Post on 01-Mar-2018
transcript
7/26/2019 Chapter 4 Discrete Fourier Transform
1/65
1All right reserved. Copyright 2013. Sharifah Saon
CHAPTER
!"SCRETE #$%R"ER TRA&S#$R'
7/26/2019 Chapter 4 Discrete Fourier Transform
2/65
#o(rier transfor)ation is (sed to de*o)pose ti)e series
signals into fre+(en*y *o)ponents ea*h having an
a)plit(de and phase.
#o(rier transfor)ation is i)ple)ented in )any !SP,!igital Signal Pro*essing- ro(tines e*a(se any
)athe)ati*al operation in the ti)e do)ain has an
e+(ivalent operation in the fre+(en*y do)ain that is
often computationally faster. Th(s/ #o(rier transfor)ation is o**asionally
i)ple)ented solely to speed up algorithms.
2All right reserved. Copyright 2013. Sharifah Saon
7/26/2019 Chapter 4 Discrete Fourier Transform
3/65
3All right reserved. Copyright 2013. Sharifah Saon
Introduction to DFT and IDFT
TheNpointDiscrete Fourier Transform,!#T-XDFTk of anNsa)ple signalxn is defined y
/ k 0/ 1/ 2/ 4/N51
TheInverse Discrete Fourier Transform,"!#T-6hi*h transfor)sXDFTk toxn is defined y
/ n 0/ 1/ 2/ 4/N51
[ ] [ ] NnkjN
nDFT
enxkX 21
0
==
[ ] [ ] NnkjN
k
DFT ekXN
nx 21
0
1
=
=
7/26/2019 Chapter 4 Discrete Fourier Transform
4/65
All right reserved. Copyright 2013. Sharifah Saon
Ea*h !#T sa)ple is fo(nd as a 6eighted s() of all thesa)ples inxn.
$ne of the )ost i)portant properties of the !#T andits inverse is i)pliedperiodi*ity.
The e7ponential/ e7p,8j2nk9N-in the definingrelations is periodi* in oth nand k6ith periodN:
the !#T and its inverse are also periodi* 6ith periodNand it is s(ffi*ient to *o)p(te the res(lts for only oneperiod ,0 toN 1-/6ith a starting inde7 of ;ero.
( ) ( ) NNknjNkNnjNknj eee ++ == 222
7/26/2019 Chapter 4 Discrete Fourier Transform
5/65
1/ 2/ 1/ 0?. #ind the !#T ofxn.
7/26/2019 Chapter 4 Discrete Fourier Transform
6/65
@All right reserved. Copyright 2013. Sharifah Saon
Solution
ithN / and/
k 0/ XDFT0
k 1/ XDFT1
k 2/ XDFT2
k 3/ XDFT3
XDFT[k] >4! "j#! $!j#%.
22
jnkNnkj ee =
[ ] 012103
0
=+++==
enxn
[ ] 2021 223
0
jeeenx jjjn
n
=+++= =
[ ] 0021 23
0
=+++=
= jjjnn
eeenx
[ ] 2021 323233
0
jeeenx jjnj
n
=+++=
=
7/26/2019 Chapter 4 Discrete Fourier Transform
7/65
BAll right reserved. Copyright 2013. Sharifah Saon
&$TE: E(lers "dentity
ej= *os+ j sin e-j= *os j sin
E7a)ple:
e5
j3
*os ,3- 12e5j392 2,e5j9 2 -3 2 ,j-3 5 2j3
5 2j,5 1- j2
12
=j je j
= 2
( )
ke jk
*os=
7/26/2019 Chapter 4 Discrete Fourier Transform
8/65
DAll right reserved. Copyright 2013. Sharifah Saon
Exercise 4.1
a- #ind the !#T of
- #ind the !#T of
?3//1/2> =
nx
?1/0/1/0/1>
=ny
7/26/2019 Chapter 4 Discrete Fourier Transform
9/65
Properties of the !#T&roperty 'ignal DFT (emar)s
Shift xn no &o *hange in )agnit(de.
