北京邮电大学：《通信系统原理》课程教学课件（PPT讲稿）第二章 信号 Signals and Spectra

Chap.2 Signals and SpectraObject:mathematical tools to describe and analyze the signalsFourier series and transformImportant function:Dirac delta function,rectangular functionperiodicfunction and sincfunction and their Fourier transformsfrequency analyze (time function and his spectrum)some properties of signal (DC value ,root mean sguare value,..)power spectral densityand autocorrelationfunctionlinear systems:linear time-invariant systems,impulseresponse,transfer function,distortionless transmissionbandwidth concept:baseband,passband and bandlimited signalsandnoise*samplingtheorem(dimensionalitytheorem)summary

Object:mathematical tools to describe and analyze the signals Fourier series and transform Important function:Dirac delta function,rectangular function, periodic function and sinc function and their Fourier transforms frequency analyze (time function and his spectrum) some properties of signal (DC value ,root mean square value,.) power spectral density and autocorrelation function linear systems:linear time-invariant systems,impulse response,transfer function,distortionless transmission bandwidth concept:baseband,passband and bandlimited signals and noise *sampling theorem (dimensionality theorem) summary Chap.2 Signals and Spectra

2-1. Properties of Signals and Noise Signal:desired part of waveforms;Noise:undesired part: Electric signal's form:voltage v(t) or current i(t)(time function): In this chapter,all signals are deterministic: But in communication systems,we will be face thestochastic waveformsDeterministic resultsstochastic results byanalogySignal analysis:first importance

• Signal:desired part of waveforms; Noise:undesired part • Electric signal’s form:voltage v(t) or current i(t) (time function) • In this chapter,all signals are deterministic. • But in communication systems,we will be face the stochastic waveforms Deterministic results stochastic results by analogy Signal analysis:first importance 2-1. Properties of Signals and Noise

Physically realizable waveformsNon zero values over a finite time intervalnon zero values over a finite frequency intervala continuous time functiona finite peak value.only real valuesIn general,the waveform is denoted by w(t)When t→±oo,we have w(t) →0,but w(t) is defined over(+80,-8)The math model of waveform can violate some or all aboveconditions.Ex. w(t)=sinot,physically this waveform can not be existed

• Non zero values over a finite time interval • non zero values over a finite frequency interval • a continuous time function • a finite peak value • only real values In general,the waveform is denoted by w(t) When t→±∞,we have w(t) →0,but w(t) is defined over (+∞,-∞) The math model of waveform can violate some or all above conditions. Ex. w(t)=sinωt,physically this waveform can not be existed. Physically realizable waveforms

The classifications of waveformsWaveforms:signal or noisedigital or analogdeterministic or nondeterministic(stochastic)physically realizable or nonphysically realizablepower type or energy typeperiodic ornonperiodicPower type:the average power of the waveform is finite(mathmodel)Energy type:the average energy of the waveform is finite(allphysically realizable signal)

Waveforms: • signal or noise • digital or analog • deterministic or nondeterministic(stochastic) • physically realizable or nonphysically realizable • power type or energy type • periodic or nonperiodic Power type:the average power of the waveform is finite(math model) Energy type:the average energy of the waveform is finite(all physically realizable signal) The classifications of waveforms

Some important math operationsTime average operator:dc(direct current) value of timefunctionDefinition: the time average operation is given by: =lim1/T /2] T2[]dt is time average operator. The operator is linear.(Why?)Definition : w(t) is periodic with period To ifw(t)=w(t+ To) for all twhere To is smallest positive number that satisfies aboverelationship.Theorem:if w(t) is periodic,the time average operation can bereduced to <[-] =1/T.T/2-a/ T/2+[ jdt1T.2where T is period of w(t)

• Time average operator:dc(direct current) value of time function Definition: the time average operation is given by: 〈[·]〉=lim1/T - T/2∫ T/2[·]dt 〈[·]〉is time average operator. The operator is linear.(Why?) Definition : w(t) is periodic with period T0 if w(t)=w(t+ T0 ) for all t where T0 is smallest positive number that satisfies above relationship. Theorem:if w(t) is periodic,the time average operation can be reduced to 〈[·]〉=1/T-T/2-a ∫ T/2+a[·]dt where T is period of w(t) Some important math operations

Dc valueDefinition:the dc value of w(t) is given by its time average,(w(t))Wde= (w(t) =lim1/T r/2J T/2w(t)dtorWdc= =i(t)+circuitv(t)

• Definition:the dc value of w(t) is given by its time average, 〈w(t)〉. Wdc=〈w(t)〉=lim1/T - T/2∫ T/2w(t)dt or Wdc=〈w(t)〉=1/(t2 -t1 )∫ w(t)dt Power • Definition:the instantaneous power is given by: p(t) = v(t)i(t) and the average power is : P== Dc value + v(t) - i(t) circuit

Rms Value and Normalized Power Definition:the root mean square (rms) value of w(t)isgiven by:Wrms=[j1/2Theorem:if a load (R) is resistive,the average power is:P= /R=R= V2rms/R=I 2rmsR: Definition:if R-1Q,the average power is called normalizedpower.Then i(t)= v(t) = w(t) and P=Energy and Power Waveforms:: Definition:w(t) is a power waveform if and only if thenormalized power P is finite and nonzero(O<P<oo).Definition:the total normalized energy is given by

• Definition:the root mean square (rms) value of w(t)is given by: Wrms=[]1/2 • Theorem:if a load (R) is resistive,the average power is: P= /R= R= V2 rms/R=I 2 rmsR • Definition:if R=1Ω,the average power is called normalized power. Then i(t) = v(t) = w(t) and P= Energy and Power Waveforms: • Definition:w(t) is a power waveform if and only if the normalized power P is finite and nonzero(0<P<∞). • Definition:the total normalized energy is given by: Rms Value and Normalized Power

T/21w?(t)dtE = limT→8 TT3Definition:w(t) is an energy waveform if and only if thetotal normalized energy is finite and nonzero (O<E<oo),Waveform: power signal or energy signalAveragePower=0Energy finitePower finiteEnergy=00Physically realizable waveform:Energy waveformPeriodic waveform:Power waveformDecibel. Definition:the decibel gain of a circuit is given bydB=10log(average power out/average power in)=10log(Pout/Pin)

• Definition:w(t) is an energy waveform if and only if the total normalized energy is finite and nonzero (0<E<∞). Waveform: power signal or energy signal Energy finite Average Power=0 Power finite Energy=∞ Physically realizable waveform:Energy waveform Periodic waveform:Power waveform Decibel • Definition:the decibel gain of a circuit is given by dB=10log(average power out/average power in) =10log(Pout/Pin)  − → = / 2 2 2 ( ) 1 lim T T T w t dt T E

For normalized power case(R=1),we have:dB=20log(Vrms out /Vrms in)= 20log(Irms out /Irms in)

For normalized power case(R=1Ω),we have: dB=20log(Vrms out /Vrms in)= 20log(Irms out /Irms in)

Fourier Transform and Spectraw(t),voltage or current,time function+analysis in time-domain. Their fluctuation as a function of time is animportant characteristic to analyze the signal'scomportment when they present in the transmissionchannel or other communication's units.-Frequencyanalysis of signal. → Tool to realize the frequencydomain analysis of signal -Fourier TransformationDefinition:The Fourier Transform (FT) of w(t) is :W(f)=F[w(t)]= -..Jw(t)exp[-j2元ft]dtf :frequency (unit:Hz if t is in sec)In general,W(f) is called a two-sided spectrum of w(t)Some properties: W(f) is a complex functionso W(f)-X(f)+jY(f)=/ W(f) / exp[jo(f))

w(t),voltage or current,time function analysis in time domain. Their fluctuation as a function of time is an important characteristic to analyze the signal’s comportment when they present in the transmission channel or other communication’s units. Frequency analysis of signal. Tool to realize the frequency domain analysis of signal Fourier Transformation • Definition:The Fourier Transform (FT) of w(t) is : W(f)=F[w(t)]= -∞∫ ∞w(t)exp[-j2πft]dt f :frequency (unit:Hz if t is in sec) In general,W(f) is called a two-sided spectrum of w(t) Some properties: W(f) is a complex function so W(f)=X(f)+jY(f)=│W(f)│exp[jθ(f)] Fourier Transform and Spectra