chapter 2 discrete fourier transform

8/20/2019 CHAPTER 2 Discrete Fourier Transform

1/75

03/04/16 1

Chapter 2Chapter 2

DiscreteDiscrete Fourier TransformFourier Transform

Instructor: TedInstructor: Ted

Email:[email protected]:[email protected]

Phone:13836034068Phone:13836034068


2/75

03/04/16 2

Three Questions aboutThree Questions about DiscreteDiscrete Fourier TransformFourier Transform

Q1:Q1: WWHAT is DFT?HAT is DFT?

Q2:Q2: WWHY is DFT?HY is DFT?

Q3: HOQ3: HO WW to DFT?to DFT?

W !T is relationshi" #et$een %&T and other 'inds o( &ourierW !T is relationshi" #et$een %&T and other 'inds o( &ourier

Trans(orm)Trans(orm)

W * $e need %&T)W * $e need %&T)

+W to realize %&T) o$ to use %&T to sol,e the "ractical+W to realize %&T) o$ to use %&T to sol,e the "ractical

"ro#lems)"ro#lems)


3/75

03/04/16 3

Basic contents of this chapterBasic contents of this chapter -.1 e,ie$ o( &ourier Trans(orm-.1 e,ie$ o( &ourier Trans(orm

-.- %iscrete &ourier /eries-.- %iscrete &ourier /eries -.3-.3 %iscrete%iscrete &ourier Trans(orm&ourier Trans(orm

-.4-.4 elationshi" #et$een %&T z Trans(orm and se2uence selationshi" #et$een %&T z Trans(orm and se2uence s

&ourier Trans(orm&ourier Trans(orm

-.-. &re2uency sam"lin5 theorem&re2uency sam"lin5 theorem

-.6-.6 om"ute se2uence s linear con,olution usin5om"ute se2uence s linear con,olution usin5 %&T%&T

-.7-.7 /"ectrum analysis #ased on %&T/"ectrum analysis #ased on %&T

-.8 e,ie$-.8 e,ie$


4/75

03/04/16 4

2 12 1 Fourier TransformFourier Transform

( ) ( )sin 2 x t f t π = × × ( )2 randn+ ×In some situation si5nal s (re2uency s"ectrum can re"resent its characteristics

more clearly.

( ) x t in (re2uency domain( ) x t in time domain

&ourier

Trans(orm

( ) ( )sin 2 500 x t t π = × ×

/i5nal !nalysis and Processin5/i5nal !nalysis and Processin5

11 Time %omain !nalysis: t !Time %omain !nalysis: t !

-- &re2uency %omain !nalysis: ( !&re2uency %omain !nalysis: ( !


5/75

03/04/16 5

2 12 1 Fourier TransformFourier Transform

/i5nal !nalysis and Processin5:/i5nal !nalysis and Processin5:

11 Time %omain !nalysisTime %omain !nalysis -- &re2uency %omain !nalysis&re2uency %omain !nalysis

&ourier Trans(orm is a #rid5e (rom time domain to (re2uency domain.&ourier Trans(orm is a #rid5e (rom time domain to (re2uency domain.

haracteristic: continuous discrete "eriodic haracteristic: continuous discrete "eriodic non"eriodicnon"eriodic ..

ontinuous "eriodic si5nalsontinuous "eriodic si5nals

ontinuous non"eriodic si5nalsontinuous non"eriodic si5nals

%iscrete "eriodic si5nals%iscrete "eriodic si5nals

%iscrete non"eriodic%iscrete non"eriodic si5nalssi5nals

Ty"e:Ty"e:


6/75

03/04/16 6

1! "ontinuous perio#ic si$na%&&Fourier 'eries1! "ontinuous perio#ic si$na%&&Fourier 'eriesIt is "ro,ed that continuous time "eriodic si5nal can #e re"resented #y aIt is "ro,ed that continuous time "eriodic si5nal can #e re"resented #y a

&ourier /eries corres"ondin5 to a sum o( harmonically related com"le9&ourier /eries corres"ondin5 to a sum o( harmonically related com"le9

e9"onential si5nal. To a "eriodic (unction $ith "eriod e9"onential si5nal. To a "eriodic (unction $ith "eriod

( ) ( )0 0202

2 1,

T jn t jn t

T n nn

f t F e F f t e dt T T

ω ω π

ω

+∞−

−=−∞= = ⇔ =∑ ∫

( ) f t T

Conclusion:Conclusion: Continuous periodicContinuous periodic function— function—

Nonperiodic discrete Nonperiodic discrete frequency impulse sequence frequency impulse sequence

Time-domain Frequency-domain

( ) f t A

t

T 2τ 2τ −0

T n

n F


7/75

03/04/16 7

2!2! "ontinuous"ontinuous nonperio#icnonperio#ic function(s Fourier Transformfunction(s Fourier Transform

∫ ∞

∞

Ω= dt et x j X t j ;;

∫ ∞

∞Ω Ωd e j X t x t j :;

-1

:;

Conclusion :Conclusion : Continuous nonperiodicContinuous nonperiodic function— function— Nonperiodic continuous Nonperiodic continuous function function


t

( ) x t A

2τ 2τ −0 Ω

( ) X jΩ


8/75

03/04/16 8

3!3! Discrete&timeDiscrete&time nonperio#icnonperio#ic se)uence(s Fourier Transformse)uence(s Fourier Transform

Conclusion :Conclusion : Discrete nonperiodic Discrete nonperiodic function— function—Continuous-time periodicContinuous-time periodic function function

; :

1; :

-

j j n

n

j j n

X e x n e

x n X e e d

∞

= ∞

=

=∑∫

; - ; -

-

; j M j M n

n

j n j nM j n

n n

X e x n e

x n e e x n e

∞

= ∞ ∞ ∞

= ∞ = ∞

=

= =

∑

∑ ∑


ω

( ) j X e ω

π − π 02 π − 2 π

n

( ) x n A

2 N 2 N −0


9/75

03/04/16 9

*! "onc%usion*! "onc%usion 11 /am"lin5 in time domain #rin5s "eriodicity in (re2uency/am"lin5 in time domain #rin5s "eriodicity in (re2uency

domain.domain.-- /am"lin5 in (re2uency/am"lin5 in (re2uency domain #rin5s "eriodicity in timedomain #rin5s "eriodicity in time

domain.domain. 33 elationshi" #et$een (re2uency domain and time domainelationshi" #et$een (re2uency domain and time domain

Time domain &re2uency domain Trans(ormTime domain &re2uency domain Trans(orm

ontinuous "eriodicontinuous "eriodic %iscrete non"eriodic &ourier series%iscrete non"eriodic &ourier series

ontinuous non"eriodic ontinuous non"eriodicontinuous non"eriodic ontinuous non"eriodic &ourier Trans(orm&ourier Trans(orm

%iscrete non"eriodic ontinuous "eriodic%iscrete non"eriodic ontinuous "eriodic /e2uence s &ourier/e2uence s &ourier

Trans(ormTrans(orm

%iscrete "eriodic %iscrete "eriodic %iscrete &ourier /eries%iscrete "eriodic %iscrete "eriodic %iscrete &ourier /eries

PeriodicPeriodic %iscrete< =on"eriodic%iscrete< =on"eriodic ontinuousontinuous


10/75

03/04/16 10

+! Basic i#ea of Discrete Fourier Transform+! Basic i#ea of Discrete Fourier Transformn !rac"ica# a!!#ica"ion, si$na# !rocessed %y com!u"er &as "'o main c&arac"eris"icsn !rac"ica# a!!#ica"ion, si$na# !rocessed %y com!u"er &as "'o main c&arac"eris"ics

;1 %iscrete;1 %iscrete ;- &inite len5th;- &inite len5thimi#ar#y, si$na#*s +requency mus" a#so &a e "'o main c&arac"eris"icsimi#ar#y, si$na#*s +requency mus" a#so &a e "'o main c&arac"eris"ics

;1 %iscrete;1 %iscrete ;- &inite len5th;- &inite len5th

IdeaIdea ::

.!and.!and +ini"e-#en$"& sequence "o !eriodic sequence+ini"e-#en$"& sequence "o !eriodic sequence , com!u"e i"s iscre"e, com!u"e i"s iscre"e

Fourier eries, so "&a" 'e can $e" "&e discre"e s!ec"rum in +requency domainFourier eries, so "&a" 'e can $e" "&e discre"e s!ec"rum in +requency domain

u" non!eriodic sequence*s Fourier Trans+orm is a con"inuous +unc"ion o+u" non!eriodic sequence*s Fourier Trans+orm is a con"inuous +unc"ion o+ ωω, and i", and i"is a !eriodic +unc"ion inis a !eriodic +unc"ion in ωω 'i"& a !eriod 2'i"& a !eriod 2 ππ o i" is no" sui"a%#e "o so# e !rac"ica# o i" is no" sui"a%#e "o so# e !rac"ica#di$i"a# si$na# !rocessin$di$i"a# si$na# !rocessin$


11/75

03/04/16 11

2 2 Discrete Fourier 'eries1 %iscrete &ourier /eries Trans(orm Pair

/imilar $ith continuous time "eriodic si5nals a "eriodic se2uence $ith"eriod = can #e re"resented #y a &ourier /eries corres"ondin5 to a sum o(

harmonically related com"le9 e9"onential se2uences such as:

( ) ( )1

0

1k

N j n

k

x n X k e

N

ω −

=

= ∑ %%

Attention: &ourier /eries (or discrete time si5nal $ith "eriod = re2uiresonly = harmonically related com"le9 e9"onentials.

( )

( )

12 , s +undamen"a# an$u#ar +requency

2, s &armonic an$u#ar +requencyk

x n N

k x n kth N

π ω

π ω

=

=

%

%

2-1

( ) x n%

$here

2 2( )

"&a" is j kn j k n N

N N k N k e e

π π

ω ω +

+= =:Q


12/75

03/04/16 12

( )( )

( )

221

20

11 1 1

01

j r k N N N j r k n

N

j r k n N

r k mN ee

otherwise N N e

π

π

π

−− −

−=

= +−= × =−∑

2-

+"er mu#"i!#yin$ %o"& sides o+ q (2-1) 'i"& , and summin$

+rom 0 -1, 'e o%"ain

j kn N e

n to N

π

=

( ) ( )

( ) ( ) ( ) ( )

2 2 21 1 1

0 0 0

2 21 1 1 1

0 0 0 0

1

1 1

N N N j kn j rn j kn N N N

n n r

N N N N j r k n j r k n N N

n r r n

x n e X r e e N

X r e X r e N N

π π π

π π

− − −− −

= = =

− − − −− −

= = = =

=

= = ÷

∑ ∑∑

∑∑ ∑ ∑

%%

% %

com!u"a"ion( ) X k %

( ) ( ) ( )21

0

N j kn N

n

X r X k x n eπ − −

=⇒ = =∑% % %


13/75

03/04/16 13

Attention:

:;>

:;> :;> :;> -1

0

;-1

0k X en x en x mN k X

kn N

j N

n

nmN k N

j N

n=

=

=

so ( ) is a#so a !eriodic sequence 'i"& !eriod

( ) and ( ) is a !eriod sequence !air in +requency domain and "ime domain

X k

X k x n

%

% %

Discrete Fourier Series for periodic sequence:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

21 1

0 0

21 1

0 0

1 1

N N j kn kn N N

n n

N N j kn kn N N

k n

X k DFS x n x n e x n W

x n IDFS X k X k e X k W N N

π

π

− −−

= =− −

−

= =

= = =

= = =∑ ∑

∑ ∑

% % % %

% % %%

N j

N eW

-

=


14/75

03/04/16 14

- Pro"erties o( %&/u!!ose ( ) ( ) ( ) ( ) DFS x n X k DFS y n Y k = = % %% %

1 ,inear,inear :;> :;> :?;> :;> @ k Y bk X n ybn x D!"

2 'e)uence 'hift'e)uence 'hift( ) ( )

( ) ( )

km N

nl N

DFS x n m W X k

IDFS X k l W x n

−+ =+ =

%%

% %

( )

1

0

1 1

1

0

( ) ( ) ,

( ) ( )

( ) ( ) ( )

N kn

N n

N m N mk r m km kr

N N N r m r m

N km kr km

N N N r

Proof DFS x n m x n m W let r n m

Then x r W W x r W

W x r W DFS x n m W X k

−

=+ − + −

− −

= =−

− −

=

+ = + = +

= =

= ⇒ + =

∑

∑ ∑

∑

% %

% %

%% %


15/75

03/04/16 15

- Pro"erties o( %&/u!!ose ( ) ( ) ( ) ( ) DFS x n X k DFS y n Y k = =% %% %

3 -erio#ic "on.o%ution-erio#ic "on.o%ution

1 1

0 0

+ ( ) ( ) ( ) "&en

( ) ( ) ( ) ( ) ( ) ( ) N N

m m

F k X k Y k

f n IDFS F k x m y n m y m x n m− −

= =

=

= = − = −∑ ∑ % % %

% % % % % %

om!ared 'i"& #inear con o#u"ion, !eriodic con o#u"ion*s maindi++erence is

T&e sum is o er "&e +ini"e in"er a# m 0 -1

:eriodic con o#u"ion


16/75

03/04/16 16

1

0

(1) ( ) (1 ) 0 N

m

f x m y m−

== − =∑% % %

1

0

(0) ( ) (0 ) 1 N

m

f x m y m−

== − =∑% % %

1

0

(2) ( ) (2 ) 1 N

m

f x m y m−

=

= − =∑% % %

1

0

( ) ( ) ( ) N

m

f n x m y n m−

=

= −∑% % %-erio#ic con.o%ution-erio#ic con.o%ution


17/75

03/04/16 17

'/mmetr/'/mmetr/ :

1 1

0 0

+ ( ) ( ) ( ) "&en1 1

( ) ( ) ( ) ( ) ( ) ( ) N N

l l

f n x n y n

F k DFS f n X l Y k l Y l X k l N N

− −

= =

== = − = −∑ ∑

%

% %

%% % % % %

0u%tip%ication of perio#ic se)uence in timeomain is0u%tip%ication of perio#ic se)uence in timeomain iscorrespon# to con.o%ution of perio#ic se)uence incorrespon# to con.o%ution of perio#ic se)uence infre)uenc/ #omainfre)uenc/ #omain


18/75

03/04/16 18

Periodic se2uence and its %&/


19/75

03/04/16 19

2 3 Discrete Fourier Transform&DFT 2 3 Discrete Fourier Transform&DFT

Periodic se2uence and its %&/


20/75

03/04/16 20

HINTS

-erio#ic se)uence is infinite %en$th

but on%/ se)uence .a%ues contain information

-erio#ic se)uence finite %en$th se)uence

e%ationship bet een these se)uences ?

4nfinite4nfinite FiniteFinite

-erio#ic-erio#ic NonperiodicNonperiodic


21/75

03/04/16 21

2 3 Discrete Fourier Transform&DFT 2 3 Discrete Fourier Transform&DFT e%ationship bet een perio#ic se)uence an# finite&%en$th se)uence

-erio#ic se)uence can be seen as perio#ica%%/ copies of finite&%en$th se)uenceFinite&%en$th se)uence can be seen as e5tractin$ one perio# fromperio#ic se)uence

Main period

Finiteuration 'e)uenceFiniteuration 'e)uence -erio#ic 'e)uence-erio#ic 'e)uence


22/75

03/04/16 22

2 3 Discrete Fourier Transform&DFT 2 3 Discrete Fourier Transform&DFT u!!ose ( ) is a +ini"e-#en$"& sequence; ( ) is a !eriodic sequence; x n x n%

