A useful approximate formula giving the decay
time-constant9.4
(in
seconds) in terms of a pole radius
is
![$\displaystyle \zbox {\tau\approx \frac{T}{1-R}} \protect$](http://www.dsprelated.com/josimages_new/filters/img1062.png) |
(9.8) |
where
![$ T$](http://www.dsprelated.com/josimages_new/filters/img96.png)
denotes the
sampling interval in seconds, and we assume
![$ T\ll\tau$](http://www.dsprelated.com/josimages_new/filters/img1063.png)
.
The exact relation between
and
is obtained by sampling an
exponential decay:
Thus, setting
![$ n=1$](http://www.dsprelated.com/josimages_new/filters/img1065.png)
yields
Expanding the right-hand side in a
Taylor series and neglecting terms
higher than first order gives
which derives
![$ R\approx 1-T/\tau$](http://www.dsprelated.com/josimages_new/filters/img1068.png)
. Solving for
![$ \tau$](http://www.dsprelated.com/josimages_new/filters/img1058.png)
then gives
Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/filters/img94.png)
(
8.8). From its derivation, we see that the approximation is
valid for
![$ T\ll\tau$](http://www.dsprelated.com/josimages_new/filters/img1063.png)
. Thus, as long as the
impulse response of a pole
![$ p$](http://www.dsprelated.com/josimages_new/filters/img48.png)
``rings'' for many samples, the formula
![$ \tau\approx T/(1-\vert p\vert)$](http://www.dsprelated.com/josimages_new/filters/img1069.png)
should well estimate the time-constant of decay in seconds. The
time-constant estimate in
samples is of course
![$ 1/(1-\vert p\vert)$](http://www.dsprelated.com/josimages_new/filters/img1070.png)
. For
higher-order systems, the approximate
decay time is
![$ 1/(1-R_{\mbox{max}})$](http://www.dsprelated.com/josimages_new/filters/img1071.png)
, where
![$ R_{\mbox{max}}$](http://www.dsprelated.com/josimages_new/filters/img873.png)
is the largest pole
magnitude (closest to the unit circle) in the (stable) system.
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