1 7.1 Discrete Fourier Transform (DFT) 7.2 DFT Properties 7.3 Cyclic Convolution 7.4 Linear...
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Transcript of 1 7.1 Discrete Fourier Transform (DFT) 7.2 DFT Properties 7.3 Cyclic Convolution 7.4 Linear...
1
• 7.1 Discrete Fourier Transform (DFT)
• 7.2 DFT Properties
• 7.3 Cyclic Convolution
• 7.4 Linear Convolution via DFT
Chapter 7 Discrete Fourier Transform
Section 8.1-8.7
2
• Definition: The Discrete Fourier Transform (DFT) of the finite
length sequence is
• Definition: The Inverse Discrete Fourier Transform (IDFT) of
is given by
• The following notation will be used:
7.1.1 Discrete Fourier Transform (DFT)
1 ,..., 1, 0 ]; [ N n n x
1 ,..., 0 for ] [ ] [ DFT ] [1
0
2
N k e n x n x k X
N
n
N
nk j
N
1 ,..., 1, 0 ]; [ N k k X
1 ,..., 0 for ] [1
] [ IDFT ] [1
0
2
N n e k XN
k X n xN
k
N
nk j
N
] [ ] [DFT
k X n x
3
7.1.2 Discrete Fourier Series (DFS)
• Periodic Extension: Given a finite-length sequence
define the periodic sequence by
• The sequence with period N is called the periodic extension
of x[n]. It has a fundamental frequency .
• does not have a Z-transform or a convergent Fourier sum
(why?). But it does have a DFS representation.
• It is actually the DFS that is the true frequency representation of
discrete periodic signals. The DFT is just one period of the DFS.
1 ,..., 0 ]; [ N n n x
n n x]; [ ~
] mod [ ] [ ] [ ~N n x n x n xN
][~ nx
N
2
][~ nx
4
7.1.3 DFT and DFS
• DFS analysis and synthesis pair is expressed as follows:
• Practical significance:
– The length-N DFT of the length-N signal contains all the information about . It is convenient to work with.
– Whenever the DFT is used, actually the DFS is being used – computations involving are affected by the true periodicity of the coefficients.
1 ,..., 0 for ] [ ~ ] [~
1
0
2
N k e n x k X
N
n
N
nk j
1 ,..., 0 for ] [~ 1
] [ ~1
0
2
N n e k XN
n xN
k
N
nk j
][kX ][nx][nx
][kX ][~
kX
][kX
][~
kX
5
7.1.4 Relation with Other Transforms
• The DFT samples the Z-transform at evenly spaced samples of
the unit circle over one revolution:
• In other words, the DFT samples on period of the Fourier
Transform at N evenly spaced frequencies
1 ,..., 0 for | ) ( ] [/ 2 N k z X k XN k je z
N
kk
2
1 ,..., 0 for | ) ( ] [/ 2 N k e X k XN kj
6
7.1.5 DFT Transformation Matrix
• The DFT can be represented in this way
• This introduces the widely-used and convenient notations:
whence
2) 1 ( ) 1 ( 3 ) 1 ( 2 1
1 3 2 1
... 1
: : : : : :
... 1
1 1 1 1 1 1
NN
NN
NN
NN
NN N N N
N
W W W W
W W W WW T
N jNe W
/ 2
) (symmetric matrix ation transform DFT
Tx X
nkN
N
n
Wn x k X1
0
] [ ] [
X T x1
1
0
] [1
] [
nkN
N
n
Wk XN
n x
]1 [
:
]1[
] 0[
N x
x
x
x
]1 [
:
]1[
] 0[
N X
X
X
X
7
7.1.6 DFT Transformation Matrix: Example
The DFT matrices of dimension 2, 3, 4 are as follows:
If we compute X by as follows:
Where we observe that the real part of X[k] is even-symmetric, and the imaginary part is odd-symmetric – the DFT of the real signal.
1 1
1 12 T
, ] 2 0 3 1[T
x
2
3 1
2
3 11
2
3 1
2
3 11
1 1 1
3
j j
j jT
j j
j j
1 1
1 1 1 1
1 1
1 1 1 1
4 T
Tx X
] 3[
] 2[
]1[
] 0[
5 1
0
5 1
2
2
0
3
1
1 1
1 1 1 1
1 1
1 1 1 1
4
X
X
X
X
j
j
j j
j jx T X
8
• Let
be length-N sequences indexed n=0,…,N-1.
