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Poles and Zeros

In the simple RC-filter example of §E.4.3, the transfer function is

$\displaystyle H(s) = \frac{1}{s+1/\tau} = \frac{RC}{RCs+1}.

Thus, there is a single pole at $ s=-1/\tau=-RC$ , and we can say there is one zero at infinity as well. Since resistors and capacitors always have positive values, the time constant $ \tau = RC$ is always non-negative. This means the impulse response is always an exponential decay--never a growth. Since the pole is at $ s=-1/\tau$ , we find that it is always in the left-half $ s$ plane. This turns out to be the case also for any complex analog one-pole filter. By consideration of the partial fraction expansion of any $ H(s)$ , it is clear that, for stability of an analog filter, all poles must lie in the left half of the complex $ s$ plane. This is the analog counterpart of the requirement for digital filters that all poles lie inside the unit circle.

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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