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