Dyadic Filter Banks

Figure 11.34: Frequency response magnitudes for a dyadic filter bank (amplitude scalings optional).

A dyadic filter bank is any octave filter bank,^12.6 as illustrated qualitatively in Figure 11.34. Note that $H_0(\omega )$ is the top-octave bandpass filter, $H_1(w) = \sqrt{2} H_0(2\omega )$ is the bandpass filter for next octave down, $H_2(w) = 2H_0(4\omega )$ is the octave bandpass below that, and so on. The optional scale factors result in the same sum-of-squares for each channel-filter impulse response.

A dyadic filter bank may be derived from the discrete wavelet filter bank by setting $a=2$ and relaxing the exact orthonormality requirement on the channel-filter impulse responses. If they do happen to be orthonormal, we may call it a dyadic wavelet filter bank.

For a dyadic filter bank, the center-frequency of the $k$ th channel-filter impulse response can be defined as

$$\omega _c(k) \eqsp \sqrt{\omega _0(\omega _0+\hbox{bandwidth})} \eqsp \sqrt{2}\omega _0$$ (12.123)

so that

$$Q \eqsp \frac{\sqrt{2}\omega _0}{2\omega _0 - \omega _0} \eqsp \sqrt{2}.$$ (12.124)

Thus, a dyadic filter bank is a special case of a constant-Q filter bank for which the $Q$ is $\sqrt{2}$ .