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biorfilt

Biorthogonal wavelet filter set

Syntax

[Lo_D,Hi_D,Lo_R,Hi_R] = biorfilt(DF,RF)
[Lo_D1,Hi_D1,Lo_R1,Hi_R1,Lo_D2,Hi_D2,Lo_R2,Hi_R2] = biorfilt(DF,RF,'8')

Description

The biorfilt command returns either four or eight filters associated with biorthogonal wavelets.

[Lo_D,Hi_D,Lo_R,Hi_R] = biorfilt(DF,RF) computes four filters associated with the biorthogonal wavelet specified by decomposition filter DF and reconstruction filter RF. These filters are

Lo_D

Decomposition low-pass filter

Hi_D

Decomposition high-pass filter

Lo_R

Reconstruction low-pass filter

Hi_R

Reconstruction high-pass filter

[Lo_D1,Hi_D1,Lo_R1,Hi_R1,Lo_D2,Hi_D2,Lo_R2,Hi_R2] = biorfilt(DF,RF,'8') returns eight filters, the first four associated with the decomposition wavelet, and the last four associated with the reconstruction wavelet.

It is well known in the subband filtering community that if the same FIR filters are used for reconstruction and decomposition, then symmetry and exact reconstruction are incompatible (except with the Haar wavelet). Therefore, with biorthogonal filters, two wavelets are introduced instead of just one:

One wavelet, ψ˜, is used in the analysis, and the coefficients of a signal s are

c˜j,k=s(x)ψ˜j,k(x)dx

The other wavelet, ψ, is used in the synthesis:

s=j,kc˜j,kψj,k

Furthermore, the two wavelets are related by duality in the following sense:
ψ˜j,k(x)ψj,k(x)dx=0 as soon as j ≠ j′ or k ≠ k′ and
ϕ˜0,k(x)ϕ0,k(x)dx=0 as soon as k ≠ k′.

It becomes apparent, as A. Cohen pointed out in his thesis (p. 110), that “the useful properties for analysis (e.g., oscillations, null moments) can be concentrated in the ψ˜ function; whereas, the interesting properties for synthesis (regularity) are assigned to the ψ function. The separation of these two tasks proves very useful.”

ψ˜ and ψ can have very different regularity properties, ψ being more regular than ψ˜.

The ψ˜, ψ, ϕ˜ and ϕ functions are zero outside a segment.

Examples

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This example shows how to obtain the decomposition (analysis) and reconstruction (synthesis) filters for the 'bior3.5' wavelet.

Determine the two scaling and wavelet filters associated with the 'bior3.5' wavelet.

wv = 'bior3.5';
[Rf,Df] = biorwavf(wv);
[LoD,HiD,LoR,HiR] = biorfilt(Df,Rf);

Plot the filter impulse responses.

subplot(2,2,1)
stem(LoD)
title(['Dec. lowpass filter ',wv]) 
subplot(2,2,2)
stem(HiD)
title(['Dec. highpass filter ',wv])
subplot(2,2,3)
stem(LoR)
title(['Rec. lowpass filter ',wv]) 
subplot(2,2,4)
stem(HiR)
title(['Rec. highpass filter ',wv])

Figure contains 4 axes. Axes 1 with title Dec. lowpass filter bior3.5 contains an object of type stem. Axes 2 with title Dec. highpass filter bior3.5 contains an object of type stem. Axes 3 with title Rec. lowpass filter bior3.5 contains an object of type stem. Axes 4 with title Rec. highpass filter bior3.5 contains an object of type stem.

Demonstrate that autocorrelations at even lags are only zero for dual pairs of filters. Examine the autocorrelation sequence for the lowpass decomposition filter.

npad = 2*length(LoD)-1;
LoDxcr = fftshift(ifft(abs(fft(LoD,npad)).^2));
lags = -floor(npad/2):floor(npad/2);
figure
stem(lags,LoDxcr,'markerfacecolor',[0 0 1])
set(gca,'xtick',-10:2:10)

Figure contains an axes. The axes contains an object of type stem.

Examine the cross correlation sequence for the lowpass decomposition and synthesis filters. Compare the result with the preceding figure.

npad = 2*length(LoD)-1;
xcr = fftshift(ifft(fft(LoD,npad).*conj(fft(LoR,npad))));
lags = -floor(npad/2):floor(npad/2);
stem(lags,xcr,'markerfacecolor',[0 0 1])
set(gca,'xtick',-10:2:10)

Figure contains an axes. The axes contains an object of type stem.

Compare the transfer functions of the analysis and synthesis scaling and wavelet filters

dftLoD = fft(LoD,64); 
dftLoD = dftLoD(1:length(dftLoD)/2+1);
dftHiD= fft(HiD,64); 
dftHiD = dftHiD(1:length(dftHiD)/2+1);
dftLoR = fft(LoR,64);
dftLoR = dftLoR(1:length(dftLoR)/2+1);
dftHiR = fft(HiR,64);
dftHiR = dftHiR(1:length(dftHiR)/2+1);
df = (2*pi)/64;
freqvec = 0:df:pi;

subplot(2,1,1)
plot(freqvec,abs(dftLoD),freqvec,abs(dftHiD),'r')
axis tight
title('Transfer modulus for dec. filters') 
subplot(2,1,2)
plot(freqvec,abs(dftLoR),freqvec,abs(dftHiR),'r') 
axis tight
title('Transfer modulus for rec. filters')

Figure contains 2 axes. Axes 1 with title Transfer modulus for dec. filters contains 2 objects of type line. Axes 2 with title Transfer modulus for rec. filters contains 2 objects of type line.

References

Cohen, A. (1992), “Ondelettes, analyses multirésolution et traitement numérique du signal,” Ph. D. Thesis, University of Paris IX, DAUPHINE.

Daubechies, I. (1992), Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics. SIAM Ed.

Introduced before R2006a