roots of the numerator polynomial and denominator polynomial, respectively. p_{k} 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: