chap3-discrete fourier transform
TRANSCRIPT
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Chapter 3
Discrete 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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Chapter 3
Discrete 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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The Family of Fourier Transform
Introduction
Fourier analysis is named afterJean Baptiste Joseph
Fourier(1768-1830), a French mathematician andphysicist.
A signal can be eithercontinuous ordiscrete, and it can
be eitherperiodicoraperiodic. The combination ofthese two features generates the four categories of
Fourier Transform.
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The Family of Fourier Transform
Aperiodic-ContinuousFourier Transform
g
g
;
g
g;
;;!
!;
dejXtx
dtetxjX
tj
tj
)(2
1)(
)()(
T
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The Family of Fourier Transform
Periodic-ContinuousFourier Series
g
g!
;
;
;!
!;
k
tjk
T
T
tjk
ejkXtx
dtetxTjkX
0
0
0
0
)()(
)(
1
)(
0
2
20
0
0
0
22
TF
TT !!;
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g
g!
!
!
T
T
[[
[[
[
T
deeXnx
enxeX
njj
n
njj
)(
2
1)(
)()(
Copyright 2005. Shi Ping CUC
The Family of Fourier Transform
Aperiodic-DiscreteDTFT
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The Family of Fourier Transform
Periodic-DiscreteDFS (DFT)
!
!
!
!
1
0
2
1
0
2
)(1
)(
)()(
N
k
nkN
j
N
n
nkN
j
ekXN
nx
enxkX
T
T
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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
T0
T
0
0
2
T
T!;
Ts
T2
!;
Ts
T2!
0
0
2
T
T!;
return
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The Discrete Fourier Series (DFS)
Definition
Periodic 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 !
g
g!
!
k
ktT
j
ekXtx
T2
)()(
!
!1
0
2
)(1
)(N
k
knN
j
ekX
N
nx
T
fundamental
frequencyt
Tj
e
T2
nN
j
e
T2fundamentalfrequency
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The Discrete Fourier Series (DFS)
!
!
!
! elsewhere,0
,1
1
1112
2
1
0
2mNr
e
e
Ne
N rN
j
rNN
jN
n
rnN
j
T
TT
)(1
)(
)(1
)(
1
0
)(1
0
1
0
1
0
1
0
rXeN
kX
eekXN
enx
N
n
nrkN
jN
k
N
n
rnN
jN
k
knN
jN
n
rnN
j
!
-
!
-
!
!
!
!
!
!
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The Discrete Fourier Series (DFS)
1
0
2
)()(N
n
knN
j
enxkX
T
)()(
)()(
1
0
1
0
)(
kXenx
enxmNkX
N
n
knN
j
N
n
nmNkN
j
!!
!
!
!
Because:
The is a periodic sequence with fundamental
period equal to N
)(kX
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The Discrete Fourier Series (DFS)
!
!
!!
!!
!
1
0
1
0
~1]~[IDFS~
~]~[DFS~
Let
N
k
kN
N
n
nk
N
j
N
WkN
knx
Wnxnxk
eW NT
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The Discrete Fourier Series (DFS)
Relation to the z-transform
ee
!elsewhere,0
10, Nnnxnx
kN
jez
zXkX T2|)()(~!!
The DFS represents N evenly spaced samples ofthe z-transform around the unit circle.
)(~
kX
)(zX
!
!
!!1
0
1
0
))(()(~
,)()(2
N
n
nkjN
n
n NenxkXznxzXT
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The Discrete Fourier Series (DFS)
Relation to the DTFT
k
j
N
n
nkN
jN
n
njj
NeXkX
enxkXenxeX
T[
[
T
[[
2|)()(
~
)()(~
)()(1
0
21
0
!
!
!
!
!!
ee
!elsewhere,0
10),(~
)(Nnnx
nx
The DFS is obtained by evenly sampling the DTFT at
intervals. It is called frequency resolution and represents the
sampling intervalin the frequency domain.
NT2
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The Discrete Fourier Series (DFS)
jIm[z]
Re[z]
0!k
k
j
N
ekX T[[
2)()(~
!!
