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

2. Outline
Random signals
Signals correlation
Power spectral density

3. 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

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

5. 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 < )

6. 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)

7. 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

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

9. 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

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

11. 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)

12. 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)

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