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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: