通信原理第二章2_Signals.ppt

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1、Peng YanniCha.2 Signals 10-2012Contents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals2.3 Characteristics of Random Signals2.4 Examples of Frequently Used Random Variables2.5 Numerical Characteristics of Random Variable2.6 Random Process2.7 Gaussian Process2.8 Narrow Band R

2、andom Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11 Brief Summary2.1 Classification of SignalsNew words:deterministic signalenergy signalrandom signalpower signalContents2.1 Classification of Signals2.2 Characteristics of Deterministic Sig

3、nals2.3 Characteristics of Random Signals2.4 Examples of Frequently Used Random Variables2.5 Numerical Characteristics of Random Variable2.6 Random Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System

4、2.11 Brief Summary2.2 Characteristics of Deterministic SignalsNew words:frequency spectrumenergy spectral densityfrequency spectral densitypower spectral densitysampling functionunit impulse functionautocorrelation functioncrosscorrelation function Frequency Spectrum of Power SignalsFor the periodic

5、 power signal:The frequency spectrum of a periodic signal is discrete,which could be showed either by the Fourier series in the time domain,or by the Fourier transform in the frequency domain.2.2.1 Characteristics in Frequency Domain periodic signals line spectraSuppose s(t)is a periodic signal with

6、 the period of T0,then it could be expressed as:where It shows that s(t)consists of infinite frequency components.On the other hand,We can make the same conclusion as the forementioned.set s(t)S(f)To integrate directly using Eq.2.2-1 By the following steps:How to obtain the C(nf0)?1.Figure out the t

7、runcation function sT(t)of s(t)2.Calculate the Fourier transform of sT(t)3.S:Example 2.1:Find the spectrum of the pulse train s(t)S:Find the spectrum of the impulse train s(t)Frequency Spectral Density of Energy SignalsFor the nonperiodic energy signal:The frequency spectral density of a nonperiodic

8、 signal is continuous,which could be found from the Fourier transform.nonperiodic signals continuous spectraFourier transform:Inverse Fourier transform:v(t)V(f)A few useful Fourier transform pairs:Several useful properties:Superposition(linearity)Time-delayedScale changeFrequency-shiftSeveral useful

9、 properties:Duality TheoremConvolution theorems if s(t)S(f)then S(t)s(-f)A few useful unit impulse properties:Energy Spectral DensityFor an energy signal s(t),the energy spectral density G(f)is defined as:where S(f)is the frequency spectral density of s(t).Parsevals Theoremwhere E is the energy of s

10、(t).Power Spectral DensityFor a power signal s(t),the power spectral density P(f)is defined as:where ST(f)is the frequency transform of sT(t)which is the truncated signal of s(t).Parsevals Theoremwhere P is the power of s(t).Autocorrelation function For the energy signal:When 2.2.2 Characteristics i

11、n Time Domain For the power signal:For the energy signals:For the power signals:Contents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals2.3 Characteristics of Random Signals2.4 Examples of Frequently Used Random Variables2.5 Numerical Characteristics of Random Variable2.6 Ra

12、ndom Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11 Brief Summary2.3 Characteristics of Random SignalsNew words:probability distributionprobability densityrandom variable2.3.1 Probability Di

13、stribution of Random Variable Probability distribution function of X:ifthen2.3.2 Probability Density of Random Variable Probability density function of X:Contents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals2.3 Characteristics of Random Signals2.4 Examples of Frequently U

14、sed Random Variables2.5 Numerical Characteristics of Random Variable2.6 Random Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11 Brief Summary2.4 Examples of Frequently Used Random VariablesNew

15、 words:normal distributionGaussian distribution uniform distributionRayleigh distributionContents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals2.3 Characteristics of Random Signals2.4 Examples of Frequently Used Random Variables2.5 Numerical Characteristics of Random Varia

16、ble2.6 Random Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11 Brief Summary2.5 Numerical Characteristics of Random VariableNew words:numerical characteristicvariancemathematical expectationmo

17、mentk-th origin momentk-th central momentstandard deviation2.5.1 Mathematical ExpectationMathematical expectation of X:if X and Y are independent2.5.2 VarianceVariance of X:if X and Y are independentContents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals2.3 Characteristics

18、of Random Signals2.4 Examples of Frequently Used Random Variables2.5 Numerical Characteristics of Random Variable2.6 Random Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11 Brief Summary2.6 Ra

19、ndom ProcessNew words:stationaryergodicitystochastic processstrict stationarywide-sense stationary,WSSgeneralized stationarywhite noise2.6.1 Basic Concept of Random Processstatistical meanvarianceFor a random process X(t),the statistic characteristics should be expressed as:autocorrelation function2

20、.6.2 Stationary Random ProcessIf the statistic characteristic of a random process is independent of the time origin,then the random process is called strict stationary random process.If the mean,the variance,and the autocorrelation function of a random process are independent of the time origin,then

21、 the random process is called generalized stationary random process.Generalized stationary is also called wide-sense stationary,WSS for short.A strict stationary random process must be a WSS random process.But a WSS random process is not always a strict stationary random process.2.6.3 ErgodicityIf a

22、 random process has ergodicity,then its statistic mean is equal to its time average.Ergodicity expresses that a realization of a stationary random process can go through all states of the process.So the“time average”may be replaced by the“statistic mean”.If a random process has ergodicity,then it mu

23、st be a strict stationary random process.But a strict stationary random process is not always ergodic.In practice,we usually assume that most communication systems are ergodic.With this assumptionis the D.C.component is the power of the normalized D.C.component is the normalized average poweris the

24、effective value of the signalis the normalized average power of the A.C.component when the signal has the average value 0is the root mean square value of the A.C.component 2.6.4 Autocorrelation function and power spectral density of stationary random process Characteristics of autocorrelation functi

25、on 1.2.3.4.5.Characteristics of power spectral density The power spectral density and the autocorrelation function of a stationary random process are a pair of Fourier transform,i.e.1.2.Exa.2.8,2.9Wiener-KinchinetheoremContents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals

26、2.3 Characteristics of Random Signals2.4 Examples of Frequently Used Random Variables2.5 Numerical Characteristics of Random Variable2.6 Random Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11

27、 Brief Summary2.7 Gaussian ProcessNew words:normal random processjoint probability densitystatistically independentalgebraic cofactorcovariancestandardized normal distributionnormal distribution functionprobability integral function Definition of Gaussian Process Gaussian process is also called norm

28、al random process.Gaussian distribution can be expressed as:Fig.2.7.1meanvariancestandard deviationStrict definition of the Gaussian process:Eq.2.7-2Generally,uncorrelatedindependentBut for Gaussian process:uncorrelatedindependent Characteristics of probability density of normal distributionset as t

29、he probability density of normal distribution1.2.3.4.standardized normal distribution Normal distribution functionset as the normal distribution functionwhere is called probability integral function Normal distribution expressed by error functionError function is defined asComplementary Error functi

30、on is defined asRelationship between F(x)and error functionRelationship between error function andContents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals2.3 Characteristics of Random Signals2.4 Examples of Frequently Used Random Variables2.5 Numerical Characteristics of Ran

31、dom Variable2.6 Random Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11 Brief Summary2.8 Narrow Band Random ProcessNew words:random enveloperandom phaseorthogonal componentinphase component2.8

32、.1 Basic Concept of Narrow Band Random ProcessThe bandwidth of the signal is narrower than its central frequency,then it is called narrow band random process.Fig.2.8.1It time domain,it looks like a sinusoidal wave with slowly vary envelope and phase.It can be expressed as:whereis random envelope,and

33、is random phase.Obviously,it also can be expresses as:whereis called inphase component,andis called orthogonal component.2.8.2 Characteristics of Narrow Band Random ProcessSince X(t)can be expressed either by its random envelope and random phase,or by its inphase component and orthogonal component,i

34、ts statistic characteristics are determined by them.On the contrary,when the statistic characteristics of X(t)are given,those components are determined.Well take the stationary narrow band Gaussian process with 0 mean for example.1.Statistic characteristics of Xc(t)and Xs(t)When X(t)is a narrow band

35、 stationary Gaussian process with 0 mean,then:Xc(t)and Xs(t)are also narrow band stationary Gaussian processXc(t)and Xs(t)are uncorrelatedXc(t)and Xs(t)are statically independentWhen X(t)is a narrow band stationary Gaussian process with 0 mean,then:The probability density of the envelope is a narrow

36、 band process obeys the Rayleigh distribution2.Statistic characteristics of aX(t)andThe probability density of the phase distribution obeys uniform distribution over 0,2Contents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals2.3 Characteristics of Random Signals2.4 Examples

37、of Frequently Used Random Variables2.5 Numerical Characteristics of Random Variable2.6 Random Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11 Brief Summary2.9 Sinusoidal Wave plus Narrow Band

38、 Random ProcessNew words:generalized Rayleigh distributionRician distributionzero-order modified Bessel functionThe sum of signal and noise in a system may be assumed approximately a sinusoidal wave plus a narrow band Gaussian noise.Assume the expression of a sinusoidal wave plus a noise is The prob

39、ability density of the envelope of r(t)obeys generalized Rayleigh distribution,or Rician distribution.The probability density of the phase of r(t)is very complicated.Contents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals2.3 Characteristics of Random Signals2.4 Examples of

40、Frequently Used Random Variables2.5 Numerical Characteristics of Random Variable2.6 Random Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11 Brief Summary2.10 Signal Transfer through Linear Sys

41、temsNew words:impulse responseconvolutioncausalitytransfer functiondistortionless transmission2.10.1 Basic Concept of Linear SystemsUsually we refer the linear systems with the following properties:a pair of passive input and outputmemorylesstime-invariantcausalitylinear relationship2.10.2 Determini

42、stic Signal Transfer through Linear Systems1.Time domain analysis methodSuppose x(t)is the input signal,y(t)is the output signal,and h(t)is the impulse response of the system,then For physically realizable systems:2.Frequency domain analysis methodSupposeExa.2.10 then 3.Conditions of distortionless

43、transmissionSince the output signal without distortion can be expressed asSo i.e.Distortionless transmission requires:Amplitude characteristic of the transfer function of the linear system is independent of frequency,and is a horizontal straight linePhase characteristic is a straight line via the or

44、igin Or,group delay is a constantFig.2.10.62.10.3 Random Signal Transfer through Linear SystemsWhen the input signal is random process instead of deterministic signal,we can make the same conclusions as forementioned.Assuming the input signal is stationary,well discuss the statistic characteristics

45、of the output.1.Mathematical expectation of the output2.Autocorrelation function of the outputIt shows that the output is generalized stationary.3.Power spectral density of the outputExa.2.114.Probability distribution of the outputAfter passing through a linear system,Gaussian random process remains

46、 Gaussian distribution,but the numerical characteristics have been changed.Contents2.1 Classification of Signals2.2 Characteristics of Deterministic Signals2.3 Characteristics of Random Signals2.4 Examples of Frequently Used Random Variables2.5 Numerical Characteristics of Random Variable2.6 Random

47、Process2.7 Gaussian Process2.8 Narrow Band Random Process2.9 Sinusoidal Wave plus Narrow Band Gaussian Process2.10 Signal Transfer through Linear System2.11 Brief Summary2.11 Brief Summary1.To calculate the frequency spectral density function of frequently used signal(including periodic and nonperio

48、dic)2.To calculate mean,variance and autocorrelation function of signal,and to calculate the powers based on these3.To judge stationary random process3.To judge stationary random process4.To understand the Wiener-Wiener-KinchineKinchine theorem theorem,and to determine the power5.To remember Gaussia

49、n distribution function6.To understand the statistic characteristics of narrow band noise and sinusoid wave plus narrow band noise8.To understand conditions of distortionless transmission7.To understand the relationships between the input signal and the output signal of linear systems(include mean,autocorrelation function,and power spectral density)