Exponential Decay and Time Constants

• Exponential functions are common in audio processing because they represent natural decay patterns in acoustical systems. When a quantity decays exponentially, it decreases in "equal" proportions relative to its current value.

• A general exponential function is given by:
    $$y(t) = A e^{-t/\tau}, \hspace{0.2in} t \geq 0,$$ (1)
  where $\tau$ is the exponential time constant and A is the peak amplitude. The time constant is the time it takes for the quantity $y(t)$ to decay by $1/e$ (i.e. $y(\tau)/y(0) = 1/e$).

• The figure below depicts a normalized exponential decay curve.

  Figure 3: An exponential decay curve.

  

• Note that a positive time constant will produce a decaying exponential while a negative time constant will produce a growing exponential.

• For audio applications, a decay of $1/e$ is not generally significant. A more common measure of sound decay is the time required for a signal to decay by 60 dB, which is represented as $t_{60}$:
  $$20 \log_{10}( y(t_{60}) / y(0) ) = -60 dB$$
  or
    $$y(t_{60}) / y(0) = 10^{-60/20} = 0.001$$ (2)

• In terms of exponential time constants and from Eqs. (1) and (2), $y(t_{60}) / y(0) = 0.001 = e^{-t_{60}/\tau},$ from which $t_{60} = -\ln(0.001) \tau = 6.91 \tau$, or about seven time constants.