Lecture 5,6,7: Random variables and signals

Aliazam Abbasfar

OutlineRandom variables overview

Random signals

Signals correlation

Power spectral density

Random variables (RV)PDF, CDF

fX(x) = d/dx [ FX(x) ]

Mean, variance, moments E[x], Var[x], E[xn]

Functions of RVs Y = g(X)

Several RVs Joint PDF, CDF Conditional probability Sum

Independent RVsCorrelation of 2 RVs E[x y]

Example : Binary communication with noise

(x)y)/f(x,fx)X|(yf XXYY

dx )(xfdy (y)f iXY

dx x)-z(x,f(z)f XYZ

x y

XYXYXY dydx y)(x,fy)(x,F y);(x,f

Binomial distributionX = # of successes in N independent trials

p : success probability (1-p : failure)

Sum of N binary RVs : X = xi

If N is large, it becomes a Gaussian PDF x =Np x

2 =Npq

Example : Error probability in binary packets

N

0k

k-NkX k)δ(xqp

kN

(x)f

Gaussian RVs and the CLT PDF (mean and variance)

CDF defined by error function (erf(•))

Central Limit Theorem: X1,…,Xn i.i.d Let Y=iXi, Z=(Y-Y)/Y As n, Z becomes Gaussian, x=0, x

2=1.

Uncorrelated Gaussian RVs are independent

]/σ)μ[(x

XX

2X

2Xe

σ2π1(x)f

x

x

N(x,x2) Z~N()

Tails decreaseexponentially

2x/erfc21Q(x)erf(u),1erfc(u),

σ2μxerf1

21(x)Fx)p(X

X

XX

Random ProcessesEnsemble of random signals (sample functions)

Deterministic signals with RVs Voltage waveforms Message signals Thermal noise

Samples of a random signal x(t) ; a random variable

E[x(t)], Var[x(t)] x(t1), x(t2) joint random variables

CorrelationCorrelation = statistic similarity

Cross correlation of two random signals RXY(t1,t2)=E[x(t1)y(t2)] Uncorrelated/Independent RSs

Autocorrelation R(t1,t2)=E[x(t1)x(t2)] RX(t,t) = E[x2(t)] = Var[x(t)]+E[x]2

Average power P = E[Pi] = E[<xi

2(t)>] = <RX(t,t)> Most of RSs are power signals ( 0< P < )

Wide Sense Stationary (WSS)A process is WSS if

E[x(t)]=X

RX(t1,t2)= E[x(t1)x(t2)]=RX(t2-t1)= RX()RX(0)=E[x2(t)]<Stationary in 1st and 2nd moments

AutocorrelationRX()= RX(-)|RX()| RX()RX()=0 : samples separated by uncorrelated

Average power P = <E[x2(t)]> = Rx(0)

Ergodic processTime average of any sample function =

Ensemble average ( any i and any g)<g(xi(t))> = E[g(x(t))]

Ensemble averages are time-independent DC : <xi(t)> = E[ x(t) ] = mx Total power : <xi

2(t)> = E[ x2(t) ] = (sx)2 + (mx)2

Average power : P = E[<xi

2(t)>] = Pi

Use one sample function to estimate signal statistics Time-average instead of ensemble average

ExamplesSinusoid with random phase

DC signal with random level

Binary NRZ signaling

Power spectral densityTime-averaged autocorrelation

Power spectral density

Average power

)](R[(f)G xxx F ]

T(f)X

lim[E](f)G[E(f)G2

T

Txx i

df(f)G P-

x

]τ)-(t(t)xxE[τ)-t(t,R)(R iixxxx

ExamplesY(t) = X(t) cos(wct)

WSS ?RY() and GY(f)

Correlations for LTI systems

If x(t) is WSS, x(t) and y(t) are jointly WSS

mY = H(0) mX RYX() = h() Rxx()RXY() = RYX(-)= h(-) Rxx()RYY() = h() h(-) Rxx()

GY(f) = |H(f)|2 GX(f)

Sum processz(t) = x(t) + y(t)

RZ() = RX() + RY() + RXY() + RXY(-) GZ(f) = GX(f) + GY(f) + 2 Re[GXY(f)]

If X and Y are uncorrelated RXY() = mX mY

GZ(f) = GX(f) + GY(f) + 2 mX mY (f)

ReadingCarlson Ch. 9.1, 9.2

Proakis&Salehi 4.1, 4.2, 4.3 4.4