Time Constant of One Pole

A useful approximate formula giving the *decay
time-constant*^{9.4}
(in
seconds) in terms of a pole radius
is

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

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University