chapter 8. the discrete fourier transform
DESCRIPTION
Chapter 8. The Discrete Fourier Transform. 8.1 Laplace, z-, and Fourier Transforms 8.2 Fourier Transform 8.3 Fourier Series 8.4 Discrete Fourier Transform (DFT) 8.5 Properties of DFS/DFT 8.6 DFT and z-Transform 8.7 Linear Convolution vs. Circular Convolution - PowerPoint PPT PresentationTRANSCRIPT
Chapter 8. The Discrete Fourier Transform
8.1 Laplace, z-, and Fourier Transforms
8.2 Fourier Transform
8.3 Fourier Series
8.4 Discrete Fourier Transform (DFT)
8.5 Properties of DFS/DFT
8.6 DFT and z-Transform
8.7 Linear Convolution vs. Circular Convolution
8.8 Discrete Cosine Transform(DCT)
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-Analog systems (continuous time)
H(s)
stesH )( ste
-Digital Systems (discrete time)
H(z)
nzzH )( nz
dehsH
esHdeeh
dtxhx(t)h(t) y(t)
sr
ststs
)()(
)( )(
)()(
k
k
nnk
k
k
zkhzH
zzH zzkh
knxkhx(n)h(n) y(n)
)()(
)( )(
)()(
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1. Laplace, z-, Fourier Transforms
planes planez j
je
1
1
1
12
z
z
Ts
d
-Laplace transform -z-transform
),( ),(
LHP inside u.c
jssHjH )()(
jez
j zHeH
)()(
Fouier transformsBGL/SNU
dejXtx
dtetxjX
tj
tj
)(2
1 )(
)()(
(1) continuous aperiodic signals
conti aper
aper conti
elsewherettxeq
,0 2/|| ,1 )( )(
)2/sin(2
)(1
)(
2/2/
2/
2/
jj
tj
eej
dtejX
2/
x(t)
2/
1
t
)( jX
2
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2. Fourier Transform
deeXnx
enxeX
njj
n
njj
)(2
1 ][
][)(
(2) Discrete aperiodic signals
conti per
aper discr
elsewhere
ntxeq
,0
|| ,1 )( )(
2sin
)21
(sin
1
1)(
22
)2
1()
2
1(
)12(
jj
jj
j
jj
n
njj
ee
ee
e
eeeeX
x(n)
1
t
12
2
)( jeX12
2ω
k
tjkk
T
tjkk
Teatx
dtttxT
a
2 , )(
)(1
00
0
(1) continuous periodic signals
discrete aper
per conti
T)x(tx(t)Tt
ttxeq
,2/|| 2/ ,0
2/|| ,1 )( )(
0
000
1(note)
)2/sin(
)(111
0
2/2/
0
2/
2/
kΩΩk
jkjktjkk
)X(jT
ak
k
eejkT
dteT
a
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3. Fourier Series
2/
X(t)
2/
1
T
t
T
T
2
2
k
(2) discrete periodic signals (*Discrete Fourier Series)
N
2 , ][
1 ][
][][
00
0
Nk
njk
Nn
njk
ekXN
nx
enxkX discrete per
per discre
ka
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0
0
0
00
)X(eX[k] note)
2sin
)21
(sin
1
1][
j
0
0)12(
k
jk
jkjk
n
njk
k
k
e
eeekX
x[n]
1
n
N
][kX
12
kN
][][ || ,0
|| ,1 ][ )(
NnxnxNn
nnxeq
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-For a numerical evaluation of Fourier transform and its inversion, (i.e,computer-aided computation), we need discrete expression of of both the time and the transform domain data.
-For this,take the advantage of discrete Fourier series(DFS, on page 4), in which the data for both domain are discrete and periodic.
periodic) : (~ ][~
][~ kXnx
discrete periodic periodic discrete
-Therefore, given a time sequence x[n], which is aperiodic and discrete, take the following approach.
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4. Discrete Fourier Transform (DFT)
X[k] ][~
[n]x~ ][ kXnx
Mip
MipTop
Top][][
~nRkX N
][][~ nRnx N DFS
DFT
][ , :
]))[(( , :
nRperiodonetakeTop
nxperiodicitmakeMip
N
N
Reminding that, in DFS
][~1
][~
][~][~
1
0
2
1
0
2
N
k
nN
jk
N
n
nN
jk
ekXN
nx
enxkX
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Define DFT as
otherwise 0,
1-N 0,1,...,n , ][1
][
otherwise ,0
1N0,1,...,k ,][ ][
1
0
2
1
0
2
N
k
nN
jk
N
n
nN
jk
ekXN
nx
enxkX
][][~][
][][~
][ (note)
n Rnxn x
kRkXkX
N
N
(eq)X[k] x[n]
N Nnk
1
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Graphical Development of DFT
)(th
t0T
)( jH
)(1 ts )(1 jS
t
t0T
T T
1
T
1
T
1
)()()( 1 jSjHjH S)()()( 1 tsthths
T
1
T
1
][~
nh
t0T
][~
kH
)(2 ts )(2 jS
t
n
0T
N
T
1T
1
)()()( 1 jSjHjH S)()()( 2 tsthth sd
0T
0
1
T
0k
N0
DFSDFS
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][nh ][kH
nN0
kN0
DFTDFT
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5. Property of DFS/DFT (8.2 , 8.6)(1) Linearity
)seq. (finite][][)seq. periodic(][
~][
~
][][:
][~][~:
kbYkaXkYbkXa
nbynaxDFT
nybnxaDFS
(2) Time shift
][]))[((:
][~
][~:
kXWmnxDFT
kXWmnxDFSkm
NN
kmN
]))[((~
][:
][~
][~:
Nnl
N
nlN
lkXnxWDFT
lkXnxWDFS
(3) Frequency shift
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(4) Periodic/circular convolution in time
(5) Periodic/circular convolution in frequency
][][]))[((][:
][~
][~
][~][~:
1
0
1
0
kYkXmnymxDFT
kYkXmnymxDFS
N
mN
N
m
1
0
1
0
]))[((][1
][][:
][~
][~1
][~][~:
N
lN
N
l
lkYlXN
nynxDFT
lkYlXN
nynxDFS
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]}[Im{][]}[~
Im{][~
]}[Re{][]}[~
Re{][~
][~
]}[Im{][~
]}[~Im{
][~
]}[Re{][~
]}[~Re{
][~
]))[((~][~
][~]))[((
~][][
~][~
****
****
kXjnxkXjnx
kXnxkXnx
kXnxjkXnxj
kXnxkXnx
kXnxkXnx
kXnxkXnx
opo
epe
opo
epe
N
N
(6) Symmetry
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DFS DFT
6. DFT and Z-Transform(1) Evaluation of from][
~kX )(zX
1
0
2
][~][~
][)(
N
n
nN
jk
n
n
enxkX
znxzX
①If length limited in time, (I.e., x[n]=0, n<0, n>=N)
then Njk
ezzXkX 2)(][
~
)2
at samplingby )(
from obtained is ][~
ly,(equivaant
)1,,2,1,2
,at
samplingby X(z) from obtained is ][~
(
NkeX
kX
NkN
kez
kX
kjω
kj k
BGL/SNU
② What if x[n] is not length-limited?
then aliasing unavoidable.
r
N
k
mnN
jk
m
N
k
nN
jk
m
mN
jk
N
k
nN
jk
m
mN
jk
ez
rNnx
eN
mx
eemxN
ekXN
nx
emxzXkX Njk
][
1][
][1
][~1
][~then
][)(][ take weIf
1
0
)(2
1
0
22
1
0
2
22
][nx ][~ nx
n0 1N 12 N 13 N
… … …
n0 1N 12 N 13 N
… … …
(2) Recovery of [or ] from
(in the length-limited case)
)( jeX ][~
kX
1
0 12
1
0
1
0
12
1
0
1
0
2
1
0
1
1][
~1
][~1
][~1
][)(X
N
k Njk
N
N
k
N
n
n
Njk
N
n
nN
k
nN
jk
N
n
n
ze
zkX
N
zekXN
zekXN
znxz
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)(zX
2sin
2sin
1
11)(
)2
(][~
1
1][
~1
1
)(1][
~1)(X
2
1
1
0
1
0 )2
(
)2
(
1
0 12
12
N
N
ee
e
Nwhere
NkkX
e
ekX
N
ze
zekX
Ne
Nj
j
Nj
N
k
N
k Nkj
NN
kj
N
k
ezN
jk
NNjk
j
j
2
