ECE 5510: Random Processes
Lecture Notes
Fall 2009
Lecture 17
Today: Random Processes: (1) Autocorrelation & Autocovari
ance, (2) Wide Sense Stationarity, (3) PSD
As a brief overview of the rest of the semester.
•
We’re going to talk about autocovariance (and autocorrela
tion, a similar topic), that is, the covariance of a random
signal with itself at a later time. The autocovariance tells us
something about our ability to predict future values (
k
in ad
vance) of
Y
k
.
The higher
C
Y
[
k
] is, the more the two values
separated by
k
can be predicted.
•
Autocorrelation is critical to the next topic: What does the
random signal look like in the frequency domain? More specif
ically, what will it look like in a spectrum analyzer (set to av
erage). This is important when there are specific limits on the
bandwidth of the signal (imposed, for example, by the FCC)
and you must design the process in order to meet those limits.
•
We can analyze what happens to the spectrum of a random
process when we pass it through a filter.
Filters are every
where in ECE, so this is an important tool.
•
We’ll also discuss new random processes, including Gaussian
random processes, and Markov chains.
Markov chains are
particularly useful in the analysis of many discretetime engi
neered systems, for example, computer programs, networking
protocols, and networks like the Internet.
1
Expectation of Random Processes
1.1
Expected Value and Correlation
Def’n:
Expected Value of a Random Process
The expected value of continuous time random process
X
(
t
) is the
deterministic function
μ
X
(
t
) =
E
X
(
t
)
[
X
(
t
)]
for the discretetime random process
X
n
,
μ
X
[
n
] =
E
X
n
[
X
n
]
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ECE 5510 Fall 2009
2
Example: What is the expected value of a Poisson process?
Let Poisson process
X
(
t
) have arrival rate
λ
. We know that
μ
X
(
t
)
=
E
X
(
t
)
[
X
(
t
)] =
∞
summationdisplay
x
=0
x
(
λt
)
x
x
!
e

λt
=
e

λt
∞
summationdisplay
x
=0
x
(
λt
)
x
x
!
=
e

λt
∞
summationdisplay
x
=1
(
λt
)
x
(
x

1)!
=
(
λt
)
e

λt
∞
summationdisplay
x
=1
(
λt
)
x

1
(
x

1)!
=
(
λt
)
e

λt
∞
summationdisplay
y
=0
(
λt
)
y
(
y
)!
=
(
λt
)
e

λt
e
λt
=
λt
(1)
This is how we intuitively started deriving the Poisson process  we
said that it is the process in which on average we have
λ
arrivals
per unit time. Thus we’d certainly expect to see
λt
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 Fall '08
 Chen,R
 Autocorrelation, Stationary process, Random Processes

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