Shift xn5 0.
7/26/2019 Chapter 4 Discrete Fourier Transform
10/65
10All right reserved. Copyright 2013. Sharifah Saon
Sy))etry
The !#T of a real se+(en*e possesses *onG(gatesy))etry ao(t the origin6ithXDFTk k.
Sin*e the !#T is periodi*/XDFTk XDFTN k.
This also i)plies *onG(gate sy))etry ao(t the inde7k 0.
7/26/2019 Chapter 4 Discrete Fourier Transform
11/65
11All right reserved. Copyright 2013. Sharifah Saon
The inde7 k 0.
7/26/2019 Chapter 4 Discrete Fourier Transform
12/65
12All right reserved. Copyright 2013. Sharifah Saon
Central $rdinates and Spe*ial !#T
al(es
The *o)p(tation of the !#T is easy to *o)p(te at
k 0 and ,for evenN- at k = N92 (sing the *entral
ordinate theore)s.
=
=1
0
0N
n
DFT nxX [ ] ( )
=
=1
02
1N
n
nNDFT nxX
==
1
0
1
0
N
kDFT kXNx [ ] ( )
= =
1
02 1
1 N
kDFT
kN
kXNx
7/26/2019 Chapter 4 Discrete Fourier Transform
13/65
13All right reserved. Copyright 2013. Sharifah Saon
Cir*(lar Shifting
The defining relation for the !#T re+(ires
signal val(es for 0 n N 1.
Iy i)plied periodi*ity/ these val(es *orrespondto one period of a periodi* signal.
To find the !#T of a ti)eshifted signalxn no/ its val(es )(st also e sele*ted over
,0/N 1- fro) its periodi* e7tension
7/26/2019 Chapter 4 Discrete Fourier Transform
14/65
1All right reserved. Copyright 2013. Sharifah Saon
generating one period ,0 J n JN- of a*ir*(larly shiftedperiodi* signal: To generatexn no: 'ove the last nosa)ples of
xn to the eginning.
To generatexnK no: 'ove the first nosa)ples of
xn to the end.
E7a)pleL
7/26/2019 Chapter 4 Discrete Fourier Transform
15/65
1
7/26/2019 Chapter 4 Discrete Fourier Transform
16/65
1@All right reserved. Copyright 2013. Sharifah Saon
Cir*(lar Sy))etry
for real periodi* signals 6ith periodN
Circular een sy!!e"ry:xn xN n
Circular o## sy!!e"ry:xn xN n
7/26/2019 Chapter 4 Discrete Fourier Transform
17/65
1BAll right reserved. Copyright 2013. Sharifah Saon
Convol(tion and Correlation
'(ltipli*ation in one do)ain *orresponds todis*rete periodi* *onvol(tion in the other.
Si)ilar *on*ept applies to the *orrelation
operation.
7/26/2019 Chapter 4 Discrete Fourier Transform
18/65
1DAll right reserved. Copyright 2013. Sharifah Saon
Periodi* Convol(tion
!#T offers an indire*t )eans of finding the periodi**onvol(tionyn xn $n of t6o se+(en*esxnand $n of e+(al lengthN.
Co)p(te theNsa)pleXDFTk and%DFTk/ )(ltiply
the) to otain
YDFTk XDFTk%DFTk and
find the inverse of YDFTk to otain the periodi**onvol(tionyn.
xn $n XDFTk%DFTk
7/26/2019 Chapter 4 Discrete Fourier Transform
19/65
1All right reserved. Copyright 2013. Sharifah Saon
Periodi* Correlation
Periodi* *orrelation *an e i)ple)ented (sing the!#T in al)ost the sa)e 6ay as periodi* *onvol(tion/e7*ept for an e7tra *onG(gation step prior to taMing the"!#T.
The periodi* *orrelation of t6o se+(en*esxn and$n of e+(al lengthNgives
rx$n xn $n XDFTk k
"fxn and $n are real/ the final rx$n )(st also ereal.
F
DFT%
7/26/2019 Chapter 4 Discrete Fourier Transform
20/65
20All right reserved. Copyright 2013. Sharifah Saon
Reg(lar Convol(tion and
Correlation
Reg(lar *onvol(tion y the !#T re+(ires ;eropadding.
"fxn and $n are of length& andN/ *reatex'n and
$'n/ ea*h ;eropadded to length&KN 1. #ind the !#T of the ;eropadded signals/ )(ltiply the
!#T se+(en*es and findyn as the inverse.
Si)ilar *on*ept applies to the *orrelation operation.
7/26/2019 Chapter 4 Discrete Fourier Transform
21/65
21All right reserved. Copyright 2013. Sharifah Saon
E7a)ple .2
a- Netyn >1/ 2/ 3/ /
7/26/2019 Chapter 4 Discrete Fourier Transform
22/65
22All right reserved. Copyright 2013. Sharifah Saon
- =iven the !#T pair
xn >1/ 2/ 1/ 0? XDFTk >/ -j2/ 0/j2?
6ithN . #ind:
i. yn xn 2 and the !#T ofyn
ii. *DFTk XDFTk 1 and its "!#T'n.iii. )n xn and its !#T.
i. n x,n and its !#T.
. $n xnxn and its !#T.
i. cn xnxn
ii. sn xnFxn
iii. x0 andXDFT0 (sing *entral ordinates.
*- Prove that (sing Parsevals relation[ ] [ ]
=
=
=1
0
221
0
1 N
k
DFT
N
n
kXN
nx
7/26/2019 Chapter 4 Discrete Fourier Transform
23/65
23All right reserved. Copyright 2013. Sharifah Saon
olu"ion
7/26/2019 Chapter 4 Discrete Fourier Transform
24/65
2All right reserved. Copyright 2013. Sharifah Saon
i. Time shift
To findyn xn 2/ the last t6o sa)ples are )oved tothe eginning to get
yn xn 2 >1/ 0/ 1/ 2?/ n 0/ 1/ 2/ 3.
To find the !#T ofyn xn 2/ (se the ti)eshiftproperty ,6ith no 2- to give
YDFTk =
>/j2/ 0/ j2?.
[ ] [ ] jkDFTknj
DFT ekXekX o =2
7/26/2019 Chapter 4 Discrete Fourier Transform
25/65
2
7/26/2019 Chapter 4 Discrete Fourier Transform
26/65
2@All right reserved. Copyright 2013. Sharifah Saon
ii. *odulation
To find*DFTk XDFTk 1/ the last one sa)ple is )oved
to the eginning to get
*DFTk XDFTk 1 >j2/ / j2/ 0?.
"ts "!#T is
'nxn ej2n.xn ejn.2 >1/j2/ 1/ 0?.
7/26/2019 Chapter 4 Discrete Fourier Transform
27/65
2BAll right reserved. Copyright 2013. Sharifah Saon
&otes:
n 0/ ,ej.2-n ,ej.2-0 1
n 1/ ,ej.2-n ,ej.2-1 G
n 2/ ,ej.2-n ,ej.2-2 ej cos,- 1
n 3/ ,ej.2-n ,ej.2-3 j3 j
7/26/2019 Chapter 4 Discrete Fourier Transform
28/65
2DAll right reserved. Copyright 2013. Sharifah Saon
iii. #olding
&ote:xn >0/ 1/ 2/ 1?
The se+(en*e)n xn is
)n >x0/x 3/x 2/x 1? >1/ 0/ 1/ 2?.
