Quadrature mirror filter

In digital signal processing, a quadrature mirror filter is a filter most commonly used to implement a filter bank that splits an input signal into two bands. The resulting high-pass and low-pass signals are often decimated by a factor of 2, giving a critically sampled two-channel representation of the original signal.

The analysis filters are related by the following formulae::$|H\_0(e^\{jOmega\})|^2\; +\; |H\_1(e^\{jOmega\})|^2\; =\; 1$

where $Omega$ is the frequency, and the sampling rate is normalized to $2pi$.

In other words, the power sum of the high-pass and low-pass filters is equal to 1. The filter responses are symmetric about $Omega\; =\; pi\; /\; 2$

:$|H\_1(e^\{jOmega\})|\; =\; |H\_0(e^\{j(pi\; -\; Omega)\}|$

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.

Perfect reconstruction

Even if the two resulting bands have been subsampled by a factor of 2, the relationship between the filters means that approximately 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.)