File:Spectral leakage from two 8-point Gaussian windows.png
Summary
| Description |
English: Comparing the discrete-time Fourier transforms of "symmetric" and "periodic" windows. The Gaussian form (σ = 0.4) was chosen for its non-zero end-points, one of which is discarded in the periodic version. |
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| Date | ||||
| Source | Own work | |||
| Author | Bob K | |||
| Permission (Reusing this file) |
I, the copyright holder of this work, hereby publish it under the following license:
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| Other versions | Derivative works of this file: Comparison of symmetric and periodic Gaussian windows.svgCategory:Files with derivative versions | |||
| PNG development | ||||
| Octave/gnuplot source | click to expand
This graphic was created by the following Octave script: <source lang='Matlab'> graphics_toolkit gnuplot pkg load signal symmetric8 = exp(-.5*(((0:7)-7/2)/(.4*7/2)).^2); % 8-points of an 8-point window periodic8 = exp(-.5*(((0:7)-8/2)/(.4*8/2)).^2); % 8-points of a 9-point window ENBW_symmetric8 = 8*sum(symmetric8.^2)/sum(symmetric8)^2 % 1.6233 ENBW_periodic8 = 8*sum(periodic8.^2) /sum(periodic8)^2 % 1.4533 %Now compute and plot the DTFTs and DFTs N = 8; % DFT size M = 64*N; % DTFT size dr = 80; % dynamic range (decibels) %------------------------------------------------------------------ % DTFT of symmetric window H = abs(fft([symmetric8 zeros(1,M-N)])); H = fftshift(H); H = H/max(H); H = 20*log10(H); H = max(-dr,H); x = N*[-M/2:M/2-1]/M; figure("position", [100 200 700 400]) plot(x, H, "color","red", "linewidth",1); hold on ylim([-dr 0]) % Compute a DFT to sample the DTFT 8 times H = abs(fft(symmetric8)); H = fftshift(H); H = H/max(H); H = 20*log10(H); H = max(-dr,H); plot(-N/2:(N/2-1), H, "color","red", ".", "markersize",14) %------------------------------------------------------------------ % DTFT of periodic window H = abs(fft([periodic8 zeros(1,M-N)])); H = fftshift(H); H = H/max(H); H = 20*log10(H); H = max(-dr,H); plot(x, H, "color","blue", "linewidth",1); % Compute a DFT to sample the DTFT 8 times H = abs(real(fft(periodic8))); % real() is redundant... just to illustrate a point H = fftshift(H); H = H/max(H); H = 20*log10(H); H = max(-dr,H); plot(-N/2:(N/2-1), H, "color","blue", ".", "markersize",14) set(gca,"XTick", -N/2:N/2-1) grid on text(-1.17,-64, "Equivalent Noise Bandwidths", "fontsize",10, "fontweight","bold") h = legend("", num2str(ENBW_symmetric8,'%5.3f'),... "", num2str(ENBW_periodic8, '%5.3f'), "location","south"); set(h, "fontsize",10) %legend boxoff text(-2.84, -11.66, {" DTFT";'symmetric8 \rightarrow'}, "color","red",... "fontsize",10, "fontweight","bold") set(gca,"XTick", -N/2:N/2-1) grid on ylabel("decibels", "fontsize",14) xlabel("DFT bins", "fontsize",12, "fontweight","bold") title('"Spectral leakage" from two 8-point Gaussian windows','FontSize', 14) % After this call, the cursor units change to a normalized ([0,1]) coordinate system annotation("textarrow", [.132 .132], [.675 .51],... "color", "blue", "string", {" DTFT";"periodic8"}, "fontsize",10,...
"linewidth",1, "headstyle","vback1", "headlength",5, "headwidth",5)
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