"ts !#T e+(als
/DFTk XDFTk k >/j2/ 0/ -j2?.FDFTX
7/26/2019 Chapter 4 Discrete Fourier Transform
29/65
2All right reserved. Copyright 2013. Sharifah Saon
iv. ConG(gate
The se+(en*en x,n is
n x,n = xn >1/ 2/ 1/ 0?.
"ts !#T is
0DFTk k >/j2/ 0/ -j2?F =>/ j2/0/j2?.FDFTX
7/26/2019 Chapter 4 Discrete Fourier Transform
30/65
30All right reserved. Copyright 2013. Sharifah Saon
v. Prod(*t
The se+(en*e $n xnxn is the point6ise prod(*t. So/
$n = >1/ / 1/ 0?.
"ts !#T is
%DFTk
>/ j2/ 0/j2?>/ j2/ 0/j2?.
eep in )ind that this is a periodi* *onvol(tionQ Th(s/
%DFTk = >2/ jl@/ D/jl@? =>@/ j/ 2/j?.
1
[ ] [ ]kXkX DFTDFT
1
1
7/26/2019 Chapter 4 Discrete Fourier Transform
31/65
31All right reserved. Copyright 2013. Sharifah Saon
&otes:>/ j2/ 0/j2?>/ j2/ 0/j2?.
k 0 1 2 3 < @
XDFT
k j2 0 j2
XDFT
k j2 0 j2
1@ jD 0 jD
jD 0
0 0 0 0
jD 0
%n 1@ j1@ j1@ D 0
7/26/2019 Chapter 4 Discrete Fourier Transform
32/65
32All right reserved. Copyright 2013. Sharifah Saon
K 0 1 2 3
#irst half ofyk 1@ j1@ j1@
rap aro(nd half ofyk D 0 0
Periodi* *onvol(tion yk 2 j1@ D j1@
7/26/2019 Chapter 4 Discrete Fourier Transform
33/65
33All right reserved. Copyright 2013. Sharifah Saon
vi. Periodi* *onvol(tion
The periodi* *onvol(tion cn xn xn
gives
cn >1/ 2/ 1/ 0?>1/ 2/ 1/ 0? >2/ / @/ ?.
"ts !#T is given y the point6ise prod(*t
CDFTkXDFTkXDFTk
>/ j2/ 0/j2?>/ j2/ 0/j2?
+1@/ / 0/ ?.
7/26/2019 Chapter 4 Discrete Fourier Transform
34/65
3All right reserved. Copyright 2013. Sharifah Saon
&otes:
n 0 1 2 3 < @
xn 1 2 1 0
xn 1 2 1 01 2 1 0
2 2 0
1 2 1 0
0 0 0 0
yn 1 @ 1 0 0
7/26/2019 Chapter 4 Discrete Fourier Transform
35/65
3
7/26/2019 Chapter 4 Discrete Fourier Transform
36/65
3@All right reserved. Copyright 2013. Sharifah Saon
vii. Reg(lar *onvol(tion
The reg(lar *onvol(tionsn= xnFxn gives
sn= xnFxn =>1/ 2/ 1/ 0?F>1/ 2/ 1/ 0?
>1/ / @/ / 1/ 0/ 0?.
Sin*exn has sa)ples ,N -/ the !#T DFTk ofsn is the prod(*t of the !#T of the ;eropadded ,tolengthN + N 1 B- signalx'n >l/ 2/ 1/ 0/ 0/ 0/ 0?and e+(als
DFTk >1@/ 2.3< jl0.2D/ 2.1D Kj1.0
7/26/2019 Chapter 4 Discrete Fourier Transform
37/65
3BAll right reserved. Copyright 2013. Sharifah Saon
&otes:
DFTk / k 0/ 1/ 2/ 4/ @.
