ifsst

Load a speech signal sampled at $$F_s=7418Hz$$. The file contains a recording of a female voice saying the word "MATLAB®." Compute the Fourier synchrosqueezed transform of the signal.

load mtlb         % To hear, type sound(mtlb,Fs)

[sst,f] = fsst(mtlb,Fs);

Invert the transform to reconstruct the signal. Plot the original and reconstructed signals, as well as the difference between them.

xrec = ifsst(sst);
 
t = (0:length(mtlb)-1)/Fs;
plot(t,mtlb,t,xrec,t,mtlb-xrec)

xlabel('Time (s)')
legend('Original','Reconstructed','Difference')

Check the accuracy of the reconstruction by computing the $${\ell }_{\infty }$$ norm of the difference between the original signal and the inverse transform.

Linf = norm(abs(mtlb-xrec),Inf)

Linf = 
1.9762e-14

% To hear, type sound(mtlb-xrec,Fs)

Generate a signal sampled at 1024 Hz for 2 seconds.

nSamp = 2048;
Fs = 1024;
t = (0:nSamp-1)'/Fs;

During the first second, the signal consists of a 400 Hz sinusoid and a concave quadratic chirp. Specify a chirp that is symmetric about the interval midpoint, starts and ends at a frequency of 250 Hz, and attains a minimum of 150 Hz.

t1 = t(1:nSamp/2);

x11 = sin(2*pi*400*t1);
x12 = chirp(t1-t1(nSamp/4),150,nSamp/Fs,1750,'quadratic');
x1 = x11+x12;

The rest of the signal consists of two linear chirps of decreasing frequency. One chirp has an initial frequency of 250 Hz that decreases to 100 Hz. The other chirp has an initial frequency of 400 Hz that decreases to 250 Hz.

t2 = t(nSamp/2+1:nSamp);

x21 = chirp(t2,400,nSamp/Fs,100);
x22 = chirp(t2,550,nSamp/Fs,250);
x2 = x21+x22;

Compute the Fourier synchrosqueezed transform of the signal. Specify a 256-sample Kaiser window with a shape parameter β = 100. Use the plotting functionality of fsst to display the result.

sig = [x1;x2];
wind = kaiser(256,120);

[sigtr,ftr,ttr] = fsst(sig,Fs,wind);

fsst(sig,Fs,wind,'yaxis')

Invert the transform to reconstruct the function. Plot the original and inverted signals and the difference between them.

x = ifsst(sigtr,wind);

plot(t,sig,t,x,t,x-sig)
legend('Original','Reconstructed','Difference')

diffnorm = norm(x-sig)

diffnorm = 
3.9026e-13

Generate a signal that consists of two chirps. The signal is sampled at 3 kHz for one second. The first chirp has an initial frequency of 400 Hz and reaches 800 Hz at the end of the sampling. The second chirp starts at 500 Hz and reaches 1000 Hz at the end. The second chirp has twice the amplitude of the first chirp.

fs = 3000;
t = 0:1/fs:1-1/fs;   

x1 = chirp(t,400,t(end),800);
x2 = 2*chirp(t,500,t(end),1000);

Compute and plot the Fourier synchrosqueezed transform of the signal. Display the time on the x-axis and the frequency on the y-axis.

[sst,f] = fsst(x1+x2,fs);
fsst(x1+x2,fs,'yaxis')

Extract the ridge corresponding to the higher-energy component of the signal, which is the chirp with the larger amplitude. Use the ridge to reconstruct the signal.

[~,iridge] = tfridge(sst,f);
 
xrec = ifsst(sst,[],iridge);

Plot the spectrogram for the higher-energy component. Divide the component into 256-sample sections and specify an overlap of 255 samples. Use 512 DFT points and a rectangular window.

spectrogram(xrec,rectwin(256),255,512,fs,'yaxis')

To extract the second chirp, specify that tfridge search for two ridges. The second column of the output is the lower-energy component of the signal.

[~,iridge] = tfridge(sst,f,'NumRidges',2);

xrec = ifsst(sst,[],iridge(:,2));

spectrogram(xrec,rectwin(256),255,512,fs,'yaxis')