Quadrature mirror filter facts for kids

Bank of QMFs

The quadrature mirror filters (QMF) are two filters with frequency characteristics symmetric about $1/4$ of sampling frequency (i.e. $\pi/2$ ). They are used especially in process of orthogonal discrete wavelet transform design.

Simple variant

In notation of Z-transform, we can create the quadrature mirror filter $H_1(z)$ to (original) filter $H_0(z)$ by substitution $z$ with $-z$ in the transfer function of $H_0(z)$ .

H_1(z) = H_0(-z)\,

By doing it, the transfer characteristic of $H_1(z)$ is shifted to $H_0(z)$ by $\pi$ .

| H_1(e^{j\omega}) | = | H_0(e^{j(\pi-\omega)}) |\,

Impulse characteristic is therefore

h_1[n] = (-1)^n h_0[n]\,

for

0 \leq n < N\,

, where

N

is filter length.

According to the picture above, the signal split and passed into these filters can be downsampled by a factor of two. Nevertheless, original signal can be still reconstructed by using reconstruction filters $G_0(z)$ and $G_1(z)$ . Reconstruction filters are given by time reversal analysis filters.