Signals and Systems Discrete Time Fourier Series

Discrete-Time Fourier Series

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

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

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

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

Symmetry Properties

Symmetry Properties Cont’d

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

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

Graphical Periodic Convolution

DTFT to DFT

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

DFT Analysis and Synthesis

DFT

DFT is Periodic with period N

Example 1

Example 1 (cont.) N=5

Example 1 (cont.) N>M

Example 1 (cont.) N=10

DFT: Matrix Form

DFT from DFS

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

Symmetry Properties

DFT Properties

Example: Circular Shift

Duality

Circular Flip

Properties: Circular Convolution

Example: Circular Convolution

illustration of the circular convolution process Example (continued)

Illustration of circular convolution for N = 8:

Example:

Example (continued)

Proof of circular convolution property:

Multiplication: