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Concept
Quadrature mirror filter
Summary
In digital signal processing, a quadrature mirror filter is a filter whose magnitude response is the mirror image around of that of another filter. Together these filters, first introduced by Croisier et al., are known as the quadrature mirror filter pair.
A filter is the quadrature mirror filter of if .
The filter responses are symmetric about :
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:
where is the frequency, and the sampling rate is normalized to .
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.
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 points is defined as
It is desirable to have it vanish for a constant, so taking the order , for example,
And to have it vanish for a linear ramp, so that
A linear filter will vanish for any , and this is all that can be done with a fourth-order wavelet. Six terms will be needed to vanish a quadratic curve, and so on, given the other constraints to be included. Next an accompanying filter may be defined as
This filter responds in an exactly opposite manner, being large for smooth signals and small for non-smooth signals.
Official source
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