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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