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iwsst

Inverse wavelet synchrosqueezed transform

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Syntax

xrec = iwsst(sst)
xrec = iwsst(sst,f,freqrange)
xrec = iwsst(sst,iridge)
xrec = iwsst(___,wav)
xrec = iwsst(___,Name=Value)

Description

xrec = iwsst(sst) returns the inverse wavelet synchrosqueezed transform of sst. The iwsst function assumes that you obtained sst using the wsst function with default parameters.

Note

  • The wavelet synchrosqueezed transform does not preserve the signal mean. After inverting the synchrosqueezed transform, you must add the original signal mean.

  • The inverse wavelet synchrosqueezed transform is an approximate inverse which is not guaranteed to provide perfect reconstruction.

example

xrec = iwsst(sst,f,freqrange) inverts the synchrosqueezed transform for a specified range of frequencies, freqrange, contained in the frequency vector, f.

example

xrec = iwsst(sst,iridge) inverts the synchrosqueezed transform along the time-frequency ridges specified by iridge, the index column vector. iridge is the output of wsstridge. The xrec output is the same size as iridge.

example

xrec = iwsst(___,wav) uses the analytic wavelet specified by wav to invert the synchrosqueezed transform. This wavelet must be the same wavelet as used in wsst. You can include any of the input arguments from previous syntaxes.

example

xrec = iwsst(___,Name=Value) returns the inverse wavelet synchrosqueezed transform with additional options specified by one or more name-value arguments.

example

Examples

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Open Live Script

Obtain the wavelet synchrosqueezed transform of a quadratic chirp using default values. Then reconstruct the signal using the inverse wavelet synchrosqueezed transform.

load quadchirp
sst = wsst(quadchirp);
xrec = iwsst(sst);
Open Live Script

Obtain the wavelet synchrosqueezed transform of a quadratic chirp sampled at 1000 Hz. Then reconstruct the chirp.

Load the chirp and obtain the synchrosqueezed transform.

load quadchirp
sstchirp = wsst(quadchirp,ExtendSignal=true);

Extract the maximum energy time-frequency ridge and reconstruct the signal mode along the ridge.

[~,iridge] = wsstridge(sstchirp);
xrec = iwsst(sstchirp,iridge);

Plot and zoom in on the original and reconstructed signal. 

plot(tquad,xrec,"r")
hold on
plot(tquad,quadchirp,"b--")
hold off
xlabel("Time")
ylabel("Amplitude")
set(gca,ylim=[-1.5 1.5])
legend("Reconstruction","Original")
grid on
title({"Reconstruction of Chirp","Along Maximum Time-Frequency Ridge"})
zoom xon
zoom(50)

Figure contains an axes object. The axes object with title Reconstruction of Chirp Along Maximum Time-Frequency Ridge, xlabel Time, ylabel Amplitude contains 2 objects of type line. These objects represent Reconstruction, Original.

Open Live Script

Obtain the inverse synchrosqueezed transform for a specified frequency range of a two-component signal. The input is a combination of an amplitude-modulated and a frequency-modulated signal.

Create the signal.

t = 0:0.001:0.1;
x1 = (2+0.5*cos(2*pi*10*t)).*cos(2*pi*200*t+10*sin(2*pi*5*t));
x2 = cos(2*pi*50*t);
sig = x1+x2;

Obtain the wavelet synchrosqueezed transform and plot the resulting two frequency components.

[sst,f] = wsst(sig,1000,ExtendSignal=true);
contour(t,f,abs(sst))
grid on
title("Wavelet Synchrosqueezed Transform")
ylabel("Frequency")
xlabel("Time")
hold on
plot(t,140*ones(size(t)),"r--")
plot(t,260*ones(size(t)),"r--")
hold off

Figure contains an axes object. The axes object with title Wavelet Synchrosqueezed Transform, xlabel Time, ylabel Frequency contains 3 objects of type contour, line.

Obtain the inverse synchrosqueezed transform for frequencies from 140 Hz to 260 Hz. Plot the result.

xrec = iwsst(sst,f,[140,260]);
tiledlayout(2,1)
nexttile
plot(t,x1)
title("Original Signal")
nexttile
plot(t,xrec,"r")
title("Reconstructed Signal")

Figure contains 2 axes objects. Axes object 1 with title Original Signal contains an object of type line. Axes object 2 with title Reconstructed Signal contains an object of type line.

Open Live Script

Obtain the wavelet synchrosqueezed transform and inverse synchrosqueezed transform of a speech sample using the analytic Morse (3,30) wavelet.

Load the speech signal and obtain the synchrosqueezed transform and inverse synchrosqueezed transform.

load mtlb
dt = 1/Fs;
t = 0:dt:numel(mtlb)*dt-dt;
Txmtlb = wsst(mtlb,"Morse",WaveletParameters=[3 30]);
xrec = iwsst(Txmtlb,"Morse",WaveletParameters=[3 30]);

Obtain the L-infinity norm of the difference between the original waveform and the reconstruction. Plot the results. 

