CE 40763 Digital Signal Processing Fall 1992 Discrete Fourier Transform (DFT)

1. CE 40763Digital Signal ProcessingFall 1992Discrete Fourier Transform (DFT) HosseinSameti Department of Computer Engineering Sharif University of Technology

2. Motivation • The DTFT is defined using an infinite sum over a discrete time signal and yields a continuous function X(ω) • not very useful because the outcome cannot be stored on a PC. • Now introduce the Discrete Fourier Transform (DFT), which is discrete and can be stored on a PC. • We will show that the DFT yields a sampled version of the DTFT. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

3. Review of Transforms C. C. D. C. D. Complex Inf. or Finite Int. D. periodic periodic D. finite finite Int.

4. Review of Transforms

5. Discrete Fourier Series (DFS) • Decompose in terms of complex exponentials that are periodic with period N. • How many exponentials? N Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

6. Discrete Fourier Series (DFS) Exponentials that are periodic with period N. arbitrary integer 1 * Proof:

7. Discrete Fourier Series (DFS) How to find X(k)? Answer: Proof : substitute X(k) in the first equation. • It can also easily be shown that X(k) is periodic with period N: arbitrary integer Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

8. DFS Pairs Analysis: Synthesis: Periodic N pt. seq. in time domain Periodic N pt. seq. in freq. domain Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

9. Example … … Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

10. Example (cont.) (eq.1) (eq.2) (eq.1) & (eq.2) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

11. Properties of DFS • Shift property: • Periodic convolution: Period N Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

12. Periodic convolution - Example Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

13. Properties of DFS • In the list of properties: = = Where: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

14. Properties of DFS Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

15. Properties of DFS Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

16. Discrete Fourier Transform N pt. N pt. DFT N pt. N pt. DTFT DFT DFS Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

17. Deriving DFT from DFS • 1) Start with a finite-length seq. x(n) with N points (n=0,1,…, N-1). • 2) Make x(n) periodic with period N to get Extracts one period of Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

18. Deriving DFT from DFS (cont.) • 3) Take DFS of • 4) Take one period of to get DFT of x(n): N pt. N pt. periodic N pt. periodic N pt. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

19. Example

20. Discrete Fourier Transform • Definition of DFT: N pt. DFT of x(n) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

21. Example

22. Example, cont’d MehrdadFatourechi, Electrical and Computer Engineering, University of British Columbia, Summer 2011

23. Relationship between DFT and DTFT DFT thus consists of equally-spaced samples of DTFT. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

24. Relationship between DFT and DTFT 8 pt. sequence 8 pt. DFT Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

25. M pt. DFT of N pt. Signal So far we calculated the N pt. DFT of a seq. x(n) with N non-zero values: Suppose we pad this N pt. seq. with (M-N) zeros to get a sequence with length M. We can now take an M-pt. DFT of the signal x(n) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

26. M pt. DFT of N pt. Signal DFT N pt. M pt. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

27. Example 4 pt. DFT: 6 pt. DFT: How are these related to each other? 8 pt. DFT: 100 pt. DFT:

28. M pt. DFT of N pt. Signal Going from N pt. to 2N pt. DFT Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

29. M pt. DFT of N pt. Signal N pt. seq. N pt. N pt. DFT 2N pt. 2N pt. DFT N pt. seq. padded with N zeros What is the minimum number of N needed to recover x(n)? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

30. Problem Statement • Assume y(n) is a signal of finite or infinite extent. • Sample at N equally-spaced points. N pt. sequence. N pt. sequence. What is the relationship between x(n) and y(n)? What happens if N is larger , equal or less than the length of y(n)? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

31. Solution to the Problem Statement • We start with x(n) and find its relationship with y(n): Change the order of summation: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

32. Solution to the Problem Statement However, we have shown that: Convolution with train of delta functions Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

33. Solution to the Problem Statement One period of the replicated version of y(n) • Examples If we sample at a sampling rate that is higher than the number of points in y(n), we should be able to recover y(n). Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

34. Properties of DFT • Shift property: N pt. seq. The above relationship is not correct, because of the definition of DFT. The signal should only be non-zero for the first N points. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

35. Properties of DFT • In the list of properties: = , = where: and where: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

36. Summery of Properties of DFT Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

37. Summery of Properties of DFT Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

38. Using DFT to calculate linear convolution Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

39. Convolution • We are familiar with, “linear convolution”. • Question: Can DFT be used for calculating the linear convolution? • The answer is: NO! (at least not in its current format) • We now examine how DFT can be applied in order to calculate linear convolution. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

40. Definitions of convolution • Linear convolution: Application in the analysis of LTI systems • Periodic convolution: A seq. with period N Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

41. Definitions of convolution(cont.) • Circular convolution: N pt. seq. • Circular convolution is closely related to periodic convolution. N pt. DFT of x3 N pt. DFT of x2 N pt. DFT of x1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

42. Example: Circular Convolution Circular convolution? Make an Npt. seq. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

43. Example: Circular Convolution

44. Circular Convolution & DFT We know from DFS properties: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

45. Circular Convolution & DFT If we multiply the DFTs of two N pt. sequences, we get the DFT of their circular convolution and not the DFT of their linear convolution. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

46. Example : Circular Convolution • Calculate N pt. circular convolution of x1 and x2for the following two cases of N: • N=L • N=2L Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

47. Case 1: N=L N pt. DFT of x1 IDFT Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

48. Case 1: N=L Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

49. Case 2: N=2L • Pad each signal with L extra zeros to get an 2L pt. seq.: N=2L pt. DFT of x1 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

50. Case 2: N=2L Same as linear convolution!!