Quadrature mirror filter

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.

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

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

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