time cross correlation
TRANSCRIPT
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Time Crosscorrelation
Ex 6-6.1
ECE 3800
Western Michigan University
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= () ,
= () ,
= = ()
= lim ( + )
= lim () +
Cross-correlation functions
Statistical Cross-correlation for
two functions with a joint density
function.
Statistical Cross-correlation for
two functions with a shared random
variable.
Time Cross-correlation for
two functions.
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Exercise 6-6.1
The two functions sharethe same random variable,
q.
Solve for the cross-
correlation using a
statistical and time basedapproach.
= 2cos(5 + ) = 10sin(5 + )
q
f(q)
0 2p
= 1 = 2 = 12
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Time Cross-Correlation
To perform the time
cross-correlation of
the two functions we
will need to solve forthe Period T of the
two functions.
= lim ( + )
5 =2
=25
Period (T)
= 2cos(5 + )
= 10sin(5 + )
= 2cos(5 + ) 10sin(5 + + )
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Time Cross-Correlation
= cos(5 + ) sin(5 + 5 + )We will need to
use a trigmanipulation toseparate the
cosine and sinefunctions.
We are now ready
to integrate thetwo sinefunctions.
= + +
= 5 + 5 + = 5 +
= 8 sin 5 + sin(10 + 5 + 2)
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Time Cross-Correlation
= 8 sin 5 + sin(10 + 5 + 2)
8 sin 5
8 sin 10 + 5 + 2
8 sin 5 |
8 sin 5 2
5 2
5
8 sin 5
45
10sin 5
8 sin
= 10 + 5+2 = 10
8cos(10 + 5+2)|
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8cos(10 + 5+2)|
Time Cross-Correlation
8[cos 10
25 + 5+2 cos 10
25 + 5+2
8[cos 4 + 5+2 cos 4 + 5+2
The 4ppushes the cosine 4 cycles CW andCCW, but doesnt change the value.
0, 2p, 4p
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Time Cross-Correlation
The final result of thetime based cross-correlation is only a
function of lag time. Ifthe function is stationarywe expect this result. Ifthe function is also
ergodic the StatisticalCross-correlation shouldbe the same as the timebased method.
=10sin 5 +0= 10sin 5
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