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Time Constant of One Pole

A useful approximate formula giving the decay time-constant9.4 $ \tau$ (in seconds) in terms of a pole radius $ R\in[0,1)$ is

$\displaystyle \zbox {\tau\approx \frac{T}{1-R}} \protect$ (9.8)

where $ T$ denotes the sampling interval in seconds, and we assume $ T\ll\tau$ .

The exact relation between $ \tau$ and $ R$ is obtained by sampling an exponential decay:

$\displaystyle e^{-t/\tau} \;\rightarrow\; e^{-nT/\tau} \;\isdef \; R^n

Thus, setting $ n=1$ yields

$\displaystyle R = e^{-T/\tau}.

Expanding the right-hand side in a Taylor series and neglecting terms higher than first order gives

$\displaystyle e^{-\frac{T}{\tau}} \eqsp 1 - \frac{T}{\tau}
+ \frac{1}{2!}\left(\frac{T}{\tau}\right)^2
- \frac{1}{3!}\left(\frac{T}{\tau}\right)^3 + \cdots
\;\approx\; 1 - \frac{T}{\tau},

which derives $ R\approx 1-T/\tau$ . Solving for $ \tau$ then gives Eq.(8.8). From its derivation, we see that the approximation is valid for $ T\ll\tau$ . Thus, as long as the impulse response of a pole $ p$ ``rings'' for many samples, the formula $ \tau\approx T/(1-\vert p\vert)$ should well estimate the time-constant of decay in seconds. The time-constant estimate in samples is of course $ 1/(1-\vert p\vert)$ . For higher-order systems, the approximate decay time is $ 1/(1-R_{\mbox{max}})$ , where $ R_{\mbox{max}}$ is the largest pole magnitude (closest to the unit circle) in the (stable) system.
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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University