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EE 4780

EE 4780. 2D Fourier Transform. 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). Fourier Transform: Concept.

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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

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