BOLETIM - 26-04 · 2020. 4. 26. · Title: BOLETIM - 26-04 Created Date: 4/26/2020 6:27:46 PM
ECE3075A-26
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Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #1 ECE 3075A Random Signals Lecture 26 Autocorrelation Functions of Random Binary Processes School of Electrical and Computer Engineering Georgia Institute of Technology Fall, 2003 Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #2 Autocorrelation of a Binary Process A A − 0 t 1 t ) (t X ) ( 1 t X τ + 1 t a t t 4 0 + a t t − 0 a t t + 0 a t τ ) ( 1 τ + t X • A discrete (equi-probable at ), stationary, zero-mean process. • State change clocked at interval with arbitrary starting time, ; that is, is considered a random variable uniformly distributed over (0, ). • X(t) in one interval is statistically independent from X(t) in another interval. • The process is very common in data communications and digital computers. A ± a t 0 t a t 0 t Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #3 Autocorrelation of a Binary Process . | | | | ) ( ) ( 2 1 2 1 a t t t t X t X > = − τ if t independen are and Therefore, due to the fact that the process is stationary, zero-mean, a X t t X E t X E t X t X E R > = + = + = | | , 0 )] ( [ )] ( [ )] ( ) ( [ ) ( τ τ τ τ . , | | 0 1 2 1 t t t t t a of value the on depending interval, same the in be not may or may and When τ τ + = < a a a a a t t t t t t t t t t t t τ τ τ τ τ − = − + − = ≤ < − + = > + )] ( [ 1 } Pr{ } , 0 , Pr{ 1 1 1 0 1 1 1 interval same the in are and a a a a a t t t t t t t t t t t t τ τ τ τ τ + = − − + = + ≤ < − = < + )] ( [ 1 } Pr{ } , 0 , Pr{ 1 1 1 0 1 1 1 interval same the in are and a a t t t t | | } Pr{ 1 1 τ τ − = + interval same the in are and 0 t a t t + 0 1 t τ + 1 t 1 t τ + 1 t or Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #4 Autocorrelation of a Binary Process When are in the same interval, the product of is always ; when they are not, are independent with zero mean and thus zero correlation. Hence, 2 1 t t and 2 1 X X and 2 A 2 1 X X and > ≤ − = − = | | , 0 | | , | | 1 | | ) ( 2 2 a a a a a X t t t A t t A R τ τ τ τ τ τ ) ( τ X R 2 A a t a t − 0 Remarks: • When the two time instances are close to each other, the two corresponding r.v.s are likely to have the same value; • When they are apart far enough, it is equally probable that they’ll have the same value as they’ll have the opposite value; • At , the autocorrelation is the same as the mean square value, representing the power of the signal. 0 = τ
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ECE3075A-26
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Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #1
ECE 3075ARandom Signals
Lecture 26Autocorrelation Functions ofRandom Binary Processes
School of Electrical and Computer EngineeringGeorgia Institute of Technology
Fall, 2003
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #2
Autocorrelation of a Binary Process
A
A
0t
1t
)(tX
)( 1tX
+1t
att 40 +att 0 att +0
at
)( 1 +tX
A discrete (equi-probable at ), stationary, zero-mean process. State change clocked at interval with arbitrary starting time, ;
that is, is considered a random variable uniformly distributed over (0, ).
X(t) in one interval is statistically independent from X(t) in another interval.
The process is very common in data communications and digital computers.
Aat 0t
at0t
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #3
Autocorrelation of a Binary Process
.||||)()( 2121 attttXtX >= if tindependen are and Therefore, due to the fact that the process is stationary, zero-mean,
aX ttXEtXEtXtXER >=+=+= ||,0)]([)]([)]()([)(
.,||
0
121
ttttta
of value the on depending interval, same the in be not may or may and When +== if zero is and of product The
b
0t att +0||
The range that the center of the time-bar ( wide) can be in for it to be totally in the shaded area is
++
2||,
2||
00 btt
||
This range has a width of .||bTherefore,
atbtt ||}Pr{ 11
=+ interval duty active same the in are and
>
=
= || ,0
||,||1||)(22
b
bbt
bAt
bARaaX
atb / is called the duty cycle.
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #6
Other Examples of Autocorrelation Functions
1. Binary process with uniformly spaced switching intervals see previous discussion.
2. Binary process with uniformly spaced switching intervals and non-zero mean.
3. Binary process with randomly spaced switching times the telegraph process.
4. Bandpass filtered signals discussion will take place with introduction of power spectrum.
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #7
discussed. previously processbinary the is where )(')('21
2)( tXtXAtX +=
Binary Process with Non-zero Mean
>
=
= || ,0
||,||1||)(22
'
a
aaa
a
X
t
tt
At
tAR
>
+=
|| ,4
||,||144)(
2
22
a
aa
X
tA
tt
AA
R
A
0 0t
1t
)(tX
)( 1tX
+1tatt 40 +att 0 att +0
at
)( 1 +tX
)(41
4)(
41
4
)](')('[41)]('[
4)]('[
44)]()([)(
'
2
'
2
2
XX
X
RARA
tXtXEtXEAtXEAAtXtXER
+=+=
+++++=+=
Therefore,
2/2A
)(XR
4/2A
0
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #8
Arrival Interval of A Poisson Process
Probability of arrival interval, , as a random variable is the same as the probability that there is no arrival during that interval.
0,1}0)(Pr{1)}(0,in arrival noPr{1
}interval arrivalPr{)(
====
=
eX
F
0,)()( == eF
ddf
=== eeX!0)(}0)(Pr{
0
Last
arrival
Arrival interval
An exponential distribution!
)(F
)(f
e e1
-
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #26, Slide #9
Binary Process with Random Switching Times
=iit }{
tA
A
0t
't
)(tX
)'(tX
+'t
2t1t 1t
)'( +tX
The switching times occur as Poisson arrivals a point process.
When is less than the switching interval, the correlation between has a value .
)'()'( +tXtX and 2A
|)|exp( )}|| (0,in arrivalor switching Pr{nointerval} switching a within fall ' and 'Pr{
==+tt
|)|exp()( 2 = ARX Therefore, is the rate of switching (average number of switching per unit time).
)(XR2A
