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Advanced MATLAB Summer 2018 Discrete Fourier Transform

Advanced MATLAB Summer 2018 Discrete Fourier Transform. Behrad TaghiBeyglou Department of Electrical Engineering Sharif University of Technology. Motivation. The DTFT is defined using an infinite sum over a discrete time signal and yields a continuous function X(ω)

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Advanced MATLAB Summer 2018 Discrete Fourier Transform

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  1. Advanced MATLABSummer 2018Discrete Fourier Transform BehradTaghiBeyglou Department of Electrical 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. BehradTaghiBeyglou, EE, SUT, 2018

  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 BehradTaghiBeyglou, EE, SUT, 2018

  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 BehradTaghiBeyglou, EE, SUT, 2018

  8. DFS Pairs Analysis: Synthesis: Periodic N pt. seq. in time domain Periodic N pt. seq. in freq. domain BehradTaghiBeyglou, EE, SUT, 2018

  9. Example … … BehradTaghiBeyglou, EE, SUT, 2018

  10. Example (cont.) (eq.1) (eq.2) (eq.1) & (eq.2) BehradTaghiBeyglou, EE, SUT, 2018

  11. Properties of DFS • Shift property: • Periodic convolution: Period N BehradTaghiBeyglou, EE, SUT, 2018

  12. Periodic convolution - Example BehradTaghiBeyglou, EE, SUT, 2018

  13. Properties of DFS • In the list of properties: = = Where: BehradTaghiBeyglou, EE, SUT, 2018

  14. Properties of DFS BehradTaghiBeyglou, EE, SUT, 2018

  15. Properties of DFS BehradTaghiBeyglou, EE, SUT, 2018

  16. Discrete Fourier Transform N pt. N pt. DFT N pt. N pt. DTFT DFT DFS BehradTaghiBeyglou, EE, SUT, 2018

  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 BehradTaghiBeyglou, EE, SUT, 2018

  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. BehradTaghiBeyglou, EE, SUT, 2018

  19. Example

  20. Discrete Fourier Transform • Definition of DFT: N pt. DFT of x(n) BehradTaghiBeyglou, EE, SUT, 2018

  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. BehradTaghiBeyglou, EE, SUT, 2018

  24. Relationship between DFT and DTFT 8 pt. sequence 8 pt. DFT BehradTaghiBeyglou, EE, SUT, 2018

  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) BehradTaghiBeyglou, EE, SUT, 2018

  26. M pt. DFT of N pt. Signal DFT N pt. M pt. BehradTaghiBeyglou, EE, SUT, 2018

  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 BehradTaghiBeyglou, EE, SUT, 2018

  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)? BehradTaghiBeyglou, EE, SUT, 2018

  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)? BehradTaghiBeyglou, EE, SUT, 2018

  31. Solution to the Problem Statement • We start with x(n) and find its relationship with y(n): Change the order of summation: BehradTaghiBeyglou, EE, SUT, 2018

  32. Solution to the Problem Statement However, we have shown that: Convolution with train of delta functions BehradTaghiBeyglou, EE, SUT, 2018

  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). BehradTaghiBeyglou, EE, SUT, 2018

  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. BehradTaghiBeyglou, EE, SUT, 2018

  35. Properties of DFT • In the list of properties: = , = where: and where: BehradTaghiBeyglou, EE, SUT, 2018

  36. Summery of Properties of DFT BehradTaghiBeyglou, EE, SUT, 2018

  37. Summery of Properties of DFT BehradTaghiBeyglou, EE, SUT, 2018

  38. Using DFT to calculate linear convolution BehradTaghiBeyglou, EE, SUT, 2018

  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. BehradTaghiBeyglou, EE, SUT, 2018

  40. Definitions of convolution • Linear convolution: Application in the analysis of LTI systems • Periodic convolution: A seq. with period N BehradTaghiBeyglou, EE, SUT, 2018

  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 BehradTaghiBeyglou, EE, SUT, 2018

  42. Example: Circular Convolution Circular convolution? Make an Npt. seq. BehradTaghiBeyglou, EE, SUT, 2018

  43. Example: Circular Convolution

  44. Circular Convolution & DFT We know from DFS properties: BehradTaghiBeyglou, EE, SUT, 2018

  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. BehradTaghiBeyglou, EE, SUT, 2018

  46. Example : Circular Convolution • Calculate N pt. circular convolution of x1 and x2for the following two cases of N: • N=L • N=2L BehradTaghiBeyglou, EE, SUT, 2018

  47. Case 1: N=L N pt. DFT of x1 IDFT BehradTaghiBeyglou, EE, SUT, 2018

  48. Case 1: N=L BehradTaghiBeyglou, EE, SUT, 2018

  49. Case 2: N=2L • Pad each signal with L extra zeros to get an 2L pt. seq.: N=2L pt. DFT of x1 BehradTaghiBeyglou, EE, SUT, 2018

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

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