1. Lecture 17: The Discrete Fourier Series Instructor: Dr. Ghazi Al Sukkar Dept. of Electrical Engineering The University of Jordan Email: ghazi.alsukkar@ju.edu.jo ghazi.alsukkar@ju.edu.jo Spring 20141

2. Outline  Discrete Fourier Series  Properties of DFS  Periodic Convolution  The Fourier Transform of Periodic Signals  Relation between Finite-length and Periodic Signals Spring 20142

3. 3 Discrete Fourier Series

4. Spring 2014 4 Discrete Fourier Series Pair

5. Cont.. Spring 2014 5

6. 6 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

7. Spring 2014 7 Example 2  DFS of an periodic rectangular pulse train  The DFS coefficients

8. Spring 2014 8 Properties of DFS  Linearity  Shift of a Sequence  Duality Proof Replace n by k

9. Spring 2014 9 Symmetry Properties

10. Spring 2014 10 Symmetry Properties Cont’d

11. Spring 2014 11 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

12. Spring 2014 12 Periodic Convolution Cont’d  Substitute periodic convolution into the DFS equation  Interchange summations  The inner sum is the DFS of shifted sequence  Substituting

13. Spring 2014 13 Graphical Periodic Convolution

14. Product of two sequences Spring 2014 14

15. Spring 2014 15 The Fourier Transform of Periodic Signals

16. Spring 2014 16 Example  Consider the periodic impulse train  The DFS was calculated previously to be  Therefore the Fourier transform is  Which is also a continuous impulse train.

17. Spring 2014 17 Relation between Finite-length and Periodic Signals

18. Cont.. Spring 2014 18

19. Spring 2014 19 Example  Consider the following sequence  The Fourier transform  The DFS coefficients  Which the same results of our previous example.