Discrete Fourier Transform

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.

Example of 1-D Fourier Transform

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')

2-D Discrete Fourier Transform

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,[ ])

2-D FFT with CenterShift

2-D FFT on Texture Images