File:DFT-even Hann window & spectral leakage.png
Summary
| Description |
English: The main point is to illustrate that the N-point DFT (discrete Fourier transform) of an N-point DFT-even Hann window function has only 3 non-zero coefficients. The other N-3 samples of the DTFT (bottom figure) coincide with zero-crossings of the DTFT. Higher-order "Cosine-sum windows" have more non-zero DFT coefficients. Wikipedia article Window function contains a link to this figure. |
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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: Odd-length, "DFT-even" Hann window & spectral leakage.pngCategory:Files with derivative versions | |||
| PNG development | ||||
| Octave/gnuplot source | click to expand
This graphic was created with the help of the following Octave script: pkg load signal
graphics_toolkit gnuplot
clc
M=4096;
window = hann(M)'; % row vector
N=16; % window size, in samples
dx = M/N; % decimation factor for 16 hops (17 samples)
periodic = window(1+(0:15)*dx); % take 16 of 17 symmetrical samples
%Plot the points
figure
plot(0:15, periodic, 'color', 'blue', '.', 'MarkerSize',14)
hold on
%Connect the dots
x = (0:M-1)*N/M;
plot(x, window, 'color', 'blue') % periodic
xlim([0 16])
set(gca, 'xgrid', 'on');
set(gca, 'ygrid', 'on');
set(gca, 'ytick', [0:.25:1]);
set(gca, 'xtick', [0:16]);
title('DFT-even Hann window function');
xlabel('\leftarrow n \rightarrow','FontSize', 14)
%Now compute and plot the DTFT
M=64*N;
dr = 80;
H = abs(fft([periodic 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
plot(x, H, 'color', 'blue');
hold on
%Plot the 3 non-zero points
plot(-1:1, H((N/2-1:N/2+1)*M/N), 'color', 'blue', '.', 'MarkerSize',14)
ylim([-dr 0])
xlim([-N/2 N/2-1])
set(gca,'XTick', -N/2:N/2-1)
grid on
ylabel('decibels','FontSize', 14)
xlabel('DFT bins','FontSize', 12)
title('Non-zero DFT coefficients of Hann window')
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![Odd-length, "DFT-even" Hann window & spectral leakage.png](./File:Odd-length,_"DFT-even"_Hann_window_&_spectral_leakage.png){kind=link}
