Biorthogonal wavelet filter set

`[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`

The `biorfilt`

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

`[Lo_D,Hi_D,Lo_R,Hi_R] = biorfilt(`

computes
four filters associated with the biorthogonal wavelet specified by
decomposition filter * DF*,

`RF`

`DF`

`RF`

`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(
```

returns
eight filters, the first four associated with the decomposition wavelet,
and the last four associated with the reconstruction wavelet. * DF*,

`RF`

`8`

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, $$\tilde{\psi}$$, is used in the
analysis, and the coefficients of a signal *s* are

$${\tilde{c}}_{j,k}={\displaystyle \int s(x){\tilde{\psi}}_{j,k}(x)dx}$$

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

$$s={\displaystyle \sum _{j,k}{\tilde{c}}_{j,k}}{\psi}_{j,k}$$

Furthermore, the two wavelets are related by duality in the
following sense:

$$\int {\tilde{\psi}}_{j,k}}(x){\psi}_{{j}^{\prime},{k}^{\prime}}(x)dx=0$$ as
soon as *j ≠ j′* or *k ≠
k′* and

$$\int {\tilde{\varphi}}_{0,k}}(x){\varphi}_{0,{k}^{\prime}}(x)dx=0$$ 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 $$\tilde{\psi}$$ function; whereas, the interesting properties for synthesis (regularity) are assigned to the ψ function. The separation of these two tasks proves very useful."

$$\tilde{\psi}$$ and ψ can have very different regularity properties, ψ being more regular than $$\tilde{\psi}$$.

The $$\tilde{\psi}$$, ψ, $$\tilde{\varphi}$$ and ϕ functions are zero outside a segment.

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

wavelet.

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

wavelet.

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

Plot the filter impulse responses.

subplot(221); stem(LoD); title('Dec. low-pass filter bior3.5'); subplot(222); stem(HiD); title('Dec. high-pass filter bior3.5'); subplot(223); stem(LoR); title('Rec. low-pass filter bior3.5'); subplot(224); stem(HiR); title('Rec. high-pass filter bior3.5');

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)

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)

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(211); plot(freqvec,abs(dftLoD),freqvec,abs(dftHiD),'r'); axis tight; title('Transfer modulus for dec. filters') subplot(212); plot(freqvec,abs(dftLoR),freqvec,abs(dftHiR),'r'); axis tight; title('Transfer modulus for rec. filters')

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.

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