Decay Time is Q Periods

Another well known rule of thumb is that the of a resonator is the number of ``periods'' under the exponential decay of its impulse response. More precisely, we will show that, for $Q\gg 1/2$ , the impulse response decays by the factor $e^{-\pi}$ in cycles, which is about 96 percent decay, or -27 dB.

The impulse response corresponding to Eq. $\,$ (E.8) is found by inverting the Laplace transform of the transfer function . Since it is only second order, the solution can be found in many tables of Laplace transforms. Alternatively, we can break it up into a sum of first-order terms which are invertible by inspection (possibly after rederiving the Laplace transform of an exponential decay, which is very simple). Thus we perform the partial fraction expansion of Eq. $\,$ (E.8) to obtain

$\displaystyle H(s) = \frac{g_1}{s-p_1} + \frac{g_2}{s-p_2}$

where

are given by Eq. $\,$ (E.9), and some algebra gives

$\displaystyle g_1$	$\displaystyle =$	$\displaystyle -g\frac{p_1}{p_2-p_1}$	(E.12)
$\displaystyle g_2$	$\displaystyle =$	$\displaystyle g\frac{p_2}{p_2-p_1}$	(E.13)

as the respective residues of the poles

The impulse response is thus

$\displaystyle h(t) = g_1 e^{p_1t} + g_2 e^{p_2t}.$

Assuming a resonator, , we have $p_2 = \overline{p}_1$ , where $p_1=\sigma_p +j\omega_p = -\alpha + j\omega_d$ (using notation of the preceding section), and the impulse response reduces to

$\displaystyle h(t) = g\,e^{-\alpha t} \left[\cos(\omega_p t) - \frac{\alpha}{\omega_p} \sin(\omega_p t)\right] = g\,e^{-\alpha t} \cos(\omega_p t + \phi),$

where $\phi\isdef \sin^{-1}(\alpha/\omega_p)$ .

We have shown so far that the impulse response decays as $e^{-\alpha t}$ with a sinusoidal radian frequency $\omega_p=\omega_d$ under the exponential envelope. After Q periods at frequency $\omega_p$ , time has advanced to

$\displaystyle t_Q = Q\frac{2\pi}{\omega_p} \approx \frac{2\pi Q}{\omega_0}, = \frac{\pi}{\alpha},$

where we have used the definition Eq. $\,$ (E.7) $Q\isdef \omega_0/(2\alpha)$ . Thus, after

periods, the amplitude envelope has decayed to

$\displaystyle e^{-\alpha t_Q} = e^{-\pi} \approx 0.043\dots$

which is about 96 percent decay. The only approximation in this derivation was

$\displaystyle \omega_p = \sqrt{\omega_0^2 - \alpha^2} \approx \omega_0$

which holds whenever $\alpha\ll\omega_0$ , or $Q\gg 1/2$ .

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
    Analog Filters
       Quality Factor (Q)
          Decay Time is Q Periods