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Introduction to Digital Filters
    Pole-Zero Analysis
       Time Constant of One Pole

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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}} = 1 - \frac{T}{\tau} + \frac{1}{2!}\left(\fr... ...{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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Previous: Bandwidth of One Pole
Next: Unstable Poles--Unit Circle Viewpoint
written by 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 Associate Professor (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 http://ccrma.stanford.edu/~jos/ for details.


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