DFT0 1 K K @ K K 1 1@
DFT1 1 K ej2.B
K @ej2,2-.B
K ej2,3-.B
K ej2,-.B
1 K ,cos29B jsin29B- K @,cos9B jsin9B- K,cos@9B jsin@9B- K ,cos29B jsin29B- K
cosD9B jsinD9B
1 K 2. j3.13 K ,1.3- j
7/26/2019 Chapter 4 Discrete Fourier Transform
38/65
3DAll right reserved. Copyright 2013. Sharifah Saon
viii. Central ordinates
x0 , j2 K 0 Kj2- ,- 1
XDFT0 1 K 2 K 1 K 0
Iy (sing &arseval,s relation/
1 K K 1 K 0 @
=>1@/ / 0/ ?/
,1@ K K 0 K - @
[ ]==
3
0
1
k
DFTkX
[ ]=
3
0n
nx
[ ] [ ]
=
=
=1
0
221
0
1 N
k
DFT
N
n
kXN
nx
[ ]23
0
=n
nx
[ ]kXDFT2
[ ]=
3
0
2
1
k
DFT kX
7/26/2019 Chapter 4 Discrete Fourier Transform
39/65
3All right reserved. Copyright 2013. Sharifah Saon
!#T of Periodi* Signals and the
!#S
The #o(rier series relations for a periodi*signalx+,"- are
( ) [ ] "(kjk
+oekX"x 2
==
[ ] ( ) #"e"xT
kX "(kj
T +
o21 =
7/26/2019 Chapter 4 Discrete Fourier Transform
40/65
0All right reserved. Copyright 2013. Sharifah Saon
"fxn is a*+(ired/n 0/ 1/ .../N 1 asNsa)ples ofx+,"-over oneperiod (sing a sa)pling rate of H; ,*orresponding to a sa)plinginterval of "s- and the integral e7pression forXk is appro7i)ated
y a s())ation (sing: #" "s " n"s TN"s (o 19T 19N"s
The +(antityXDFk defines the !is*rete #o(rier Series ,!#S- as anappro7i)ation to the #o(rier series *oeffi*ients of a periodi* signaland e+(alsNti)es the !#T
[ ] [ ] [ ] 1&/1/0/M/11 2
1
0
21
0
===
=
= NknjN
n
s
"n(kjN
ns
DF- enxN
"enxN"
kX so
7/26/2019 Chapter 4 Discrete Fourier Transform
41/65
1All right reserved. Copyright 2013. Sharifah Saon
The "nverse !#S ,"!#S-
The #o(rier series re*onstr(*tion relation is (sed to re*overxn fro) one period ofXDF-k 6hose s())ation inde7*overs one period ,fro) k 0 to kN 1- to otain
This relation des*ries theInverse Discrete Fourier'eries,"!#S-.
The sa)pling interval "sdoes not enter into the *o)p(tation
of the !#S or its inverse. The !#T of a sa)pled periodi* signalx,"-is related to its
#o(rier series *oeffi*ients
[ ] [ ] [ ] 1&/1/0/n/2
1
0
21
0 ===
=
=
NknjN
kDF
"n(kjN
nDF ekXenXnx
so
7/26/2019 Chapter 4 Discrete Fourier Transform
42/65
2All right reserved. Copyright 2013. Sharifah Saon
#ast #o(rier Transfor)
The !#T des*ries a set ofNe+(ations/ ea*h 6ithNprod(*t ter)s and th(s re+(ires a total ofN2 )(ltipli*ationsfor its *o)p(tation.
Co)p(tationally effi*ient algorith)s to otain the !#T go
y the generi* na)e ##T ,#ast #o(rier Transfor)-andneed far fe6er )(ltipli*ations
7/26/2019 Chapter 4 Discrete Fourier Transform
43/65
3All right reserved. Copyright 2013. Sharifah Saon
Sy))etry and Periodi*ity
All ##T algorith)s taMe advantage of the sy))etryand periodi*ity of the e7ponential N ej2n9N/ as listed
elo6
7/26/2019 Chapter 4 Discrete Fourier Transform
44/65
All right reserved. Copyright 2013. Sharifah Saon
Choi*e of Signal Nength
The signal lengthN is *hosen as a n()er that is the prod(*t of)any s)aller n()ers rks(*h thatN r1r2... r!.
