Bank of QMFs

The quadrature mirror filters (QMF) are two filters with frequency characteristics symmetric about ${\displaystyle 1/4}$ of sampling frequency (i.e. ${\displaystyle \pi /2}$). 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 ${\displaystyle H_{1}(z)}$ to (original) filter ${\displaystyle H_{0}(z)}$ by substitution ${\displaystyle z}$ with ${\displaystyle -z}$ in the transfer function of ${\displaystyle H_{0}(z)}$.

${\displaystyle H_{1}(z)=H_{0}(-z)\,}$

By doing it, the transfer characteristic of ${\displaystyle H_{1}(z)}$ is shifted to ${\displaystyle H_{0}(z)}$ by ${\displaystyle \pi }$.

${\displaystyle |H_{1}(e^{j\omega })|=|H_{0}(e^{j(\pi -\omega )})|\,}$

Impulse characteristic is therefore

${\displaystyle h_{1}[n]=(-1)^{n}h_{0}[n]\,}$ for ${\displaystyle 0\leq n, where ${\displaystyle N}$ 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 ${\displaystyle G_{0}(z)}$ and ${\displaystyle G_{1}(z)}$. Reconstruction filters are given by time reversal analysis filters.

${\displaystyle G_{0}(z)=H_{0}(z^{-1})\,}$
${\displaystyle G_{1}(z)=H_{1}(z^{-1})\,}$

## Orthogonal filter banks

For orthogonal discrete wavelet transform ${\displaystyle H_{1}(z)}$ is given by

${\displaystyle H_{1}(z)=z^{-N}H_{0}(-z^{-1})\,}$, where ${\displaystyle N}$ is filter length.

Impulse characteristic is

${\displaystyle h_{1}[n]=(-1)^{n}h_{0}[N-1-n]\,}$ for ${\displaystyle 0\leq n.

Reconstruction filters are still given by same equations.

${\displaystyle G_{0}(z)=H_{0}(z^{-1})\,}$
${\displaystyle G_{1}(z)=H_{1}(z^{-1})\,}$