Parseval’sTheorem:

4.4EnergyandPowerinFrequencyDomain

Foraperiodicpowersignalx(t)

1T0

x(t) 2 dt = ck2

k=−∞

∞

∑t0

t0+T0∫

Foranon-periodicpowersignal

WhereCkistheExponen.alFourierSeries(EFS)Coefficients

x(t) 2 dt =−∞

∞∫ X( f ) 2 df

−∞

∞∫

PowerSpectralDensity:

4.4EnergyandPowerinFrequencyDomain

Foraperiodicpowersignalx(t)

1T0

x(t) 2 dt = ck2

k=−∞

∞

∑t0

t0+T0∫

Normalizedaveragepower Poweratthefrequencycomponentatf=kf0

PowerSpectralDensity:

Sx ( f ) = ck2δ( f − kf0 )

k=−∞

∞

∑ Sx (w) = 2π ck2δ( f − kw0 )

k=−∞

∞

∑

EnergySpectralDensity:

4.4EnergyandPowerinFrequencyDomain

Foranon-periodicpowersignal

x(t) 2 dt =−∞

∞∫ X( f ) 2 df

−∞

∞∫

Normalizedsignalenergy Energydensity

PowerSpectralDensity:

4.4EnergyandPowerinFrequencyDomain

Foraperiodicpowersignalx(t)

Sx ( f ) = ck2δ( f − kf0 )

k=−∞

∞

∑ Sx (w) = 2π ck2δ( f − kw0 )

k=−∞

∞

∑

Foranon-periodicpowersignal

Sx ( f ) = lim1TXT ( f )

2⎡

⎣⎢⎤

⎦⎥T− >∞

WhereXT(f)istheFouriertransformofthetruncatedsignalx(t)

xT (t) =x(t), −T / 2 < x < T / 2

0, otherwise

⎧

⎨⎪

⎩⎪

AutocorrelaEonFuncEon:

4.4EnergyandPowerinFrequencyDomain

Foraperiodicpowersignalx(t)withperiodofT0

Foranon-periodicpowersignal

T0− >∞

Foraenergysignalx(t)theautocorrelaEonfuncEonis

Rxx (τ ) =< x(t)x(t +τ )>= x(t)x(t +τ )dt−∞

∞∫

Rxx (τ ) =< x(t)x(t +τ )>=1T0

x(t)x(t +τ )dt−T0 /2T0 /2∫

Rxx (τ ) =< x(t)x(t +τ )>= lim1T0

x(t)x(t +τ )dt−T0 /2T0 /2∫

⎡

⎣⎢

⎤

⎦⎥

FourierPair

4.4EnergyandPowerinFrequencyDomain

Powerspectraldensity<--------->AutocorrelaEonfuncEon

Energyspectraldensity<--------->AutocorrelaEonfuncEon

T0Periodicsignal

LimT->∞

Non-periodicsignal

ProperEesofAutocorrelaEonFuncEon:

4.4EnergyandPowerinFrequencyDomain

Evensymmetry

Periodic

Lagzerohasthemaximumvalue

Rxx (0) ≥ Rxx (τ )

Rxx (−τ ) = Rxx (τ )

x(t) = x(t + kT ) Rxx (τ ) = Rxx (τ + kT )

ForallintegersK

SystemfuncEon(frequencyresponse)

4.5SystemFuncEonConcept

Impulseresponse(h(t)) Systemfunc.on(H(w))FourierTransform

H (w) =ℑ h(t){ }= h(t)e− jwt dt−∞

∞∫

Ingeneral,H(w)isacomplexfunc.onofw,andcanbewriIeninpolarformas:

H (w) = H (w) e jΘ(w)

SystemfuncEon(frequencyresponse)

4.5SystemFuncEonConcept

1010

PowerTransferFuncEon

Wealsocanobtaintherela.onshipbetweenthepowerspectraldensity(PSD)attheinput,andtheoutputforalinear.me-invariantnetworkas:

Py ( f ) = lim1TYT ( f )

2

T→∞

Y ( f ) = X( f )H ( f )using

weget Py ( f ) = H ( f ) 2 lim 1TXT ( f )

2

T→∞

or Py ( f ) = H ( f ) 2 Px ( f )

PowertransferfuncEon Gh ( f ) =Py ( f )Px ( f )

= H ( f ) 2

4.5SystemFuncEonConcept

4.5SystemFuncEonConcept

4.5SystemFuncEonConcept