## Time Constant of One Pole

A useful approximate formula giving the decay time-constant9.4 (in seconds) in terms of a pole radius is

 (9.8)

where denotes the sampling interval in seconds, and we assume .

The exact relation between and is obtained by sampling an exponential decay:

Thus, setting yields

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

which derives . Solving for then gives Eq.(8.8). From its derivation, we see that the approximation is valid for . Thus, as long as the impulse response of a pole rings'' for many samples, the formula should well estimate the time-constant of decay in seconds. The time-constant estimate in samples is of course . For higher-order systems, the approximate decay time is , where is the largest pole magnitude (closest to the unit circle) in the (stable) system.
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Bandwidth of One Pole