%EXAMP_MIXDUAL  How to use MIXDUAL%%   This script illustrates how one can produce dual windows%   using MIXDUAL%%   The example constructs a dual window that is more concentrated in%   the time domain by mixing the original Gabor window by one that is%   extremely well concentrated. The result is somewhat in the middle%   of these two.% %   The lower framebound of the mixing Gabor system is horrible,%   but this does not carry over to the mixdual.%%   FIGURE 1 Mixdual of two Gaussians.%%    The first row of the figure shows the canonical dual window%    of the input window, which is a Gaussian function perfectly%    localized in the time and frequency domains.%%    The second row shows the canonical dual window of the window we%    will be mixing with: This is a Gaussian that is 10 times more%    concentrated in the time domain than in the frequency domain.%    The resulting canonical dual window has rapid decay in the time domain.%%    The last row shows the mixdual of these two. This is a non-canonical%    dual window of the first Gaussian, with decay resembling that of the%    second.%%    SEE ALSO: MIXDUAL, CANDUALdisp('Type "help examp_mixdual" to see a description of how this example works.');L=120;a=10;M=12;% Compute frequency shift.b=L/M;% Optimally centered Gaussiang1=pgauss(L);% Compute and print framebounds.[A1,B1]=gfbounds(g1,a,M);disp('');disp('Framebounds of initial Gabor system:');A1, B1% Narrow Gaussiang2=pgauss(L,.1);% Compute and print framebounds.[A2,B2]=gfbounds(g2,a,M);disp('');disp('Framebounds of mixing Gabor system:');A2, B2% Create a mixdual. The window gd is a dual window to g1gd=mixdual(g1,g2,a,M);% Compute and print framebounds.[Am,Bm]=gfbounds(gd,a,M);disp('');disp('Framebounds of mixdual Gabor system:');Am, Bm% Create canonical duals, for plotting.gc1=candual(g1,a,M);gc2=candual(g2,a,M);% Standard note on plotting:%% - The windows are all centered around zero, but this%   is not visually pleasing, so the window must be%   shifted to the middle by an FFTSHIFTfigure(1);subplot(3,2,1);plot(fftshift(gc1));title('Canonical dual window.');  legend('off');subplot(3,2,2);semilogy(abs(fftshift(gc1)));title('Decay of canonical dual window.');  legend('off');subplot(3,2,3);plot(fftshift(gc2));title('Can. dual of mix. window.');legend('off');subplot(3,2,4);semilogy(abs(fftshift(gc2)));title('Decay of can.dual of mix. window.')legend('off');subplot(3,2,5);plot(fftshift(gd));title('Mixdual');legend('off');subplot(3,2,6);semilogy(abs(fftshift(gd)));title('Decay of mixdual');legend('off');