IUB Dr. Abdur Razzak 1 Digital Signal Processing Objectives To learn and understand DFT - the frequency domain sampling & reconstruction of discrete-time signal. FFT – an efficient computation technique of DFT. Lecture – 8 Discrete Fourier Transform & Fast Fourier Transform ECR 305_L8 nk N N n W n x n x k X ~ ~ DSF ˆ ~ 1 0 nk N N k W k X N k X n x ~ 1 ~ IDFS ˆ ~ 1 0
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IUB Dr. Abdur Razzak 1
Digital Signal Processing
Objectives
To learn and understand DFT - the frequency domain
sampling & reconstruction of
discrete-time signal.
FFT – an efficient computation
technique of DFT.
Lecture – 8
Discrete Fourier Transform
& Fast Fourier Transform
ECR 305_L8
nk
N
N
n
WnxnxkX ~~DSFˆ~ 1
0
nk
N
N
k
WkXN
kXnx
~1~
IDFSˆ~1
0
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Introduction
In lecture-4 & 5, we have studied transform-domain
representations of discrete signals. The discrete-time Fourier transform (DTFT) provides the frequency-
domain () representation for absolutely summable sequences (see
lecture-5).
The z-transform provides a generalized frequency-domain (z)
representation for arbitrary sequences (see lecture-4).
These transforms have two features in common: First, the transforms are defined for infinite-length sequences.
Second, and the most important, they are functions of continuous
variables ( or z).
From the numerical computation viewpoint (or from MATLAB's
viewpoint), these two features are troublesome because one has
to evaluate infinite sums at uncountably infinite frequencies
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Introduction (contd..)
To use MATLAB, we have to truncate sequences and then
evaluate the expressions at finitely many points, which are the
approximations to the exact calculations. In other words, the
discrete-time Fourier transform and the z-transform are not
numerically computable transforms.
Therefore we need a numerically computable transform, which is
obtained by sampling the discrete-time Fourier transform in the
frequency domain. This is known Discrete Fourier Transform.
From Fourier analysis we know that a periodic sequence can
always be represented by a linear combination of harmonically
related complex exponentials (which is a form of sampling). This
gives us the Discrete Fourier Series (DFS) to finite-duration
sequence which leads to a new transform, called DFT.
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Discrete Fourier Series
Let us consider a periodic sequence
where N is the fundamental period of the sequence.
From Fourier analysis, we know that the periodic function can be
synthesized as a linear combinations of complex exponentials
whose frequencies are multiples (or harmonics) of the
fundamental frequency (which in our case is 2/N).
From the frequency-domain periodicity of the discrete-time
Fourier transform, we conclude that there are a finite number of
harmonics; the frequencies are { , k = 0, 1, 2,……, N–1}.
kNnxnx ~~
kN
2
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Discrete Fourier Series (contd..)
Therefore, the discrete Fourier series representation of a periodic
sequence can be expressed as
where are called the discrete
Fourier series coefficients, which are given by
1,...,2,1,0,~~21
0
NkenxkXnk
Nj
N
n
1,...,2,1,0,~1~
21
0
NnekXN
nxkn
Nj
N
k
nx~
1,...,2,1,0,~
NkkX
ECR 305_L8
IDFS
DFS
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Discrete Fourier Series (contd..)
Using , previous 2 equations can be expressed as
Analysis or a DFS equation:
Synthesis or an inverse DFS equation:
Nj
N eW
2
nk
N
N
n
WnxnxkX ~~DFSˆ~ 1
0
nk
N
N
k
WkXN
kXnx
~1~
IDFSˆ~1
0
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Discrete Fourier Series (contd..)
In matrix form, the DFS and IDFS equations can be rewritten as
where the matrix called a DFS matrix given by
NNN xWX NNNNN
NXWXWx
*1- 1
211
11
10
1
1
111
N
N
N
N
N
NN
kn
NnkNN
WW
WWk
n
W
...
...::
...
...
ˆW ,
NW
ECR 305_L8
1-
1
0
Nx
x
x
N:
x
1
1
0
N
X
X
N
X
:X
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Example-1
Find DFS representation of .
Solution: Given N = 4. So
Now
Hence
Similarly,
,.....3,2,1,0,3,2,1,0,3,2,1,0....,~ nx
32104
3
0
,,,,~~
kWnxkX nk
n
jjeeW jj 2/sin2/cos2/4/2
4
632103~2~1~0~~0~ 0
3
0
xxxxjnxXn
n
jjjjjnxXn
n
223210~1~ 32
3
0
23210~2~ 6422
3
0
jjjjnxXn
n
jjjjjnxXn
n
223210~3~ 9633
3
0
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Relation to z-transform
Construct a periodic sequence by periodically repeating x(n)
with a period N,
The DFS of is given by,
Comparing with z-transform, ,we have
which means that DFS represents N evenly spaced samples
of the z-transform X(z) around the unit circle.
nx~
nx~
elsewhere
10
,0
,~
Nnnx
nx
n
kN
jN
n
nkN
jN
n
enxenxkX
21
0
21
0
~~
nN
nznxzX
~1
0
kN
j
ezzXkX 2
~
kX~
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Relation to DTFT
The DTFT of x(n) is given by
Comparing DFS & IDFS equations, we have
Let and , then the DFS
which means that the DFS is obtained by evenly sampling the
DTFT with the sampling interval .
njN
n
enxX
1
0
k
Nk
N
njN
n
nkN
jN
n
XenxenxkX
2
2
1
0
21
0
~~~
N
2ˆ1 1
2ˆ
kk
Nk
1 jkjeXeXkX k
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Discrete Fourier transform
The discrete Fourier series provided us a mechanism for numerically
computing the discrete-time Fourier transform. It also alerted us to a potential
problem of aliasing in the time domain.
Mathematics dictates that the sampling of the discrete-time Fourier transform
result in a periodic sequence x(n). But most of the signals in practice are not
periodic. They are likely to be of finite duration. How can we develop a
numerically computable Fourier representation for such signals?
Theoretically, we can take care of this problem by defining a periodic signal
whose primary shape is that of the finite-duration signal and then using the
DFS on this periodic signal.
Practically, we define a new transform called the Discrete Fourier Transform
(DFT), which is the primary period of the DFS. This DFT is the ultimate
numerically computable Fourier transform for arbitrary finite-duration
sequences.
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Discrete Fourier transform (contd..)
Discrete Fourier transform (DFT) of an N-point sequence:
or
Inverse discrete Fourier transform (IDFT) of an N-point
sequence:
elsewhere
10
,0
,~
DFTˆ
NkkX
nxkX
10,1
0
NkWnxkX nk
N
N
n
10,1~
1
0
NnWkXN
nx nk
N
N
k
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Properties of DFT
Linearity:
Circular folding: If an N-point sequence is folded, then the
result would not be an N-point sequence, and it would not be
possible to compute DFT. Therefore, we use the modulo-N
operation on the argument (-n) and define folding by
The DFT of a circular folding is given by
nxbnxanbxnax 2121 DFTDFTDFT
11
0
,
,0
Nn
n
nNx
xnx N
11
0
,
,0DFT
Nk
k
kNX
XkXnx NN
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Properties of DFT (contd..)
Circular shifting: An N-point DFT of a finite duration sequence
x(n) of length N is equivalent to the N-point DFT of a periodic
sequence xp(n), which is obtained by periodically extending x(n),
Now suppose that we shift xp(n) by k units to the right,