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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.

Previous: Frequency Response Analysis Problems
Next: Filter Order = Transfer Function Order

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


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