1.3 con(nuous-time signals - siueyadwang/ece351_lec2.pdf1.3.2 basic building blocks for...
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1.3Con(nuous-TimeSignals
Considerx(t),amathema/calfunc/onof/mechosentoapproximatethestrengthofthephysicalquan/tyatthe/meinstantt.inthisrela/onship,tistheindependentvariable,andxisthedependentvariable.Thesignalx(t)isreferredtoasacon-nuous--mesignalorananalogsignal.
1.3Con(nuous-TimeSignals
Somesignalscanbedescribedanaly-cally.Forexamplex(t)=5sin(12t)
1.3Con(nuous-TimeSignals
Matlabcodeofx(t)=5sin(12t):
clearcloseall%Script:matex_1_1a%%Constructavectorof/meinstants.t=linspace(0,5,1000);%Computethesignalat/meinstantsinvector"t".x1=5*sin(12*t);hh=plot(t,x1);set(hh,'LineWidth',3,'Color','r');hh=xlabel('Time(sec)');set(hh,'FontSize',26,'FontWeight','bold');hh=ylabel('x_1(t)');set(hh,'FontSize',26,'FontWeight','bold');set(gca,'FontSize',26,'FontWeight','bold');grid
1.3.1SignalOpera(onsArithme(cOpera(ons
g(t)=x(t)+A
Addi-onofaconstantoffsetAtothesignalx(t)
g(t)=Bx(t)
Mul-plica-onofaconstantgainBtothesignalx(t)
1.3.1SignalOpera(onsArithme(cOpera(ons
g(t)=x1(t)+x2(t)
Summa(onoftwosignalsx1(t)andx2(t)
1.3.1SignalOpera(onsArithme(cOpera(ons
g(t)=x1(t)*x2(t)
Mul(plica(onoftwosignalsx1(t)andx2(t)
1.3.1SignalOpera(ons
Arithme(cOpera(ons
g(t)=x(t-td)
A-meshiCedversionofthesignalx(t)canbeobtainedthrough
1.3.1SignalOpera(onsTimeshiAing
g(t)=x(at)
A-mescalingversionofthesignalx(t)canbeobtainedthrough
1.3.1SignalOpera(onsTimescaling
g(t)=x(-t)
A-meshiCedversionofthesignalx(t)canbeobtainedthrough
1.3.1SignalOpera(ons
Timereversal
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Basicbuildingblocks
² Unit-impulsefunc(on
² Unit-stepfunc(on
² Unit-pulsefunc(on
² Unit-rampfunc(on
² Unit-trianglefunc(on
² Sinusoidalsignals
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
unit-impulsefunc(on
Anarrowisusedtoindicatetheloca(onofthatundefinedamplitude.
0
1
t
δ(t)
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
unit-impulsefunc(on
δ(t) =0 if t ≠ 0
undefined if t = 0
"
#$
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(1.16)
δ(t)dt =1−∞
∞
∫ (1.17)and
Note:Eqn.1.16byitselfrepresentsanincompletedefini/onofthefunc/onsincetheamplitudeofitisdefinedonlywhen,andisundefinedatthe/meinstantt=0.TheEqn.1.17fillsthisvoid.
δ(t) t ≠ 0
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
unit-impulsefunc(on
aδ(t − t1)dt = a−∞
∞
∫and
aδ(t − t1) =0 if t ≠ t1
undefined if t = t1
#
$%
&%
Scalingand(meshiAing
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
unit-impulsefunc(on
Anarrowisusedtoindicatetheloca(onofthatundefinedamplitude.
0
1
t
δ(t)
0
a
t
aδ(t − t1)
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
unit-impulsefunc(onObtainingunit-impulsefunc/onfromarectangularpulse
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Theimpulsefunc/onhastwofundamentalproper/esthatareuseful
² Samplingpropertyoftheimpulsefunc/on
f (t)δ(t − t1) = f (t1)δ(t − t1)
² SiCingpropertyoftheimpulsefunc/on
f (t)−∞
∞
∫ δ(t − t1)dt = f (t1)
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Samplingpropertyoftheunit-impulsefunc/on
f (t)δ(t − t1) = f (t1)δ(t − t1)
Thefunc/onf(t)mustbecon/nuousatt=t1
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
SiCingpropertyoftheunit-impulsefunc/on
Thefunc/onf(t)mustbecon(nuousatt=t1.Also,
f (t)−∞
∞
∫ δ(t − t1)dt = f (t1)
f (t)t1−Δt
t1+Δt
∫ δ(t − t1)dt = f (t1)
Δt > 0
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
unit-stepfunc(on
u(t) =1 if t > 0
0 if t < 0
!
"#
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(1.30)
t
u(t)
1
u(t − t1) =1 if t > t1
0 if t < t1
"
#$
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(1.31)
t
u(t)
1
TimeshiCoftheunit-stepfunc/on
t1
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Usingtheunit-stepfunc(ontoturnasignalonataspecified(meinstant
x(t) = sin(2π f0t)u(t − t1) =sin(2π f0t) if t > t1
0 if t < t1
"
#$
%$
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Usingtheunit-stepfunc(ontoturnasignaloffataspecified(meinstant
x(t) = sin(2π f0t)u(−t + t1) =sin(2π f0t) if t < t1
0 if t > t1
"
#$
%$
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Therela(onshipbetweenunit-stepandunit-impulsefunc/ons
u(t) = δ(λ)−∞
∞
∫ dλ δ(t) = dudt
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
unit-pulsefunc(on
∏(t) =1, | t |<1/ 2
0, | t |>1/ 2
"
#$
%$
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Construc(ngaunit-pulsefunc(onfromunit-stepfunc(ons
∏(t) = u(t + 12)−u(t − 1
2)
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Construc(ngaunit-pulsefunc(onfromunit-impulsefunc(ons
∏(t) = u(t + 12)−u(t − 1
2) = δ(λ)
−∞
t+1/2
∫ dλ − δ(λ)−∞
t−1/2
∫ dλ = δ(λ)t−1/2
t+1/2
∫ dλ
δ(λ)t−1/2
t+1/2
∫ dλ =1, t − 1
2< 0,and, t + 1
2> 0
0, otherwise
#
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=1, −
12< t < 1
2
0, otherwise
"
#$$
%$$
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Construc(ngaunit-pulsefunc(onfromunit-impulsefunc(ons
∏(t) = δ(λ)t−1/2
t+1/2
∫ dλ
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
unit-rampfunc(on
r(t) =t, t ≥ 0
0, t < 0
"
#$
%$
r(t) = tu(t)or,equivalently
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Construc(ngaunit-rampfunc(onfromaunit-step
r(t) = u(λ)dλ−∞
∞
∫
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
unit-trianglefunc(on
Λ(t) =
t +1, −1≤ t < 0
−t +1, 0 ≤ t <1
0, otherwise
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%
&&
'
&&
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Construc(ngaunit-triangleusingunit-rampfunc(ons
Λ(t) = r(t +1)− 2r(t)+ r(t +1)
1.3.2BasicBuildingBlocksforCon(nuous-(meSignals
Sinusoidalfunc(onx(t) = Acos(ω0t +θ )
WhereAistheamplitudeofthesignal,andistheradianfrequencywhichhastheunitofradianspersecond,abbreviatedasrad/s.Theparameteristheini-alphaseangleinradians.Theradianfrequencycanbeexpressedaswheref0isthefrequencyinHz.
ω0
θ
ω0 = 2π f0
−θ2π f0
2π −θ2π f0
T0=1/f0
TheAcontrolsthepeakvalueofthesignal,andtheaffectsthepeaksloca(onsθ
1.3.3Impulsedecomposi(onforcon(nuous-(mesignals
roughapproxima/ontothesignalx(t)
x̂(t) = x(nΔ)n=−∞
∞
∑ (t − nΔΔ
)∏
Δ→ 0Takethelimitas x(t) = lim x̂(t)[ ]Δ→ 0
= x(λ)−∞
∞
∫ δ(t −λ)dλ
1.3.4SignalClassifica(onRealvs.complexsignals
Ø Arealsignalisoneinwhichtheamplitudeisreal-valueatall/meinstants.x(t)=uwhereuisthevoltage
Ø Acomplexsignalisoneinwhichtheamplitudemayalsohaveanimaginarypart.x(t)=xr(t)+xi(t)Cartesianform
x(t) = x(t) e j∠x ( t )orPolarform
xr (t) = x(t) cos ∠x(t)[ ] xi (t) = x(t) sin ∠x(t)[ ]and
x(t) = xr2 (t)+ xi
2 (t)!" #$1/2
and ∠x(t) = tan−1 xi (t)xr (t)#
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'(
1.3.4SignalClassifica(onPeriodicvs.non-periodicsignals
Asignalissaidtobeperiodicifitsa/sfies
x(t+T0)=x(t)Forall/meinstantst,andforaspecificvalueof.TheT0isreferredastheperiodofthesignal
T0 ≠ 0
IfasignalisperiodicwithperiodT0,thenitisalsoperiodicwithperiodsof2T0,3T0,….,kT0,….,wherekisanyinteger
1.3.4SignalClassifica(on
Example1.4:Workingwithacomplexperiodicsignal
real
imaginary
magnitude
phase
1.3.4SignalClassifica(on
Example1.6.Discusstheperiodicityofthesignalsx(t) = sin(2π1.5t)+ sin(2π2.5t)
Forthissignal,thefundamentalfrequencyisf0=0.5Hz.Thetwosignalfrequenciescanbeexpressedas
f1=1.5Hz=3f0andf2=2.5Hz=5f0
Theresul/ngfundamentalperiodisT0=1/f0=2seconds.Withinoneperiodofx(t)therearem1=3fullcyclesofthefirstsinusoidandm2=5cyclesofthesecondsinusoid.Thisisillustratedinfollowingfigure
1.3.4SignalClassifica(on
Example1.6.Discusstheperiodicityofthesignalsy(t) = sin(2π1.5t)+ sin(2π2.75t)
Forthissignal,thefundamentalfrequencyisf0=0.25Hz.Thetwosignalfrequenciescanbeexpressedas
f1=1.5Hz=6f0andf2=2.75Hz=11f0
Theresul/ngfundamentalperiodisT0=1/f0=4seconds.Withinoneperiodofx(t)therearem1=6fullcyclesofthefirstsinusoidandm2=11cyclesofthesecondsinusoid.Thisisillustratedinfollowingfigure
1.3.5EnergyandPowerDefini(onsEnergyofasignal
Withphysicalsignalsandsystems,theconceptofenergyisassociatedwithasignalthatisappliedtoaload.
+-
i(t)
v(t) Rv(t)i(t) R
+
-
E = v(t)i(t)dt = v2 (t)R
dt−∞
∞
∫−∞
∞
∫
Ifwewantedtousethevoltagev(t)asourbasisinenergycalcula/ons:
E = v(t)i(t)dt = Ri2 (t)dt−∞
∞
∫−∞
∞
∫Alterna/vely,wewantedtousethecurrenti(t)asourbasisinenergycalcula/ons:
1.3.5EnergyandPowerDefini(onsNormalizedenergyofasignal
E = x2 (t)dt−∞
∞
∫
Thenormalizedenergyofareal-valuedsignalx(t):
E = | x(t) |2 dt−∞
∞
∫
Thenormalizedenergyofacomplexsignalx(t):
Iftheintegralcanbecomputed
Iftheintegralcanbecomputed
1.3.5EnergyandPowerDefini(onsTimeaveragingoperator
Weusetheoperator<>toindicate/meaverage.
x(t) = 1T0
x(t)−T0 /2
T0 /2∫ dt
² Ifthesignalx(t)isperiodicwithperiodT0,its(meaveragecanbecomputedas.
² Ifthesignalx(t)isnon-periodic,its(meaveragecanbecomputedas.
x(t) = lim 1T
x(t)−T /2
T /2
∫ dt#
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'(T→∞
1.3.5EnergyandPowerDefini(onsTimeaveragingoperator
Px =1T0
x2(t)−T0 /2T0 /2∫ dt
² ThenormalizedaveragepowerforaperiodicsignalwithperiodT0
Px = lim1T
x2(t)−T /2T /2∫ dt
#
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'(T→∞
² Thenormalizedaveragepowerforanon-periodicsignal
1.3.5EnergyandPowerDefini(onsExample1.8.Timeaverageofapulsetrain
Computethe/meaverageofaperiodicpulsetrainwithanamplitudeofAandaperiodofT0=1s,definedbytheequa/ons
x(t) =A, 0 < t < d
0, d < t <1
!
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andx(t+kT0)=x(t+k)=x(t)forallt,andallintegersk,thesignalx(t)isshownasbelow
1.3.5EnergyandPowerDefini(onsExample1.8.Timeaverageofapulsetrain
Solu(on:
x(t) = 1T0
x(t)−T0 /2
T0 /2∫ dt
The/meaverageofx(t)canbecalculatedas
whereT0=1
x(t) = x(t)0
1
∫ dt = (A)dt + (0)dt = Add
1
∫0
d
∫
1.3.5EnergyandPowerDefini(onsPowerofasignal
Theinstantaneouspowerdissipatedintheloadresistorwouldbe. pinst (t) = v(t)i(t)
Iftheloadischosetohaveavalueof,thenormalizedinstantaneouspowercanbe
R =1Ω
pnorm (t) = x2 (t)
1.3.5EnergyandPowerDefini(ons
EnergySignalsvs.Powersignals
² Energysignalsarethosethathavefiniteenergyandzeropower
Ex <∞ and Px = 0
² Powersignalsarethosethathavefinitepowerandinfiniteenergy
Ex →∞ and Px <∞
1.3.5EnergyandPowerDefini(ons
RMSvalueofasignal
Theroot-mean-square(RMS)valueofasignalx(t)isdefinedas
XRMS = x2 (t)!" #$1/2
1.3.5EnergyandPowerDefini(onsExample1.11.RMSvalueofasinusoidalsignal
Solu(on:
Px =A2
2TheRMSvalueofthissignalis
DeterminetheRMSvalueofthesignal
x(t) = Asin(2π f0t +θ )
Recallexample1.9,thenormalizedaveragepowerofx(t)is
XRMS = Px =A2
1.3.6SymmetryProper(es
² Somesignalshavecertainsymmetryproper(esthatcouldbeu/lizedinavarietyofwaysintheanalysis.
² Moreimportantly,asignalthatmaynothaveanysymmetryproper(escans-llbewriRenasalinearcombina(onofsignalswithcertainsymmetryproper(es
1.3.6SymmetryProper(esevenandoddsymmetry
² Areal-valuesignalissaidtohaveevensymmetryifithastheproperty
x(−t) = x(t)
² Areal-valuesignalissaidtohaveoddsymmetryifithastheproperty
x(−t) = −x(t)
1.3.6SymmetryProper(es
Decomposi(onintoevenandoddcomponents
² Evencomponent
xe(t) =x(t)+ x(−t)
2
² oddcomponent
x(t) = xe(t)+ xo(t)
xe(−t) = xe(t)
xo(t) =x(t)− x(−t)
2xo(−t) = −xo(t)
1.3.6SymmetryProper(esExample1.13.Evenandoddcomponentofarectangularpulse
Determinetheoddandevencomponentsoftherectangularpulsesignal
x(t) =∏(t − 12) =
1, 0 < t <1
0, otherwise
#
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&%
Solu(on:
xe(t) =∏(t − 1
2)+∏(−t − 1
2)
2=12∏( t2)
xo(t) =∏(t − 1
2)−∏(−t − 1
2)
2
1.3.6SymmetryProper(esExample1.13.Evenandoddcomponentofasinusoidalsignal
Determinetheoddandevencomponentsoftherectangularpulsesignal
x(t) = 5cos(10t +π / 3)
Solu(on:
xe(t) = 2.5cos(10t)
1.3.6SymmetryProper(esExample1.13.Evenandoddcomponentofasinusoidalsignal
Determinetheoddandevencomponentsoftherectangularpulsesignal
x(t) = 5cos(10t +π / 3)
Solu(on:
xo(t) = −4.3301sin(10t)
1.3.6SymmetryProper(esSymmetryproper(esforcomplexsignals
² Acomplex-valuesignalissaidtohaveconjugatesymmetryifitsa/sfies
x(−t) = x*(t)
² Acomplex-valuesignalissaidtohaveconjugatean(symmetryifitsa/sfies
forallt.
x(−t) = −x*(t) forallt.
x(t) = xE (t)+ xO (t)
Conjugatesymmetriccomponent
xE (t) =x(t)+ x*(−t)
2
Conjugatean(symmetriccomponent
xO (t) =x(t)− x*(−t)
2
1.3.6SymmetryProper(esExample1.15.SymmetryofacomplexexponenNalsignal
Considerthecomplexexponen/alsignalx(t) = Ae jωt
Solu(on:
A:realWhatsymmetrypropertydoesthissignalhave,ifany?
Timereversethesignal: x(−t) = Ae− jωt
Conjugatethesignal: x*(t) = (Ae jωt )* = Ae− jωt
x(−t) = x*(t)Since,thesignalisconjugatesymmetric
1.3.7Graphicalrepresenta(onofsinusoidalsignalsusingphasor
x(t) = Acos(2π f0t +θ )
LetthephasorXbedefinedas
X = Ae jθΔ
sothat
x(t) = Re Ae j (2π f0t+θ ){ }= Re Xe j2π f0t{ }
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