Exponentials

The canonical form of an exponential function, as typically used in signal processing, is

$$a(t) = A e^{-t/\tau}, \quad t\geq 0$$

where $\tau$ is called the time constant of the exponential. $A$ is the peak amplitude, as before. The time constant is the time it takes to decay by $1/e$, i.e.,

$$\frac{a(\tau)}{a(0)} = \frac{1}{e}.$$

A normalized exponential decay is depicted in Fig.4.7.

Figure 4.7: The decaying exponential $Ae^{-t/\tau }$, normalized to unit amplitude ( $e^{-t/\tau }$).