1. 10 Intro. to Random Processes
A random process is a family of random variables – usually an infinite family; e.g.,
{ Xn , n=1,2,3,... }, { Xn , n=0,1,2,... },
{ Xn , n=...,-3,-2,-1,0,1,2,3,... } or
{ Xt , t ≥ 0 }, { Xt , 0 ≤ t ≤ T }, { Xt , -∞ < t < ∞ }.

2. ω, the sequence of numbers Xn(ω) or the wave-
Recalling that a random variable is a function of
the sample space Ω, note that Xn is really Xn(ω)
and Xt is really Xt(ω). So, each time we change
ω, the sequence of numbers Xn(ω) or the wave-
form Xt(ω) changes...
A particular sequence or waveform is called a
realization, sample path, or sample function.

3. Xn(ω) for different ω

4. Zn(ω) for different ω

5. 5sin(2πfn) + Zn(ω) for different ω

6. Yn(ω) for different ω

7. Xt(ω) = cos(2πft+Θ(ω)) for different ω

8. Nt(ω) for different ω

9. Brownian Motion = Wiener Process

10. 10.2 Characterization of Random Process
Mean function
Correlation function

11. Properties of Correlation Fcns
symmetry: RX(t1,t2)=RX(t1,t2)

12. Properties of Correlation Fcns
symmetry: RX(t1,t2)=RX(t1,t2) since

13. Properties of Correlation Fcns
symmetry: RX(t1,t2)=RX(t1,t2) since
RX(t,t) ≥ 0

14. Properties of Correlation Fcns
symmetry: RX(t1,t2)=RX(t1,t2) since
RX(t,t) ≥ 0 since

15. Properties of Correlation Fcns
symmetry: RX(t1,t2)=RX(t1,t2) since
RX(t,t) ≥ 0 since
Bound:

16. Properties of Correlation Fcns
symmetry: RX(t1,t2)=RX(t1,t2) since
RX(t,t) ≥ 0 since
Bound:
follows by Cauchy-Schwarz inequality:

17. Second-Order Process
A process is second order if

18. Second-Order Process A process is second order if
Such a process has finite mean by the Cauchy-Schwarz inequality:

20. How It Works
You can interchange expectation and integration. If
then

21. How It Works
You can interchange expectation and integration. If
then

22. Example 10.12 If
then

23. Similarly,
and then

31. SX(f) must be real and even:

32. SX(f) must be real and even:
integral of odd function
between symmetic limits
is zero.

33. SX(f) must be real and even:

34. SX(f) must be real and even:
This is an even function of f.

35. 10.4 WSS Processes through LTI Systems

36. 10.4 WSS Processes through LTI Systems

37. 10.4 WSS Processes through LTI Systems

38. Recall
What if Xt is WSS?

39. Recall
What if Xt is WSS? Then
which depends only on the time difference!

40. Since

46. 10.5 Power Spectral Densities for WSS Processes

47. 10.5 Power Spectral Densities for WSS Processes