1. Signals and Systems Discrete Time Fourier Series

2. Discrete-Time Fourier Series

4. The conventional (continuous-time) FS represent a periodic signal using an infinite number of complex exponentials, whereas the DFS represent such a signal using a finite number of complex exponentials

5. Example 1  DFS of a periodic impulse train  Since the period of the signal is N  We can represent the signal with the DFS coefficients as

6. Example 2  DFS of an periodic rectangular pulse train  The DFS coefficients

7. Properties of DFS  Linearity  Shift of a Sequence  Duality

8. Symmetry Properties

9. Symmetry Properties Cont’d

10. Periodic Convolution  Take two periodic sequences  Let’s form the product  The periodic sequence with given DFS can be written as  Periodic convolution is commutative

11. Periodic Convolution Cont’d  Substitute periodic convolution into the DFS equation  Interchange summations  The inner sum is the DFS of shifted sequence  Substituting

12. Graphical Periodic Convolution

13. DTFT to DFT

14. Sampling the Fourier Transform  Consider an aperiodic sequence with a Fourier transform  Assume that a sequence is obtained by sampling the DTFT  Since the DTFT is periodic resulting sequence is also periodic  We can also write it in terms of the z-transform  The sampling points are shown in figure  could be the DFS of a sequence  Write the corresponding sequence

15. DFT Analysis and Synthesis

16. DFT

17. DFT is Periodic with period N

18. Example 1

19. Example 1 (cont.) N=5

20. Example 1 (cont.) N>M

21. Example 1 (cont.) N=10

22. DFT: Matrix Form

23. DFT from DFS

24. Properties of DFT  Linearity  Duality  Circular Shift of a Sequence

25. Symmetry Properties

26. DFT Properties

27. Example: Circular Shift

30. Duality

31. Circular Flip

32. Properties: Circular Convolution

33. Example: Circular Convolution

35. illustration of the circular convolution process Example (continued)

36. Illustration of circular convolution for N = 8:

37. Example:

38. Example (continued)

39. Proof of circular convolution property:

40. Multiplication: