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 < ∞ }.
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 waveform Xt(ω) changes...
A particular sequence or waveform is called a
realization, sample path, or sample function.
Xn(ω) for different ω
Zn(ω) for different ω
5sin(2πfn) + Zn(ω) for different ω
Yn(ω) for different ω
Xt(ω) = cos(2πft+Θ(ω)) for different ω
Nt(ω) for different ω
Brownian Motion = Wiener Process
10.2 Characterization of Random
Process

Mean function

Correlation function
Properties of Correlation Fcns

symmetry: RX(t1,t2)=RX(t1,t2)
Properties of Correlation Fcns

symmetry: RX(t1,t2)=RX(t1,t2) since
Properties of Correlation Fcns

symmetry: RX(t1,t2)=RX(t1,t2) since

RX(t,t) ≥ 0
Properties of Correlation Fcns

symmetry: RX(t1,t2)=RX(t1,t2) since

RX(t,t) ≥ 0 since
Properties of Correlation Fcns

symmetry: RX(t1,t2)=RX(t1,t2) since

RX(t,t) ≥ 0 since

Bound:
Properties of Correlation Fcns

symmetry: RX(t1,t2)=RX(t1,t2) since

RX(t,t) ≥ 0 since

Bound:
follows by Cauchy-Schwarz inequality:
Second-Order Process

A process is second order if
Second-Order Process


A process is second order if
Such a process has finite mean by the
Cauchy-Schwarz inequality:
How It Works
You can interchange expectation and
integration. If
then
How It Works
You can interchange expectation and
integration. If
then
Example 10.12
If
then
Similarly,
and then
SX(f) must be real and even:
SX(f) must be real and even:
integral of odd function
between symmetic limits
is zero.
SX(f) must be real and even:
SX(f) must be real and even:
This is an even function of f.
10.4 WSS Processes through LTI
Systems
10.4 WSS Processes through LTI
Systems
10.4 WSS Processes through LTI
Systems
Recall
What if Xt is WSS?
Recall
What if Xt is WSS? Then
which depends only on the time difference!
Since
10.5 Power Spectral Densities for
WSS Processes
10.5 Power Spectral Densities for
WSS Processes