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In [[digital signal processing]], a '''quadrature mirror filter''' is a filter whose magnitude response is mirror image about <math>\pi/2</math> of that of another filter. Together these filters are known as the Quadrature Mirror Filter pair.
 
A filter <math>H_1(z)</math> will be quadrature mirror filter of <math>H_0(z)</math> if <math>H_1(z) = H_0(-z)</math>
 
The filter responses are symmetric about <math>\Omega = \pi / 2</math>
 
:<math>|H_1(e^{j\Omega})| = |H_0(e^{j(\pi - \Omega)})|</math>
 
 
In audio/voice codecs, a quadrature mirror filter pair is often used to implement a [[filter bank]] that splits an input [[signal processing|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 formulae in addition to quadrate mirror property:
:<math>|H_0(e^{j\Omega})|^2 + |H_1(e^{j\Omega})|^2 = 1</math> where <math>\Omega</math> is the [[frequency]], and the sampling rate is normalized to <math>2\pi</math>.
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]]s -- the [[Haar wavelet]]s and related [[Daubechies wavelet]]s, [[Coiflet]]s, and some developed by [[Stéphane Mallat|Mallat]], are generated by [[Wavelet#Scaling_function|scaling functions]] which, with the wavelet, satisfy a quadrature mirror filter relationship.
 
==Further Description==
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 <math>N</math> points <math>x_n</math> is defined as:
:<math>y_n = \sum_{i=0}^{M-1} b_i x_{n-i}</math>
 
It is desirable to have it vanish for a constant, so taking the order <math>m = 4</math> for example:
:<math>b_0 \centerdot 1 + b_1 \centerdot 1 + b_2 \centerdot 1 + b_3 \centerdot 1 = 0</math>
And to have it vanish for a linear ramp so that:
:<math>b_0 \centerdot 0 + b_1 \centerdot 1 + b_2 \centerdot 2 + b_3 \centerdot 3 = 0</math>
 
A linear filter will vanish for any <math>x = \alpha n + \beta</math>, 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:
:<math>z_n = \sum_{i=0}^{M-1} c_i x_{n-i}</math>
 
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:
:<math>\sum_{i=0}^{M-1} c_i b_i = 0</math>
 
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.<ref>
{{Citation | last1=Gershenfeld
| first1=Neil
| title=The Nature of Mathematical Modeling
| pages=132–135
| publisher=Cambridge University Press
| location=Cambridge, England
| isbn=0521570956
| year=1998}}.</ref>
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==
* Johnston, JD, A Filter Family Designed for use in Quadrature Mirror Filter Banks.  [http://www.info490b.ece.mcgill.ca/Data/Exp4/Johnston.pdf], Acoustics, Speech and Signal Processing,IEEE International Conference, 5, 291-294, April, 1980.
* Mohlenkamp, M. J, A Tutorial on Wavelets and Their Applications.  [http://www.ohio.edu/people/mohlenka/20044/PASIII/waveletIPAM.pdf], University of Colorado, Boulder, Dept. of Applied Mathematics, 2004.
* Polikar, R, Multiresolution Analysis: The Discrete Wavelet Transform.  [http://users.rowan.edu/~polikar/WAVELETS/WTpart4.html], Rowan University, NJ, Dept. of Electrical and Computer Engineering
 
==References==
{{reflist}}
 
[[Category:Digital signal processing]]
[[Category:Filter theory]]
[[Category:Wavelets]]

Latest revision as of 01:39, 29 October 2014

Do we learn why diets pill usually never help you lose fat? Its because the makers of diet pills do not have your best interests at heart. They depend on making we worry that should you don't take diet pills which you will not lose weight or gain ever more fat. However, no diet medications are going to make we lose weight, provide you power, or make we feel happy. The diet which is a life-style change plus not a fad is one that truly understands certain master principles to a vital life: One of these master principles is the force of essential oils.

We all desire fad diets quick plus are efficient, otherwise, what's the point. If you will spend any time at all on a diet like this, you will experience hunger, psychological deprivation plus an anxiousness for results. The truth is, we can lose temporary weight on fad diets that work swiftly, but the results usually won't last.

Fast weight loss plus crash diets clearly go hand in hand. After all, the serious diet restrictions involved inside them are sure to cause rapid fat loss. Crash diets could conduce to quick weight reduction, but, they are deleterious ways of losing weight, because they deal with nutrition deprivation. Following crash diet plans will conduce to rapid weight reduction, but, the weight lost is not fat. All which happens is the fact that the body's limited supply of reserved glycogen (carbohydrate form) is employed up, along with water, that is why one appears to lose weight certainly fast. However, as soon because you start eating usually again, we will put on weight. In fact, most folks on crash diets begin binging, thereby gaining more fat than before.

If no one else in the home is "dieting", do NOT feel obligated to cook one meal for oneself and another for the rest of the family. The "No More Diet Diet" is NOT about special or forbidden foods. It's about reducing the calories. This means minimize the portions, serve the cheese sauce found on the side instead of mixed inside with all the broccoli plus limit yourself to one scoop of ice cream rather of 3.

How can you discover much more about how lose fat fast diets never function plus are not the method to forever lose weight? What you need to do to achieve fat reduction without being hungry and without to purchase fat loss goods that never function?

Their are numerous variations of this diet, the basis of which is to consume a specific type of homemade cabbage soup for seven days inside balance with alternative fruits and greens. At least for cups of water plus a multi-vitamin daily is furthermore piece of the diet.

As you can see, these five healthy eating facts for kids usually create it much simpler for we to get started ensuring a own child is eating healthy for a better future.

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