In digital signal processing, a quadrature mirror filter is a filter whose magnitude response is the mirror image around π / 2 {\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 H 1 ( z ) {\displaystyle H_{1}(z)} is the quadrature mirror filter of H 0 ( z ) {\displaystyle H_{0}(z)} if H 1 ( z ) = H 0 ( − z ) {\displaystyle H_{1}(z)=H_{0}(-z)} . The filter responses are symmetric about Ω = π / 2 {\displaystyle \Omega =\pi /2} :
| H 1 ( e j Ω ) | = | H 0 ( e j ( π − Ω ) ) | . {\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:
| H 0 ( e j Ω ) | 2 + | H 1 ( e j Ω ) | 2 = 1 , {\displaystyle {\big |}H_{0}{\big (}e^{j\Omega }{\big )}{\big |}^{2}+{\big |}H_{1}{\big (}e^{j\Omega }{\big )}{\big |}^{2}=1,}
where Ω {\displaystyle \Omega } is the frequency, and the sampling rate is normalized to 2 π {\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.
Relationship to other filter banks The earliest wavelets were based on expanding a function in terms of rectangular steps, the Haar wavelets. This is usually a poor approximation, whereas Daubechies wavelets are among the simplest but most important families of wavelets. A linear filter that is zero for “smooth” signals, given a record of N {\displaystyle N} points x n {\displaystyle x_{n}} is defined as
y n = ∑ i = 0 M − 1 b i x n − i . {\displaystyle y_{n}=\sum _{i=0}^{M-1}b_{i}x_{n-i}.}
It is desirable to have it vanish for a constant, so taking the order m = 4 {\displaystyle m=4} , for example,
b 0 ⋅ 1 + b 1 ⋅ 1 + b 2 ⋅ 1 + b 3 ⋅ 1 = 0. {\displaystyle b_{0}\cdot 1+b_{1}\cdot 1+b_{2}\cdot 1+b_{3}\cdot 1=0.}
And to have it vanish for a linear ramp, so that
b 0 ⋅ 0 + b 1 ⋅ 1 + b 2 ⋅ 2 + b 3 ⋅ 3 = 0. {\displaystyle b_{0}\cdot 0+b_{1}\cdot 1+b_{2}\cdot 2+b_{3}\cdot 3=0.}
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