ece3075a-27
DESCRIPTION
ECE3075A-27TRANSCRIPT
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Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #27, Slide #1
ECE 3075ARandom Signals
Lecture 27Crosscorrelation Functions
School of Electrical and Computer EngineeringGeorgia Institute of Technology
Fall, 2003
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #27, Slide #2
Crosscorrelation Functions
The correlation function between two different random processes is called a crosscorrelationfunction.
The crosscorrelation function measures how coherently two processes behave together, at various points in time.
Crosscorrelation functions are very much of interest in system analysis; for example, in studying the relationship between the input and the output of an electronic system, the relationship between the interest rate and various market indices, etc.
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #27, Slide #3
Jointly W-S Stationary Processes
A process X(t) is wide sense stationary if
Two processes are jointly wide sense stationary if
.difference time the only of function a 2.
and ,stationary sense wideare and Both 1.
),()]()([),(
)()( XYXY RtYtXEttR
tYtX=+=+
.difference time the only of function aand time), of nt(independe constant
),()]()([)]([
XXRtXtXEXtXE
=+==
)()( tYtX and
)(),(
),()]()([),(
2211
2212112121 21
tYYtXX
dyyxfyxdxtYtXEttR YXXY==
==
notation the used have wewhere
Recall the original definition:
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #27, Slide #4
Crosscorrelation of W-S Stationary Processes
are jointly stationary in wide sense. are two random variables,
The crosscorrelation function is defined as
The temporal order is of significance:
In the above, .
)()( tYtX and
21 YX and )(( 1211 +== tYY tXX and)
22121121 ),(][)( 21 dyyxfyxdxYXER YXXY
==
22121121 ),(][)( 21 dxxyfxydyXYER XYYX
==
)(( 1211 +== tXX tYY and)
).()(, 1221 =+== YXXY RRtttt and that follows then it Let
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Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #27, Slide #5
Time Crosscorrelation Functions
Define
If the processes are jointly ergodic,
+=T
TTXYdttytx
T)()(
21lim)( R
+=T
TTYXdttxty
T)()(
21lim)( R
)()(),()( YXYXXYXY RR == RR and Example:Two jointly random processes are of the form
where is a random variable uniformly distributed in (0, 2). Find the crosscorrelation of the two processes
)5sin(10)()5cos(2)( +=+= ttYttX and
)5sin(10)]2510sin(10[)5sin(10)]55sin(10)5cos(2[)(
=+++=+++=
tEttERXY
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #27, Slide #6
Properties of Crosscorrelation Functions
only measures the correlation of the two processes at synchronous points.
The maximum of a crosscorrelation function can occur anywhere, not necessarily at .
If the two random processes are independent,
And if any of the process has zero mean, the crosscorrelation function vanishes everywhere.
)0()0( YXXY RR and
)()();0()0( == YXXYYXXY RRRR2/1)]0()0([|)(| YXXY RRR
0=
)(][][][)( 212121 YXXY RYXYEXEYXER ====
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #27, Slide #7
Applications of Crosscorrelation Functions
)()()( tYtXtZ =
221112
111111
)()()()()()(
YXtYtXtZZYXtYtXtZZ
=++=+====
)()()()(][
)])([(][)(
21212121
221121
YXXYYX
Z
RRRRXYYXYYXXE
YXYXEZZER
+=+=
== The autocorrelation function of the sum is the sum of all the autocorrelation functions plus the sum of all the crosscorrelation functions.
If the two random processes are statistically independent and one of them has zero mean, the cross terms varnish, resulting inthe autocorrelation function of the sum is the sum of the autocorrelation functions.
.000][][][)( 212121 ===== YXYEXEYXERXY or if
Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #27, Slide #8
Sinusoid Plus Noise
).2,0()cos()(
over ddistribute
uniformly variable random a is where Let += tAtX cos
21)( 2ARX =
:),()(
function ationautocorrel withsignal, the of tindependen llystatistica process, noise mean zero a be Let
tXtV
||2)( = eBRV
:function ationautocorrel anhas which is process observed The )()cos()( tVtAtZ ++=
||22 cos21)( += eBARZ
Note that . as ,0)( ||2 = eBRVIt is thus possible to recover a sinusoid from noise contamination as long as we measure the autocorrelation at sufficiently long time lags.
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Fall 2003 ECE 3075A B. H. Juang Copyright 2003 Lecture #27, Slide #9
Use of Crosscorrelation - Example
A radar sends out a signal X(t) which is returned from a target, after a round-trip delay of . The receiver receives Y(t),which includes the delayed and attenuated signal and the noise :
1)( 1tX
)(tV1a where)()()( 1 XYa RtXtYtXt thus mean, zero has and tindependen are and , If
.01.0||136
01.0||1612
21)()()(
)12()(6)()(||
=
=
+=+=
XY
a
RtYtX
tXtYt
Thus 0. is and negative is
time, of half other the time; the of half of half is , If