A )ore (sef(l *hoi*e res(lts 6hen the fa*tors are e+(al/ s(*h that
N r!.
The fa*tor r is *alled theradix. Iy far the )ost pra*ti*ally i)ple)ented *hoi*e for ris 2/ s(*h that
N 2!and leads to theradix"###T algorith)s. "n parti*(lar/radix"###T algorith)s re+(ire the n()er of sa)ples
N to e a po6er of 2 ,N 2!/ integer !- and the !#T is *o)p(ted(sing onlyN log2N )(ltipli*ations
7/26/2019 Chapter 4 Discrete Fourier Transform
45/65
7/26/2019 Chapter 4 Discrete Fourier Transform
46/65
@All right reserved. Copyright 2013. Sharifah Saon
#(nda)ental Res(lts
Consider t6o trivial (t e7tre)ely i)portantres(lts.
1"point transform-The !#T of a single n()er2is the n()er2itself.
#"point transform-The !#T of a 2point se+(en*eis easily fo(nd to e
XDFT0 x0 Kx1
XDFT1 x0 x1
7/26/2019 Chapter 4 Discrete Fourier Transform
47/65
BAll right reserved. Copyright 2013. Sharifah Saon
The single )ost i)portant res(lt in the develop)ent of a radi72##T algorith) is that anNsa)ple !#T *an e 6ritten as the s() oft6o N92sa)ple !#Ts for)ed fro) the eveninde7ed and oddinde7ed sa)ples of the original se+(en*e
&ote:
[ ] [ ] [ ] [ ] ( )knNn
nk
N
n
nk
N
N
n
DFT 1nx1nx1nxkX
NN
12
1
0
2
1
0
1
0
22
122 +
=
=
= ++==
[ ] [ ] [ ] nkNn
k
N
nk
N
n
DFT 1nx11nxkX
NN
2
1
0
2
1
0
22
122
=
=
++=
[ ] [ ] [ ] nkN
n
k
N
nk
N
n
DFT 1nx11nxkX
NN
2
1
0
2
1
0
22
122
=
=
++=
2
2
NN 11 = nk
N
nk
N 11
2
2=
Th ! i ti i # ##T
7/26/2019 Chapter 4 Discrete Fourier Transform
48/65
DAll right reserved. Copyright 2013. Sharifah Saon
The !e*i)ation in #re+(en*y ##T
Algorith)
The de*i)ation in fre+(en*y ,!"#- ##T algorith) starts yred(*ing the singleNpoint transfor) at ea*h s(**essive stage/ (ntilarrive at 1point transfor)s that *orrespond to the a*t(al !#T.
,e.g. D 2 1/ 6ith 3 stages-
ith the inp(t se+(en*e in nat(ral order/ *o)p(tations *an e done/(t the !#T res(lt is in itreversed order and )(st e reordered.,001 100-
#or a point inp(t/ inary indi*es :>00/ 01/ 10/ 11? it order :>x0/x1/x2/x3?
Dpoint inp(t se+(en*e inary indi*es :>000/ 001/ 010/ 011/ 100/ 101/ 110/ 111? it order :>x0/x1/x2/x3/x/x
7/26/2019 Chapter 4 Discrete Fourier Transform
49/65
All right reserved. Copyright 2013. Sharifah Saon
Separating even and odd indi*es/ and lettingxn = xaandxnKN92=x3
[ ] [ ] 1/42/1/0/M/22
21
0
2
+=
=
Nnk
N
n
3a
DFT 1xxkXN
[ ] [ ] 1/42/1/0/M/122
2
1
0
2
=+
=
Nnk
N
n
Nn
3a
DFT 11xxkX
N
7/26/2019 Chapter 4 Discrete Fourier Transform
50/65
xn Kxn K ?
