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# blscalf

Best-localized Daubechies scaling filter

Since R2022b

## Description

example

scalf = blscalf(wname) returns the best-localized Daubechies scaling filter corresponding to wname.

## Examples

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Obtain the scaling filter corresponding to the best-localized Daubechies wavelet with 10 vanishing moments. Confirm the sum of the filter coefficients nearly equals $\sqrt{2}$ and the L2 norm of the filter nearly equals 1.

scalf = blscalf("bl10");
sum(scalf)-sqrt(2)
ans = -2.2204e-16

norm(scalf,2)
ans = 1.0000

Use orthfilt to obtain the scaling and wavelet filters corresponding to the wavelet.

[LoD,HiD,LoR,HiR] = orthfilt(scalf);

Confirm the filters form an orthonormal perfect reconstruction wavelet filter bank.

[tf,checks] = isorthwfb(LoD)
tf = logical
1

checks=7×3 table
Pass-Fail    Maximum Error    Test Tolerance
_________    _____________    ______________

Equal-length filters                    pass                 0                 0
Even-length filters                     pass                 0                 0
Unit-norm filters                       pass        1.7665e-10        1.4901e-08
Filter sums                             pass        7.2923e-15        1.4901e-08
Even and odd downsampled sums           pass        3.7748e-15        1.4901e-08
Zero autocorrelation at even lags       pass        7.3088e-11        1.4901e-08
Zero crosscorrelation at even lags      pass        1.3089e-17        1.4901e-08

Create a discrete wavelet transform filter bank using the wavelet. Plot the frequency responses of the wavelet filters and the final resolution scaling filter for the default signal length.

fb = dwtfilterbank(Wavelet="bl10");
freqz(fb)

Plot the wavelet at the coarsest scale.

[psi,t] = wavelets(fb);
plot(t,psi(end,:))
grid on
title("Wavelet")

Plot the scaling function at the coarsest scale.

[phi,t] = scalingfunctions(fb);
plot(t,phi(end,:))
grid on
title("Scaling Function")

## Input Arguments

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Best-localized Daubechies wavelet, specified as one of these:

• "bl7" — Best-localized Daubechies wavelet with seven vanishing moments

• "bl9" — Best-localized Daubechies wavelet with nine vanishing moments

• "bl10" — Best-localized Daubechies wavelet with 10 vanishing moments

## Output Arguments

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Scaling filter corresponding to wname, returned as a vector. scalf should be used in conjunction with orthfilt to obtain scaling and wavelet filters with the proper normalization. The scaling filters agree exactly with Doroslovački [1]. The sum of filter coefficients is nearly √2 and the L2 norm is nearly 1.0.

Data Types: double

## References

[1] Doroslovački, M.L. “On the Least Asymmetric Wavelets.” IEEE Transactions on Signal Processing 46, no. 4 (April 1998): 1125–30. https://doi.org/10.1109/78.668562.

## Version History

Introduced in R2022b