Lecture 17: The Discrete Fourier Series Instructor: Dr. Ghazi Al Sukkar Dept. of Electrical Engineering The University of Jordan Spring 20141

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 Discrete Fourier Series

Spring Discrete Fourier Series Pair

Cont.. Spring

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

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

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

Spring Symmetry Properties

Spring Symmetry Properties Cont’d

Spring 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

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

Spring Graphical Periodic Convolution

Product of two sequences Spring

Spring The Fourier Transform of Periodic Signals

Spring 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.

Spring Relation between Finite-length and Periodic Signals

Cont.. Spring

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