Quadrature mirror filter

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Bank of QMFs

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

Simple variant[change | change source]

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

By doing it, the transfer characteristic of is shifted to by .

Impulse characteristic is therefore

for , where 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 and . Reconstruction filters are given by time reversal analysis filters.

Orthogonal filter banks[change | change source]

For orthogonal discrete wavelet transform is given by

, where is filter length.

Impulse characteristic is

for .

Reconstruction filters are still given by same equations.