1 convergence of fourier series can we get fourier series representation for all periodic signals....
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1
Convergence of Fourier Series
• Can we get Fourier Series representation for all periodic signals.
• I.e. are the coefficients from eqn 3.39 finite or in other words the integrals do not diverge.
• If the coefficients are finite and substituted in the synthesis eqn, will the resulting infinite series converge or not to the original signal x(t).
2
Convergence of Fourier Series
N
Nk
tjkkeat 0)(x
-:form theof series finite aby is,that -:lsexponentiacomplex relatedly harmonical of
number finite a ofn combinatiolinear aby x(t)signal periodicgiven theingApproximat
N
3
Convergence of Fourier Series
t.coefficien seriesFourier our is which .)(1a
-:ist coefficien thefor choice particular theso, be tofor this shown that becan It
error. in theenergy theminimise tois objective The
dt. |(t)e |E
period. oneover error in theenergy theuse togoing are We
-x(t)(t)x-x(t)(t)e isthat
error.ion approximat thedenote (t)eLet
Tk
2N
T
NN
N
0
0
dtetxT
ea
tjk
N
Nk
tjkk
4
Convergence of Fourier SeriesIf x(t) has a Fourier Series representation, the best approximation is obtained by truncating the Fourierseries to the desired term
.|x(t)|
-:period single aover energy finite havemust signal theseriesFourier by the blerepresenta be tosignal periodic aFor
T
2
5
Dirichlet Conditions
Series.Fourier by the blerepresenta becannot signal theso violated,being conditions 3 theseof examples theshows 199 page 3.8 Figure
finite. is itiesdiscontinu theseofeach e,Furthermor ities.discontinu ofnumber finite aonly are there time,of interval finiteany In 3)
interval. in the minima and maxima ofnumber infinitean havenot should
function The signal. theof period singleany during minima and maxima ofnumber finite a than more no are there, isthat ; variationbounded is x(t) time,of interval finiteany In )2
.dt|x(t)| i.e.
,integrable absolutely bemust x(t)period,any Over 1)
T
6
Gibbs Phenomenon• The partial sum in the vicinity of the
discontinuity exhibits ripples.• The peak amplitudes of these ripples does not
seem to decrease with increasing N.• As N increases, the ripples in the partial sums
become compressed towards the discontinuity, but for any finite value of N, the peak amplitude of the ripples remains constant. Fig 3.9 pg 201 illustrates this phenomenon.
7
Properties of Continuous-time Fourier Series.
kkk
k
k
k
BbAa cThen ,c z(t)
By(t).Ax(t) z(t) T. period with periodic be also willsignals two theofn combinatiolinear Any
ion.represenat SeriesFourier thedenotes where,b y(t)
a t) x( T. period same with thesignals periodic twodenotes y(t) and Let x(t)
-:Linearity (1)
8
Properties of Continuous-time Fourier Series.
kT
2jk-
kjk-
0
2jk-
jk-)(
0
0k
k,
k
0
aeae )t-x(ty(t)Then
,e
)(T1e)(
T1
-:have weT,duration of intervalanover range also will and ,t-t Letting
.)t-x(tT1b
b y(t) ,a t) x(
).t-x(t y(t) preserved. is signal
theof T period the x(t),signal periodic a toapplied isshift timeaWhen -:Shifting Time (2)
000
000
00000
0
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k
tT
jk
kt
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T
tjk
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aea
dexdex
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9
Properties of Continuous-time Fourier Series.
.a ) x(-t ,a x(t)if is,That
.b
ax(-t)y(t)
-:have we-m,k ngSubstituti
a x(-t)
b y(t) ,a t) x( ).x(-t y(t)
preserved. is signal theof T period the x(t),signal periodic a toapplied is reversal timeaWhen
-:Reversal Time (3)
k-
k
k
-m
2
m-
-k
2
k
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k
k
tT
jm
tT
jk
a
e
e
10
Properties of Continuous-time Fourier Series.
-k
)(k
0
0
0at) x(
same. remain the tscoefficienFourier The . positivefor frequency lfundamenta and
,
T period with periodic is t)then x(
,2
frequency lfundamenta and period thebeing T with periodic is x(t)Ifsignal. theof period thechanges scaling timeofoperation The
-:Scaling Time (4)
tjke
T
11
Properties of Continuous-time Fourier Series.
-:tionMultiplica (5)
domain.frequency in then convolutio toequivalent isdomain timein thetion Multiplica
.h)()(
b y(t),a x(t)
k
k,
k
llklbatytx
12
Properties of Continuous-time Fourier Series.
-:symmetry conjugate andn Conjugatio (6)
*k-
*
**
k
a
-:symmetry conjugate be willtscoefficien seriesFourier ).(x x(t)real, is x(t)
)(
,a x(t)
k
k
a
ThetIf
atxthen
13
Properties of Continuous-time Fourier Series.
-:Relation sParseval' (7)
k
kTadttx
T.|||)(|1 22
The total average power in a periodic signalequals the sum of the average powers in all of its harmonic components.
14
The Discrete-time Fourier Series •In the previous lectures you have learned about:-
–Fourier Series Pair Equations for the continuous-time periodic signals.–And also their properties.
•The derivation is through using:-–Signals as represented by linear combinations of basic signals with the following 2 properties.–The set of basic signals can be used to construct a broad and useful class of signals.–The response of an LTI system is a combination of the responses to these basic signals at the input.
15
Parallel Between The Continous-time & The Discrete-time.
LTI
x(t) y(t)x[n] y[n]
Sai ......aaayThen
.......aaa x-:asInput Decompose
33221
332211
Fi
16
Parallel Between The Continous-time & The Discrete-time.
)(
)(
)(
-:Time-Continuouscompute. easy to s' toResponse -
d.constructe becan Signals of Class Broad -: thatso ][or )( Choose
k
k
ktj
jtjtj
tj
k
He
dethee
et
nt
k
kkk
k
17
Discrete-time
ValueEigen Function xEigen )(
][
][
knj
r
rjnjnj
njk
He
erhee
en
k
kkk
k
18
Linear Combinations of Harmonically Related Complex Exponentials,
1).-(N.2,........1,0,k ,][
-:bygiven is N period withsignals lexponentiacomplex time-discrete all ofSet
N. with periodic is e lexponentiacomplex theE.g.
frequency. lfundamenta theis 2N,integer positivesmallest theis period lfundementa The
N].x[n x[n]ifN period with periodic is x[n]signal time-discreteA
)/2(k
)/2(
0
0
nNjknjk
nNj
een
N
19
Linear Combinations of Harmonically Related Complex Exponentials.
ts.coefficien seriesFourier theare s'alues.integer va Nany of
nscombinatioany on can takesummation thewhere
aa][ax[n]
-:bygiven is SeriesFourier time-Discretefor Notation 1.-.N0,1,2,....k where
aa][ax[n]
-:follows as ,][
lsexponentiacomplex relatedly harmonical of nscombinatiolinear aby drepresente becan x[n]sequence periodic general moreA
k
)/2(
)(k
)(kk
(N)kk
)/2(kkk
kk
)/2(k
0
0
0
nNjk
Nk
njk
Nk
nNjk
k
njk
k
nNjknjk
een
een
een
20
Discrete-time Fourier Series
njk
N
njk
Nkk
njk
kk
n
nNkjnjk
enxN
ea
eanx
eeN
0
0
0
00
][1a
unknowns N with equations N
x[n]
1-0,1,2,...Nk ,][
integeran isn since ,1e where
frequency.in periodic is
,2frequency lFundamenta
N period with in time periodic x[n]
k
N2jN
)(
0
21
Discrete-time Fourier series pair
.][1][1a
,aax[n]
. frequencyfor symbol same back the Using
)/2(k
)/2(kk
00
0
0
Nn
nNjk
Nn
njk
nNjk
Nk
njk
Nk
enxN
enxN
ee
22
Continuous-time & Discrete-time Fourier Series
njk
Nn
njk
Nkk
T
tjkk
k
tjkk
enxN
Analysis
ea
dtetxT
a
eatx
0
0
0
0
0
][1a
:
x[n]
-:Synthesis-:FS time-Discrete
.)(1
,)(
-:FS time-Continuous
k
0
23
True for Discrete-time case
kin periodic akin periodic e
nin periodic e
nin periodic ][
k
jk
jk
0
0
n
n
nx
Not true for C-T
True also for C-T
24
Convergence of Fourier Series
• Continuous time:– x(t) square integrable– or Dirichlet conditions.
][][ N,p ,][
x[n]
time-Discrete
^
termsp
^0
0
nxnxeanx
ea
njkk
njk
Nkk
25
Fourier Series Representation of Discrete-time Periodic Signals
• Fourier series of continuous-time periodic signals are infinite series.
• Fourier series of discrete-time periodic signals are of finite series in nature.
• So for the fourier series of discrete-time periodic signals, mathematical issues of convergence do not arise.
26
.2/1a and 2/1a thusNh repeat wit
,21
21
21
21 x[n]ls,exponentiacomplex twoof sum a as signal the
expanding and iprelationsheigen theUsing.sin x(t)time-continuous toanalogousexactly isresult and N period lfundamenta with periodic is x[n],/2when
i.e. Ninteger an is/2 when caseFor integers. of ratio aor integer an is /2 ifonly periodic is ][ ,sin x[n]signal heConsider t
3.10 Example
1-N1N
11
)/2()/2(
0
0
0
00
jjtsCoefficienj
aandj
aequationsynthesisFrom
ej
ej
tN
nxn
nNjnNj
27
.)21()
21(
)21
23()
21
23(1x[n]
-: termsCollecting
].[21
][23][
211 x[n]
ls,exponentiacomplex as signal theexpanding and iprelationsheigen theUsing
N. period lfundamenta with periodic is x[n],/2when
)2/2cos(cos3sin1 x[n]signal heConsider t3.11 Example
)/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
28
ts.coefficienFourier theseof phase magnitude/or imaginary real/ showsbook in the 15.321 ,
21),
21
23(a
),21
23(a and ,1a thusNh repeat wit
k. of seother valufor 0
,21
,21
)21
23(
)21
23(
,1:
3.11 Example
222211N
11-N0N
2
2
1
1
0
Figure
jaajaaj
a
jaatsCoefficien
awith
ja
ja
ja
ja
aequationsynthesisFrom
NN
k
29
0-N N
0 N-N
Re(ak)
Im (ak)
13/2
1/2
k
k
Example 3.11
30
0-N N
0 N-N
|(ak)|
phase(ak)
1
/2
k
k
2/10
1/2
2
Example 3.11
31
0-N N
|
n
Example 3.12
1
N1-N1
11
11
2
0
)/2()/2(
2
0
))(/2(
1
.1
.1
.Nnm
N
m
mNjkNNjkk
N
m
NmNjkk
eeN
a
eN
a
Let
1
1
)/2(k
1a
-:equation analysis FromNn
Nn
nNjkeN
32
0-N N
|
n
Example 3.12
1
N1-N1
2N,....N,0,k ,12
,...2,N0,k ,)/sin(
]/)2/1(2sin[1
1
1
NNa
and
NNk
NNkN
a
k
k
1
1
)/2(k
1a
-:equation analysis FromNn
Nn
nNjkeN
33
| Example 3.12
20,....,100,k ,2/110
12*2
,...20,100,k ,)10/sin(
]10/)2/12(2sin[101
k
k
a
andk
ka
04 7
-1 12 3
5 6 8 9 10
Case N=10, 2N1+1=5.1/2 1/2
34
| Example 3.12
0,....4,200,k ,4/120
12*2
,...40,200,k ,)20/sin(
]20/)2/12(2sin[201
k
k
a
andk
ka
0 47
-1 1 2 356 8
20
Case N=20, 2N1+1=5.1/4 1/4
-2 9101112
19
35
| Example 3.12
0,....8,400,k ,8/140
12*2
,...80,400,k ,)40/sin(
]40/)2/12(2sin[401
k
k
a
andk
ka
0 4 7-1 123 56 8 40
Case N=40, 2N1+1=5.1/8 1/8
-29
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