N
T2
N=8frequency resolution
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The properties of DFS
The Discrete Fourier Series (DFS)
Linearity
)(
~
)(
~
)](~
)(~
[DFS 11 kXbkXanxbnxa
Shift of a sequence
)(~
)(~
)](~[DFS kXekXWmnxmk
Nj
mk
N
T
!!
Modulation
)(~
)](~[DFS lkXnxWlnN
!
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The Discrete Fourier Series (DFS)
Periodic convolution
!
!
!
!!
!
1
012
1
0
21
21
)(~)(~
)(~)(~)](~
[IDFS)(~then
)(~
)(~
)(~if
N
m
N
m
mnxmx
mnxmxkYny
kXkXkY
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!
!
!
!
!
!
!
!!
-
!
-
!
!!
1
0
12
1
0
21
1
0
1
0
)(21
1
0
2
1
0
1
1
0
2121
)(~)(~)(~)(~
)(~1)(~
)(~
)(~1
)(~
)(~1
)](~
)(~
[IDFS)(~
N
m
N
m
N
m
N
k
kmnN
N
k
nk
N
N
m
mk
N
N
k
nk
N
mnxmxmnxmx
WkXN
mx
WkXWmxN
WkXkX
N
kXkXny
return
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Introduction
The Discrete Fourier Transform (DFT)
The DFS provided us a mechanism fornumerically
computingthe 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
periodicsignal whoseprimaryshape 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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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),(~)(
nRnxnxelsewhere
Nnnxnx
N!
ee!
))(()(~
)()(~
N
r
nxnx
rNnxnx
!
! g
g!
Window operation
Periodic extension
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The definition of DFT
The Discrete Fourier Transform (DFT)
10,)(1
)]([ID T)(
10,)()]([D T)(
1
0
1
0
ee
ee
NnWkXN
kXnx
NkWnxnxkX
N
n
nk
N
N
n
nk
N
)()(~)()(1
)(
)()(~)()()(
1
0
1
0
nRnxnRWkXN
nx
kRkXkRWnxkX
N
N
n
N
nk
N
N
N
n
NnkN
!!
!!
!
!
return
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The Properties of DFT
Linearity
)()()]()([DFT 11 kbkanbxnax
N3-point DFT, N3=max(N1,N2)
Circular shift of a sequence
)()]())(([ kXWRmkm
!
)())(()]([ kRlkXWl
!
Circular shift in the frequency domain
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The Properties of DFT
The sum of a sequence
!!
!!
!!1
0
0
1
00
)()()(N
n
N
n
n
NnxWnxX
The first sample of sequence
!
!1
0
)(1
)0(N
k
kXN
x
)())(()]([
)()]([
kRkNNxnXDFT
kXnxDFT
NN!
!
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The Properties of DFT
Circular convolution
)()()())(()(
)())(()()()(
12
1
0
12
1
0
2121
nxnxnRnxmx
nRmnxmxnxnx
N
N
m
N
N
N
m
N
!
-
!
-
!
!
!
N
N
)()()]()([DFT2121
kXkXnxnx !N
)()(1
)]()([DFT 2121 kXkXN
nxnx ! N
Multiplication
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The Properties of DFT
Circular correlation
g
g!
g
g! !! nnxy nymnxmnynxmr )(*)()(*)()(
Linear correlation
Circular correlation
)()(*))((
)())((*)()(
1
0
1
0
mRymx
mRmyxmr
N
N
N
N
N
Nxy
!
!
!
!
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The Properties of DFT
)()(*))((
)())((*)(
)]([IDFT)(
)()()(
1
0
1
0
*
mRnymnx
mRmnynx
kRmrthen
kYkXkRif
N
N
n
N
N
N
n
N
xyxy
xy
!
!
!
!
!
!
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The Properties of DFT
Parsevals 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
NN
n
NN
n
Nnx
N
nxnx
nynx
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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
nNx
nx
nx
nx
nx
nNxnxnxnxnx
!!
!!
It can be proved that
)(
~
)(
~
)(~)(~
*
*
nx
nx
nxnx
oo
ee
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The Properties of DFT
? A
? A )())(())((2
1
)()(~)(
)())(())((2
1
)()(~)(
nRnNn
nRnn
nRnNn
nRnn
NNN
Noop
NNN
Np
!
