1.3 con(nuous-time signals - siueyadwang/ece351_lec2.pdf1.3.2 basic building blocks for...

1.3Con(nuous-TimeSignals

Considerx(t),amathema/calfunc/onof/mechosentoapproximatethestrengthofthephysicalquan/tyatthe/meinstantt.inthisrela/onship,tistheindependentvariable,andxisthedependentvariable.Thesignalx(t)isreferredtoasacon-nuous--mesignalorananalogsignal.


Somesignalscanbedescribedanaly-cally.Forexamplex(t)=5sin(12t)


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)


g(t)=x1(t)+x2(t)

Summa(onoftwosignalsx1(t)andx2(t)


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


unit-impulsefunc(on

Anarrowisusedtoindicatetheloca(onofthatundefinedamplitude.

0

1

t

δ(t)


unit-impulsefunc(on

δ(t) =0 if t ≠ 0

undefined if t = 0

"

#$

%$

(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


unit-impulsefunc(on

aδ(t − t1)dt = a−∞

∞

∫and

aδ(t − t1) =0 if t ≠ t1

undefined if t = t1

#

$%

&%

Scalingand(meshiAing


unit-impulsefunc(on

Anarrowisusedtoindicatetheloca(onofthatundefinedamplitude.

0

1

t

δ(t)

0

a

t

aδ(t − t1)


unit-impulsefunc(onObtainingunit-impulsefunc/onfromarectangularpulse


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)


Samplingpropertyoftheunit-impulsefunc/on

f (t)δ(t − t1) = f (t1)δ(t − t1)

Thefunc/onf(t)mustbecon/nuousatt=t1


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


unit-stepfunc(on

u(t) =1 if t > 0

0 if t < 0

!

"#

$#

(1.30)

t

u(t)

1

u(t − t1) =1 if t > t1

0 if t < t1

"

#$

%$

(1.31)

t

u(t)

1

TimeshiCoftheunit-stepfunc/on

t1


Usingtheunit-stepfunc(ontoturnasignalonataspecified(meinstant

x(t) = sin(2π f0t)u(t − t1) =sin(2π f0t) if t > t1

0 if t < t1

"

#$

%$


Usingtheunit-stepfunc(ontoturnasignaloffataspecified(meinstant

x(t) = sin(2π f0t)u(−t + t1) =sin(2π f0t) if t < t1

0 if t > t1

"

#$

%$


Therela(onshipbetweenunit-stepandunit-impulsefunc/ons

u(t) = δ(λ)−∞

∞

∫ dλ δ(t) = dudt


unit-pulsefunc(on

∏(t) =1, | t |<1/ 2

0, | t |>1/ 2

"

#$

%$


Construc(ngaunit-pulsefunc(onfromunit-stepfunc(ons

∏(t) = u(t + 12)−u(t − 1

2)


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

#

$%%

&%%

=1, −

12< t < 1

2

0, otherwise

"

#$$

%$$


Construc(ngaunit-pulsefunc(onfromunit-impulsefunc(ons

∏(t) = δ(λ)t−1/2

t+1/2

∫ dλ


unit-rampfunc(on

r(t) =t, t ≥ 0

0, t < 0

"

#$

%$

r(t) = tu(t)or,equivalently


Construc(ngaunit-rampfunc(onfromaunit-step

r(t) = u(λ)dλ−∞

∞

∫


unit-trianglefunc(on

Λ(t) =

t +1, −1≤ t < 0

−t +1, 0 ≤ t <1

0, otherwise

$

%

&&

'

&&


Construc(ngaunit-triangleusingunit-rampfunc(ons

Λ(t) = r(t +1)− 2r(t)+ r(t +1)


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)#

$%

&

'(

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


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


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#

$%&

'(T→∞

1.3.5EnergyandPowerDefini(onsTimeaveragingoperator

Px =1T0

x2(t)−T0 /2T0 /2∫ dt

²  ThenormalizedaveragepowerforaperiodicsignalwithperiodT0

Px = lim1T

x2(t)−T /2T /2∫ dt

#

$%&

'(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

!

"#

$#

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

#

$%

&%

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


x(t) = 5cos(10t +π / 3)

Solu(on:

xe(t) = 2.5cos(10t)

1.3.6SymmetryProper(esExample1.13.Evenandoddcomponentofasinusoidalsignal


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{ }

1.3 con(nuous-time signals - siueyadwang/ece351_lec2.pdf1.3.2 basic building blocks for...

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