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Understanding Discrete Fourier Transform (DFT) & Applications with FFTSHIFT

Learn about 1D and 2D FFT in MATLAB, shifting zero-frequency, Fourier spectrum, and FFT on images. Practical examples included.

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Understanding Discrete Fourier Transform (DFT) & Applications with FFTSHIFT

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  1. Discrete Fourier Transform

  2. FFT and Its Applications FFTSHIFT Shift zero-frequency component to the center of spectrum. For vectors, FFTSHIFT(X) swaps the left and right halves of X. For matrices, FFTSHIFT(X) swaps the first and third quadrants and the second and fourth quadrants. For N-D arrays, FFTSHIFT(X) swaps "half-spaces" of X along each dimension.

  3. Example of 1-D Fourier Transform

  4. fftBox.m – Plot Fourier Spectrum • % • % Script file: fftBox.m • % Fourier Spectrum Plot of Box function • % • X1=linspace(0,1,17); • Y1=ones(1,length(X1)); • X2=linspace(1,16,241); • Y2=zeros(1,length(X2)); • X=[X1 X2]; Y=[Y1 Y2]; • W=abs(fftshift(fft(Y))); • subplot(2,1,1) • plot(X,Y,'r'); axis([0 16, 0,1.2]); title('Box function') • subplot(2,1,2) • plot(W,'b-'); • title('Fourier Spectrum of Box function')

  5. 2-D Discrete Fourier Transform

  6. Example of 2-D FFT Matlab Code • % Script file: fourier.m - 2D Fourier Transform • % Pictures on P.113 of Gonzalez, Woods, Eddins • m=128; n=128; • f=zeros(m,n); • f(56:71,48:79)=255; • F0=fft2(f); S0=abs(F0); • Fc=fftshift(fft2(f)); Sc=abs(Fc); • Fd=fft2(fftshift(f)); Sd=log(1+abs(Fc)); • subplot(2,2,1) • imshow(f,[]) • subplot(2,2,2) • imshow(S0,[]) • subplot(2,2,3) • imshow(Sc,[ ]) • subplot(2,2,4) • imshow(Sd,[ ])

  7. 2-D FFT with CenterShift

  8. 2-D FFT on Texture Images

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