copyright © 2005. shi ping cuc chapter 3 discrete fourier transform review features in common we...
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![Page 1: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/1.jpg)
Copyright © 2005. Shi Ping CUC
Chapter 3Discrete Fourier Transform
Review
Features in common
We need a numerically computable transform, that is
Discrete Fourier Transform (DFT)
The DTFT provides the frequency-domain ( ) representation for absolutely summable sequences.
The z-transform provides a generalized frequency-domain ( ) representation for arbitrary sequences.
z
Defined for infinite-length sequences. Functions of continuous variable ( or ). They are not numerically computable transform.
z
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Copyright © 2005. Shi Ping CUC
Chapter 3Discrete Fourier Transform
Content
The Family of Fourier Transform
The Discrete Fourier Series (DFS)
The Discrete Fourier Transform (DFT)
The Properties of DFT
The Sampling Theorem in Frequency Domain
Approximating to FT (FS) with DFT (DFS)
Summary
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Copyright © 2005. Shi Ping CUC
The Family of Fourier Transform
Introduction
Fourier analysis is named after Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist.
A signal can be either continuous or discrete, and it can be either periodic or aperiodic. The combination of these two features generates the four categories of Fourier Transform.
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Copyright © 2005. Shi Ping CUC
The Family of Fourier Transform
Aperiodic-Continuous - Fourier Transform
dejXtx
dtetxjX
tj
tj
)(2
1)(
)()(
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Copyright © 2005. Shi Ping CUC
The Family of Fourier Transform
Periodic-Continuous - Fourier Series
k
tjk
T
T
tjk
ejkXtx
dtetxT
jkX
0
0
0
0
)()(
)(1
)(
0
2
20
0
00
22
TF
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Copyright © 2005. Shi Ping CUC
The Family of Fourier Transform
Aperiodic-Discrete - DTFT
deeXnx
enxeX
njj
n
njj
)(2
1)(
)()(
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Copyright © 2005. Shi Ping CUC
The Family of Fourier Transform
Periodic-Discrete - DFS (DFT)
1
0
2
1
0
2
)(1
)(
)()(
N
k
nkNj
N
n
nkNj
ekXN
nx
enxkX
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Copyright © 2005. Shi Ping CUC
The Family of Fourier Transform
Summary
Time function Frequency function
Continuous and Aperiodic Aperiodic and Continuous
Continuous and Periodic( ) Aperiodic and Discrete( )
Discrete ( ) and Aperiodic Periodic( ) and Continuous
Discrete ( ) and Periodic ( )
Periodic( )
and Discrete( )
0T
T
T 0T
00
2
T
Ts
2
Ts
2
00
2
T
return
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Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
DefinitionPeriodic time functions can be synthesized as a linear combination of complex exponentials whose frequencies are multiples (or harmonics) of the fundamental frequency
Periodic continuous-time function )()( rTtxtx
Periodic discrete-time function )()( rNnxnx
k
ktTjekXtx
2
)()(
1
0
2
)(1
)(N
k
knNjekX
Nnx
fundamental frequency
tTje
2
nNje
2fundamental frequency
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Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
elsewhere ,0
,1
1
1112
21
0
2 mNr
e
e
Ne
N rNj
rNNj
N
n
rnNj
)(1
)(
)(1
)(
1
0
)(21
0
1
0
21
0
21
0
2
rXeN
kX
eekXN
enx
N
n
nrkNjN
k
N
n
rnNjN
k
knNjN
n
rnNj
![Page 11: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/11.jpg)
Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
1
0
2
)()(N
n
knNj
enxkX
)()(
)()(
1
0
2
1
0
)(2
kXenx
enxmNkX
N
n
knNj
N
n
nmNkNj
Because:
The is a periodic sequence with fundamental period equal to N
)(kX
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Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
1
0
1
0
)(~1
)](~
[IDFS)(~
)(~)](~[DFS)(~
Let2
N
k
nkN
N
n
nkN
jN
WkXN
kXnx
WnxnxkX
eW N
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Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
Relation to the z-transform
elsewhere,0
10),(~)(
Nnnxnx
kNjez
zXkX 2|)()(~
The DFS represents N evenly spaced samples of the z-transform around the unit circle.
)(~kX
)(zX
1
0
1
0
))(()(~
, )()(2
N
n
nkjN
n
n NenxkXznxzX
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Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
Relation to the DTFT
kj
N
n
nkNjN
n
njj
NeXkX
enxkXenxeX
2|)()(~
)()(~
)()(1
0
21
0
,
elsewhere,0
10),(~)(
Nnnxnx
The DFS is obtained by evenly sampling the DTFT at
intervals. It is called frequency resolution and represents the
sampling interval in the frequency domain.
N2
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Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
jIm[z]
Re[z]
0k
kj
NeXkX
2|)()(~
N
2
N=8 frequency resolution
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Copyright © 2005. Shi Ping CUC
The properties of DFS
The Discrete Fourier Series (DFS)
Linearity
)(~
)(~
)](~)(~[DFS 2121 kXbkXanxbnxa
Shift of a sequence
)(~
)(~
)](~[DFS2
kXekXWmnxmk
Nj
mkN
+
Modulation
)(~
)](~[DFS lkXnxW lnN
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Copyright © 2005. Shi Ping CUC
The Discrete Fourier Series (DFS)
Periodic convolution
1
012
1
021
21
)(~)(~
)(~)(~)](~
[IDFS)(~ then
)(~
)(~
)(~
if
N
m
N
m
mnxmx
mnxmxkYny
kXkXkY
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Copyright © 2005. Shi Ping CUC
1
012
1
021
1
0
1
0
)(21
1
02
1
01
1
02121
)(~)(~)(~)(~
)(~1
)(~
)(~
)(~1
)(~
)(~1
)](~
)(~
[IDFS)(~
N
m
N
m
N
m
N
k
kmnN
N
k
nkN
N
m
mkN
N
k
nkN
mnxmxmnxmx
WkXN
mx
WkXWmxN
WkXkXN
kXkXny
return
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Copyright © 2005. Shi Ping CUC
Introduction
The Discrete Fourier Transform (DFT)
The DFS provided us a mechanism for numerically computing the discrete-time Fourier transform. But most of the signals in practice are not periodic. They are likely to be of finite length.
Theoretically, we can take care of this problem by defining a periodic signal whose primary shape is that of the finite length signal and then using the DFS on this periodic signal.
Practically, we define a new transform called the Discrete Fourier Transform, which is the primary period of the DFS.
This DFT is the ultimate numerically computable Fourier transform for arbitrary finite length sequences.
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Copyright © 2005. Shi Ping CUC
Finite-length sequence & periodic sequence
The Discrete Fourier Transform (DFT)
)(nx Finite-length sequence that has N samples
)(~ nx periodic sequence with the period of N
)()(~)(
,0
10 ),(~)(
nRnxnx
elsewhere
Nnnxnx
N
))(()(~
)()(~
N
r
nxnx
rNnxnx
Window operation
Periodic extension
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Copyright © 2005. Shi Ping CUC
The definition of DFT
The Discrete Fourier Transform (DFT)
10 ,)(1
)]([IDFT)(
10 ,)()]([DFT)(
1
0
1
0
NnWkXN
kXnx
NkWnxnxkX
N
n
nkN
N
n
nkN
)()(~)()(1
)(
)()(~
)()()(
1
0
1
0
nRnxnRWkXN
nx
kRkXkRWnxkX
N
N
nN
nkN
N
N
nN
nkN
return
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
Linearity
)()()]()([DFT 2121 kbXkaXnbxnax N3-point DFT, N3=max(N1,N2)
Circular shift of a sequence
)()]())(([DFT kXWnRmnx kmNNN
)())(()]([DFT kRlkXnxW NNnl
N
Circular shift in the frequency domain
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
The sum of a sequence
1
00
1
00
)()()(N
nk
N
n
nkNk
nxWnxkX
The first sample of sequence
1
0
)(1
)0(N
k
kXN
x
)())(()]([
)()]([
kRkNNxnXDFT
kXnxDFT
NN
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
Circular convolution
)( )()())(()(
)())(()()( )(
12
1
012
1
02121
nxnxnRmnxmx
nRmnxmxnxnx
N
N
mN
N
N
mN
N
N
)()()]( )([DFT 2121 kXkXnxnx N
)( )(1
)]()([DFT 2121 kXkXN
nxnx N
