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SIMULATION OF QMF USING SIMULATION PACKAGE

AIM:

APPARATUS REQUIRED:

THEORY:

This filter is used in audio or signal processing to split the wideband (WB) signal into two

narrowbands (NB) using analysis filters, and then these NBs are recombined using synthesis

filters. Furthermore, this concept can be extended for multiple narrowbands such as 2,4,8,162n.

Indigital signal processing,a quadrature mirror filter is a filter whose magnitude response is

mirror image about of that of another filter. Together these filters are known as theQuadrature Mirror Filter pair.

A filter will be quadrature mirror filter of if

The filter responses are symmetric about

Following is the block diagram of QMF banks. The output of the Synthesis Bank should be theperfect reconstruction of the input signal of the Analysis Bank.

These filter banks are made of lowpass, bandpass and highpass filters.

In audio/voice codecs, a quadrature mirror filter pair is often used to implement afilter

bank that splits an inputsignal into two bands. The resulting high-pass and low-pass signals areoften 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:

http://en.wikipedia.org/wiki/Digital_signal_processinghttp://en.wikipedia.org/wiki/Filter_bankhttp://en.wikipedia.org/wiki/Filter_bankhttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Signal_processinghttp://en.wikipedia.org/wiki/Filter_bankhttp://en.wikipedia.org/wiki/Filter_bankhttp://en.wikipedia.org/wiki/Digital_signal_processing

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where is thefrequency,and the sampling rate isnormalized 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 filter banks

For orthogonal discrete wavelet transform is given by

, where is filter length.

Impulse characteristic is

for .Reconstruction filters are still given by same equations.

Orthogonalwavelets -- theHaar wavelets and relatedDaubechies wavelets,Coiflets,and somedeveloped byMallat,are generated byscaling functions which, with the wavelet, satisfy a

quadrature mirror filter relationship.

CODE

% Load scaling filter associated with an orthogonal wavelet.

load db10;

subplot(321); stem(db10); title('db10 low-pass filter');

% Compute the quadrature mirror filter.qmfdb10 = qmf(db10);

subplot(322);

stem(qmfdb10);

title('QMF db10 filter');

% Check for frequency condition (necessary for orthogonality):

% abs(fft(filter))^2 + abs(fft(qmf(filter))^2 = 1 at each

% frequency.

m = fft(db10);

mt = fft(qmfdb10);

freq = [1:length(db10)]/length(db10);

subplot(323);

plot(freq,abs(m));

title('Transfer modulus of db10')subplot(324);

plot(freq,abs(mt));

title('Transfer modulus of QMF db10')

subplot(325);

plot(freq,abs(m).^2 + abs(mt).^2);

title('Check QMF condition for db10 and QMF db10')

xlabel(' abs(fft(db10))^2 + abs(fft(qmf(db10))^2 = 1')

% Check for orthonormality.

http://en.wikipedia.org/wiki/Frequencyhttp://en.wikipedia.org/wiki/Wavelethttp://en.wikipedia.org/wiki/Haar_wavelethttp://en.wikipedia.org/wiki/Daubechies_wavelethttp://en.wikipedia.org/wiki/Coiflethttp://en.wikipedia.org/wiki/St%C3%A9phane_Mallathttp://en.wikipedia.org/wiki/Wavelet#Scaling_functionhttp://en.wikipedia.org/wiki/Wavelet#Scaling_functionhttp://en.wikipedia.org/wiki/St%C3%A9phane_Mallathttp://en.wikipedia.org/wiki/Coiflethttp://en.wikipedia.org/wiki/Daubechies_wavelethttp://en.wikipedia.org/wiki/Haar_wavelethttp://en.wikipedia.org/wiki/Wavelethttp://en.wikipedia.org/wiki/Frequency

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df = [db10;qmfdb10]*sqrt(2);

id = df*df'

OUTPUT:% Editing some graphical properties,

% the following figure is generated.

Orthonormality OUTPUT

id =

1.0000 0.00000.0000 1.0000

RESULT: