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EE354 : Communications System I

EE354 : Communications System I. Lecture 5,6,7: Random variables and signals Aliazam Abbasfar. Outline. Random variables overview Random signals Signals correlation Power spectral density. Random variables (RV). PDF, CDF f X (x) = d/ dx [ F X (x) ]

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EE354 : Communications System I

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  1. EE354 : Communications System I Lecture 5,6,7: Random variables and signals Aliazam Abbasfar

  2. Outline • Random variables overview • Random signals • Signals correlation • Power spectral density

  3. 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 RVs • Correlation of 2 RVs E[x y] • Example : Binary communication with noise

  4. Binomial distribution • X = # of successes in N independent trials • p : success probability (1-p : failure) • Sum of N binary RVs : X = Sxi • If N is large, it becomes a Gaussian PDF • mx=Np • sx2=Npq • Example : Error probability in binary packets

  5. Z~N(0,1) Tails decrease exponentially Gaussian RVs and the CLT • PDF (mean and variance) • CDF defined by error function (erf(•)) • Central Limit Theorem: X1,…,Xni.i.d • Let Y=iXi, Z=(Y-mY)/sY • As n, Z becomes Gaussian, mx=0, sx2=1. • Uncorrelated Gaussian RVs are independent N(mx,sx2) sx mx

  6. Random Processes • Ensemble 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

  7. Correlation • Correlation = 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[<xi2(t)>] = <RX(t,t)> • Most of RSs are power signals ( 0< P < )

  8. Wide Sense Stationary (WSS) • A process is WSS if • E[x(t)]=mX • RX(t1,t2)= E[x(t1)x(t2)]=RX(t2-t1)= RX(t) • RX(0)=E[x2(t)]< • Stationary in 1st and 2nd moments • Autocorrelation • RX(t)= RX(-t) • |RX(t)| RX(0) • RX(t)=0 : samples separated by t uncorrelated • Average power • P = <E[x2(t)]> = Rx(0)

  9. Ergodic process • Time 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 : <xi2(t)> = E[ x2(t) ] = (sx)2 + (mx)2 • Average power : • P = E[<xi2(t)>] = Pi • Use one sample function to estimate signal statistics • Time-average instead of ensemble average

  10. Examples • Sinusoid with random phase • DC signal with random level • Binary NRZ signaling

  11. Power spectral density • Time-averaged autocorrelation • Power spectral density • Average power

  12. Examples • Y(t) = X(t) cos(wct) • WSS ? • RY(t) and GY(f)

  13. Correlations for LTI systems • If x(t) is WSS, x(t) and y(t) are jointly WSS • mY = H(0) mX • RYX(t) = h(t) Rxx(t) • RXY(t) = RYX(-t)= h(-t) Rxx(t) • RYY(t) = h(t)  h(-t) Rxx(t) • GY(f) = |H(f)|2 GX(f)

  14. Sum process • z(t) = x(t) + y(t) • RZ(t) = RX(t) + RY(t) + RXY(t) + RXY(-t) • GZ(f) = GX(f) + GY(f) + 2 Re[GXY(f)] • If X and Y are uncorrelated • RXY(t) = mXmY • GZ(f) = GX(f) + GY(f) + 2 mXmYd(f)

  15. Reading • Carlson Ch. 9.1, 9.2 • Proakis&Salehi 4.1, 4.2, 4.3 4.4

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