EE354 : Communications System I Lecture 6-8: Random signals Aliazam Abbasfar

Outline Random signals Signals correlation Power spectral density

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

Examples Sinusoid with random phase DC signal with random level Binary signaling

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 < )

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)

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

Examples Sinusoid with random phase DC signal with random level Binary signaling

Power spectral density Time-averaged autocorrelation Power spectral density Average power E, P are obtained by integration of ESD and PSD PSD measurement : use narrowband filters

Examples X(t) = A cos(wct + f) Y(t) = X(t) cos(wct) WSS ? RY(t) and GY(f)

Sum process z(t) = x(t) + y(t) x(t) and y(t) are WSS 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) = mX mY GZ(f) = GX(f) + GY(f) + 2 mX mY d(f)

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)

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