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Poles and Zeros

In our simple RC-filter example, the transfer function is

$\displaystyle H(s) = \frac{\tau}{s+\tau}.
$

Thus, there is a single pole at $ s=-\tau$, and we can say there is one zero at infinity as well. Since resistors and capacitors always have positive values, the time constant $ \tau =
1/(RC)$ is always non-negative. This means the impulse response is always an exponential decay--never a growth. Since the pole is at $ s=-\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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[How to cite this work] [Order a printed hardcopy]

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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