Discrete time fourier series.pptx

7/28/2019 Discrete time fourier series.pptx

1/27

Periodicity of Discrete time signal

njwenx 0][ Periodic signal with fundamental frequency w0

][][ 000 2)2(

1 nxeeeenxnjwnjwnjnwj

Signal with fundamental frequency w0 is identical to signal with

fundamental frequency w0+2

Frequency .......6,4,2 0000 wwww


2/27

Plot of X[n]=cos(nw)

W=5 W=5+2

W=5+4 W=5+6


3/27

Periodicity in time

njwenx 0][ This is periodic with time period N=

NjwnjwNnjwnjweeeenx 0000

)(][

2

wly,equivalentor

integermwhere2w,requiresthis

1periodicbetofor this

0

0

0

N

m

vemN

eNjw

Rational

number

2w 0


4/27

x[n]= cos (2n/12)

rationalw

Nw

,12

1

2

12,12

2

0

0


5/27

x[n]= cos (n/6)


6/27

][][][

2wwithperiodic][2kforwith wperiodic][1kfor

.........2,1,0kfor][onsider

N.periodandfrequencylFundamentawithperiodic:

kNk

2

2

)(

Nk

0

2

2

01

k

0

0

00000

0

0

0

0

neneeeeeeen

enen

enc

we

njkw

njnjkwn

NjN

njkwnjNwnjkwnwNkj

nwj

njw

njkw

njw


7/27

Nkso,si naluniqueNonlyhas][:conclusion

][][][][

][][][][

][][][][

][][][

][][][

][][][

][][

][][

][][

][][.........2,1,0kand3Nconsider

][][

.........2,1,0kfor][

N.periodandfrequencylFundamentawithperiodic:

0

0

0

0

k

25811

14710

0369

258

147

036

25

14

03

k3k

kNk

k

0

njkw

njkw

njkw

njw

en

nnnn

nnnn

nnnn

nnn

nnn

nnn

nn

nn

nn

nn

nen

en

we

For N=3 , only 3 distinct signals are defined

In general only N values of K exist


8/27

8

Fourier series of discrete time signal

ts.coefficienseriesFouriertheares'alues.integer vaNanyof

nscombinatioanyoncan takesummationthewhere

aa][ax[n]

-:bygivenisSeriesFouriertime-DiscreteforNotation

1.-.N0,1,2,....kwhere

aa][ax[n]

k

)/2(

)(

k

)(

kk

(N)k

k

)/2(

kkkk

k

0

0

nNjk

Nk

njkw

Nk

nNjk

k

njkw

k

een

een


9/27

9

0r-kor

know,we

ax[n]

summation,oforderinginterchang

ax[n]

also

.....2,,0

0

also,

aax[n]

)(

)/2)((

)(

)/2)((

)(

k

)/2(

)(

)(

)/2)((

)(

k

)/2(

)(

)(

)/2(

)/2(

)(

k

)(

k0

fNe

ee

ee

NNk

otherwise

forNe

ee

Nn

nNrkj

Nk

nNrkj

Nn

nNjr

Nn

Nn

nNrkj

Nk

nNjr

Nn

Nn

nNjk

nNjk

Nk

njkw

Nk


10/27

equationAnalysisx[n]1

equationsynthesisaax[n]

NOW,

x[n]1

x[n]

,rkfor

)/2(

)(

)/2(

)(

k

)(

k

)/2(

)(

)(

)/2(0)/2(

)(

0

nNjK

Nn

K

nNjk

Nk

njkw

Nk

nNjr

Nn

r

rNn

nNj

r

nNjr

Nn

eN

a

ee

e

N

a

Naeae

ak=ak+N


11/27

11

.2/1aand2/1athusNhrepeat wit

,

2

1

2

1

21

21

21

21x[n]

sinx[n]signalheConsider t

3.10Example

1-N1N

11

)/2()/2(

0

00

jjtsCoefficienj

aand

j

aequationsynthesisFrom

ej

ej

ej

ej

n

njnjnNjnNj

3 10E l


12/27

1/2j

-1/2j

k

ak

0 1 2 3 4 5-1-2

.2/1aand2/1athusNhrepeat wit

,2

1

2

1

2

1

2

1

x[n]

)5

2sin(sinx[n]signalheConsider t

3.10Example

1-N1N

11

)5/2()5/2(

0

jjtsCoefficien

jaand

jaequationsynthesisFrom

ejej

nn

njnj


13/27

13

.)2

1()

2

1(

)2

1

2

3()

2

1

2

3(1x[n]

-:termsCollecting

].[2

1

][2

3

][2

1

1x[n]

ls,exponentiacomplexassignalthe

expandingandiprelationsheigentheUsingN.periodlfundamentawithperiodicisx[n],/2when

)2/2cos(cos3sin1x[n]signalheConsider t

3.11Example

)/2(22/)/2(22/

)/2()/2(

)2//4()2//4(

)/2()/2()/2()/2(

0

000

nNjjnNjj

nNjnNj

NnjNnj

nNjnNjnNjnNj

eeee

ej

ej

ee

eeeej

N

nnn

1313


14/27

14

jaajaaj

a

jaatsCoefficien

awith

ja

ja

ja

ja

a

eeee

ej

ej

NN

k

nNjjnNjj

nNjnNj

2

1,

2

1),

2

1

2

3(a

2

1

2

3(aand,1athusNhrepeat wit

k.ofseother valufor0

,2

1

,2

1

)2

1

2

3(

)2

1

2

3(

,1

.)2

1()

2

1(

)2

1

2

3()

2

1

2

3(1x[n]

222211N

11-N0N

2

2

1

1

0

)/2(22/)/2(22/

)/2()/2(

( )


15/27

15

0-N N

0 N-N

Re(ak)

Im (ak)

1

3/2

1/2

k

k

Example 3.11

N=10

|( )| xamp e


16/27

16

0-N N

0 N-N

|(ak)|

phase(ak)

1

/2

k

k

2/10

1/2

/2

xamp e .

xamp e


17/27

17

0-N N n

xamp e .

1

N1-N1

1

1

11

2

0

)/2()/2(

2

0

))(/2(

1

.1

.1

.Nnm

N

m

mNjkNNjk

k

N

m

NmNjk

k

eeN

a

eN

a

Let

1

1

)/2(

k

1a

-:equationanalysisFrom

Nn

Nn

nNjkeN

Discrete time

square wave

| xamp e


18/27

18

0-N N

|

n

xamp e .

1

N1-N1

2N,....N,0,k,12

,...2,N0,k,)/sin(

]/)2/1(2sin[1

1

1

N

Na

and

NNk

NNk

Na

k

k

1

1

)/2(

k

1a

-:equationanalysisFrom

Nn

Nn

nNjkeN

| xamp e1 2 1 2


19/27

19

| xamp e .

20,....,100,k,2/110

12*2

,...20,100,k,)10/sin(

]10/)2/12(2sin[101

k

k

a

and

kka

04 7

-1 1

2 3

5 6 8 9 10

Case N=10, 2N1+1=5.

1 2 1 2


20/27

| xamp e1 8 1 8


21/27

21

| xamp e .

0,....8,400,k,8/140

12*2

,...80,400,k,)40/sin(

]40/)2/12(2sin[401

k

k

a

and

kka

0 4 7-1 1 23 5 6 8 40

Case N=40, 2N1+1=5.

1 8 1 8

-2

9


22/27


23/27

23

Nl

lkl

kkNr

banynx

bNarnyrx

][][

tionMultiplica6.

][][y[n]*x[n]

ion5.Convoluta[-n]x

reversal4.time

k-


24/27

oddandimaginaryareFSCthenoddandrealisx[n]if

evenandrealaevenandrealx[n]if,

},{aIm}{aI

},Re{a}Re{a,a

-:symmetryconjugatebewilltscoefficienseriesFourier

real,isx[n]

][

,ax[n[

symmetryconjugateande7.conjugat

k

k-k

k-k

*

k

**

k

kkkk

k

k

aaaa

m

a

The

If

anxthen

R l tiP l'


25/27

25

-:RelationsParseval'

Nk

k

Nn

anx

N

.||.|][|1 22

Average power in one period

2

kaAverage power in kth harmonic component is

Parsevals relation :Total average power in a periodic signal equals sum

of average powers in all of its harmonic components

2

1

2

12

21 ][1

1nintervaloversignalofpoweraverage

n

nn

nxnn

nn

lf di hi li di[ ]bd[ ]L


26/27

numberrationalNKm

Km

Km

m

mk

N

N

21

221121

2211

22222

11111

21

21

21

2

1NNhavemustwethus,

]N[nx]N[nxN][nxN][nxN]x[n

,Nperiodwithperiodicbefor x[n]to

]N[nx]N[nxx[n]integervek,]N[nx]N[nx[n]x

integervem,]N[nx]N[nx[n]x

:solution

periodic?isitifx[n]ofperiodlfundamenta

theiswhatandperiodic,[n]x[n]x[n]xsumtheis

conditionshatly.Under wrespective,Nand,Nperiods

lfundamentawithsignalsperiodic[n]bexand[n]Let x


27/27

24

4433

123

123

124

124

123

123

124

124

4/4/3/3/

0

2

1

2

1,

2

1,

2,

2,

222

1

2

1][

2

1

2

1

2

1

2

1][

][2

1][

2

1

4

nsin

3

ncosx[n]

formula,eulerusing

12wand24Nofperiodhas][

8Nperiodwithperiodicis4

nsin

6Nperiodwithperiodicis3

ncos

4

nsin

3

ncosx[n]ofFSfind

kk

njnjnjnj

njnjnjnj

njnjnjnj

aa

aaj

aj

athus

ej

ej

eenx

ej

ej

eenx

eej

ee

nx

Discrete time fourier series.pptx

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Transcript of Discrete time fourier series.pptx