2.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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
3.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
3
Sampling the Fourier Transform Cont’d
The only assumption made on the sequence is that DTFT exist
Combine equation to get
Term in the parenthesis is
So we get
4.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
4
Sampling the Fourier Transform Cont’d
5.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
5
Sampling the Fourier Transform Cont’d
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
6.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
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 of the DTFT of x[n]
Since x[n] is of length N there is no overlap between terms of x[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]
7.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
7
The Discrete Fourier Transform Cont’d
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
8.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
8
Example
The DFT of a rectangular pulse
x[n] is of length 5
We can consider x[n] of any length greater than 5
Let’s pick N=5
Calculate the DFS of the periodic form of x[n]
9.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
9
Example Cont’d
If we consider x[n] of length 10
We get a different set of DFT coefficients
Still samples of the DTFT but in different places
10.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
10
Properties of DFT
Linearity
Duality
Circular Shift of a Sequence
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351M Digital Signal Processing
11
Example: Duality
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351M Digital Signal Processing
12
Symmetry Properties
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351M Digital Signal Processing
13
Circular Convolution
Circular convolution of of two finite length sequences
14.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
14
Example
Circular convolution of two rectangular pulses L=N=6
DFT of each sequence
Multiplication of DFTs
And the inverse DFT
15.Copyright (C) 2005 Güner Arslan
351M Digital Signal Processing
15
Example
We can augment zeros to each sequence L=2N=12
The DFT of each sequence
Multiplication of DFTs