Linf = norm(abs(mtlb-xrec),Inf);
plot(t,mtlb)
hold on
plot(t,xrec,"r")
hold off
xlabel("Seconds")
ylabel("Amplitude")
title({"Reconstruction of Wavelet Synchrosqueezed Transform", ...
    "Largest Absolute Difference: "+num2str(Linf,"%1.2f")})

Figure contains an axes object. The axes object with title Reconstruction of Wavelet Synchrosqueezed Transform Largest Absolute Difference: 0.09, xlabel Seconds, ylabel Amplitude contains 2 objects of type line.

Open Live Script

This example shows how to invert the wavelet synchrosqueezed transform using a specified number of frequency bins for a quadratic chirp. The chirp is sampled at 1000 Hz. 

load quadchirp
sstchirp = wsst(quadchirp,1000,ExtendSignal=true);

Extract the maximum energy time-frequency ridge using 10 bins on each side of the iridge index and reconstruct the signal mode along the ridge.

[~,iridge] = wsstridge(sstchirp);
xrec = iwsst(sstchirp,iridge,NumFrequencyBins=10);

Plot the original and reconstructed signal.

plot(tquad,xrec,"r")
hold on
plot(tquad,quadchirp,"b--")
hold off
xlabel("Time")
ylabel("Amplitude")
set(gca,ylim=[-1.5 1.5])
legend("Reconstruction","Original")
grid on
title({"Reconstruction of Chirp","Along Maximum Time-Frequency Ridge"})

Figure contains an axes object. The axes object with title Reconstruction of Chirp Along Maximum Time-Frequency Ridge, xlabel Time, ylabel Amplitude contains 2 objects of type line. These objects represent Reconstruction, Original.

Input Arguments

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Wavelet synchrosqueezed transform, specified as a matrix. sst is the output from the wsst function.

Data Types: single | double

Synchrosqueezed transform frequencies corresponding to the rows of the synchrosqueezed transform sst, specified as a vector. f is the output of wsst. If you specify f, you must also specify freqrange.

Frequency range for which to return inverse synchrosqueezed transform values, specified as a two-element vector. The values of freqrange must be in the range of the values of the frequencies, f. The first and second elements of freqrange define the start and end of the frequency range, where the frequency values in that range must be positive and strictly increasing. If you specify freqrange, you must also specify f.

Time-frequency ridge row indices of the synchrosqueezed transform, specified as a vector or matrix. iridge is the output of the wsstridge function. If iridge is a matrix, iwsst inverts the synchrosqueezed transform along the first column of iridge. Then, it iteratively reconstructs along subsequent columns of iridge. The sizes of iridge and the xrec output are the same.

Analytic wavelet used to compute the inverse synchrosqueezed transform, specified as one of the following:

  • "amor" — Analytic Morlet wavelet

  • "Morse" — Analytic Morse wavelet (since R2024a)

  • "bump" — Bump wavelet

You must use the same wavelet in the reconstruction that you used to compute the synchrosqueezed transform, sst.

For use in the inverse synchrosqueezed transform, the analytic Morlet or Morse (3,30) wavelet generally provide the best results.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: NumBins=16

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'NumBins',16

Since R2024a

Morse wavelet parameters, specified as a two-element vector. The first element is the symmetry parameter (gamma), which must be greater than or equal to 1. The second element is the time-bandwidth product, which must be greater than or equal to gamma. The ratio of the time-bandwidth product to gamma cannot exceed 40. Note the ratio of the time-bandwidth parameter to gamma is often referred to as the beta parameter in the literature on Morse wavelets.

When gamma is equal to 3, the Morse wavelet is perfectly symmetric in the frequency domain. The skewness is equal to 0. Values of gamma greater than 3 result in positive skewness, while values of gamma less than 3 result in negative skewness.

If you are using synchrosqueezing for mode reconstruction, the Morse (3,30) wavelet generally provides superior inverse synchrosqueezing results.

WaveletParameters is only valid if you specify the analytic Morse wavelet.

For more information, see Morse Wavelets.

Number of additional frequency bins to include on either side of each iridge index bin, specified as a positive integer. If the number of additional bins exceeds the number of frequency bins available at a particular time step, iwsst truncates the reconstruction at the first or last frequency bin. The default, 16, is one half the default number of voices per octave.

To specify this argument, you must also specify iridge, which is the output of wsstridge. You cannot include a frequency f and frequency range freqrange, if you include the number of frequency bins.

Output Arguments

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Inverse synchrosqueezed transform, returned as a vector or matrix. If you do not specify an iridge input, xrec is a column vector with the same number of rows as sst. If you specify an iridge input, xrec is the same size as iridge.

Data Types: single | double

References

[1] Daubechies, I., J. Lu, and H. T. Wu. "Synchrosqueezed Wavelet Transforms: an Empirical Mode Decomposition-like Tool." Applied and Computational Harmonic Analysis, Vol. 30, Number 2, 2011, pp. 243–261.

[2] Thakur, G., E. Brevdo, N. S. Fučkar, and H. T. Wu. "The Synchrosqueezing algorithm for time-varying spectral analysis: robustness properties and new paleoclimate applications." Signal Processing, Vol. 93, Number 5, 2013, pp. 1079–1094.

Version History

Introduced in R2016a

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