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EE 4780

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EE 4780

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  1. EE 4780 2D Fourier Transform

  2. Fourier Transform • What is ahead? • 1D Fourier Transform of continuous signals • 2D Fourier Transform of continuous signals • 2D Fourier Transform of discrete signals • 2D Discrete Fourier Transform (DFT)

  3. Fourier Transform: Concept • A signal can be represented as a weighted sum of sinusoids. • Fourier Transform is a change of basis, where the basis functions consist of sines and cosines (complex exponentials).

  4. Fourier Transform • Cosine/sine signals are easy to define and interpret. • However, it turns out that the analysis and manipulation of sinusoidal signals is greatly simplified by dealing with related signals called complex exponential signals. • A complex number has real and imaginary parts: z = x + j*y • A complex exponential signal: r*exp(j*a) =r*cos(a) + j*r*sin(a)

  5. Fourier Transform: 1D Cont. Signals • Fourier Transform of a 1D continuous signal “Euler’s formula” • Inverse Fourier Transform

  6. Fourier Transform: 2D Cont. Signals • Fourier Transform of a 2D continuous signal • Inverse Fourier Transform • Fandfare two different representations of the same signal.

  7. Fourier Transform: Properties • Remember the impulse function (Dirac delta function) definition • Fourier Transform of the impulse function

  8. Fourier Transform: Properties • Fourier Transform of 1 Take the inverse Fourier Transform of the impulse function

  9. Fourier Transform: Properties • Fourier Transform of cosine

  10. Examples Magnitudes are shown

  11. Examples

  12. Fourier Transform: Properties • Linearity • Shifting • Modulation • Convolution • Multiplication • Separable functions

  13. Fourier Transform: Properties • Separability 2D Fourier Transform can be implemented as a sequence of 1D Fourier Transform operations.

  14. Fourier Transform: Properties • Energy conservation

  15. Fourier Transform: 2D Discrete Signals • Fourier Transform of a 2D discrete signal is defined as where • Inverse Fourier Transform

  16. Fourier Transform: Properties • Periodicity: Fourier Transform of a discrete signal is periodic with period 1. 1 1 Arbitrary integers

  17. Fourier Transform: Properties • Linearity, shifting, modulation, convolution, multiplication, separability, energy conservation properties also exist for the 2D Fourier Transform of discrete signals.

  18. Fourier Transform: Properties • Linearity • Shifting • Modulation • Convolution • Multiplication • Separable functions • Energy conservation

  19. Fourier Transform: Properties • Define Kronecker delta function • Fourier Transform of the Kronecker delta function

  20. Fourier Transform: Properties • Fourier Transform of 1 To prove: Take the inverse Fourier Transform of the Dirac delta function and use the fact that the Fourier Transform has to be periodic with period 1.

  21. Impulse Train • Define a comb function (impulse train) as follows where M and N are integers

  22. Impulse Train • Fourier Transform of an impulse train is also an impulse train:

  23. Impulse Train

  24. Impulse Train • In the case of continuous signals:

  25. Impulse Train

  26. Sampling

  27. Sampling No aliasing if

  28. Sampling If there is no aliasing, the original signal can be recovered from its samples by low-pass filtering.

  29. Sampling Aliased

  30. Sampling Anti-aliasing filter

  31. Sampling • Without anti-aliasing filter: • With anti-aliasing filter:

  32. Anti-Aliasing a=imread(‘barbara.tif’);

  33. Anti-Aliasing a=imread(‘barbara.tif’); b=imresize(a,0.25); c=imresize(b,4);

  34. Anti-Aliasing a=imread(‘barbara.tif’); b=imresize(a,0.25); c=imresize(b,4); H=zeros(512,512); H(256-64:256+64, 256-64:256+64)=1; Da=fft2(a); Da=fftshift(Da); Dd=Da.*H; Dd=fftshift(Dd); d=real(ifft2(Dd));

  35. Sampling

  36. Sampling No aliasing if and

  37. Interpolation Ideal reconstruction filter:

  38. Ideal Reconstruction Filter