Signals and Systems Discrete Time Fourier Series.
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Transcript of Signals and Systems Discrete Time Fourier Series.
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: