signals and systems discrete time fourier series
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Signals and Systems
Discrete Time Fourier Series
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Discrete-Time Fourier Series
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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
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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~
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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
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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
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Symmetry Properties
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Symmetry Properties Cont’d
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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~
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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
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Graphical Periodic Convolution
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DTFT to DFT
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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~
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DFT Analysis and Synthesis
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DFT
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DFT is Periodic with period N
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Example 1
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Example 1 (cont.) N=5
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Example 1 (cont.) N>M
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Example 1 (cont.) N=10
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DFT: Matrix Form
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DFT from DFS
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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
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Symmetry Properties
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DFT Properties
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Example: Circular Shift
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Example: Circular Shift
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Example: Circular Shift
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Duality
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Circular Flip
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Properties: Circular Convolution
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Example: Circular Convolution
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Example: Circular Convolution
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illustration of the circular convolution process
Example (continued)
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Illustration of circular convolution for N = 8:
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•Example:
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•Example (continued)
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•Proof of circular convolution property:
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•Multiplication: