Chapter 4 Discrete Fourier Transform

7/26/2019 Chapter 4 Discrete Fourier Transform

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CHAPTER

!"SCRETE #$%R"ER TRA&S#$R'


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#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.



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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

=

=


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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


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1/ 2/ 1/ 0?. #ind the !#T ofxn.


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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

=+++=

=


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&$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=


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Exercise 4.1

a- #ind the !#T of

- #ind the !#T of

?3//1/2> =

nx

?1/0/1/0/1>

=ny


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Properties of the !#T&roperty 'ignal DFT (emar)s

Shift xn no &o *hange in )agnit(de.

Shift xn5 0.


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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.


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The inde7 k 0.


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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


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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


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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


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1


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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


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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.


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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


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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%


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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.


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E7a)ple .2

a- Netyn >1/ 2/ 3/ /


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- =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


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olu"ion


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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


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2


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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?.


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&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


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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


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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


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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


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&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


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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@


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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/ ?.


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&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


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3


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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


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&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


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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


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!#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 =


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"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


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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


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#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


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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


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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


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#(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


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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


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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


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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


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xn Kxn K ?

XDFT2kK 1 >xn xn K ?

2N

2N

2

N n

N


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=ny

The !e*i)ation in Ti)e ##T


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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


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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


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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


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The se*ond stage of de*i)ation in ti)e


'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


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The third stage of de*i)ation in ti)e


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


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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 +=


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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*.

Chapter 4 Discrete Fourier Transform

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Transcript of Chapter 4 Discrete Fourier Transform