!
!
!
Circular
conjugate
symmetric
component
Circular
conjugate
antisymmetriccomponent
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The Properties of DFT
)()()( nxnxnxopep
!
)())(()(
)())(()(*
*
nRnNxnx
nRnNxnx
NNopop
NNepep
!
!
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The Properties of DFT
)()()(kXkXkX
opp
!
)())(()(
)())(()(
*
*
kRkNXkX
kRkNXkX
NNopop
NNepep
and)(kXep )(kXop
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The Properties of DFT
)]())((Im[)](Im[
)]())((Re[)](Re[
kRkNXkX
kRkNXkX
NNepep
NNepep
!
!
)]())((Im[)](Im[
)]())((Re[)](Re[
kRkNXkX
kRkNXkX
NNopop
NNopop
!
!
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The Properties of DFT
)())(()(then
)())(()(if
kRkNXkX
nRnNxnx
NN
NN
!
!
Circular even sequences
Circular odd sequences
)())(()(then)())(()(ifkRkNXkX
nRnNxnx
NN
NN
!
!
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The Properties of DFT
)()())((
)())(()]([DFT**
**
kNXkRkNX
kRkXnx
NN
NN
!!
!
Conjugate sequences
)()]())(([D
)]())(([D
**
*
knRnNx
nRnx
NN
NN
!!
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The Properties of DFT
_ a
? A )())(())((21
)()](Re[DFT
* kRkNXkX
kXnx
NNN
ep
!
!
Complex-value sequences
_ a? A )())(())((
2
1)()](Im[DFT
* kRkNXkX
kXnxj
NNN
op
!
!
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The Properties of DFT
!
!
)(]))(())(([2
1
DFT
)](Re[)]([DFT
*
nnn
kXnp
!
!
)(]))(())(([2
1DFT
)](Im[)]([DFT
*nRnNxnx
kXjnx
NNN
op
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The Properties of DFT
)())(()(then
sequencevalue-realis)(i*
kRkNXkX
nx
NN!
Real-value sequences
Imaginary-value sequences
)())(()(tpartimagi aryaso ly)(if
*kRkNXkX
nx
NN!
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The Properties of DFT
Summary
)()()(
)](Im[)](Re[)(
kXkXkX
nxjnxnx
opep !
!
DDD
)](Im[)](Re[)(
)()()(
kXjkXkX
nxnxnx opep
!
!
DDD
example
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The Properties of DFT
Linear convolution & circular convolution
!
g
g!
!!
!
1
0
2121
211
)()()()(
)()()(N
mm
l
mnxmxmnxmx
nxnxny
Linear convolution
)(1 nx
)(nx
N1 point sequence, 0n N1-1
N2 point sequence, 0n N2-1
)(nyl L point sequence, L= N1+N2-1
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The Properties of DFT
Circular convolution
ee
ee!
1,0
10),()(
1
11
1LnN
Nnnn
We have to make both and L-point
sequences by padding an appropriate number of zeros
in order to make L point circular convolution.
)(nx )(2 nx
ee
ee!
,
),()(
LnN
Nnnxnx
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The Properties of DFT
)()(
)()()(
)()()(
)())(()()()()(
2
1
0
1
1
0
21
1
0
2121
nRrLny
nRmrLnxmx
nRmrLnxmx
nRmnxmxnxnxny
L
r
l
L
r
L
m
L
L
m r
L
L
m
Lc
-
!
-
!
- !
-
!!
g
g!
g
g!
!
!
g
g!
!
L
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The Properties of DFT
)()()( nRrnynyr
lc
-
!
g
g!
)()()()(isthat
)()(then1if
2121
21
nxnxnxnx
nynyNNL
lc
!
!u
L
return
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The Sampling Theorem in Frequency Domain
Sampling in frequency domain
g
g!!
!!
m
km
NWzWmxzXkX k
N
)(|)()(~
g
g!
g
g!
!
!
g
g!
!
!
-
!
-
!
