Linear Phase Quadrature Mirror Filter Banks

It is generally desirable to use linear phase filters whenever possible in audio work. This is because linear phase filters delay all frequencies by equal amounts.

A filter phase response is linear in $\omega$ whenever its impulse response $h_0(n)$ is symmetric, i.e.,

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

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 () 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. Since FIR QMF filters are constrained to the two-tap case, this is best accomplished using IIR filters. See Vaidyanathan for details.