Quadrature mirror filter

In digital signal processing, a quadrature mirror filter is a filter whose magnitude response is the mirror image around ${\displaystyle \pi /2}$ of that of another filter. Together these filters, first introduced by Croisier et al., are known as the quadrature mirror filter pair.

A filter ${\displaystyle H_{1}(z)}$ is the quadrature mirror filter of ${\displaystyle H_{0}(z)}$ if ${\displaystyle H_{1}(z)=H_{0}(-z)}$.

The filter responses are symmetric about ${\displaystyle \Omega =\pi /2}$:

${\displaystyle {\big |}H_{1}{\big (}e^{j\Omega }{\big )}{\big |}={\big |}H_{0}{\big (}e^{j(\pi -\Omega )}{\big )}{\big |}.}$

In audio/voice codecs, a quadrature mirror filter pair is often used to implement a filter bank that splits an input signal into two bands. The resulting high-pass and low-pass signals are often reduced by a factor of 2, giving a critically sampled two-channel representation of the original signal. The analysis filters are often related by the following formula in addition to quadrate mirror property:

${\displaystyle {\big |}H_{0}{\big (}e^{j\Omega }{\big )}{\big |}^{2}+{\big |}H_{1}{\big (}e^{j\Omega }{\big )}{\big |}^{2}=1,}$

where ${\displaystyle \Omega }$ is the frequency, and the sampling rate is normalized to ${\displaystyle 2\pi }$. This is known as power complementary property. In other words, the power sum of the high-pass and low-pass filters is equal to 1.

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.