Multiplication
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
Circular correlation
nn
xy nymnxmnynxmr )(*)()(*)()(
Linear correlation
Circular correlation
)()(*))((
)())((*)()(
1
0
1
0
mRnymnx
mRmnynxmr
N
N
nN
N
N
nNxy
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
)()(*))((
)())((*)(
)]([IDFT)(
)()()(
1
0
1
0
*
mRnymnx
mRmnynx
kRmrthen
kYkXkRif
N
N
nN
N
N
nN
xyxy
xy
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
Parseval’s theorem
1
0
*1
0
* )()(1
)()(N
k
N
n
kYkXN
nynx
1
0
21
0
2
1
0
*1
0
*
)(1
)(
)()(1
)()( then
)()( let
N
k
N
n
N
k
N
n
kXN
nx
kXkXN
nxnx
nynx
![Page 28: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/28.jpg)
Copyright © 2005. Shi Ping CUC
The Properties of DFT
Conjugate symmetry properties of DFT
and)(nxep )(nxop
Let be a N-point sequence)(nx Nnxnx ))(()(~
]))(())(([2
1)](~)(~[
2
1)(~
]))(())(([2
1)](~)(~[
2
1)(~
NNo
NNe
nNxnxnxnxnx
nNxnxnxnxnx
It can be proved that
)(~)(~)(~)(~
*
*
nxnx
nxnx
oo
ee
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
)())(())((2
1
)()(~)(
)())(())((2
1
)()(~)(
nRnNxnx
nRnxnx
nRnNxnx
nRnxnx
NNN
Noop
NNN
Neep
Circular conjugate symmetric
component
Circular conjugate
antisymmetriccomponent
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
)()()( nxnxnx opep
)())(()(
)())(()(*
*
nRnNxnx
nRnNxnx
NNopop
NNepep
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
)()()( kXkXkX opep
)())(()(
)())(()(*
*
kRkNXkX
kRkNXkX
NNopop
NNepep
and)(kX ep )(kX op
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
)]())((Im[)](Im[
)]())((Re[)](Re[
kRkNXkX
kRkNXkX
NNepep
NNepep
)]())((Im[)](Im[
)]())((Re[)](Re[
kRkNXkX
kRkNXkX
NNopop
NNopop
![Page 33: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/33.jpg)
Copyright © 2005. Shi Ping CUC
The Properties of DFT
)())(()( then
)())(()( if
kRkNXkX
nRnNxnx
NN
NN
Circular even sequences
Circular odd sequences
)())(()( then
)())(()( if
kRkNXkX
nRnNxnx
NN
NN
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
)()())((
)())(()]([DFT**
**
kNXkRkNX
kRkXnx
NN
NN
Conjugate sequences
)()]())(([DFT
)]())(([DFT**
*
kXnRnNx
nRnx
NN
NN
![Page 35: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/35.jpg)
Copyright © 2005. Shi Ping CUC
The Properties of DFT
)())(())((2
1
)()](Re[DFT
* kRkNXkX
kXnx
NNN
ep
Complex-value sequences
)())(())((2
1
)()](Im[DFT
* kRkNXkX
kXnxj
NNN
op
![Page 36: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/36.jpg)
Copyright © 2005. Shi Ping CUC
The Properties of DFT
)(]))(())(([2
1DFT
)](Re[)]([DFT
* nRnNxnx
kXnx
NNN
ep
)(]))(())(([2
1DFT
)](Im[)]([DFT
* nRnNxnx
kXjnx
NNN
op
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
)())(()( then
sequence value-real is )( if* kRkNXkX
nx
NN
Real-value sequences
Imaginary-value sequences
)())(()( then
part imaginary has only )( if* kRkNXkX
nx
NN
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
Summary
)( )( )(
)](Im[ )](Re[)(
kXkXkX
nxjnxnx
opep
)](Im[ )](Re[)(
)( )( )(
kXjkXkX
nxnxnx opep
example
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
Linear convolution & circular convolution
1
02121
21
1
)()()()(
)()()(N
mm
l
mnxmxmnxmx
nxnxny
Linear convolution
)(1 nx
)(2 nx
N1 point sequence, 0≤n≤ N1-1
N2 point sequence, 0≤n≤ N2-1
)(nyl L point sequence, L= N1+N2-1
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Copyright © 2005. Shi Ping CUC
The Properties of DFT
Circular convolution
1 ,0
10 ),()(
1
111 LnN
Nnnxnx
We have to make both and L-point
sequences by padding an appropriate number of zeros
in order to make L point circular convolution.
