Quadrature mirror filter

digital signal filter

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.

Bank of QMFs

Simple variantEdit

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 banksEdit

For orthogonal discrete wavelet transform   is given by

 , where   is filter length.

Impulse characteristic is

  for  .

Reconstruction filters are still given by same equations.