Linear Phase Quadrature Mirror Filter Banks

Linear phase filters delay all frequencies by equal amounts, and this is often a desirable property in audio and other applications. A filter phase response is linear in $\omega$ whenever its impulse response $h_0(n)$ is symmetric, i.e.,

$$h_0(-n) = h_0(n)$$

in which case the frequency response can be expressed as

$$H_0(e^{j\omega}) = e^{-j\omega N/2}\left\vert H_0(e^{j\omega})\right\vert$$

Substituting this into the QMF perfect reconstruction constraint (10.7) gives

$$\hbox{constant} = e^{-j\omega N}\left[ \left\vert H_0(e^{j\omega})\right\vert^2 - (-1)^N\left\vert H_0(e^{j(\pi-\omega)})\right\vert^2\right].$$

When $N$ is even, the right hand side of the above equation is forced to zero at $\omega=\pi/2$ . Therefore, we will only consider odd $N$ , for which the perfect reconstruction constraint reduces to

$$\hbox{constant} = e^{-j\omega N}\left[ \left\vert H_0(e^{j\omega})\right\vert^2 + \left\vert H_0(e^{j(\pi-\omega)}\right\vert^2\right]$$

We see that perfect reconstruction is obtained in the linear-phase case whenever the analysis filters are power complementary. See [249] for further details.