)(1 nx )(2 nx
1 ,0
10 ),()(
2
222 LnN
Nnnxnx
![Page 41: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/41.jpg)
Copyright © 2005. Shi Ping CUC
The Properties of DFT
)()(
)()()(
)()()(
)())(()()( )()(
2
1
01
1
021
1
02121
nRrLny
nRmrLnxmx
nRmrLnxmx
nRmnxmxnxnxny
Lr
l
Lr
L
m
L
L
m r
L
L
mLc
L
![Page 42: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/42.jpg)
Copyright © 2005. Shi Ping CUC
The Properties of DFT
)()()( nRrLnyny Lr
lc
)()()( )( isthat
)()( then
1 if
2121
21
nxnxnxnx
nyny
NNL
lc
L
return
![Page 43: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/43.jpg)
Copyright © 2005. Shi Ping CUC
The Sampling Theorem in Frequency Domain
Sampling in frequency domain
m
kmNWz
WmxzXkX kN
)(|)()(~
rm
N
k
nmkN
N
k
knN
m
kmN
N
k
knNN
rNnxWN
mx
WWmxN
WkXN
kXnx
)(1
)(
)(1
)(~1
)](~
[IDFS)(~
1
0
)(
1
0
1
0
![Page 44: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/44.jpg)
Copyright © 2005. Shi Ping CUC
The Sampling Theorem in Frequency Domain
r
N rNnxnx )()(~
Frequency Sampling TheoremFor M point finite duration sequence, if the frequency sampling number N satisfy:
MN then
)()()(~)( nxnRnxnx NNN
![Page 45: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/45.jpg)
Copyright © 2005. Shi Ping CUC
The Sampling Theorem in Frequency Domain
Interpolation formula of )(zX
1
01
1
01
1
0
1
0
11
0
1
0
1
0
1
0
1
0
1
)(1
1
1)(
1
)(1
)(1
)(1
)()(
N
kk
N
NN
kk
N
NNkN
N
k
N
n
nkN
N
k
N
n
nnkN
N
n
nN
k
nkN
N
n
n
zW
kX
N
z
zW
zWkX
N
zWkXN
zWkXN
zWkXN
znxzX
![Page 46: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/46.jpg)
Copyright © 2005. Shi Ping CUC
The Sampling Theorem in Frequency Domain
1
1
0
1
01
1
11)(
)()(1
)(1)(
zW
z
Nz
zkXzW
kX
N
zzX
kN
N
k
N
kk
N
kk
N
N
Interpolation function
![Page 47: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/47.jpg)
Copyright © 2005. Shi Ping CUC
The Sampling Theorem in Frequency Domain
1
0
1
0
)2
()()()()(N
k
N
k
jK
jw kN
kXekXeX
2
1
2
2
sin
sin)(
Nj
N
eN
Interpolation function
Interpolation formula of )( jeX
return
![Page 48: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/48.jpg)
Copyright © 2005. Shi Ping CUC
Approximating to FT (FS) with DFT (DFS)
Approximating to FT of continuous-time aperiodic signal with DFT
dejXtx
dtetxjX
tj
tj
)(2
1)(
)()(
CTFT
![Page 49: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/49.jpg)
Copyright © 2005. Shi Ping CUC
n
TdtTdtnTt ,,
Sampling in time domain
n
nTjtj enTxTdtetxjX )()()(
S
dejXntx
dejXtx
nTj
tj
0)(
2
1)(
)(2
1)(
Tf ss
22
Approximating to FT (FS) with DFT (DFS)
![Page 50: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/50.jpg)
Copyright © 2005. Shi Ping CUC
Truncation in time domain
)1~0(:,,)~0(: 00 NnNTTTt
1
0
)()(N
n
nTjenTxTjX
S
dejXnTx nTj
0)(
2
1)(
Approximating to FT (FS) with DFT (DFS)
![Page 51: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/51.jpg)
