1. ELEC 303 – Random Signals Lecture 20 – Random processes
Dr. Farinaz Koushanfar
ECE Dept., Rice University
Nov 11, 2010

2. Lecture outline Basic concepts Random processes and linear systems
Power spectral density of stationary processes
Power spectra in LTI systems
Power spectral density of a sum process
Gaussian processes

3. RP and linear systems
When a RP passes a linear time-invariant system the output is also a RP
Assuming a stationary process X(t) is input, the linear time-invariant system with the impulse response h(t), output process Y(t)
Under what condition the output process would be stationary?
Under what conditions will the input/output jointly stationary?
Find the output mean, autocorrelation, and crosscorrelation
h(t)
X(t)
Y(t)

4. Linear time invariant systems
If a stationary RP with mean mX and autocorrelation function RX()
Linear time invariant (LTI) system with response h(t)
Then, the input and output process X(t) and Y(t) will be jointly stationary with
h(t)
X(t)
Y(t)

5. The response mean
Using the convolution integral to relate the output Y(t) to the input X(t), Y(t)=X()h(t-)d
This proves that mY is independent of t
h(t)
X(t)
Y(t)

6. Cross correlation
The cross correlation function between output and the input is
This shows that RXY(t1,t2) depends only on =t1-t2

7. Output autocorrelation
The autocorrelation function of the output is
This shows that RY and RXY depend only on =t1-t2,
 Output process is stationary, and input/output are jointly stationary

8. Power spectral density of a stationary process
If the signals in the RP are slowly varying, then the RP would mainly contain the low frequencies in its power concentration
If the signal changes very fast, most of the power will be concentrated at high frequency
The power spectral density of a RP X(t) is denoted by SX(f) showing the strength of the power in RP as a function of frequency
The unit for SX(f) is Watts/Hz

9. Wiener-Khinchin theorem
For a stationary RP X(t), the power spectral density is the Fourier transform of the autocorrelation function, i.e.,

10. Example 2 Randomly choose a phase  ~ U[0,2]
Generate a sinusoid with fixed amplitude (A) and fixed freq (f0) but a random phase 
The RP is X(t)= A cos(2f0t + )
From the previous lecture, we know

11. Example 3 X(t)=X Random variable X~U[-1,1] In this case Thus,
For each realization of the RP, we have a different power spectrum

12. Power spectral density
The power content of a RP is the sum of the powers at all frequencies in that RP
To find the total power, need to integrate the power spectral density across all frequencies
Since SX(f) is the Fourier transform of RX(), then RX() will be the inverse Fourier transform of SX(f), Thus
Substituting =0, we get

13. Example 4
Find the power in the process of example 2

14. Translation to frequency domain
For the LTI system and stationary input, find the translation of the relationships between the input/output in frequency domain
Compute the Fourier transform of both sides to obtain
Which says the mean of a RP is its DC value. Also, phase is irrelevant for power. Only the magnitude affects the power spectrum, i.e., power dependent on amplitude, not phase

15. Example 5 If a RP passes through a differentiator H(f)=j2f
Then, mY=mX H(0) = 0
Also, SY(f) = 42 f2 SX(f)

16. Cross correlation in frequency domain
Let us define the cross spectral density SXY(f)
Since RYX() = RXY(-), we have
Although SX(f) and SY(f) are real nonnegative functions, SXY(f) and SYX(f) can generally be complex functions

17. Example 6 Randomly choose a phase  ~ U[0,2]
Generate a sinusoid with fixed amplitude (A) and fixed freq (f0) but a random phase 
The RP is X(t)= A cos(2f0t + )
The X(t) goes thru a differentiator H(f)=j2f

18. Example 7 X(t)=X Random variable X~U[-1,1]
If this goes through differentiation, then
SY(f) = 42 f2 ((f)/3) = 0
SXY(f) = -j2f ((f)/3) = 0

19. Power spectral density of a sum process
Z(t) = X(t)+Y(t)
X(t) and Y(t) are jointly stationary RPs
Z(t) is a stationary process with
RZ() = RX() + RY() + RXY() + RYX()
Taking the Fourier transform from both sides:
SZ(f) = SX(F) + SY(f) + 2 Re[SXY(f)]
The power spectral density of the sum process is the sum of the power spectral of the individual processes plus a term, that depends on the cross correlation
If X(t) and Y(t) are uncorrelated, then RXY()=mXmY
If at least one of the processes is zero mean, RXY()=0, and we get: SZ(f) = SX(F) + SY(f)

20. Example 8 X(t)=X Random variable X~U[-1,1]
Z(t) = X(t) + d/dt X(t), then
SXY(f) = jA2f0 /2 [(f+f0) - (f-f0)]
Thus,
Re[SXY(f)] = 0
SZ(f)= SX(f)+SY(f) = A2(1/4+2f02)[(f+f0)+(f-f0)]

21. Gaussian processes Widely used in communication
Because thermal noise in electronics is produced by the random movement of electrons closely modeled by a Gaussian RP
In a Gaussian RP, if we look at different instances of time, the resulting RVs will be jointly Gaussian:
Definition 1: A random process X(t) is a Gaussian process if for all n and all (t1,t2,…,tn), the RVs {X(ti)}, i=1,…,n have a jointly Gaussian density function.

22. Gaussian processes (Cont’d)
Definition 2: The random processes X(t) and Y(t) are jointly Gaussian if for all n and all (t1,t2,…,tn), and (1,2,…,m)the random vector {X(ti)}, i=1,…,n, {Y(j}, j=1,…,m have an n+m dimensional jointly Gaussian density function.
It is obvious that if X(t) and Y(t) are jointly Gaussian, then each of them is individually Gaussian
The reverse is not always true
The Gaussian processes have important and unique properties

23. Important properties of Gaussian processes
Property 1: If the Gaussian process X(t) is passed through an LTI system, then the output process Y(t) will also be a Gaussian process. Y(t) and X(t) will be jointly Gaussian processes
Property 2: For jointly Gaussian processes, uncorrelatedness and independence are equivalent