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The z-transform of an finite impulse response (FIR) filter is defined as follows:

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:

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.

Pole-Zero Plots

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:

• You can determine if a digital filter is stable by determining if the output of the digital filter is bounded for all possible bounded inputs. The necessary and sufficient condition for IIR filters to be stable is that all poles are inside the unit circle. In contrast, FIR filters are always stable because the FIR filters do not have poles.
• You can determine if pole-zero pairs are close enough to cancel out each other effectively. Try deleting close pairs and then check the resulting frequency response. Fewer pole-zero pairs means fewer computations.

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