!!
rm
N
mk
N
N
k
kn
N
m
km
N
N
k
kn
NN
rNnxWN
mx
WWmxN
WkXN
kXnx
)()(
)(
)()]([IDFS)(
)(
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The Sampling Theorem in Frequency Domain
g
g!
!r
NrNnxnx )()(~
Frequency Sampling TheoremForM point finite duration sequence, if the frequency
sampling number N satisfy:
MN uthen
)()()(~)( nxnRnxnxNNN
!!
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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
11)(1
)(1
)(1
)(1
)()(
N
kk
N
NN
kk
N
NNk
N
N
k
N
n
nk
N
N
k
N
n
nnk
N
N
n
n
N
k
nk
N
N
n
n
zWkX
N
zzWzWkX
N
zWkXN
zWkXN
zWkXN
znxzX
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The Sampling Theorem in Frequency Domain
1
1
0
1
01
1
11)(
)()(1
)(1)(
!
!
!*
*!
!
zW
z
Nz
zkXzW
kX
N
zzX
k
N
N
k
N
k
k
N
kk
N
N
Interpolation
function
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The Sampling Theorem in Frequency Domain
!
!
*!*!1
0
1
0
)2
()()()()(N
k
N
k
j
K
jwk
NkXekXeX
T
[[
2
1
2
2
sin
sin)(
!*
NjN
eN
[
[
[
[
Interpolation
function
Interpolation formula of )( [jeX
return
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Approximating to FT (FS) with DFT (DFS)
Approximating to FT of continuous-time aperiodic
signal with DFT
g
g;
g
g
;
;;!
!;
dejXtx
dtetxjX
tj
tj
)(21)(
)()(
T
CTFT
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g
g!
g
gppp
n
TdtTdtnTt ,,
Sampling in time domain
g
g!
;g
g
; }!;n
nTjtjenTxTdtetxjX )()()(
;;
g
g
;
;;}
;;!S
dejXntx
dejXtx
nTj
tj
0)(
2
1)(
)(2
1
)(
T
T
Tfss
T
T
22 !!;
Approximating to FT (FS) with DFT (DFS)
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Truncation in time domain
)1~0(:,,)~0(:00 ! NNTTTt
!
;};1
0
)()(N
n
nTjenTxTjX
S
dejXnTxnTj
0)(
21)(T
Approximating to FT (FS) with DFT (DFS)
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Sampling in frequency domain
NT
TT
TFNT
fN
FT
ddk
s
N
n
s
TT
TT
22
22,1
,,
0
0
0
00
0
0
1
0
00
00
!!;
!!;!!!
;p;;p;;p;
!
;
)]([T
)()()(1
0
21
0
0
0
nxT
enxTenTxTjkXN
n
nkN
jN
n
nTjk
!
!};
!
!
;T
Approximating to FT (FS) with DFT (DFS)
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)]([I T1
)]([I T
)(1
)(1
)(2
)(
00
1
0
2
0
1
0
2
00
1
0
0
0 0
;!;!
;!
;!
;;
}
!
!
!
;
jkXT
jkXf
ejkXN
f
ejkXN
NF
ejkXnTx
s
N
k
nkN
j
s
N
k
nkN
j
N
k
nTjk
T
T
T
demo
Approximating to FT (FS) with DFT (DFS)
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Approximating to FS of continuous-time periodic
signal with DFS
g
g!
;
;
;!
!;
k
tjk
Ttjk
ejkXtx
dtetxT
jkX
0
00
)()(
)(1
)(
0
00
0
0
00
22
TF
TT !!;
Approximating to FT (FS) with DFT (DFS)
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!
!1
00
0
0
,,N
n
T
TdNTTTdnTt
Sampling in time domain
)]([1
)(1
)()(1
0
21
00
00
nxN
enxN
enTxT
TjkX
N
n
nkN
jN
n
nTjk
!
!};
!
!
;T
Approximating to FT (FS) with DFT (DFS)
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Truncating in frequency domain
),(:let,, !@! NkNFfNTT s3
)]([IDFS)(1
)()()(
)()(
0
1
0
2
0
1
0
2
0
1
0
0
0
0
0
;!;!
;!;}
;!
!
!
!
;
g
g!
