通信原理(英文版).ppt

1Chapter 2 Signals2.1 2.1 Classification of Signals Classification of Signals2.1.1 Deterministic signals and random signals2.1.1 Deterministic signals and random signalsWhat is deterministic signal?What is deterministic signal?What is random signal?What is random signal?2.1.2 Energy signals and power signals2.1.2 Energy signals and power signalsSignal power:Let Signal power:Let R R=1,then =1,then P P=V V2 2/R R=I I2 2R R=V V2 2=I I2 2Signal energySignal energy：Let Let S S represent represent V V or or I I，if if S S varies with time varies with time，then S can then S can be rewritten as be rewritten as s s(t t),),Hence,the signal energy Hence,the signal energy E E=s s2 2(t t)d)dt tEnergy signal satisfies Energy signal satisfies Average power:Average power:,then ,then P P=0 for energy signal.=0 for energy signal.For power signal:For power signal:P P 0,i.e.,power signal has infinite duration.0,i.e.,power signal has infinite duration.Energy signal has finite energy,but its average power equals 0.Energy signal has finite energy,but its average power equals 0.Power signal has finite average power,but its energy equals infinity.Power signal has finite average power,but its energy equals infinity.22.2 Characteristics of deterministic signals2.2.1 Characteristics in frequency domain2.2.1 Characteristics in frequency domainl lFrequency spectrum of power signal:let Frequency spectrum of power signal:let s s(t t)be a periodic)be a periodic power signal,its period is power signal,its period is T T0 0,then we have,then we havewhere where 0 0=2=2 /T T0 0=2=2 f f0 0 C C(j(jn n 0 0)is a complex function,)is a complex function,C C(j(jn n 0 0)=|)=|C Cn n|e|ej j n nwhere|where|C Cn n|amplitude of the component with frequency amplitude of the component with frequency nfnf0 0 n n phase of the component with frequency phase of the component with frequency nfnf0 0Fourier series of signal Fourier series of signal s s(t t):):3【Example 2.1Example 2.1】Find the spectrum of a periodic rectangular Find the spectrum of a periodic rectangular wave.wave.Solution:Assume the period of a periodic Solution:Assume the period of a periodic rectangular rectangular wave is wave is T T,the width is the width is ,and the amplitude is,and the amplitude isV,V,thenthenIts frequency spectrum isIts frequency spectrum is 4Frequency spectrum figureFrequency spectrum figure5【Example 2.2Example 2.2】Find the frequency spectrum of a sinusoidal wave Find the frequency spectrum of a sinusoidal wave after full-wave rectification.after full-wave rectification.SolutionSolution：Assume the expression of the signal isAssume the expression of the signal isIts frequency spectrum:Its frequency spectrum:The Fourier series of the signal is:The Fourier series of the signal is:1f(t)t6l lFrequency spectral density of energy signalsFrequency spectral density of energy signalsLet an energy signal be Let an energy signal be s s(t t),then its frequency spectral density),then its frequency spectral density is isThe inverse Fourier transform of The inverse Fourier transform of S S()is the original signal:)is the original signal:【Example 2.3Example 2.3】Find the frequency spectral density of a Find the frequency spectral density of a rectangular pulse.rectangular pulse.Solution:Let the expression of the rectangular pulse beSolution:Let the expression of the rectangular pulse beThen its frequency spectral density isThen its frequency spectral density is its Fourier transform:its Fourier transform:7【Example 2.4Example 2.4】Find the waveform and the frequency spectral Find the waveform and the frequency spectral density of a sample function.density of a sample function.Solution:The definition of the sample function isSolution:The definition of the sample function isthe frequency spectral density the frequency spectral density SaSa(t t)is:)is:From the above equation,we see that From the above equation,we see that SaSa()is a gate function.)is a gate function.【Example 2.5Example 2.5】Find the unit impulse function and its frequency Find the unit impulse function and its frequency spectral density.spectral density.Solution:Unit impulse function is usually called Solution:Unit impulse function is usually called function function (t t).).Its definition isIts definition isThe frequency spectral density of The frequency spectral density of (t):(t):8 (t t)and its frequency spectral density:)and its frequency spectral density:Physical meaning of Physical meaning of functionfunction:It is a pulse with infinite height,infinitesimal width,and unit It is a pulse with infinite height,infinitesimal width,and unit area.area.Sa(t)has the following property:Sa(t)has the following property:WhenWhen k k k k ,amplitudeamplitude ,andand the zero-spacing of the waveform the zero-spacing of the waveform 0 0，Hence,Hence,tttf(f)10t(t)09Characterisitics of Characterisitics of (t t)n n n n (t)is an even function:(t)is an even function:n n (t t)is the derivative of unit step function:)is the derivative of unit step function:Difference between frequency spectral density S(f)of energy Difference between frequency spectral density S(f)of energy signal and frequency spectrum of periodic power signal:signal and frequency spectrum of periodic power signal:n nS S(f f)continuous spectrumcontinuous spectrum；C C(j(jn n 0 0)discretediscreten nUnit of Unit of S S(f f):V/Hz):V/Hz；Unit of Unit of C C(j(jn n 0 0):V):Vn nAmplitude of Amplitude of S S(f f)at a frequency point)at a frequency point infinitesimal infinitesimal u(t)=(t)t10Fig.2.2.6 Unit step function10【Example 2.6Example 2.6】Find the frequency spectral density of a Find the frequency spectral density of a cosinusoidal wave with infinite length.cosinusoidal wave with infinite length.Solution:Let the expression of a cosinusoidal wave be Solution:Let the expression of a cosinusoidal wave be f f(t t)=)=coscos 0 0t,t,then according to eq.(2.2-10),then according to eq.(2.2-10),F F()can be written as)can be written asReferencing eq.(2.2-19),the above equation can be written as:Referencing eq.(2.2-19),the above equation can be written as:Introducing Introducing (t t),the concept of frequency spectral density),the concept of frequency spectral density can be generalized to power signal.can be generalized to power signal.t000(b)频谱密度(a)波形11l lEnergy spectral densityLet the energy of an energy signal Let the energy of an energy signal s s(t t)be)be E E,then the energy,then the energy of the signal is decided byof the signal is decided byIf its frequency spectral density is If its frequency spectral density is S S(f f),then from Parsevals),then from Parsevals theorem we havetheorem we havewhere|where|S S(f f)|)|2 2 is called energy spectral density.is called energy spectral density.The above equation can be rewritten asThe above equation can be rewritten as：where where G G(f f)|S(f)|S(f)|2 2 （J/HzJ/Hz）is energy spectral density.is energy spectral density.Property of Property of G G(f f):Since):Since s s(t t)is a real function,|)is a real function,|S S(f f)|)|2 2 is an is an even function,even function,12l lPower spectral densityPower spectral densityLet the truncated signal of s(t)is Let the truncated signal of s(t)is s sT T(t t)，-T T/2 /2 t t T T/2,then/2,thenTo define the power spectral density of the signal as:To define the power spectral density of the signal as:obtain the signal power:obtain the signal power:132.2.2 2.2.2 Characteristics in time domainCharacteristics in time domainl lAutocorrelation functionAutocorrelation functionDefinition of the autocorrelation function for energy signal:Definition of the autocorrelation function for energy signal:Definition of the autocorrelation function for power signal:Definition of the autocorrelation function for power signal:Characteristics:Characteristics:n nR R()is only dependent on)is only dependent on ,but independent of but independent of t.t.n nWhen When =0,=0,R R()of energy signal equals the energy of)of energy signal equals the energy of the signal,and the signal,and R R()of power signal equals the average)of power signal equals the average power of the signal.power of the signal.14l lCross-correlation functionCross-correlation functionDefinition of the cross-correlation function for energy Definition of the cross-correlation function for energy signal:signal:Definition of the cross-correlation function for power signal:Definition of the cross-correlation function for power signal:Characteristics:Characteristics:n n1 1.R.R1212()is dependent on)is dependent on ,and independent of,and independent of t.t.n n2.2.Proof:Let Proof:Let x x=t t+，then then 152.3 Characteristics of random signals2.3.12.3.1 Probability distribution of random variable Probability distribution of random variablel lConcept of random variable:If the random outcome of a trial Concept of random variable:If the random outcome of a trial A A is expressed by is expressed by X X,then we call,then we call X X a random variable,and let its a random variable,and let its value be value be x x.For example,the number of calls received within a given For example,the number of calls received within a given period of time at the telephone exchange is a random variable.period of time at the telephone exchange is a random variable.l lDistribution function of random variableDistribution function of random variableDefinition:Definition:F FX X(x x)=)=P P(X X x x)Characteristics:Characteristics:P P(a a X X b b)+)+P P(X X a a)=)=P P(X X b b),),P P(a a X X b b)=)=P P(X X b b)P P(X X a a)，P P(a a X X b b)=)=F FX X(b b)F FX X(a a)16Distribution function of discrete random variable:Distribution function of discrete random variable:n nLet the values of Let the values of X X be:be:x x1 1 x x2 2 x xi i x xn n，their their probabilities are respectively probabilities are respectively p p1 1,p p2 2,p pi i,p pn n,then,thenP P(X X x x1 1)=0,)=0,P P(X X x xn n)=1)=1n n P P(X X x xi i)=)=P P(X X=x x1 1)+)+P P(X X=x x2 2)+)+P P(X X=x xi i),),n nCharacteristics:Characteristics:pp F FX X(-(-)=0)=0pp