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digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
, a quadrature mirror filter is a filter whose magnitude response is the mirror image around \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) is the quadrature mirror filter of H_0(z) if H_1(z) = H_0(-z). The filter responses are symmetric about \Omega = \pi / 2: : \big, H_1\big(e^\big)\big, = \big, H_0\big(e^\big)\big, . In audio/voice codecs, a quadrature mirror filter pair is often used to implement a filter bank that splits an input
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
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: : \big, H_0\big(e^\big)\big, ^2 + \big, H_1\big(e^\big)\big, ^2 = 1, where \Omega is the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, and the sampling rate is normalized to 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
wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
s – the
Haar wavelet In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be repres ...
s and related
Daubechies wavelet The Daubechies wavelets, based on the work of Ingrid Daubechies, are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support. With each wavelet type ...
s,
Coiflet Coiflets are discrete wavelets designed by Ingrid Daubechies, at the request of Ronald Coifman, to have scaling functions with vanishing moment Moment or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums ...
s, 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 points x_n is defined as : y_n = \sum_^ b_i x_. It is desirable to have it vanish for a constant, so taking the order m = 4, for example, : 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 \cdot 0 + b_1 \cdot 1 + b_2 \cdot 2 + b_3 \cdot 3 = 0. A linear filter will vanish for any x = \alpha n + \beta, 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 : z_n = \sum_^ c_i x_. This filter responds in an exactly opposite manner, being large for smooth signals and small for non-smooth signals. A linear filter is just a convolution of the signal with the filter’s coefficients, so the series of the coefficients is the signal that the filter responds to maximally. Thus, the output of the second filter vanishes when the coefficients of the first one are input into it. The aim is to have : \sum_^ c_i b_i = 0. Where the associated time series flips the order of the coefficients because the linear filter is a convolution, and so both have the same index in this sum. A pair of filters with this property are defined as quadrature mirror filters. . 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.)


Further reading

* A. Croisier, D. Esteban, C. Galand. ''Perfect channel splitting by use of interpolation/decimation tree decomposition techniques''. First International Conference on Sciences and Systems, Patras, August 1976, pp. 443–446. * Johnston, J. D
''A Filter Family Designed for use in Quadrature Mirror Filter Banks''
, Acoustics, Speech and Signal Processing, IEEE International Conference, 5, 291–294, April, 1980. *
Binomial QMF A binomial QMF – properly an orthonormal binomial quadrature mirror filter – is an orthogonal wavelet developed in 1990. The binomial QMF bank with perfect reconstruction (PR) was designed by Ali Akansu, and published in 1990, using the family ...
, also known as
Daubechies wavelet The Daubechies wavelets, based on the work of Ingrid Daubechies, are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support. With each wavelet type ...
filters.
NJIT Symposia on Subbands and Wavelets 1990, 1992, 1994, 1997
* Mohlenkamp, M. J
''A Tutorial on Wavelets and Their Applications''
University of Colorado, Boulder, Dept. of Applied Mathematics, 2004. * Polikar, R

Rowan University, NJ, Dept. of Electrical and Computer Engineering.


References

{{reflist Digital signal processing Filter theory Wavelets