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
7/28/2019 Discrete time fourier series.pptx
2/27
Plot of X[n]=cos(nw)
W=5 W=5+2
W=5+4 W=5+6
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
4/27
x[n]= cos (2n/12)
rationalw
Nw
,12
1
2
12,12
2
0
0
7/28/2019 Discrete time fourier series.pptx
5/27
x[n]= cos (n/6)
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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(
( )
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
20/27
| xamp e1 8 1 8
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
22/27
7/28/2019 Discrete time fourier series.pptx
23/27
23
Nl
lkl
kkNr
banynx
bNarnyrx
][][
tionMultiplica6.
][][y[n]*x[n]
ion5.Convoluta[-n]x
reversal4.time
k-
7/28/2019 Discrete time fourier series.pptx
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'
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
7/28/2019 Discrete time fourier series.pptx
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
Top Related