XDFT2kK 1 >xn xn K ?
2N
2N
2
N n
N
7/26/2019 Chapter 4 Discrete Fourier Transform
51/65
7/26/2019 Chapter 4 Discrete Fourier Transform
52/65
7/26/2019 Chapter 4 Discrete Fourier Transform
53/65
7/26/2019 Chapter 4 Discrete Fourier Transform
54/65
7/26/2019 Chapter 4 Discrete Fourier Transform
55/65
7/26/2019 Chapter 4 Discrete Fourier Transform
56/65
=ny
The !e*i)ation in Ti)e ##T
7/26/2019 Chapter 4 Discrete Fourier Transform
57/65
00/ 01/ 10/ 11? reverse :>00/ 10/ 01/ 11? and itreversed order :>x0/x2/xl/x3?
#or an Dpoint inp(t se+(en*e/ the inary indi*es :>000/ 001/ 010/ 011/ 100/ 101/ 110/ 111?/ reversed se+(en*e :>000/ 100/ 010/ 110/ 001/ 101/ 011/ 111? itreversed order :>x0/x/x2/x@/xl/x
7/26/2019 Chapter 4 Discrete Fourier Transform
58/65
7/26/2019 Chapter 4 Discrete Fourier Transform
59/65
7/26/2019 Chapter 4 Discrete Fourier Transform
60/65
@0All right reserved. Copyright 2013. Sharifah Saon
As 6ith the de*i)ation in fre+(en*y algorith)/the t6iddle fa*tors "at ea*h stage appear onlyin the otto) 6ing of ea*h (tterfly.
The e7ponents "also have a definite ,andal)ost si)ilar- order des*ried y &()er0of distin*t t6iddle fa*tors "at ith stage:
P #i " 1.
al(es of "in the t6iddle fa*tors0t- t #m "iQ6ith 0/ 1/ 2/ .../0 1.
Th fi t t f d i ti i ti
7/26/2019 Chapter 4 Discrete Fourier Transform
61/65
@1All right reserved. Copyright 2013. Sharifah Saon
The first stage of de*i)ation in ti)e
,!"T- ##T algorith) forN D
'tage 1:
0 2i 1 21 1 1. Th(s 0.
i 1/ " 2! i 23 1,0- 0
Th d t f d i ti i ti
7/26/2019 Chapter 4 Discrete Fourier Transform
62/65
@2All right reserved. Copyright 2013. Sharifah Saon
The se*ond stage of de*i)ation in ti)e
,!"T- ##T algorith) forN D
'tage #:
0 2i 1 22 1 2. Th(s 0/ 1.
i 2/ " 2! i 23 2,0- 0
" 23 2,1- 2
Th thi d t f d i ti i ti
7/26/2019 Chapter 4 Discrete Fourier Transform
63/65
@3All right reserved. Copyright 2013. Sharifah Saon
The third stage of de*i)ation in ti)e
,!"T- ##T algorith) forN D
Stage 35
0 = 6i - 1= 67 - 1= 4. T$us = 89 19 69 7.
i = 79 " = 6! - i = 67 - 7:8; = 8
" = 67 - 7:1; = 1
" = 67 - 7:6; = 6
" = 67 - 7:7; = 7
7/26/2019 Chapter 4 Discrete Fourier Transform
64/65
@All right reserved. Copyright 2013. Sharifah Saon
E7a)ple .
The !#T of a dis*rete signal/yn is given y
Apply !e*i)ation in Ti)e ,!"T- #ast #o(rier
Transfor)ation ,##T- algorith) to deter)ine its
dis*rete signal/yn.
[ ] ?3/D/3/@> jjkYDFT +=
7/26/2019 Chapter 4 Discrete Fourier Transform
65/65
Success Is Not The Key To Hainess.
Hainess Is The Key To Success. I! "ou#o$e %hat "ou &'e Doing( "ou %ill )e
Success!ul*.