F FX X(+(+)=1)=1pp If If x x1 1 x x2 2,then,then F FX X(x x1 1)F FX X(x x2 2)-monotonic -monotonic increasing function.increasing function.17Distribution function of continuous random variable:When x is continuous,from the definition of distribution function FX(x)=P(X x)we know that we know that FX(x)is a continuous monotonic increasing function.182.3.22.3.2 Probability density of random variable Probability density of random variablel lProbability density of continuous random variable Probability density of continuous random variable p pX X(x x)Definition of Definition of p pX X(x x):):Meaning of Meaning of p pX X(x x):):n np pX X(x x)is the derivative of)is the derivative of F FX X(x x),and is the slope of the),and is the slope of the curve of curve of F FX X(x x)n nP P(a a 0,0,a a=const.const.Probability density curve:Probability density curve:21l lRandom variable with uniform distributionRandom variable with uniform distributionDefinition:probability densityDefinition:probability densitywhere where a a,b b are constants.are constants.Probability density curve:Probability density curve:bax0pA(x)22l lRandom variable with Rayleigh distributionRandom variable with Rayleigh distributionDefinition:Probability densityDefinition:Probability densitywhere where a a 0,and is a constant.0,and is a constant.Probability density curve:Probability density curve:232.5 Numerical characteristics of random variable2.5.1 Mathematical expectationl lDefinition:for continuouse random variableDefinition:for continuouse random variablel lCharacteristics:Characteristics:If X and Y are independent of each other,and If X and Y are independent of each other,and E E(X X)and)and E E(Y Y)exist,then)exist,then 242.5.2 VariancelDefinition:where Variance can be rewritten as:Proof:For discrete variable:For continuous variable:lCharacteristics:D(C)=0 D(X+C)=D(X)，D(CX)=C2D(X)D(X+Y)=D(X)+D(Y)D(X1+X2+Xn)=D(X1)+D(X2)+D(Xn)252.5.3 Momentl lDefinition:the Definition:the k k-th moment of a random variable-th moment of a random variable X X is isk-k-th origin moment is the moment when th origin moment is the moment when a a=0:=0:k-k-th central moment is the moment when :th central moment is the moment when :l lCharacteristics:Characteristics:The first origin moment is the mathematical expectation:The first origin moment is the mathematical expectation:The second central moment is the variance:The second central moment is the variance:262.6 Random process2.6.1 Basic concept of random processl lX X(A A,t t)ensumble consisting of all possible ensumble consisting of all possible“realizations”of an event“realizations”of an event A Al lX X(A Ai i,t),t)a realization of event a realization of event A A,it is a determined,it is a determined time functiontime functionl lX X(A A,t tk k)value of the function at the given time value of the function at the given time t tk kDenote for short:Denote for short:X X(A A,t t)X X(t t)X X(A Ai i,t t)X Xi i (t t)27l lExample:receiver noiseExample:receiver noisel lNumerical characteristics of random process:Numerical characteristics of random process:Statistical mean:Statistical mean:Variance:Variance:Autocorrelation function:Autocorrelation function:282.6.2 Stationary random processlDefinition of stationary random process:A random process whose statistical characteristics is independent of the time origin is called a stationary random process.(or,strict stationary random process)lDefinition of generalized stationary random process:The random process whose mean,variance and autocorrelation function are independent of the time originlCharacteristics of generalized stationary random process:lA strict stationary random process must be a generalized stationary random process;but a generalized stationary random process is not always a strict stationary random process.292.6.3 Ergodicityl lSignificance of ergodicitySignificance of ergodicityA realization of a stationary random process can go through A realization of a stationary random process can go through all states of the process.all states of the process.l lCharacteristic of ergodicity:time average may be replaed by Characteristic of ergodicity:time average may be replaed by statistical mean.For example,statistical mean.For example,Statistical mean of ergodic process Statistical mean of ergodic process mmX X：Autocorrelation function of ergodic process RAutocorrelation function of ergodic process RX X():):If a random process has ergodicity,then it must be a strict If a random process has ergodicity,then it must be a strict stationary random process.However,a strict stationary stationary random process.However,a strict stationary random process is not always ergodic.random process is not always ergodic.30l lErgodicity of stationary communication systemErgodicity of stationary communication system If the signal and the noise are both ergodic,then If the signal and the noise are both ergodic,thenFirst origin moment First origin moment mmX X =E E X X(t t)D.C.component of D.C.compon