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.