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.,

$$h_0(-n) \eqsp h_0(n)$$ (12.35)

in which case the frequency response can be expressed as

$$H_0(\ejo ) \eqsp e^{-j\omega N/2}\left\vert H_0(\ejo )\right\vert.$$ (12.36)

Substituting this into the QMF perfect reconstruction constraint (11.34) gives

$$g\,e^{-j\omega d} \eqsp e^{-j\omega N}\left[ \left\vert H_0(\ejo )\right\vert^2 - (-1)^N\left\vert H_0(e^{j(\pi-\omega)})\right\vert^2\right].$$ (12.37)

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

$$g\,e^{-j\omega d} \eqsp e^{-j\omega N}\left[ \left\vert H_0(\ejo )\right\vert^2 + \left\vert H_0(e^{j(\pi-\omega)}\right\vert^2\right].$$ (12.38)

We see that perfect reconstruction is obtained in the linear-phase case whenever the analysis filters are power complementary. See [287] for further details.