( ) ( ) ( ) N x n x n R n= %1, 0 1

( ) is a square-'a e sequence ( )0, o"&er'ise N N

n N R n R n

≤ ≤ −=

'e use (( )) "o deno"e (n modu#o ) N n

( )( ) ( )12

5 5 x x=

(0) x(1) x

(2) x

(3) x

(4) x

(6) x(7) x

(8) x

(11) x(10) x

(9) x

(5) x

12 N =

( )( ) ( )12

20 8 x x=

( )( ) ( )12

1 11 x x− =

( )( )( ) ( )r

N r x n x n rN x n

=∞

=−∞= + =∑%


23/75

03/04/16 23

( ) x n

2 3 Discrete Fourier Transform2 3 Discrete Fourier Transform

Get DFT by extracting one period of DFS

DFS of periodic sequence

( ) N R k

om"utation o( %&T #y e9tractin5 one "eriod o( %&/

To a (inite len5th se2uence :

Periodicalcopies

( )( ) N

x n( ) X k ⇔ DFS

Attention DFT is ac)uire# b/ e5tractin$ one perio# of

DF'6 it is not a ne 7in# of Fourier Transform

DFT


24/75

03/04/16 24

1

0

1

0

1

0

+ini"e-#en$"& sequence can %e e.!ressed in "&is 'ay

( ) ( ) ( )

( ) ( ) ( ) (( )) ( )

"&en ( ) ( ) 0 1

1"&en ( ) ( ) 0 1

N

N kn

N N N N n

N kn

N n

N kn

N k

x n x n R n

X k X k R k x n W R k

X k x n W k N

x n X k W n N N

−

=−

=−

−

=

=

∴ = =

= ≤ ≤ −

= ≤ ≤ −

∑∑

∑

%

%

DFT TransforPair

=

=

10;1

;

10;;

1

0

1

0

N nW k X N n x

N k W n x k X

N

k kn N

N

n

kn N

( ) ( ) ( ) ( ) X k DFT x n x n IDFT X k = =abbr .

4n.erse Transform4n.erse Transform


25/75

03/04/16 25

Pro"erty o( %&Tu!!ose ( ) ( ) ( ) ( ) X k DFT x n Y k DFT y n= =

;1 Ainearity( ) ( ) ( ) ( ) , are coe++icien" DFT ax n by n aX k bY k a b+ = +

;- ircular /hi(t

ircular shi(t o( x#n$ can #e de(ined:( ) (( )) ( )m N N x n x n m R n= +

( ) ( ) ( ) x n x n x n m+% %

( )m x n

Periodi

!o"ies

Shift #xtra t

$ain "eriod (( )) N x n m= +


26/75

03/04/16 26

2x (n)

"ircu%ar shift of se)uence"ircu%ar shift of se)uence ,inear shift of se)uence,inear shift of se)uence

N - %


27/75

03/04/16 27

( ) ( ) (( )) ( ) ( )mk m m N N N X k DFT x n DFT x n m R n W X k −= = + =

(( )) ( ) ( )nl N N N IDFT X k l R k W x n+ =

'/mmetric bet een DFT an# 4DFT '/mmetric bet een DFT an# 4DFT

(( )) ( ) ( ) ( ) N N N DFT x n m R n DFT x n m R n+ = +%( ) ( ) N DFS x n m R k = +%

( ) ( )

( )

mk

N N mk

N

W X k R k

W X k

−

−

==

%


28/75

03/04/16 28

3 -arse.a%(s Theorem-arse.a%(s Theorem1 1

0 01 1

2 2

0 0

1( ) ( ) ( ) ( )

1


29/75

03/04/16 29

4 ircular con,olution

1

0

1

0

u!!ose ( ) ( ) ( )

"&en ( ) ( ) ( ) (( )) ( )

or ( ) ( ) (( )) ( )

N

N N m

N

N N m

F k X k Y k

f n IDFT F k x m y n m R n

f n y m x n m R n

−

=−

=

=

= = −

= −

∑∑

Periodic con,olution is con,olution o( t$o se2uences $ith "eriod= in one "eriod so it is also a "eriodic se2uence $ith "eriod =.

ircular con,olution is ac2uired #y e9tractin5 one "eriod o(

"eriodic con,olution e9"ressed #y ⊗.

ircularcon,olution


30/75

03/04/16 30

f#n$

1 N y m & n%

- N y m & n%

0 N y m & n%

"ircu%ar con.o%ution"ircu%ar con.o%ution -erio#ic con.o%ution-erio#ic con.o%ution

0 f

1 f

- f


31/75

03/04/16 31

( ) ( )

( ) ( )( ) ( ) ( ) ( )

DFT x n X k

DFT y n Y k IDFT X k Y k x n y n

==

= ⊗

Circular convolution can be used to

compute two sequence !s linear

convolution.


32/75

03/04/16 32

onBu5ate symmetric "ro"erties

a %&T o( conBu5ate se2uence

2

1 1( )

0 0

2

u!!ose ( ) is com!#e. con>u$a"e sequence o+ ( ) "&en

( ) ( ),0 1

:roo+ ( ) ( ) ( ) ,0 1

1

(

N

N N nk n k

N N n n

j nN nN j n N

x n x n

DFT x n X N k k N

DFT x n x n W x n W k N

W e e

DFT x n

π π

∗

∗ ∗

− −∗ ∗ − ∗

= =−

−

∗

= − ≤ ≤ −

= = ≤ ≤ −

= = =

∑ ∑

Q1 1

( ) ( )

0 0

) ( ) ( )

( ),0 1

N N nN n k n N k

N N N n n

x n W W x n W

X N k k N

− −− ∗ − ∗

= =∗

= =

= − ≤ ≤ −

∑ ∑

Attention C;' has only ' ,alid ,alues 0≤' ≤= 1 %u" '&en 0 (( 0)) (0) no" ( ),

so ( ) (( )) ( ) in a s"ric" 'ay N

N N

k X N X X N

DFT x n X N k R k

∗ ∗ ∗

∗ ∗

= − == −


33/75

03/04/16 33

# %&T o( se2uence s real and ima5inary "art

u!!ose ( ) s rea# and ima$inary !ar" are ( ) and ( )

"&en ( ) ( ) ( )1 1

( ) ( ) ( ) , ( ) ( ) ( )2 2

( ) and ( ) s /FT are ( ) and ( ), "&en

1( ) ( ) ( )2

r i

r i

r i

r i e o

e r

x n x n jx n

x n x n jx n

x n x n x n jx n x n x n

x n jx n X k X k

X k DFT x n DFT x n

∗ ∗= +

= + = −

= =

'

1( ) ( ) ( )2

1 1( ) ( ) ( ) ( ) ( ) ( )

2 2o i

x n X k X N k

X k DFT jx n DFT x n x n X k X N k

∗ ∗

∗ ∗

+ = + −

= = − = − −

( ) ( ) ( )e o X k X k X k = +


34/75

03/04/16 34

na#ysis o+ ( ) and ( ) s symme"ric

1( ) ( ) ( )

21 1

( ) ( ) ( ) ( ) ( )2 2

"&en ( ) ( )

e o

e

e

e e

X k X k

X k X k X N k

X N k X N k X N N k X N k X k

X k X N k

∗

∗ ∗ ∗ ∗

∗

= + −

− = − + − + = − +

= −

Q

CC ee;' is e,en com"onents o( C;' C;' is e,en com"onents o( C;' C ee;' is conBu5ate symmetric


35/75

03/04/16 35

C e;' conBu5ate e,en "art

conBu5ate symmetric<

real "art is e2ual ima5inary

"art is o""osite.

C e;' s real "art

C e;' s ima5inary "art

C o;' con"ugate odd part con"ugate asy etric#real part is opposite$ i aginarypart is equal%

C o;' s real "art

C o;' s ima5inary "art


36/75

03/04/16 36

"onc%usion1 %&T o( se2uence s real "art is corres"ondin5 to C;' s conBu5ate symmetric "art.

- %&T o( se2uence s ima5inary "art is corres"ondin5 to C;' s conBu5ate asymmetric

"art.

3 /u""ose 9;n is a real se2uence that is 9;n D9 r ;n

then C;' only has conBu5ate symmetric "art that is C;' DC e;'

sa"is+y ( ) ( )

(0) (0) ( ) ( )2 2

X k X N k N N

X X X X

∗

∗ ∗

= −= =:

/o: I( $e 5et hal( X#k$' $e can ac2uire all X#k$ usin5 symmetric"ro"erties.


37/75

03/04/16 37

01

0 1 1

1

( ( (

N N N ( ) N-)

N N N

( N-) #N-)$#N-)$ N N N

W W W X

W W W X

x x x N

W W W

X N

÷÷ ÷÷ ÷=÷ ÷÷ ÷÷÷÷

K K M O M

M K

1

0( ) ( ) 0 1

N kn

N n X k x n W k N

−

== ≤ ≤ −

∑ k N →0 1

n N →0 1

DFT -ro$rammin$ 85amp%e

DFT 0atri5


38/75

03/04/16 38

function &'()*dft+xn,

N*lengt +xn,# .lengt of sequencen*/:N-1# . ti e sa ple(*/:N-1#

0N*exp+-" 2 pi N,#n(*n3 (#0Nn(*0N%4n(# .calculate t e DFT 5atrix

'(*xn 0Nn(# .co pute DFT

5ore effecti6e et od%


39/75

03/04/16 39

Fs * 7//# . Get t e analy8ed signal

T * 1 Fs#9 * 1///#t * +/:9-1, T#x * /% sin+2 pi ;/ t,#

plot+1/// t+1:2//,$x+1:2//,,#

< * dft+x, 9# . Discrete Fourier Transfor

f * Fs 2 linspace+/$1$9 2=1,#ste +f$2 abs+


40/75

03/04/16 40


41/75

03/04/16 41

2 * DFT6 'e)uence(s Fourier Transform an# 9&transform

; : x t X j Ω

; : x n x n* j+ X e

%&/

/am"lin5

N

x n

Periodic o"ies

X k %

x n

E9tract +ne "eriod E9tract +ne "eriod

X k %&T

/e2uence s &ourierTrans(orm

&ourier Trans(ormontinuoustime

%iscretetime


42/75

03/04/16 42

Three #ifferent fre)uenc/omain representations of a finite&%en$th #iscrete&time se)uence

2 'e)uence(s Fourier Transform2 'e)uence(s Fourier Transform

3 Discrete Fourier Transform3 Discrete Fourier Transform DFT!DFT!

1 9&Transform1 9&Transform

-1

0

-1

0

; : ; : 0 1

1; : ; : 0 1

N j kn kn N

N n

N j kn N

k

X k x n e W k N

x n X k e n N N

=

=

= ≤ ≤

= ≤ ≤

∑

∑

11

0

1; ; ; ;

-

N n n

C n

X , x n , x n X , , d, =

= = ∫Ñ

1

0

1; ; ; ;

-

N j+ j+n j+ j+n

n

X e x n e x n X e e d+

=

= = ∫

j+ , e}


43/75

03/04/16 43

j I m5?6

Re5?6

C;' and C;z )C;' and C;z )

C;' and C;eC;' and C;e j+ j+ ) )

0 N W

-

N W

1 N W

$ N #

N W -

$ N # N W

1

k N

j k

N

W ,

eW ,

, X k X k N

-

E:;:;

=

=

=

:;:;-

k N

j e X k X

=

101

0

∑= N k W $n# x $k # X N

n

kn N

1

0

( ) ( ) N

n

n

X % x n % −

−

==∑

1

0

( ) ( ) N

jw jwn

n

X e x n e−

−

=

=∑


44/75

03/04/16 44

; : ; : j X k X e andelationshi" #et$een


45/75

03/04/16 45

( ) ( ) or ( ) j X k X e X % ω ⇒ 2 + Fre)uenc/ samp%in$ theorem2 + Fre)uenc/ samp%in$ theorem

o$ to realize) Prere2uisite (or im"lementation)o$ to realize) Prere2uisite (or im"lementation)What is inter"olation (ormula)What is inter"olation (ormula)

11 /am"lin5/am"lin5 9;n s z trans(orm:9;n s z trans(orm:

∑ ∞ =

n

n

, n x , X :;:;

2

( ) ( ) = ( )k N

kn N N % W

n

j k k N N

X k X % x n W

% W eπ

−

∞

==−∞

−

= =

= =

∑e5ular inter,al sam"lin5 on unit circle:e5ular inter,al sam"lin5 on unit circle:

,oss after samp%in$?,oss after samp%in$?

!(t "li 5 i ( 2 d i $ 2 i 2!(ter sam"lin5 in (re2uency domain can $e ac2uire se2uence


46/75

03/04/16 46

!(ter sam"lin5 in (re2uency domain can $e ac2uire se2uence!(ter sam"lin5 in (re2uency domain can $e ac2uire se2uencere"resentin5re"resentin5 x x ;; nn #y in,erse trans(ormin5 (rom C #y in,erse trans(ormin5 (rom C == ;; k k ))

is "eriodical co"ies o(is "eriodical co"ies o( x x ;; nn that is sam"lin5 in (re2uency that is sam"lin5 in (re2uency

domain causes "eriodical co"ies o( se2uence in time domain.domain causes "eriodical co"ies o( se2uence in time domain.:;> n x N

I( $e $ant to reco,er the (inite len5th se2uenceI( $e $ant to reco,er the (inite len5th se2uence x x ;; nn $ith no $ith no

loss a(ter sam"lin5 in (re2uency domain then it must #e satis(ied:loss a(ter sam"lin5 in (re2uency domain then it must #e satis(ied:

/u""ose:/u""ose: FF is num#er o( "oints in time domain


47/75

03/04/16 47

-- Inter"olation (ormulaInter"olation (ormula

( ) N X k ( ), 1 x n n $ =

( ) N X k 1

0

( ) ( ) $

n

n

X % x n % −

−

=

= ∑

( )1

0

( ) $

j jn

n

X e x n eω ω −

==∑

1

0

2( ) ( ) ,

N j

N k

X e X k k N

ω π

ω

−

=

= Φ − ÷ ∑ ω

ω

ω ω

−−

=Φ 21

)2/sin()2/sin(1

)( N

j

e N

N

1

0

( ) ( ) ( ), N

N k k

X % X k % −

== Φ∑ 11

11)( −−

−

−−=Φ

% W %

N % k

N

N

k


48/75

03/04/16 48

Ob;ecti.e DFT or 4DFT can be use# to compute t o se)uence(s circu%arcon.o%ution6 an# DFT6 4DFT ha.e their fast a%$orithm 'o if ecan bui%# the re%ationship bet een t o se)uences( circu%ar

con.o%ution an# %inear con.o%ution6 e can impro.e computationspee# of %inear con.o%ution b/ fast Fourier Transform a%$orithm

2 < "omputin$ se)uence(s %inear con.o%ution ith DFT

( ) ( )i $ & @ d i $ &h


49/75

03/04/16 49

Circular Convolution

Linear Convolution

( ) ( )u!!ose is a #en$"& @ sequence and is a #en$"& sequence,"&eir #inear con o#u"ion is

x n h n

( ) ( )1

0

( ) ( ) ( ) $

l m

y n x n h n x m h n m−

=

= ∗ = −∑

What relationship between and ?( )l y n ( ) y n

[ ]Aero !addin$ ( ) and ( ) "o "&e same #en$"& , ma. , , "&en x n h n & & N $ ≥( ), 0 1 ( ),0 1