;
jkXNejkXN
N
ejkXejkXnTx
ejkXtx
N
k
nkN
j
N
k
nkN
jN
k
nTjk
k
tjk
T
T
Approximating to FT (FS) with DFT (DFS)
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Some problems
Aliasing
Otherwise, the aliasing will occur in frequency domainhs
hs
ffff
2,2 !"Sampling in time domain:
Sampling in frequency domain:
0
0
1
FT !
Period in time domain0T Frequency resolution0F
NT
T
F
fs!!
0
0
and
is contradictoryhf 0F
Approximating to FT (FS) with DFT (DFS)
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Spectrum leakage
sequencelength-finite),()()(
sequencelength-infinite),(
12
1
nRnxnx
nx
N!
)()()(j
R
jjeWeXeX !
Spectrum extension (leakage)
Spectrum aliasing
Approximating to FT (FS) with DFT (DFS)
demo
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Fence effect
N
fF
f
F
fN
s
ss
!!;
!! 000
0 ,22 TT
[
Frequency resolution
00
11
TT
fF
s!!!
demo
Approximating to FT (FS) with DFT (DFS)
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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 obtainmore data from the experiment or observation.
example
Approximating to FT (FS) with DFT (DFS)
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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 frequencydomain
ZT
DTFTDFT
kN
T2!
[jz! kjz
2
!
interpolation
interpolation
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Illustration of the four Fourier transforms
Discrete Fourier SeriesSignals that are discrete and
periodic
DTFTSignals that are discrete andaperiodic
Fourier SeriesSignals that are continuousand periodic
FourierTransformSignals that are continuousand aperiodic
~
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)(~ ny
n0 1 2 3 4 5 6
0!n m
)(~1 mx
0
m
)(~2 mnx
0
1!n2!n
3!n4!n5!n
6!n
return
? A1
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Copyright 2005. Shi Ping CUCreturn
? A)())(()(2
1)( nRnNxnxnx NNep !
n
)(nx
0
n
N
nNx
))((
0 55
n
)(nxep
05
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Copyright 2005. Shi Ping CUCreturn
n
)(nxep
0 5
)())(()( * nRnNxnx NNepep !
n
epn))((
0 5
)(nRN
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0 1 2 3 4 5 6 7 8 9 10
0
5
10
O r ig i n a l s e q u e n c e
n
x
(n
)
0 1 2 3 4 5 6 7 8 9 10
0
5
10
C i r c u l ar c o n j u g a t e s y m m e t r ic c o m p o n e n t
n
xep(n
)
0 1 2 3 4 5 6 7 8 9 10 -4
-2
0
2
4C i r c u l ar c o n j u g a t e an t is y m m e t r ic c o m p o n e n t
n
xo
p
(n
)
return
)()8.(1 11 nRn
v
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0 1 2 3 4 5 6 7 8 9 1 0
0
5
1 0
Ci
l
v
0 1 2 3 4 5 6 7 8 9 1 0
0
2 0
4 0
T h e D FT
f
k
0 1 2 3 4 5 6 7 8 9 1 0
0
2 0
4 0
k
return
)(kX
)())(( nkNX NN
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0 1 2 3 4 5 6 7 8 9 1 0 -4
-2
0
2
4Ci
l
d d
e q u e n c e
n )
n
0 1 2 3 4 5 6 7 8 9 1 0
-1 0
0
1 0
T h e i !
in a r y "
a rt
f D FT [x (n ) ]
k
0 1 2 3 4 5 6 7 8 9 1 0
-1 0
0
1 0
k
return
)(kXd
)())(( nRkNXNN
d
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Copyright 2005. Shi Ping CUCreturn
)())(()(*
kkNXkXNN
!
? Anumberrealais)0(
)0()())(()0( *0
*
X
XkRkNXXk
NN
@
!!!
? A
numberrealais)2
(
)
2
()())(()
2
(
evenisif
*
2
*
NX
NXRNX
NX
N
NNN
@
!!!
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Copyright 2005. Shi Ping CUCreturn
)())(()(*
kRkNXkXNN
!
numb rimaginaryanis)0(
)0()())(()0( *0
*
X
XRNXXNN
@
!!!
