Scaling and Wavelet Filter
Y = qmf(
Y = qmf(
Y = qmf(X,0)
Y = qmf( changes
the signs of the even index elements of the reversed vector filter
1, the signs
of the odd index elements are reversed. Changing
the phase of the Fourier transform of the resulting wavelet filter
by π radians.
Y = qmf( is
Y = qmf(X,0).
x be a finite energy signal. Two filters F0 and F1 are
quadrature mirror filters (QMF) if, for any x,
where y0 is a decimated version of the signal x filtered with F0 so y0 defined by x0 = F0(x) and y0(n) = x0(2n), and similarly, y1 is defined by x1 = F1(x) and y1(n) = x1(2n). This property ensures a perfect reconstruction of the associated two-channel filter banks scheme (see Strang-Nguyen p. 103).
For example, if F0 is
a Daubechies scaling filter and F1 =
then the transfer functions F0(z)
of the filters F0 and F1 satisfy
the condition (see the example for db10):
% 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') % Editing some graphical properties, % the following figure is generated.
% Check for orthonormality. df = [db10;qmfdb10]*sqrt(2); id = df*df' id = 1.0000 0.0000 0.0000 1.0000
This example shows the effect of setting the phase parameter of the
Obtain the decomposition low-pass filter associated with a Daubechies wavelet.
lowfilt = wfilters('db4');
qmf function to obtain the decomposition low-pass filter for a wavelet. Then, compare the signs of the values when the
qmf phase parameter is set to 0 or 1. The reversed signs indicates a phase shift of
radians, which is the same as multiplying the DFT by
p0 = qmf(lowfilt,0) p1 = qmf(lowfilt,1)
p0 = Columns 1 through 7 0.2304 -0.7148 0.6309 0.0280 -0.1870 -0.0308 0.0329 Column 8 0.0106 p1 = Columns 1 through 7 -0.2304 0.7148 -0.6309 -0.0280 0.1870 0.0308 -0.0329 Column 8 -0.0106
Compute the magnitudes and display the difference between them. Unlike the phase, the magnitude is not affected by the sign reversals.
ans = 0 0 0 0 0 0 0 0
Strang, G.; T. Nguyen (1996), Wavelets and Filter Banks, Wellesley-Cambridge Press.
Usage notes and limitations:
Use in a MATLAB Function block is not supported.