• DFT Properties:
– Linearity: For constant a, b:
– Even Sequences: If x[n] is even:
– Odd Sequences: If x[n] is odd:
– Real Sequences: If x[n] is real:
7.2.1 DFT Properties - I
] [ ] [ ], [ ] [ ], [ ] [k H n h k Y n y k X n xDFT DFT DFT
] [ ] [ ] [ ] [k Y k aX n by n axDFT
. 1 ,..., 0 , ; ] [ ] [ ] [ ] [ N k n k N X k X n N x n x
. 1 ,..., 0 , ; ] [ ] [ ] [ ] [ N k n k N X k X n N x n x
] [ ] [ and ] [ ] [k N X k X k N X k X
9
7.2.2 DFT Properties - II
– Circular Shift:
– Duality:
– Parseval’s Theorem
(DFT conserves energy)
– Cyclic (circular convolution): If
] [ ] )) [((/ 2
k X e m n xN m j DFT
N
] [ ] [ DFTn N Nx k X N
1
0
2 21
0
2] [ ] [
N
n
N
k
n x N k X
. 1 ,..., 1, 0 ]; [ ] [ ] [ N k k H k X k Y
1
0
] [ ] [ ] [N
mN m n h mx n ythen
1
0
] [ ] [N
mNmh m n x
] [ ] [n h n xN
10
7.2.3 DFT Properties – Circular Shift Example
). 2 ( ] [ ] )) [((/ 2
m k X e m n xN m j DFT
N
] [ ] [k X n xDFT ] [ ] [1 1k X n x
DFT
] [ ] [) / 2(
1k X e k Xm N k j
11
• The cyclic convolution is not the same as the
linear convolution of linear system theory. It
is a by-product of the periodicity of DFS/DFT.
• When the DFT X[k] is used, the periodic interpretation of the
signal x[n] is implicit: if
then for any integer m:
• Thus, just as the DFT X[k] is implicitly period-N (i.e., is the
DFS), the inverse DFT is also implicitly period-N — the periodic
extension of x[n].
7.3.1 Cyclic Convolution – What?
] [ ] [ ] [n h n x n yN
] [ *] [ ] [n h n x n y
1 ,..., 0 for ] [1
] [ IDFT ] [1
0
2
N n e k XN
k X n xN
k
N
nk j
N
1
0
22 1
0
) ( 2
] [ ] [1
] [1
N
k
km j N
kn j N
k
N
mN n k j
n x e e k XN
e k XN
12
7.3.2 Cyclic Convolution – Why?
• Why is cyclic convolution not true linear convolution?
• Because a wraparound effect occurs at the “ends”:
• The procedure of each pair are summed around the circle.
• In a while, it will be seen that can be computed
using
] [ *] [ ] [n h n x n y]. [ ] [ ] [n h n x n y
N
13
7.3.3 Cyclic Convolution – Example 1
0] [ ] [ ] [1 0 1
knN
DFTW n X n n n x
] [ ] [ ] [ ] [2 2 1 30
k X W k X k X k Xkn
N
] )) [(( ] [ ] [ ] [0 2 2 1 3N n n x n x n x n x
14
7.3.4 Cyclic Convolution – Example 2
15
7.4.1 Linear Convolution by DFT
• Of course, linear convolution is desired. Fortunately, the linear convolution can be computed via the DFT, with a minor modification.
• Method: To compute the linear convolution
of a sequence x[n] of length- N1 and a sequence h[n] of length- N2.
via the DFT, form the length N1 + N2 -1 zero-padded sequences
and then] [ *] [ ] [n h n x n y
2 0
1 0 ]; [] [ˆ
2 1 1
1
N N n N
N n n xn x
2 0
1 0 ]; [] [ˆ
2 1 2
2
N N n N
N n n hn h
] [ˆ ] [ˆ ] [ˆ1 2 1
n h n x n yN N
1
0
] [ˆ ] [ˆN
mN m n h mx
2 0 ]; [ *] [2 1 N N n n h n x
16
7.4.2 Cyclic Convolution Example -1
The linear convolution is computed as the time instants (in this example) 0 n 4. This can be regarded as a form of time-aliasing – resulting from the sampling of the Fourier Transform.
17
7.4.2 Cyclic Convolution Example -1
The linear convolution is computed as the time instants (in this example) 0 n 9. Aliasing is eliminated, so the result is the same as the linear convolution of the non-extended sequence.
18
7.4.3 Cyclic Convolution Example -2k
N
LkN
W
Wk X k X
1
1] [ ] [2 1
2
31
1] [
kN
LkN
W
Wk X
). 2 (L N