Wide sense stationary process in digital communication

Non

1. cStanley Chan 2020. All Rights Reserved. ECE 302: Lecture A.5 Wide Sense Stationary Processes
Prof Stanley Chan
School of Electrical and Computer Engineering
Purdue University
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2. cStanley Chan 2020. All Rights Reserved.
Wide Sense Stationary Processes
Denition
A random processX(t) iswide sense stationary (W.S.S.)if:
1
X(t) = constant;for allt,
2
RX(t1;t2) =RX(t1t2)for allt1;t2.
Remark 1: WSS processes can also be dened using the autocovariance
function
CX(t1;t2) =CX(t1t2):
Remark 2: Because a WSS is completely characterized by the dierence
t1t2, there is no need to keep track of the absolute indicest1andt2.
We can rewrite the autocorrelation function as
RX() =E[X(t+)X(t)]: (1)
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3. cStanley Chan 2020. All Rights Reserved.
Visualizing WSS Processes -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
t
1
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
t
2 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
t
1
-1
-0.5
0
0.5
1
t
2
= -0.13
t
2
= 0
t
2
= 0.3
Figure:
RX(t1;t2) =
1
2
cos

!(t1t2)

.
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4. cStanley Chan 2020. All Rights Reserved.
Physical Interpretation ofRX()
Consider the following function:
b
RX()
def
=
1
2T
Z
T
T
X(t+)X(t)dt: (2)
This function is thetemporal averageofX(t+)X(t)
How do we understand
b
RX()?
b
RX() is the \un-ipped convolution", orcorrelation, ofX() and
X(t+).
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5. cStanley Chan 2020. All Rights Reserved.
Correlation vs convolution-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-10 -5 0 5 10 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-10 -5 0 5 10
(a) Convolution (b) Correlation
Figure:
functionX(t) is ipped before we compute the result. For correlation, there is no
ip.
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6. cStanley Chan 2020. All Rights Reserved.
So what?
Lemma
Let
b
RX()
def
=
1
2T
R
T
T
X(t+)X(t)dt. Then,
E
h
b
RX()
i
=RX(): (3)
Proof.
E
h
b
RX()
i
=
1
2T
Z
T
T
E[X(t+)X(t)]dt
=
1
2T
Z
T
T
RX()dt
=RX()
1
2T
Z
T
T
dt
=RX():
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7. cStanley Chan 2020. All Rights Reserved.
Example
Example 1. Consider a random processX(t) such that for everyt,X(t)
is an i.i.d. Gaussian random variance with zero mean and unit variance.
FindRX().
Solution.
RX() =E[X(t+)X(t)] =
(
E[X
2
(t)]; = 0;
E[X(t+)]E[X(t)]; 6= 0
Using the fact thatX(t) is i.i.d. Gaussian for allt, we can show
E[X
2
(t)] = 1 for anyt, andE[X(t+)]E[X(t)] = 0. Therefore, we have
RX() =
(
1; = 0;
0; 6= 0
:
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8. cStanley Chan 2020. All Rights Reserved.
Visualization0 200 400 600 800 1000
-4
-3
-2
-1
0
1
2
3
4 0 500 1000 1500 2000
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
correlation of sample 1
correlation of sample 2
auto-correlation function
(a)X(t) (b)
b
RX()
Figure:
functionX(t) is ipped before we compute the result. For correlation, there is no
ip.
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9. cStanley Chan 2020. All Rights Reserved.
Properties ofRX()
Corollary
RX(0) =average power ofX(t)
Proof. SinceRX(0) =E[X(t+ 0)X(t)] =E[X(t)
2
], and sinceE[X(t)
2
] is
the average power, we have thatRX(0) is the average power ofX(t).
Corollary
RX()is symmetric. That is,RX() =RX().
Proof. Note thatRX() =E[X(t+)X(t)]. By switching the order of
multiplication in the expectation, we have
E[X(t+)X(t)] =E[X(t)X(t+)] =RX().
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10. cStanley Chan 2020. All Rights Reserved.
Properties ofRX()
Corollary
P(jX(t+)X()j> )
2(RX(0)RX())

2
This result says that ifRX() is slowly decaying fromRX(0), then the
probability of having a large deviationjX(t+)X()jis small.
Proof.
P(jX(t+)X()j> )E[(X(t+)X())
2
]=
2
=

E[X(t+)
2
]2E[X(t+)X(t)] +E[X(t)
2
]

=
2
=

2E[X(t)
2
]2E[X(t+)X(t)]

=
2
= 2

RX(0)RX()

=
2
:
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11. cStanley Chan 2020. All Rights Reserved.
Properties ofRX()
Corollary
jRX()j RX(0), for all.
Proof. By Cauchy inequalityE[XY]
2
E[X
2
]E[Y
2
], we can show that
RX()
2
=E[X(t)X(t+)]
2
E[X(t)
2
]E[X(t+)
2
]
=E[X(t)
2
]
2
=RX(0)
2
:
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12. cStanley Chan 2020. All Rights Reserved.
Ergodic theorem
Under what conditions will
b
RX()!RX() asT! 1?
Theorem (Mean-Square Ergodic Theorem)
LetY(t)be a W.S.S. process, with meanE[Y(t)] =and autocorrelation
functionRY(). Dene
MT
def
=
1
2T
Z
T
T
Y(t)dt: (4)
Then,E
h

MT


2
i
!0asT! 1if and only if
lim
T!1

1
2T
Z
T
T

1
jj
2T

RY()d

= 0: (5)
ProofOptional. See eBook.
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13. cStanley Chan 2020. All Rights Reserved.
Summary
Everything you need to know about a WSS process.
The mean of a WSS process is a constant (does not need to be
zero)
The correlation function only depends on the dierence, so
RX(t1;t2) is toeplitz.
You can writeRX(t1;t2) asRX(), where=t1t2.
RX() tells you how much correlation you have with someone
located at a time instantfrom you.
You can think ofRX() as the temporal correlation
b
RX().
Under certain regularity conditions,
b
RX() is a good
approximation ofRX().
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14. cStanley Chan 2020. All Rights Reserved.
Questions?
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