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

else0

rNn1rNn]n[x~

r

1ee]n[e]n[x~kX~ 0kN/2j

1N

0n

knN/2j1N

0n

knN/2j

1N

0k

knN/2j

r

eN1

rNn]n[x~

Example 2 DFS of an periodic rectangular pulse train

The DFS coefficients

10/ksin2/ksin

ee1e1

ekX~ 10/k4j

k10/2j

5k10/2j4

0n

kn10/2j

Properties of DFS Linearity

Shift of a Sequence

Duality

kX~

bkX~

anx~bnx~a

kX~

nx~kX

~nx~

21DFS

21

2DFS

2

1DFS

1

mkX~

nx~e

kX~

emnx~kX

~nx~

DFSN/nm2j

N/km2jDFS

DFS

kx~NnX

~kX

~nx~

DFS

DFS

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

kX

~nx~

kX~

nx~

2DFS

2

1DFS

1

kX~

kX~

kX~

213

1N

0m213 mnx~mx~nx~

1N

0m123 mnx~mx~nx~

Periodic Convolution Cont’d

Substitute periodic convolution into the DFS equation

Interchange summations

The inner sum is the DFS of shifted sequence

Substituting

1N

0m213 mnx~mx~nx~

1N

0n

knN2

1N

0m13 W]mn[x~]m[x~kX

~

1N

0m

knN

1N

0n213 W]mn[x~]m[x~kX

~

kX~

WW]mn[x~ 2kmN

knN

1N

0n2

kX~

kX~

kX~

W]m[x~W]mn[x~]m[x~kX~

21

1N

0m2

kmN1

1N

0m

knN

1N

0n213

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

kN/2j

kN/2

j eXeXkX~

jDTFT eX]n[x

kN/2j

ezeXzXkX

~kN/2

kX~

1N

0k

knN/2jekX~

N1

]n[x~

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

kbXkaXnbxnax

kXnx

kXnx

21DFT

21

2DFT

2

1DFT

1

mN/k2jDFT

N

DFT

ekX1-Nn0 mnx

kXnx

N

DFT

DFT

kNxnX

kXnx

Symmetry Properties

DFT Properties

Example: Circular Shift

Example: Circular Shift

Example: Circular Shift

Duality

Circular Flip

Properties: Circular Convolution

Example: 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: