File:Sampling the Discrete-time Fourier transform.svg

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English: Definitions: DTFT=discrete-time Fourier transform; DFT=discrete Fourier transform

Nine symmetric samples of a cosine function are shifted from the finite Fourier transform domain [-4,4] to the DFT domain [0,8], causing its DTFT to become complex-valued, except at the frequencies of an 8-length DFT. Pictured here are the real and imaginary parts of the DTFT after the cosine is multiplied by a symmetric Gaussian window function. Also shown (in red) are the estimates obtained by deleting the 9th data sample (equivalent to applying an 8-length DFT-even (aka periodic) window) and performing an 8-length DFT. The imaginary parts of the estimates exactly match the zero-valued DTFT function, which Harris[1] attributes to "DFT-even symmetry". But the real parts have a negative bias. Increasing the gain of the window function (the sum of its coefficients) improves the one positive-value estimate, but degrades the four negative ones.

The exact way to compute the estimates with an 8-length DFT is to combine the first and last samples by addition (instead of truncation), called periodic summation, with period 8. Then the outcomes (blue circles) exactly match the DTFT graphs.
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Author Bob K
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Other versions Also see File:Comparison_of_symmetric_and_periodic_Gaussian_windows.svg.
Usage
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Additional information can be found at Window_function#DFT-symmetry. And Sampling the DTFT links to this image.
SVG development
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This W3C-invalid vector image was created with GNU Octave.
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This graphic was created by the following Octave script: 

pkg load signal graphics_toolkit gnuplot %======================================================= function out=gauss(M2,sigma)     % window function out = exp(-.5*(((0:M2)-M2/2)/(sigma*M2/2)).^2); endfunction %======================================================= % Dimensions of figure   x1 = .07;                     % left margin   x2 = .02;                     % right margin   y1 = .07;                     % bottom margin for annotation   y2 = .07;                     % top margin for title    width = 1-x1-x2;   height= 1-y1-y2;    x_origin = x1;   y_origin = 1;                 % start at top of graph area %======================================================= set(0, "DefaultAxesFontsize",12) set(0, "DefaultTextFontsize",14) figure("position",[50 100 800 600]);  y_origin = y_origin -y2 -height;        % position of top row subplot("position",[x_origin y_origin width height])  N   = 8*9*10; M   = 4;                                % finite Fourier transform domain is [-M,M] M2  = 2*M; M21 = M2+1;                             % sequence length symmetric = gauss(M2,1); symmetric = symmetric/sum(symmetric); periodic  = symmetric(1:M2); periodic  = periodic/sum(periodic);  % A similar window is: % window = kaiser(M21+2, pi*.75)'; % window = window(2:end-1);  % Remove zero-valued end points  x = 0:M2; y      = cos(2*pi*x/4); y_sym  = y.*symmetric; y_even = y(1:end-1).*periodic;  DTFT    = fft(y_sym,N); DFT     = fft([y_sym(1)+y_sym(end) y_sym(2:end-1)]); % periodic summation DFTeven = fft(y_even);                               % truncation (aka "DFT-even")  x = 0:N/2; plot(x, real(DTFT(1+x)), "color","blue") hold on plot(x, imag(DTFT(1+x)), "color","blue", "linestyle","--") set(gca, "xaxislocation","origin") xlim([0 N/2]) ylim([-0.4 0.6])  x = (0:M); DFT     = DFT(1+x); DFTeven = DFTeven(1+x); x = x*N/M2; plot(x, real(DFT),     "color","blue", "o", "markersize",8, "linewidth",2) plot(x, real(DFTeven), "color","red",  "*", "markersize",4, "linewidth",2) plot(x, imag(DFT),     "color","blue", "o", "markersize",8, "linewidth",2) plot(x, imag(DFTeven), "color","red",  "*", "markersize",4, "linewidth",2)  h = legend("DTFT real",...            "DTFT imaginary",...            "periodic summation",...            "DFT-even (truncation)", "location","northeast"); set(h, "fontsize",10) %legend boxoff  set(gca, "xtick", x, "xgrid","on", "xticklabel",[0 1 2 3 4]) xlabel("DFT bins", "fontsize",14) ylabel("amplitude") title("Sampling the Discrete-time Fourier transform", "fontsize",14);
References
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  1. ↑ Harris, Fredric J. (1978-01). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform". Proceedings of the IEEE 66 (1): 52. DOI:10.1109/PROC.1978.10837.
  2. Sampling the DTFT

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Date/TimeThumbnailDimensionsUserComment
current16:02, 19 August 2020Thumbnail for version as of 16:02, 19 August 2020754 × 562 (46 KB)Bob Kmake some objects darker
13:49, 19 August 2020Thumbnail for version as of 13:49, 19 August 2020754 × 562 (53 KB)Bob KNormalize DFT-even window coefficients. Change graph symbols.
13:32, 28 March 2020Thumbnail for version as of 13:32, 28 March 2020754 × 562 (47 KB)Bob Kreplace Hamming with a Gaussian to emphasize the difference between summation and truncation
16:06, 25 March 2020Thumbnail for version as of 16:06, 25 March 2020754 × 562 (41 KB)Bob KUploaded own work with UploadWizard
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