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7/28/2019 Discret FT
1/15
The Discrete Fourier Transform
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice HallInc.
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Spring 2008 2
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/2jkN/2
j eXeXkX~
jDTFT eX]n[x
kN/2jez eXzXkX~
kN/2
kX~
1N
0k
knN/2jekX~
N
1]n[x
~
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Spring 2008 3
Sampling the Fourier Transform Contd
The only assumption made on the sequence is that DTFT exist
Combine equation to get
Term in the parenthesis is
So we get
m
mjj emxeX
1N
0k
knN/2j
ekX
~
N
1
]n[x
~
kN/2jeXkX
~
mm
1N
0k
mnkN/2j
1N
0k
knN/2j
m
kmN/2j
mnp
~
mxeN
1
mx
eemxN
1]n[x
~
r
1N
0k
mnkN/2j rNmneN
1mnp
~
rr
rNnxrNnnx]n[x~
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Spring 2008 4
Sampling the Fourier Transform Contd
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Sampling the Fourier Transform Contd
Samples of the DTFT of an aperiodic sequence
can be thought of as DFS coefficients
of a periodic sequence
obtained through summing periodic replicas of original sequence
If the original sequence
is of finite length
and we take sufficient number of samples of its DTFT
the original sequence can be recovered by
It is not necessary to know the DTFT at all frequencies
To recover the discrete-time sequence in time domain Discrete Fourier Transform
Representing a finite length sequence by samples of DTFT
else0
1Nn0nx~nx
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Spring 2008 6
The Discrete Fourier Transform
Consider a finite length sequence x[n] of length N
For given length-N sequence associate a periodic sequence
The DFS coefficients of the periodic sequence are samples ofthe DTFT of x[n]
Since x[n] is of length N there is no overlap between terms ofx[n-rN] and we can write the periodic sequence as
To maintain duality between time and frequency
We choose one period of as the Fourier transform of x[n]
1Nn0ofoutside0nx
r
rNnxnx~
NkXNmodkXkX~
kX~
else0
1Nk0kX~
kX
NnxNmodnxnx~
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Spring 2008 7
The Discrete Fourier Transform Contd
The DFS pair
The equations involve only on period so we can write
The Discrete Fourier Transform
The DFT pair can also be written as
1N
0k
knN/2jekX~
N
1]n[x
~
1N
0n
knN/2je]n[x~kX~
else0
1Nk0e]n[x~
kX
1N
0n
knN/2j
else0
1Nk0ekX~N1
]n[x
1N
0k
knN/2j
1N
0n
knN/2je]n[xkX
1N
0k
knN/2jekX
N
1]n[x
]n[xkX DFT
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Spring 2008 8
Example
The DFT of a rectangular pulse
x[n] is of length 5
We can consider x[n] of anylength greater than 5
Lets pick N=5
Calculate the DFS of theperiodic form of x[n]
else0
,...10,5,0k5
e1
e1
ekX~
5/k2j
k2j
4
0n
n5/k2j
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Spring 2008 9
Example Contd
If we consider x[n] of length 10
We get a different set of DFTcoefficients
Still samples of the DTFT but indifferent places
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Spring 2008 10
Properties of DFT
Linearity
Duality
Circular Shift of a Sequence
kbXkaXnbxnax
kXnx
kXnx
21DFT
21
2DFT
2
1DFT
1
mN/k2jDFTN
DFT
ekX1-Nn0mnx
kXnx
N
DFT
DFT
kNxnX
kXnx
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Example: Duality
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Symmetry Properties
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Circular Convolution
Circular convolution of of two finitelength sequences
1N
0mN213 mnxmxnx
1N
0m
N123 mnxmxnx
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Spring 2008 14
Example
Circular convolution of two rectangular pulses L=N=6
DFT of each sequence
Multiplication of DFTs
And the inverse DFT
else0
1Ln01nxnx 21
else0
0kNekXkX
1N
0n
knN
2j
21
else0
0kNkXkXkX
2
213
else0
1Nn0Nnx3
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Spring 2008 15
Example
We can augment zeros toeach sequence L=2N=12
The DFT of each sequence
Multiplication of DFTs
N
k2j
N
Lk2j
21
e1
e1kXkX
2
N
k2j
N
Lk2j
3
e1
e1kX