( ) ( )0, 1 0, 1

x n n $ h n n N

x n and h n $ n & N n &

≤ ≤ − ≤ ≤ − = = ≤ ≤ − ≤ ≤ −

( ) ( ) ( ) ( ) ( ) ( )( ) DFT x n X k DFT h n ' k Y k X k ' k = = =

( )0

( ) ( ) ( ) ( ) (( )) ( ) &

N &m

y n IDFT Y k x n h n x m h n m R n=

′ ′ ′ ′= = = −

∑


50/75

03/04/16 50

" can %e !ro ed "&a" "&a" "&e re#a"ions&i! %e"'een ( ) and ( ) isl y n y n

( )( ) ( ) l &(

y n y n (& R n∞

=−∞

= +∑'&ic& means !oin"s circu#ar con o#u"ion ( ) is !eriodic co!ies

o+ ( ) %y e."rac"in$ "&e main !eriodl

& y n

y n

nd ( ) s #en$"& is 1l y n N $ + −

so '&en -1, "'o sequence s !oin"s circu#ar con o#u"ion

( ) is e.ac"#y "&eir #inear con o#u"ion ( ) l

& N $ &

y n y n

∴ ≥ +

( ) ( ) ( ) y n x n h n′ ′= ( ) ( )( )l y n x n h n= ∗=


51/75

03/04/16 51

( ), 3h n N =

( ), 5 x n $ =

( )l y n

5 & =

6 & =

7 & =

8 & =

rocess


52/75

03/04/16 52

-rocess( ), 0 1

#e" ( )0, 1

( ), 0 1( ) 0, 1

x n n $ x n

$ n &

h n n N

h n N n &

≤ ≤ −= ≤ ≤ −

≤ ≤ −= ≤ ≤ −

"onc%usion:We can compute %inear con.o%ution usin$ circu%ar con.o%ution if%en$th of DFTs satisf/ -1 & N $ ≥ +

x #n$

. #n$

Gero "addin5

Gero "addin5 ⊗

C;'

;'

C;' ;' x

#n$ .

#n$ x#n$ .#n$

FT

FT

FT∗

Pro#lems:


53/75

03/04/16 53

After FFT a%$orithm6 o.er%ap&a## metho# an# o.er&%ap sa.e

metho# i%% be %earne#

Pro#lems:4n practica% app%ication: / n!=5 n!>h n!6suppose 5 n!(s %en$th is 0 h n!( %en$th is

sua%%/6 0@@ 6 4f ,= 0&16 then:For short se)uence: man/ 9eros pa##e# into h n!

For %on$ se)uence: compute after a%% se)uence inputDifficu%ties ,ar$e memor/6 %on$ computation time6

so rea%&time propert/ can not be satisfie#'o%ution: #ecomposition computation on %on$ se)uence

Di.i#e# an# "on)uer Di.i#e# an# "on)uer


54/75

03/04/16 54

2 'pectrum ana%/sis usin$ DFT 2 'pectrum ana%/sis usin$ DFT

1! Appro5imation process1! Appro5imation process

; : x t

X j Ω

'amp%e ; : x n x n* ⇒

1

0

N nk

N n

X k x n W

=

= ∑

1! -rocess of spectrum ana%/sis usin$ DFT 1! -rocess of spectrum ana%/sis usin$ DFT

%&T

2! 8rror ana%/sis2! 8rror ana%/sis3! 4mportant parameters3! 4mportant parameters

/"ectrum analysis/"ectrum analysis %&T om"utation%&T om"utation%iscretization in time and%iscretization in time and

(re2uency domain(re2uency domain


55/75

03/04/16 55

Basic theor/ of Fourier TransformB &inite duration si5nal H In(inite $idth (re2uency s"ectrum


56/75

03/04/16 56

-rocess of spectrum ana%/sis usin$ DFT -rocess of spectrum ana%/sis usin$ DFT

22 8rrors of spectrum ana%/sis usin$ DFT8rrors of spectrum ana%/sis usin$ DFT

3! Fence effect3! Fence effect

; : x t

X j Ω

'amp%in$ x n

j X e

⇒ ⇒

- j

X k

X e k N

=

′ =

⇒

x n x n + n=

K j j j X e X e W e ′ ="on.o%ution

1

3

2

1!1! A%iasin$A%iasin$

2! "utoff effect2! "utoff effect

Win#o in$


57/75

03/04/16 57

1

; : x t

X j Ω

'amp%in$ ; : x n x n* ⇒

1

0

N nk

N n

X k x n W

=

=

∑

%&T



1! A%iasin$1! A%iasin$ I( condition is not met: there $ill #e s"ectrum distortion at (sJ-


58/75

03/04/16 58

22 "utoff effect of DFT


; : x t

X j Ω

x n

j

X e

⇒ ⇒

- j

X k

X e k N

=

′ =

⇒

x n x n + n=

K j j j

X e X e W e

′ =

"on.o%ution

1 2

Win#o in$



59/75

03/04/16 59

"utoff effect of DFT

!m"litude o( s2uare $a,e (unctions

s s"ectrum #e(ore and a(ter$indo$in5 #y s2uare $a,e (unction.cos; :4 n

Aea'a5e%istur#ance

'o%ution: increase 'amp%in$ points 6 or usin$ other 7in# of'o%ution: increase 'amp%in$ points 6 or usin$ other 7in# of in#o functionin#o function

2 f $8 f %/ i i $ DFT


60/75

03/04/16 60

3

; : x t

X j Ω

x n

j X e

⇒

- j

X k

X e k

N

=

=

%&T



3! Fence effect3! Fence effect = %&TH= e2ual inter,al sam"lin5 o( &T.= %&TH= e2ual inter,al sam"lin5 o( &T. /"ectrum (unction ,alue is omitted #et$een sam"lin5 "oints = inter,als./"ectrum (unction ,alue is omitted #et$een sam"lin5 "oints = inter,als.

