Time Constant of One Pole

A useful approximate formula giving the decay time-constant^9.4 $\tau$ (in seconds) in terms of a pole radius $R\in[0,1)$ is

$$\zbox {\tau\approx \frac{T}{1-R}} \protect$$ (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:

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

Thus, setting $n=1$ yields

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

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

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

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.