)0(~X
)(
)2
(N
)1(
~X
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7. Linear Convolution vs. Circular Convolution
(1) Definition
NlengthNn
knhkxnhnx
NnNlengthnh
NnNlengthnx
N
k
2,12,,1,0
,][][][][
1,,1,0,],[
1,,1,0,],[
1
0
① Linear convolution
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② Circular convolution
NlengthNn
nRknhkxnhnx N
N
kNN
,1,,1,0
][]))[((]))[((][][1
0
N
Rectangular window of length
N
]))[((][
]))[((][
NMip
NMip
nhnh
nxnx
Periodic
convolution
1
0
]))[((]))[((N
kNN knhkx
][][ nhnx N
TopNperiodWith
nhnh N ][~
]))[((
BGL/SNU
(2) Comparison
][][ 21 nxnx
][][ 21 nxnx
0
1
N
0
N
N2
][][ nhnx N
N
0
N
N2][][ nhnx 2N
H[n]][nx ][][][ nhnxny
)2(
][
N
kX
)2(
][
N
kH][][][ kHkXkY 2N
)2( N
) period zero N (
2Nlength a of ],[ ,][for 21 nxnx
Omit chap. 8.7
- Effects of Energy compaction
8. Discrete cosine transform (DCT)Definition
otherwise
kke
NnN
knkXkenx
NkN
knnxke
NkX
N
k
N
n
,1
0,2
1
][
1,,1,0,2
)12(cos][][][
1,,1,0,2
)12(cos][][
2][
1
0
1
0
BGL/SNUTest signal for computing DFT and DCT
(a) Real part of N-point DFT; (b) Imaginary part of N-point DFT; (c) N-point DCT-2 of the test signal
BGL/SNU
Comparison of truncation errors for DFT and DCT-2
BGL/SNU
Appendix: Illustration of DFTs for Derived Signals
]1[][1 nNxng
][)1(][2 nxng n
otherwise
NnNNnx
Nnnx
ng
,0
12],[
10],[
][3
n1N
n
n1N
1N 12 N
A
A
A
A
][1 ng
][2 ng
][3 ng
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otherwise
NnNnxnxng
,0
12/0],2/[][][4
otherwise
NnN
Nnnx
ng
,0
12,0
10],[
][5
odd ,0
even ],2
[][6
n
nn
xng
n1
2
N
A2
][4 ng
][5 ng
A
1Nn
n
][6 ng
A
12 N
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]2[][7 nxng
][7 ng
A
n1
2
N
)(][
)(][
)(][
)(][
)]()([5.0][
)(][
oddk ,0
evenk ),(][
)(][
)(][
/29
)2/)(/2(8
/2/27
/46
/)2/(2/25
)12/(24
/2
3
2/22
/21
Nkj
NkNj
NkjNkj
Nkj
NNkjNkj
Nkj
Nkj
Nkj
Nkj
eXkH
eXkH
eXekH
eXkH
eXeXkH
eXkH
eXkH
eXkH
eXkH
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][)( ][
][ ]1[][
7
2221
0
2
1
0
)1(1
01
kHeXeeemx
WmxWnNxkG
kN
jN
kjmk
NjN
m
N
kj
N
m
mNkN
N
n
knN
][)( ][
][ ][)1(][
8
)2
(21
0
)2
(2
1
0
2
1
02
kHeXenx
WWnxWnxkG
Nk
NjN
n
nN
kN
j
N
n
knN
Nn
N
N
n
knN
n
][)(])1(1[ ]1[][
]][[ ][ ][][
32
21
022
)(2
1
02
12
2
1
023
kHeXWWnx
WWnxWNnxWnxkG
N
kj
kN
n
NkN
nkN
knNN
N
n
nkN
N
Nn
knN
N
n
knN
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][)( ][
][][])2
[][(][
6
41
0 2
1
2
)2
(
2
12
0 2
12
0 2
4
kHeXWnx
WnxWnxWN
nxnxkG
NjN
n
nkN
N
Nn
kN
nN
N
n
knN
N
n
knN
][)(X ][][ 2
12
025 kHeWnxkG N
kjN
n
knN
][)(X ][][ 1
21
0
226 kHeWnxkG N
kjN
n
knN
][))()([2
1 ]][[
2
1
]2
)1(1][[ ]2[][
5
)2
(221
0
)2
(
1
0
2
2
12
0 2
7
kHeXeXWWnx
WnxWnxkG
Nk
N
kj
N
kjN
n
Nkn
Nnk
N
N
n
nk
N
nN
n
knN
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H.W. of Chapter 8
Text : [1] 8.10 [2] 8.29 [3] 8.32 [4] 8.37
Ref : [5] Project 3.4 DFT Properties
Q3.31, Q3.33, Q3.36, Q3.38, Q3.40, Q3.44
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