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| Keywords: From Continuous To Discrete Fourier Transform.gif Fourier transform and the discrete Fourier transform <u>Left column </u> A continuous function top and its Fourier transform bottom <u>Center-left column </u> Periodic summation of the original function top Fourier transform bottom is zero except at discrete points The inverse transform is a sum of sinusoids called Fourier series <u>Center-right column </u> Original function is discretized multiplied by a Dirac comb top Its Fourier transform bottom is a periodic summation DTFT of the original transform <u>Right column </u> The DFT bottom computes discrete samples of the continuous DTFT The inverse DFT top is a periodic summation of the original samples The FFT algorithm computes one cycle of the DFT and its inverse is one cycle of the inverse DFT 2011-12-05 own Sbyrnes321 <pre> Source code written in Mathematica 6 0 by Steve Byrnes 2011 I release this code into the public domain ClearAll Global` SetOptionsPlot Frame -> True FrameTicks -> None Axes -> False PlotRange -> -8 8 All ; SetOptionsListPlot Frame -> True FrameTicks -> None Axes -> False Filling -> Axis PlotStyle -> None PlotRange -> -8 8 All ; fx_ Exp- 4/3 \Pi x 2; gx_ Exp- 3/4 \Pi x 2; repeatedfx_ p_ Sumfx + n p n -10 10 ; repeatedgx_ p_ Sumgx + n p n -10 10 ; plotf Plotfx x -10 10 PlotStyle -> DarkerBlue; plotg Plotgx x -10 10 PlotStyle -> DarkerRed; plotrepeatedf Plotrepeatedfx 5 x -10 10 PlotStyle -> DarkerBlue; discreteg Table x gx x -10 10 1/5 ; plotdiscreteg ListPlotdiscreteg FillingStyle -> DarkerRed; discretef Table x fx x -10 10 1/3 ; plotdiscretef ListPlotdiscretef FillingStyle -> DarkerBlue; plotrepeatedg Plotrepeatedgx 3 x -10 10 PlotStyle -> DarkerRed; discreterepeatedf Table x repeatedfx 11/4 x -12 12 1/4 ; plotdiscreterepeatedf ListPlotdiscreterepeatedf FillingStyle -> DarkerBlue; discreterepeatedg Table x repeatedgx 4 x -12 12 4/11 ; plotdiscreterepeatedg ListPlotdiscreterepeatedg FillingStyle -> DarkerRed; finalimg ShowGraphicsGrid plotf plotrepeatedf plotdiscretef plotdiscreterepeatedf plotg plotdiscreteg plotrepeatedg plotdiscreterepeatedg ImageSize -> 800 SetDirectory C \\Users\\Steve\\Desktop ; Export test gif finalimg</pre> cc-zero math Uploaded with UploadWizard Fourier analysis Images with Mathematica source code | ||||