Copyright © 2005. Shi Ping CUC
Sampling in frequency domain
NT
TT
TFNT
f
N
FT
ddk
s
N
n
s
22
22,
1
,,
00
000
00
1
00000
)]([DFT
)()()(1
0
21
00
0
nxT
enxTenTxTjkXN
n
nkNjN
n
nTjk
Approximating to FT (FS) with DFT (DFS)
![Page 52: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/52.jpg)
Copyright © 2005. Shi Ping CUC
)]([IDFT1
)]([IDFT
)(1
)(1
)(2
)(
00
1
0
2
0
1
0
2
00
1
00
0 0
jkXT
jkXf
ejkXN
f
ejkXN
NF
ejkXnTx
s
N
k
nkNj
s
N
k
nkNj
N
k
nTjk
demo
Approximating to FT (FS) with DFT (DFS)
![Page 53: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/53.jpg)
Copyright © 2005. Shi Ping CUC
Approximating to FS of continuous-time periodic signal with DFS
k
tjk
T tjk
ejkXtx
dtetxT
jkX
0
00
)()(
)(1
)(
0
00
0
000
22
TF
Approximating to FT (FS) with DFT (DFS)
![Page 54: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/54.jpg)
Copyright © 2005. Shi Ping CUC
1
000
0
,,N
n
TTdtNTTTdtnTt
Sampling in time domain
)]([DFS1
)(1
)()(1
0
21
000
0
nxN
enxN
enTxT
TjkX
N
n
nkNjN
n
nTjk
Approximating to FT (FS) with DFT (DFS)
![Page 55: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/55.jpg)
Copyright © 2005. Shi Ping CUC
Truncating in frequency domain
)1,0(: let, , 00 NkNFfNTT s
)]([IDFS)(1
)()()(
)()(
0
1
0
2
0
1
0
2
0
1
00
0
0
0
jkXNejkXN
N
ejkXejkXnTx
ejkXtx
N
k
nkNj
N
k
nkNjN
k
nTjk
k
tjk
Approximating to FT (FS) with DFT (DFS)
![Page 56: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/56.jpg)
Copyright © 2005. Shi Ping CUC
Some problems Aliasing
Otherwise, the aliasing will occur in frequency domainhs
hs ffTff
2
11,2 Sampling in time domain:
Sampling in frequency domain:0
0
1
FT
Period in time domain0T Frequency resolution 0F
NT
T
F
f s 0
0
andis contradictory
hf 0F
Approximating to FT (FS) with DFT (DFS)
![Page 57: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/57.jpg)
Copyright © 2005. Shi Ping CUC
Spectrum leakage
sequence length-finite ),()()(
sequence length-infinite ),(
12
1
nRnxnx
nx
N
)()()( 12 j
Rjj eWeXeX
Spectrum extension (leakage)
Spectrum aliasing
Approximating to FT (FS) with DFT (DFS)
demo
![Page 58: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/58.jpg)
Copyright © 2005. Shi Ping CUC
Fence effect
N
fF
f
F
fNs
ss
000
0 ,22
Frequency resolution
00
11
TNTN
fF s
demo
Approximating to FT (FS) with DFT (DFS)
![Page 59: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/59.jpg)
Copyright © 2005. Shi Ping CUC
Comments
return
demo
Zero-padding is an operation in which more zeros are appended to the original sequence. It can provides closely spaced samples of the DFT of the original sequence.
The zero-padding gives us a high-density spectrum and provides a better displayed version for plotting. But it does not give us a high-resolution spectrum because no new information is added.
