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Pole-Zero Analysis

This chapter discusses pole-zero analysis of digital filters. Every digital filter can be specified by its poles and zeros (together with a gain factor). Poles and zeros give useful insights into a filter's response, and can be used as the basis for digital filter design. This chapter additionally presents the Durbin step-down recursion for checking filter stability by finding the reflection coefficients, including matlab code.

Going back to Eq.$ \,$ (6.5), we can write the general transfer function for the recursive LTI digital filter as

$\displaystyle H(z) = g\frac{1 + \beta_1 z^{-1}+ \cdots + \beta_M z^{-M}}{1 + a_1 z^{-1}+ \cdots + a_N z^{-N}} \protect$ (9.1)

which is the same as Eq.$ \,$ (6.5) except that we have factored out the leading coefficient $ b_0$ in the numerator (assumed to be nonzero) and called it g. (Here $ \beta_i \isdeftext b_i/b_0$ .) In the same way that $ z^2 + 3z + 2$ can be factored into $ (z + 1)(z + 2)$ , we can factor the numerator and denominator to obtain

$\displaystyle H(z) = g\frac{(1-q_1z^{-1})(1-q_2z^{-1})\cdots(1-q_Mz^{-1})}{(1-p_1z^{-1})(1-p_2z^{-1})\cdots(1-p_Nz^{-1})}. \protect$ (9.2)

Assume, for simplicity, that none of the factors cancel out. The (possibly complex) numbers $ \{q_1,\ldots,q_M\}$ are the roots, or zeros, of the numerator polynomial. When $ z$ is set to any of these values, the transfer function evaluates to 0. For this reason, the numerator roots $ q_i$ are called the zeros of the filter. In other words, the zeros of the numerator of an irreducible transfer-function are called the zeros of the transfer-function. Similarly, when $ z$ approaches any root of the denominator polynomial, the magnitude of the transfer function approaches infinity. Consequently, the denominator roots $ \{p_1, \ldots,
p_N\}$ are called the poles of the filter.

The term ``pole'' makes sense when one plots the magnitude of $ H(z)$ as a function of z. Since $ z$ is complex, it may be taken to lie in a plane (the $ z$ plane). The magnitude of $ H(z)$ is real and therefore can be represented by distance above the $ z$ plane. The plot appears as an infinitely thin surface spanning in all directions over the $ z$ plane. The zeros are the points where the surface dips down to touch the $ z$ plane. At high altitude, the poles look like thin, well, ``poles'' that go straight up forever, getting thinner the higher they go.

Notice that the $ M+1$ feedforward coefficients from the general difference quation, Eq.$ \,$ (5.1), give rise to $ M$ zeros. Similarly, the $ N$ feedback coefficients in Eq.$ \,$ (5.1) give rise to $ N$ poles. Recall that we defined the filter order as the maximum of $ N$ and $ M$ in Eq.$ \,$ (6.5). Therefore, the filter order equals the number of poles or zeros, whichever is greater.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2017-10-19 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University