Gaussian Random Variables and Processes

1 Gaussian Random Variables and ProcessesSaravanan of electrical EngineeringIndian Institute of Technology BombayAugust 1, 20121 / 33 Gaussian Random VariablesGaussian Random VariableDefinitionA continuous Random variable with pdf of the formp(x) =1 2 2exp( (x )22 2), <x< ,where is the mean and 2is the variance. 4 2024 (x)3 / 33 Notation N( , 2)denotes a Gaussian distribution with mean andvariance 2 X N( , 2) Xis a Gaussian RV with mean andvariance 2 X N(0,1)is termed a standard Gaussian RV4 / 33 Affine Transformations Preserve GaussianityTheoremIf X is Gaussian , then aX+b is Gaussianfor a,b IfX N( , 2), thenaX+b N(a +b,a2 2). IfX N( , 2), thenX N(0,1).5 / 33 CDF and CCDF of Standard Gaussian Cumulative distribution function (x) =P[N(0,1) x] = x 1 2 exp( t22)dt Complementary cumulative distribution functionQ(x) =P[N(0,1)>x] = x1 2 exp( t22)dtxtp(t)Q(x) (x)6 / 33 Properties ofQ(x) (x) +Q(x) =1 Q( x) = (x) =1 Q(x) Q(0) =12 Q( ) =0 Q( ) =1 X N( , 2)P[X> ] =Q( )P[X< ] =Q( )7 / 33 Jointly Gaussian Random VariablesDefinition (Jointly Gaussian RVs) Random variablesX1,X2.

2 ,Xnare jointly Gaussian if anynon-trivial linear combination is a Gaussian Random + +anXnis Gaussian for all(a1,..,an) Rn\0 Example (Not Jointly Gaussian )X N(0,1)Y={X,if|X|>1 X,if|X| 1Y N(0,1)andX+Yis not / 33 Gaussian Random VectorDefinition ( Gaussian Random Vector)A Random vectorX= (X1,..,Xn)Twhose components arejointly N(m,C)wherem=E[X],C=E[(X m)(X m)T]Definition (Joint Gaussian Density)IfCis invertible, the joint density is given byp(x) =1 (2 )mdet(C)exp( 12(x m)TC 1(x m))9 / 33 Uncorrelated Random VariablesDefinitionX1andX2are uncorrelated if cov(X1,X2) =0 RemarksFor uncorrelated Random variablesX1,..,Xn,var(X1+ +Xn) =var(X1) + +var(Xn).IfX1andX2are independent,cov(X1,X2) = coefficient is defined as (X1,X2) =cov(X1,X2) var(X1)var(X2).10 / 33 Uncorrelated Jointly Gaussian RVs are IndependentIfX1,..,Xnare jointly Gaussian and pairwise uncorrelated,then they are (x) =1 (2 )mdet(C)exp( 12(x m)TC 1(x m))=n i=11 2 2iexp( (xi mi)22 2i)wheremi=E[Xi]and 2i=var(Xi).}

3 11 / 33 Uncorrelated Gaussian RVs may not be IndependentExample X N(0,1) Wis equally likely to be +1 or -1 Wis independent ofX Y=WX Y N(0,1) XandYare uncorrelated XandYare not independent12 / 33 Complex Gaussian Random VectorsComplex Gaussian Random VariableDefinition (Complex Random Variable)A complex Random variableZ=X+jYis a pair of real The pdf of a complex RV is the joint pdf of its real andimaginary parts. E[Z] =E[X] +jE[Y] var[Z] =E[|Z|2] |E[Z]|2=var[X] +var[Y]Definition (Complex Gaussian RV)IfXandYare jointly Gaussian ,Z=X+jYis a complexGaussian / 33 Complex Random VectorsDefinition (Complex Random Vector)A complex Random vector is defined asZ=X+jYwhereXandYare real Random vectors having dimensionn 1. There are four matrices associated withXandYCX=E[(X E[X])(X E[X])T]CY=E[(Y E[Y])(Y E[Y])T]CXY=E[(X E[X])(Y E[Y])T]CYX=E[(Y E[Y])(X E[X])T] The pdf ofZis the joint pdf of its real and imaginary the pdf of Z=[XY]15 / 33 Covariance and Pseudocovariance of ComplexRandom Vectors Covariance ofZ=X+jYCZ=E[(Z E[Z])(Z E[Z])H]=CX+CY+j(CYX CXY) Pseudocovariance ofZ=X+jY CZ=E[(Z E[Z])(Z E[Z])T]=CX CY+j(CXY+CYX) A complex Random vectorZis called proper if itspseudocovariance is zeroCX=CYCXY= CYX16 / 33 Motivating the Definition of Proper Random Vectors Forn=1, a proper complex RVZ=X+jYsatisfiesvar(X) =var(Y)cov(X,Y)= cov(Y,X) Thus cov(X,Y) =0 IfZis a proper complex Gaussian Random variable, its realand imaginary parts areindependent17 / 33 Proper Complex Gaussian Random VectorsFor Random vectorZ=X+jYand Z=[X Y]T, the pdf isgiven byp(z) =p( z) =1(2 )n(det(C Z))12exp( 12( z m)TC 1 Z( z m))