To get a high-resolution spectrum, one has to obtain more data from the experiment or observation.
example
Approximating to FT (FS) with DFT (DFS)
![Page 60: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/60.jpg)
Copyright © 2005. Shi Ping CUC
Summary
return
The frequency representations of x(n)
)(nx
)(kX
)(zX
)( jeX
Time sequence
z-transform of x(n)
Complex frequency
domain
DTFT of x(n)Frequency
domainDFT of x(n)
Discrete frequency domain
ZT
DTFT
DFTk
N
2
jez kNjez
2
interpolation
interpolation
![Page 61: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/61.jpg)
Copyright © 2005. Shi Ping CUC
Illustration of the four Fourier transforms
Discrete Fourier SeriesSignals that are discrete and periodic
DTFTSignals that are discrete and aperiodic
Fourier SeriesSignals that are continuous and periodic
Fourier TransformSignals that are continuous and aperiodic
![Page 62: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/62.jpg)
Copyright © 2005. Shi Ping CUC
)(~ ny
n0 1 2 3 4 5 6
0n m
)(~1 mx
0
m
)(~2 mnx
0
1n2n3n4n5n6n
return
![Page 63: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/63.jpg)
Copyright © 2005. Shi Ping CUCreturn
)())(()(2
1)( nRnNxnxnx NNep
n
)(nx
0 5
n
NnNx ))((
0 55
n
)(nxep
0 5
![Page 64: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/64.jpg)
Copyright © 2005. Shi Ping CUCreturn
n
)(nxep
0 5
)())(()( * nRnNxnx NNepep
n
Nep nNx ))((
0 5
)(nRN
![Page 65: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/65.jpg)
Copyright © 2005. Shi Ping CUC
0 1 2 3 4 5 6 7 8 9 10
0
5
10
Original sequence
n
x(n
)
0 1 2 3 4 5 6 7 8 9 10
0
5
10
Circular conjugate symmetric component
n
xe
p(n
)
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4Circular conjugate antisymmetric component
n
xo
p(n
)
return
)()8.0(10 11 nRn
![Page 66: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/66.jpg)
Copyright © 2005. Shi Ping CUC
0 1 2 3 4 5 6 7 8 9 10
0
5
10
Circular even sequence x(n)
n
0 1 2 3 4 5 6 7 8 9 100
20
40
The DFT of x(n)
k
0 1 2 3 4 5 6 7 8 9 100
20
40
k
return
)(kX
)())(( nRkNX NN
![Page 67: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/67.jpg)
Copyright © 2005. Shi Ping CUC
0 1 2 3 4 5 6 7 8 9 10-4
-2
0
2
4Circular odd sequence x(n)
n
0 1 2 3 4 5 6 7 8 9 10
-10
0
10
The imaginary part of DFT[x(n)]
k
0 1 2 3 4 5 6 7 8 9 10
-10
0
10
k
return
)(kX
)())(( nRkNX NN
![Page 68: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/68.jpg)
Copyright © 2005. Shi Ping CUCreturn
)())(()( * kRkNXkX NN
number real a is )0(
)0()())(()0( *
0
*
X
XkRkNXXkNN
number real a is )2
(
)2
()())(()2
(
even is if
*
2
*
NX
NXkRkNX
NX
N
NkNN
![Page 69: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/69.jpg)
Copyright © 2005. Shi Ping CUCreturn
)())(()( * kRkNXkX NN
number imaginary an is )0(
)0()())(()0( *
0
*
X
XkRkNXXkNN
number imginary an is )2
(
)2
()())(()2
(
even is if
*
2
*
NX
NXkRkNX
NX
N
NkNN
![Page 70: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/70.jpg)
Copyright © 2005. Shi Ping CUCreturn
)( , )( 21 nxnx N-point real-value sequences
)]([DFT)( )],([DFT)( 2211 nxkXnxkX
)()()]([DFT)]([DFT
)]()([DFT)]([DFT)(
)()()(
2121
21
21
kjXkXnxjnx
njxnxnykY
njxnxny
)())(()(2
1)()](Re[DFT)(1 kRkNYkYkYnykX NNep
)())(()(2
1)(
1)](Im[DFT)(2 kRkNYkY
jkY
jnykX NNop
![Page 71: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/71.jpg)
Copyright © 2005. Shi Ping CUC
0 1 2 3 4 5 6 7 8 9
0
2
4
6
8
10
12
