1. 2 ship encountering the superposition of 3 waves
TRANSCRIPT
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2
Ship encountering the superposition of 3 waves.
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Fourier Series Representation of Periodic Signals
•Signals can be represented as 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.
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Fourier Analysis
• Both properties are provided for by the complex exponential signals in the continuous and discrete time
• I.e. Signal of the form:-
numbers.complex are z & s where
time.discrete in the z
time.continuousin n
ste
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The principle of superposition for linear systems.
LTI Systemx(t)
x[n]
y(t)
y[n]
case. time-discrete for the
...)(a)(ay(t) :Then
linear is system and
)( )(
......)(a)(a x(t)if
2211
k
2211
Similarly
tt
tt
tt
k
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Criteria for choosing a set of basic signals in terms of which to decompose the input
compute easy to are ' toresponses -
.' of
ncombinatiolinear a as dconstructe
becan signals of class broad a-
: that so ][or )( Choose
...)(a)(ay(t) :Then
linear is system and
)( )(
......)(a)(a x(t)if
k
k
2211
k
2211
s
s
nt
tt
tt
tt
kk
k
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Choice for basic signals that led to the convolution integral and convolution sum for
LTI Systems
Sumn Convolutio
][][
][][ :T-D
Integraln Convolutio
)()(
)()( :T-C
k
k
knhn
knn
ktht
ktt
k
k
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Complex exponentials as a set of basic signals
.transforms-z complex z
s transformLaplace complex s
][ 1|z| :T-D
)( s :T-C
:AnalysisFourier
complex z ][
complex s )(
k
k
kk
kk
kk
k
nj
tjk
nk
tsk
k
k
k
en
etj
zn
et
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Eigenfunction (est , zn) & Eigenvalues (H(s) , H(z))
LTI Systemx(t)
x[n]
y(t)
y[n]
amplitude.in change aonly with
lexponentiacomplex same theisinput lexponentia
complex a tosystem LTIan of response the
fact that thefrom stems systems LTI ofstudy
in the lsexponentiacomplex of importance The
H(z)zy[n] z x[n]: timediscrete
)()(e x(t): timenn
st
stesHtycontinuous
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Showing complex exponential as eigenfunction of system
.)(constantcomplex a is H(s)
whereH(s)ey(t) form theof is e toreponse the
converge, RHSon integral that thessuming
.)(
.)(
)()()(
einput x(t) with integraln convolutio theFrom
stst
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)(
st
deh
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dehe
deh
dtxhty
s
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ts
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Showing complex exponential as eigenfunction of system
k
kn
kn
n
zkh
A
zkhz
zkh
knxkhny
z
][constantcomplex a is H(z)
whereH(z)zy[n] form theof is z toreponse the
converges, RHSon summation that thessuming
][
][
][][][
,input x[n] with sumn convolutio theFrom
-:case Time-D for theSimilarly
nn
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Example x(t) to be linear combination of 3 complex exponentials
tsts
tsts
tsts
tststs
esHe
esHe
esHe
eee
33
22
11
321
)(aa
)(aa
)(aa
-:is separatelyeach toresponse the
property,ion eigenfunct theFrom
aaax(t)
333
222
111
321
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Example x(t) is linear combination of 3 complex exponentials
kk
kk
kk
kk
332211
321
.)(ay[n] be loutput wil then the
,ainput x[n] if time-Dfor similarly
)(ay(t) be loutput wil then the
,ainput x(t) if Generally,
)(a)(a)(ay(t)
-:responses theof sum theis sum the toresponse the
property,ion superposit theFrom
aaax(t)
321
321
nkk
nk
tsk
ts
tststs
tststs
zzH
z
esH
e
esHesHesH
eee
k
k
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Fourier Series Representation of Continuous-time Periodic Signals.
A signal is periodic if x(t)=x(t+T), for all t.T is the fundamental period.
,e x(t)lexponentiacomplex and
)cos( x(t)sinusoidal signals,
periodic basic 2 before studied have We
frequency.radian lfundamenta theis ,2
0j
0
0
t
t
T
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.components harmonics second theis 2k component,
harmonicfirst theis 1k T. period with periodic also is and series
Fourier theis, x(t)lexponentia
complex relatedly harmonical ofn combinatiolinear a Thus,
T). offraction a is period lfundamenta the
2,kfor ( T with periodic iseach and of multiple a is
thatfrequency lfundamenta a has signals theseofEach
2,...1,0,k ,)(
ls,exponentiacomplex relatedly harmonical ofset the
is signal lexponentiacomplex basic with thisAssociated
-k
2
-k
,0
2
k
00
0
tT
jk
ktjk
k
tT
jktjk
eaea
eet
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above. x(t)signal peroidic aconstruct toup added becan
waveformscosine thesehow sillustrate OWN 188 pg 3.4 Figure
.6cos3
24cos2cos
2
11)(
-:below form cosine get the to
),e(e2
1cos ip,relationshEuleruse
.3
1,
2
1,
4
1,1 where
,)(
3.2 Example
j-j
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3
3
2
ttttx
aaaaaaa
eatxk
tjkk
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Trigonometric forms of Fourier Series.
][)( -:3.25Eqn From
.or ,
3.25, eqn. with comparing and (t)
k,-by k replacing ,(t)
x(t).(t) x x(t),realFor ,(t)
-:have we3.25, eqn. of sideboth ofn conjugatio Taking
. SeriesFourier the
of Form lExponentiaComplex theis
Equation) (Synthesis 25.3.........)(
1k0
**k-
*
*
***
00
0
0
0
0
tjkk
tjkk
kkk
k
tjkk
k
tjkk
k
tjkk
k
tjkk
eaeaatx
aaaa
eax
eax
eax
eqneatx
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Trigonometric forms of Fourier Series.
].sincos[2)(
,a lettingby formother the
, )cos(A2ax(t)
}.2)( ,a Expressing
}.2][)(
01
00
k
1k0k0
1
)(0k
10
1k
*0
0
000
tkCtkBatx
jCB
tk
eRe{AatxeA
eRe{aaeaeaatx
kk
k
kk
k
k
tkjk
jk
k
tjkk
tjktjkk
kk
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Solving for Fourier Series Coefficients(Analysis)
TTT tnkj
k
T tnkjk
tjn
k
T tjntjkk
tjn
tjntjk
kk
tjn
tjntjk
kk
tjn
tjn
tdtnkjtdtnkdte
dteadtetx
dteeadtetx
dteeadtetx
eeaetx
e
0 00 00
)(
0
)(T
0
0
T
0
T
0
T
0
0
)sin()cos(However
3.34quation .........e ][)(
][)(
)(
have we,2
T to0 fromequation of sidesboth gIntegratin
.)(
-:by eqn Synthesis SeriesFourier theof sideboth gMultiplyin
0
00
000
000
000
0
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Solving for Fourier Series Coefficients(Analysis)
T0 dt
t0
jnx(t)e
T
1n
a
.
-:nk taking3.34Equ From
nk 0.
nk T,T0 dt
t0
n)j(ke
-:have weOverall
TT0 0dtjT
0 1dtT0 dt
t0
n)j(ke
0t0
n)sin(k 1andt 0
n)cos(k n,kFor
zeros. are termcosine and sinefor
periods lfundamenta of numbersinteger eover whol nsintegratio The
).|n-k|
T( period lfundamenta with
sinusoids periodic aret n)sin(k andt n)cos(k n,kFor 00
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Summarizing for Fourier Series Pair Representation
tT
jk
kk
tjk
kk eaeatx
2
0)(
TT
dtt
T
2jk
x(t)eT
1dt
t0
jkx(t)e
T
1k
a
Fourier Series Synthesis:- eqn 3.38
Fourier Series Analysis or F.S. Coefficients 0r Spectral Coefficents:- eqn 3.39
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.2||0,2
1,
2
1
),2
11(),
2
11(,1
-:are tscoefficien seriesFourier
)2
1)
2
1)
2
11()
2
11(1)(
][2
1][][
2j
11x(t)
-:)e(e2
1sin and
)e(e2
1cosfor iprelationshEuler use
)4
2cos(cos2sin1)(
3.4 Example
)4/(2
)4/(2
110
2)4/(2)4/(
4/24/2
j-j
j-j
000
0000
000000
kforaeaea
ja
jaa
eeeeej
ej
tx
eeeeee
j
ttttx
kjj
tjjtjjtjtj
tjtjtjtjtjtj
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0
0.2
0.4
0.6
0.8
1
1.2
k= -3 -2 -1 0 1 2 3 4
Magnitude of Fourier Coefficients.
|| ka
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-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
k= -3 -2 -1 0 1 2 3 4
Phase of Fourier Coefficients.
kaangle
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Example 3.5
T/2T1-T/2 -T1
This periodic signal x(t) repeats every T seconds.x(t)=1, for |t|<T1 , and x(t)=0, for T1 <|t|< T/2
Fundamental period= T,Fundamental frequency Choosing the period of integration to be between-T/2 and +T/2. Use eqn 3.39 to get at Fourier Series Coefficients.
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Example 3.5 continued
T
Tdt
T
T
TT
12
T
10
a
0k i.e. first, period, aover
valueaverageor ermconstant t dc, get the usLet
dtt
T
2jk
x(t)eT
1dt
t0
jkx(t)e
T
1k
a
1
1
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Example 3.5 continued
k
Tk
Tk
Tk
j
ee
Tk
eTjk
TjkTjk
TT
tjkT
T
TT
)sin()sin(2]
2[
2k
a
,|1
dtt
0jk
eT
1k
a
0.k have we
-:harmonicsorder higher andorder first lfundamentaFor
dtt
T
2jk
x(t)eT
1dt
t0
jkx(t)e
T
1k
a
10
0
10
0
0
1010
1
1
01
1
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Example 3.5 Continued
• In this example the coefficients are real values.
• Generally the coefficients are complex.
• In this case we can represent a single plot of magnitude of coefficient against k.
• Generally we will have the magnitude plot and the phase plot of the coefficients.
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Example 3.5 Continued
.2
1a 3.42eqn from and
zeros. all are sa' thesk'even For
,5
1,
3
1,
1
0,k ,)2/sin(
a -: 3.44eqn From
,2
2 ,T4TFor
0
553311
k
1101
aaaaaa
k
k
TT
T
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Plot of the Coefficients with T1 Fixed and T varied.
2
1x
a) T=4T1
b) T=8T1
c) T=16T1