Quadrature mirror filter

From Wikipedia, the free encyclopedia

In digital signal processing, a quadrature mirror filter is a filter bank which splits an input signal into two bands which are usually then subsampled by a factor of 2.

The filters are related by the following formula:

$|\hat{h}(\xi)| + |\hat{h}(\xi + \frac{\pi}{2})| = 1$

where $ξ$ is the frequency, and the sampling rate is normalized to $2π$ .

In other words, the sum of the magnitude response of the high-pass and low-pass filters is equal to 1 at every frequency.

Orthogonal wavelets -- the Haar wavelets and related Daubechies wavelets, Coiflets, and some developed by Mallat, are generated by scaling functions which, with the wavelet, satisfy a quadrature mirror filter relationship.

Also called a conjugate mirror filter.

[edit] Perfect reconstruction

Even if the two resulting bands have been subsampled by a factor of 2, the relationship between the filters means that perfect reconstruction is possible. That is, the two bands can then be upsampled, filtered again with the same filters and added together, to reproduce the original signal exactly (but with a small delay). (In practical implementations, numeric precision issues in floating-point arithmetic may affect the perfection of the reconstruction.)