Biorthogonal wavelet filter set
LoD — Decomposition
HiD — Decomposition
LoR — Reconstruction
HiR — Reconstruction
This example shows how to obtain the decomposition (analysis) and reconstruction (synthesis) filters for the
Obtain the two scaling and wavelet filters associated with the
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])
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) title('Autocorrelation') xlabel('Lag')
Examine the cross-correlation sequence for the lowpass decomposition and synthesis filters. Compare the result with the preceding figure. At even lags, the cross-correlation is zero.
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) title('Cross-correlation') xlabel('Lag')
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 - Dec. Filters') subplot(2,1,2) plot(freqvec,abs(dftLoR),freqvec,abs(dftHiR),'r') axis tight title('Transfer Modulus - Rec. Filters')
DF— Decomposition scaling filter
Decomposition scaling filter associated with a biorthogonal wavelet, specified as a vector.
RF— Reconstruction scaling filter
Reconstruction scaling filter associated with a biorthogonal wavelet, specified as a vector.
LoD,HiD— Decomposition filters
Wavelet decomposition filters, returned as a
pair of even-length real-valued vectors.
LoD is the lowpass
decomposition filter, and
is the highpass decomposition filter.
LoR,HiR— Reconstruction filters
Wavelet reconstruction filters, returned as
a pair of even-length real-valued vectors.
LoR is the lowpass
reconstruction filter, and
is the highpass reconstruction filter.
Filters associated with the decomposition (analysis) wavelet, returned as even-length real-valued vectors.
LoD1 — Decomposition
HiD1 — Decomposition
LoR1 — Reconstruction
HiR1 — Reconstruction
Filters associated with the reconstruction (synthesis) wavelet, returned as even-length real-valued vectors.
LoD2 — Decomposition
HiD2 — Decomposition
LoR2 — Reconstruction
HiR2 — Reconstruction
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, , is used in the analysis, and the coefficients of a signal s are
The other wavelet, ψ, is used in the synthesis:
Furthermore, the two wavelets are related by duality in
the following sense:
as soon as j ≠ j′ or k ≠ k′ and
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 function; whereas, the interesting properties for synthesis (regularity) are assigned to the ψ function. The separation of these two tasks proves very useful.”
and ψ can have very different regularity properties, ψ being more regular than .
The , ψ, and ϕ functions are zero outside a segment.
 Cohen, Albert. "Ondelettes, analyses multirésolution et traitement numérique du signal," Ph. D. Thesis, University of Paris IX, DAUPHINE. 1992.
 Daubechies, Ingrid. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. Philadelphia, Pa: Society for Industrial and Applied Mathematics, 1992.