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 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.