Linear convolution
n 0 1 2 3 4 5 6 7 8 9
0
2
4
6
8
10
12
Circular convolution N = 6
n
0 1 2 3 4 5 6 7 8 9
0
2
4
6
8
10
12
Circular convolution N = 7
n 0 1 2 3 4 5 6 7 8 9
0
2
4
6
8
10
12
Circular convolution N = 5
n
return
],2,3,2,1[)( ],2,2,1[)( 21 nxnx
![Page 72: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/72.jpg)
Copyright © 2005. Shi Ping CUC
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Magnitude Response, N = 8
frequency in pi units
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.5
0
0.5
1Phase Response
frequency in pi units
pi
)(
return
N
2N
4
![Page 73: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/73.jpg)
Copyright © 2005. Shi Ping CUC
0 1 2 3 4 5 6 70
1
2
3
4
5
6
X(k),N = 8
k
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
frequency in pi units
return
)()0( X )2
()1(N
X
)4
()2(N
X
)6
()3(N
X
![Page 74: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/74.jpg)
Copyright © 2005. Shi Ping CUC
0 5 10 15 20 250
2
4
6
8
10
t-1 -0.5 0 0.5 10
10
20
30
40
50
rad
0 5 10 15 20 250
2
4
6
8
10
n-2 -1 0 1 2
0
10
20
30
40
50
pi
ta tx )8.0(10)( )( jX a
FT
DTFT
)(nx )( jeX
![Page 75: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/75.jpg)
Copyright © 2005. Shi Ping CUC
-10 -5 0 5 100
2
4
6
8
10
n-2 -1 0 1 2
0
10
20
30
40
50
pi
-10 0 100
2
4
6
8
10
n-10 0 10
0
10
20
30
40
50
k
)()( 11 nRnx )()( jj eReX
)(~ nxN )(~kX N
DTFT
DFS
![Page 76: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/76.jpg)
Copyright © 2005. Shi Ping CUC
-10 0 100
2
4
6
8
10
n-10 0 10
0
10
20
30
40
50
k
return
)(nxN )(kX N
DFT
![Page 77: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/77.jpg)
Copyright © 2005. Shi Ping CUCreturn
0
)( jeR
0
)(2jeX
n
)(nRN
0
n
)(2 nx
0
n
)(1 nx
0 0
)(1jeX
![Page 78: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/78.jpg)
Copyright © 2005. Shi Ping CUC
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
pi
pi
pi
DTFT DFT
DTFT DFT
DTFT DFT
return
]1,1,1,1[)( nx
]0,0,0,0,1,1,1,1[)( nx
]0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1[)( nx
![Page 79: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/79.jpg)
Copyright © 2005. Shi Ping CUC
0 2 4 6 8 10 12 14 16 18 20-2
-1
0
1
2signal x(n), 0<=n<=19
n
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
pi
)(20 kX
)52.0cos()48.0cos()( nnnx
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Copyright © 2005. Shi Ping CUC
0 10 20 30 40 50 60 70 80 90 100-2
-1
0
1
2signal x(n), 0<=n<=19+80 zeros
n
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
pi
)(100 kX
![Page 81: Copyright © 2005. Shi Ping CUC Chapter 3 Discrete Fourier Transform Review Features in common We need a numerically computable transform, that is Discrete](https://reader034.vdocuments.net/reader034/viewer/2022051820/56649de55503460f94addacc/html5/thumbnails/81.jpg)
Copyright © 2005. Shi Ping CUC
0 10 20 30 40 50 60 70 80 90 100-2
-1
0
1
2signal x(n), 0<=n<=99
n
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
pi
)(100 kX
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0 50 100 150 200 250 300 350 400-2
-1
0
1
2signal x(n), 0<=n<=99+300 zeros
n
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
20
40
60
pi
return
)(400 kX
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Copyright © 2005. Shi Ping CUCreturn
Suppose
kHz 4 Hz, 100 hfF
Determine
N , ,0 TT
Solution sF
T 1.010
11
00
msff
Ths
125.01042
1
2
113
102422
80010125.0
1.0
10
30
mN
T
TN