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.

$G_0(z) = H_0(z^{-1})\,$
$G_1(z) = H_1(z^{-1})\,$

## Orthogonal filter banks

For orthogonal discrete wavelet transform $H_1(z)$ is given by

$H_1(z) = z^{-N} H_0(-z^{-1})\,$, where $N$ is filter length.

Impulse characteristic is

$h_1[n] = (-1)^n h_0[N-1-n]\,$ for $0 \leq n < N\,$.

Reconstruction filters are still given by same equations.

$G_0(z) = H_0(z^{-1})\,$
$G_1(z) = H_1(z^{-1})\,$