## 11.3. Exercises

EXERCISE 11.6.

Consider again the signal generated in exercise 11.4, whose WVD contains interference terms. Write a MATLAB function to calculate the SWVD of this signal and choose the appropriate length for windows *h* and *g*. Verify the effect of these windows on the time and frequency resolutions. Note also the failure of the WVD theoretical properties, such as the time and frequency support conservation.

Note: modify the loop of the function wigner (see exercise 11.4) using the code lines below to obtain the new time-frequency distribution:

% Defining the windows length_FFT=N;nh=N;NG=16; g=hamming(2*NG+1)'; h=hamniing(nh); % Variables initialization ECH=t_s;inc=1; length_time=t_s/inc; WX=zeros(length_FFT, length_time); A=zeros(1,ng); X=linspace(0,ng-1,ng); coef_norm=length_FFT*sum(g)*h(nh/2)/4; for t=1:length_time, % Calculation of the first term for tau=0 ind=X+ECH-fix(ng/2); % g window centred on the first signal value A=s(ind) .*s_conj(ind) .*g; R=zeros(1,nh); R(1)=sum(A)*h(nh/2) for tau=1:fix(nh/2); A=s(ind+tau).*s_conj(ind-tau) .*g; R(tau+1)=sum(A)*h(tau+nh/2) ; R(fix(nh+1-tau))=conj(R(tau+1)) ; end WX(:,t)=fft(R,length_FFT)'/coef_norm; ECH=ECH+inc; end

Other weighting windows may also be used (functions boxcar, triang, hanning, etc.). Remember that the multiplication by window *h* leads to a spectral smoothing and removes the frequency interferences if its length is smaller than the time distance between the time-frequency atoms. In the same way, window

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