/olution: Gero "addin5 or chan5e se2uence s len5th increase =./olution: Gero "addin5 or chan5e se2uence s len5th increase =.

3 DFT important parameters


61/75

03/04/16 61

p*

L−L

( ) BMC e

M

1 1 s

"

f F N NT T = = =

3 DFT important parametersTime-domain +s, T, , T!; +requency domainTime-domain +s, T, , T!; +requency domain FF

n

2L

:a$e 46 ana#o$ +requency +s/2

di$i"a# +requency L


62/75

03/04/16 62

3 4mportant parameter of DFT

ome im!or"an" conc#usionome im!or"an" conc#usion1 1 s

"

f F

N NT T = = =

1 1(1)

"

F T NT

= =

;- I( N unc. n/ed & incensement can only #e ac2uired #y

lo$erin5 f s 0/o s"ectrum analysis sco"e $ill #e small.

;3 f s unchan5ed & incensement can only #e ac2uired #y

increase N *p1N* that is increase sam"lin5 len5th.


63/75

03/04/16 63

3 4mportant parameter of DFT

ome im!or"an" conc#usionome im!or"an" conc#usion1 1 s

"

f F

N NT T = = =

1 1(1)

"

F T NT

= =

;- I( N unc. n/ed & incensement can only #e ac2uired #y

lo$erin5 f s 0/o s"ectrum analysis sco"e $ill #e small.

;3 f s unchan5ed & incensement can only #e ac2uired #y

increase N *p1N* that is increase sam"lin5 len5th.


64/75

03/04/16 64

%etermine sam"lin5 rate #y si5nal s hi5hest%etermine sam"lin5 rate #y si5nal s hi5hest(re2uency .(re2uency .

c f s f

-roce#ure of spectrum ana%/sis usin$ DFT

!dBust "arameters #y %&T results.!dBust "arameters #y %&T results.

%etermine e9tractin5 len5th = #y (re2uency%etermine e9tractin5 len5th = #y (re2uency

resolution.resolution.

DFT ro$rammin$ 85amp%e


65/75

03/04/16 65

01

0 1 1

1

( ( ( N N N

( ) N-) N N N

( N-) #N-)$#N-)$ N N N

W W W X W W W X

x x x N

W W W X N

÷÷ ÷÷ ÷=÷ ÷÷ ÷÷ ÷÷

K K M O M

M K

1

0

( ) ( ) 0 1 N

kn

N n

X k x n W k N −

== ≤ ≤ −

∑ k N →0 1

n N →0 1

DFT -ro$rammin$ 85amp%e

DFT 0atri5


66/75

03/04/16 66

function &'()*dft+xn,

N*lengt +xn,# . lengt of sequencen*/:N-1# . ti e sa ple(*/:N-1#

0N*exp+-" 2 pi N,#

n(*n3 (#0Nn(*0N%4n(# .calculate t e DFT 5atrix

'(*xn 0Nn(# .co pute DFT

0ore effecti.e metho#?


67/75

03/04/16 67

Fs * 7//# . Get t e analy8ed signal

T * 1 Fs#9 * 1///#t * +/:9-1, T#x * /% sin+2 pi ;/ t,#

plot+1/// t+1:2//,$x+1:2//,,#

< * dft+x, 9# . Discrete Fourier Transfor

f * Fs 2 linspace+/$1$9 2=1,#ste +f$2 abs+


68/75

03/04/16 68

l b E l #2l b E l #2


69/75

03/04/16 69

atlab Example #2atlab Example #2

f f 11 2C?2C? f f 22 2 05C?2 05C?

f f ss 10C?10C?

x)n*+sin), x)n*+sin), π π f f -- n.f n.f s s */ sin), */ sin), π π f f ,, n.f n.f s s * * !"!" #$%&'()*+,#$%&'()*+,

-. /0-. /0

2 1 0 05 f f f '% ∆ = − =

0 05 F '% ≤

min 200 s f N N

F = ⇒ =

(1)(1) 128 128 11 .(n).(n) 2323 D(E) D(E)

(2)(2) 256 256 11 .(n).(n) 2323 D(E) D(E)

(3)(3) 512 512 11 .(n).(n) 2323 D(E) D(E)

1 1 s "

f F

N NT T = = =


70/75

03/04/16 70

45-

9;n 0NDnN 6


71/75

03/04/16 71

9;n 0NDnN- 6

0

100

1 0

)*-


72/75

03/04/16 72

)*

-

9;n 0NDnN10 4


73/75

03/04/16 73

9;n 0NDnN10-4


74/75

03/04/16 74

Three Questions aboutThree Questions about DiscreteDiscrete Fourier TransformFourier Transform

Q1:Q1:

WW

HAT is DFT?HAT is DFT?

Q2:Q2: WWHY is DFT?HY is DFT?

Q3: HOQ3: HO WW to DFT?to DFT?

W !T is relationshi" #et$een %&T and other 'inds o( &ourierW !T is relationshi" #et$een %&T and other 'inds o( &ourier

Trans(orm)Trans(orm)

W * $e need %&T)W * $e need %&T)

+W to realize %&T) o$ to use %&T to sol,e the "ractical+W to realize %&T) o$ to use %&T to sol,e the "ractical

"ro#lems)"ro#lems)


75/75

HO08WO CHO08WO C

6771 +1 + !! 11 "#"#

1

chapter 2 discrete fourier transform

Documents