Decay Time is Q Periods

Another well known rule of thumb is that the $Q$ 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 $Q$ 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 $H(s)$. 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

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

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

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

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

as the respective residues of the poles $p_i$.

The impulse response is thus

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

Assuming a resonator, $Q>1/2$, 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

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

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

We have shown so far that the impulse response $h(t)$ 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

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

where we have used the definition Eq.$\,$(E.7) $Q\isdef \omega_0/(2\alpha)$. Thus, after $Q$ periods, the amplitude envelope has decayed to

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

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

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

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