1. The basis of Fourier transform Recall DFT: Exercise#1: M=8, plot cos(2πu/M)x at each frequency u=0,…,7 x=0:7; u=0; plot(x, cos(2*pi*u/8*x));

2. Half of the DFT frequencies is redundant x=0,1, …,M-1, u=0,1, …,M-1 u<2/M u |F(u)| 01M2M2 … M-1 … M2M2 …

3. For real signal Forward DFT: That is,Magnitude: |F(M-u)| = |F(u)|, u<2/M Phase: F(M-u) = - F(u)

4. Test for 1-D DFT Use fast Fourier transform function a=[1 2 3 4 5 6]; fft(a ’ ) Exercise#2: plot the magnitude and phase of the FFT of the step signal x=zeros(1,50); x(25:40)=1; Get magnitude and angle of a complex variable: use abs and angle

5. DFT example for real signal f(x) |F(u)| angle(F(u))

6. Reconstruct signal using inverse FFT Exercise#3: apply ifft to your previous FFT results with increasing number of frequency component.

7. 2-D DFT Recall 2-D DFT: Exercise#4: Let M=N=8, draw the images of cos((2x+1)πu/2M)cos((2y+1)πv/2N) for each u=0…7, v=0,…7 64 images This is the 8x8 basis of discrete cosine transform (DCT)

8. 2-D Translation M N M/2 N/2 v u

9. M N M/2 N/2 0

10. DFT of images a=[zeros(256,128) ones(256, 128)]; af=fftshift(fft2(a)); imshow(log(1+abs(af)), []); % apply log transform to increase contrast Exercise#5: show the magnitude of DFT of the following images a=zeros(256, 256); a(78:178, 78:178)=1; [x,y]=meshgrid(-128:127, - 128:127); z=sqrt(x.^2+y.^2); c=(z<15); cameraman.tif