4 IfZis proper, the pdf is given byp(z) =1 ndet(CZ)exp( (z m)HC 1Z(z m))18 / 33 Random ProcessesRandom ProcessDefinitionAn indexed collection of Random Variables {X(t) :t T}.Discrete-time Random ProcessT=ZorNContinuous-time Random ProcessT=RStatisticsMean functionmX(t) =E[X(t)]Autocorrelation functionRX(t1,t2) =E[X(t1)X (t2)]Autocovariance functionCX(t1,t2) =E[(X(t1) mX(t1)) (X(t2) mX(t2)) ]20 / 33 Crosscorrelation and CrosscovarianceCrosscorrelationRX1,X2(t1 ,t2) =E[X1(t1)X 2(t2)]CrosscovarianceCX1,X2(t1,t2) =E[(X1(t1) mX1(t1))(X2(t2) mX2(t2)) ]=RX1,X2(t1,t2) mX1(t1)m X2(t2)21 / 33 Stationary Random ProcessDefinitionA Random process which is statistically indistinguishable from adelayed version of For anyn N,(t1,..,tn) Rnand R,(X(t1),..,X(tn))has the same joint distribution as(X(t1 ),..,X(tn )). mX(t) =mX(0) RX(t1,t2) =RX(t1 ,t2 ) =RX(t1 t2,0)22 / 33 Wide Sense Stationary Random ProcessDefinitionA Random process is WSS ifmX(t) =mX(0)for alltandRX(t1,t2)=RX(t1 t2,0)for allt1, function is expressed as a function of =t1 t2asRX( ).

5 Definition (Power Spectral Density of a WSS Process)The Fourier transform of the autocorrelation (f) =F(RX( ))23 / 33 Energy Spectral DensityDefinitionFor a signals(t), the energy spectral density is defined asEs(f) =|S(f)| (t)through an ideal narrowband filter with responseHf0(f) ={1,iff0 f2<f<f0+ f20,otherwiseOutput isY(f) =S(f)Hf0(f).Energy in output is given by |Y(f)|2df= f0+ f2f0 f2|S(f)|2df |S(f0)|2 f24 / 33 Power Spectral DensityMotivationPSD characterizes spectral content of Random signals whichhave infinite energy but finite powerExample (Finite-power infinite-energy signal)Binary PAM signalx(t) = n= bnp(t nT)25 / 33 Power Spectral Density of a RealizationTime windowed realizations have finite energyxTo(t) =x(t)I[ To2,To2](t)STo(f)=F(xTo(t)) Sx(f)=|STo(f)|2To(PSD Estimate)Definition (PSD of a realization) Sx(f) =limTo |STo(f)|2To26 / 33 Autocorrelation Function of a RealizationMotivation Sx(f) =|STo(f)|2To 1To xTo(u)x To(u )du=1To To2 To2xTo(u)x To(u )du= Rx( )(Autocorrelation Estimate)Definition (Autocorrelation function of a realization) Rx( ) =limTo 1To To2 To2xTo(u)x To(u )du27 / 33 The Two Definitions of Power Spectral DensityDefinition (PSD of a WSS Process)SX(f) =F(RX( ))whereRX( ) =E[X(t)]}

6 X (t )].Definition (PSD of a realization) Sx(f) =F( Rx( ))where Rx( ) =limTo 1To To2 To2xTo(u)x To(u )duBoth are equal for ergodic processes28 / 33 Ergodic ProcessDefinitionA stationary Random process is ergodic if time averages equalensemble averages. Ergodic in meanlimT 1T T2 T2x(t)dt=E[X(t)] Ergodic in autocorrelationlimT 1T T2 T2x(t)x (t )dt=RX( )29 / 33 Gaussian Random ProcessesGaussian Random ProcessDefinitionA Random process{X(t) :t T}is Gaussian if its samplesX(t1),..,X(tn)are jointly Gaussian for anyn The mean and autocorrelation functions completelycharacterize a Gaussian Random process. Gaussian WSS Processes are stationary. If the input to an LTI system is a Gaussian RP, the output isalso a Gaussian / 33 White Gaussian NoiseDefinitionA zero mean WSS Gaussian Random process with powerspectral densitySn(f) = Rn( ) =N02 ( ) N02is termed the two-sided PSD and has units Watts / 33 Thanks for your attention33 / 33