? A
numberimginaryanis)2
(
)
2
()())(()
2
(
evenisif
*
2
*
NX
NXkRkNX
NX
N
Nk
NN
@
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Copyright 2005. Shi Ping CUCreturn
)(,)( 21 nxnx N-point real-value sequences
)]([DFT)()],([DFT)(2211
nxnx !!
)()()]([)]([
)]()([)]([)()()()(
kjkj
jykYjy
!!
!!!
_ a )())(()(2
1)()](Re[DFT)(1 kRkNYkYkYykX NNep !!!
_ a ? A )())(()(2
1)(
1)](Im[DFT)(2 kRkNYkY
jkY
jnykX NNop !!!
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0 1 2 3 4 5 6 7 8 9
0
2
4
68
1 0
1 2
L in e a r c o n vo lut io n
n 0 1 2 3 4 5 6 7 8 9
0
2
4
68
1 0
1 2
Circ ula rc o n vo lut io n N = 6
n
0 1 2 3 4 5 6 7 8 9
0
2
46
8
1 0
1 2
Circ ula rc o n vo lut io n N = 7
n 0 1 2 3 4 5 6 7 8 9
0
2
46
8
1 0
1 2
Circ ula rc o n vo lut io n N = 5
n
return
],2,3,2,1[)(],2,2,1[)( 21 !! nn
)([*
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-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
M a g n i tude R e s p o n s e , N = 8
fr e q u e n c y in p iu n i ts
-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
1P h a s e R e s p o n s e
fr e q u e n c y in p iu n i ts
p
i
)([*
return
N
T2N
T4
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0 1 2 3 4 5 6 7 0
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
fr e q u e n c y in p iu n i ts
return
)()0( [*X )()(N
X T[*
)4
()2(N
T[ *
)6()3(N
X T[ *
ttx )80(0)( v )( ;jX
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0 5 1 0 1 5 2 0 2 5 0
2
4
6
8
1 0
t
-1 -0 .5 0 0 .5 10
1 0
2 0
3 0
4 0
5 0
ra d
0 5 1 0 1 5 2 0 2 5 0
2
4
6
8
1 0
n
-2 -1 0 1 20
1 0
2 0
3 0
4 0
5 0
p i
atx )8.0(0)( v! )( ;jXa
FT
DTFT
)(nx )([jeX
)()( R )()([[ jj
eReX
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-1 0 -5 0 5 1 0 0
2
4
6
8
1 0
n
-2 -1 0 1 20
1 0
2 0
3 0
4 0
5 0
p i
-1 0 0 1 0 0
2
4
6
8
1 0
n
-1 0 0 1 0 0
1 0
2 0
3 0
4 0
5 0
k
)()( nRnx )()(jj
eReX
)(~ nxN )(~
kN
DTFT
DF
)( )(kX
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-1 0 0 1 0 0
2
4
6
8
1 0
n
-1 0 0 1 0 0
1 0
2 0
3 0
4 0
5 0
k
return
)(nxN
)(kXN
DFT
)(1 nx )([j
eX
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Copyright 2005. Shi Ping CUCreturn
[0
)([j
eR
[0
)(2[j
X
n
)(nR
0
n
)(nx
0
n
)(1
0[0
)(eX
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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
p i
p i
p i
DTFTDFT
DTFT
DFT
DTFT
DFT
return
],,,[)( !nx
],,,,1,1,1,1[)( !nx
],,,,,,,,,,,,1,1,1,1[)( !nx
s ig n a l (n) 0 < n < 1 9
)50cos()480cos()( TT
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0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 -2
-1
0
1
2s ig n a lx(n),0
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0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 -2
-1
0
1
2s ig n a lx(n),0
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0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 -2
-1
0
1
2s ig n a lx(n),0
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0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 -2
-1
0
1
2s ig n a lx(n),0
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Suppose kHz4Hz,100 ee hfF
Determine N,,0
TT
Solutions
FT .!u!
msThs
125.01042
1
2
113
!vv
!e!
102422
80010125.0
1.0
10
3
0
!!!
!vu!
mN
T
TN