%EXAMP_PGAUSS  How to use PGAUSS%%  This script illustrates various properties of the%  Gaussian function.%%  FIGURE 1 Window+Dual+Tight%%   This figure shows an optimally centered Gaussian for a %   given Gabor system, its canonical dual and tight windows%   and the DFTs of these windows.%%  FIGURE 2 Framebounds sweep.%%   This figure shows how the framebounds behave for a Gabor frame%   with a fixed lattice, but with Gaussian windows of different%   widths.%disp('Type "help examp_pgauss" to see a description of how this example works.');% A quick test: If the second input parameter to% pgauss is not specified, the output will be% invariant under an unitary DFT. Matlabs FFT is does not% preserve the norm, so it must be scaled a bit.L=128;g=pgauss(L);disp('');disp('Test of DFT invariance: Should be close to zero.');norm(g-dft(g))% Setup parameters and length of signal.% Note that it must hold that L=M*b=N*a for some integers% b and N, and that a <= ML=72;  % Length of signal.a=6;   % Time shift.M=9;   % Number of modulations.% Calculate the frequency shift.b=L/M;% For this Gabor system, the optimally concentrated Gaussian% is given byg=pgauss(L,a/b);% This is not invarient with respect to a DFT, but it is still% real and whole point evendisp('');disp('The function is WP even. The following should be 1.');iseven(g)disp('Therefore, its DFT is real.');disp('The norm of the imaginary part should be close to zero.');norm(imag(dft(g)))% Calculate the canonical dual.gdual=candual(g,a,M);% Calculate the canonical tight window.gtight=cantight(g,a,M);% Plot them:% Standard note on plotting:%% - All windows have real DFTs, but Matlab does not%   always recoqnize this, so we have to filter away%   the small imaginary part by calling REAL(...)%% - The windows are all centered around zero, but this%   is not visually pleasing, so the window must be%   shifted to the middle by an FFTSHIFT%gf_plot      = fftshift(real(dft(g)));gdual_plot   = fftshift(gdual);gdualf_plot  = fftshift(real(dft(gdual)));gtight_plot  = fftshift(gtight);gtightf_plot = fftshift(real(dft(gtight)));figure(1);subplot(3,2,1);x=(1:L).';plot(x,fftshift(g),'-',...     x,circshift(fftshift(g),a),'-',...     x,circshift(fftshift(g),-a),'-');title('g=pgauss(72,6/8)');legend('off');subplot(3,2,2);plot(gf_plot);title('g, frequency domain');legend('off');subplot(3,2,3);plot(gdual_plot);title('Dual window of g');legend('off');subplot(3,2,4);plot(gdualf_plot);title('dual window, frequency domain');legend('off');subplot(3,2,5);plot(gtight_plot);title('Tight window generated from g');legend('off');subplot(3,2,6);plot(gtightf_plot);title('tight window, frequency domain');legend('off');%% Plot the behaviour of the framebounds when the parameter% of pgauss is swept over a range of values. % Number of plotting points.npoints=100;% The range of parameterw=logspace(-1,1,npoints)*a/b;% Calculate the framebounds for each plotting point.ar=zeros(npoints,1);br=zeros(npoints,1);for ii=1:npoints  [ar(ii),br(ii)]=gfbounds(pgauss(L,w(ii)),a,M);end;figure(2);loglog(w,ar,'-',w,br,'-');title('Framebounds');xlabel('Parameter for PGAUSS');legend('off');