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Augustin
Augustin
8:27 PM
So what's wrong with the covariance matrix?
 
sepideh
sepideh
I mean the definition of being $WSS$
is WSS random processes only require that 1st moment and autocovariance do not vary with respect to time
 
Augustin
Augustin
It's just two things.
1. E(X_t) does not depend on t.
2. The autocovariance depends only on the lag.
 
sepideh
sepideh
yeah
so why don't we prove this in order to show that a process is WSS
and instead
we prove
 
Augustin
Augustin
This is exaclty what you should prove if you want to prove that a process is WSS
 
sepideh
sepideh
1.E(X_t) dows not depend on t
2.autocorrelation depends only on the lag
 
Augustin
Augustin
8:31 PM
yes that's all
 
sepideh
sepideh
you mean using either of them
is true?
 
Augustin
Augustin
No no no, you have to prove both
 
sepideh
sepideh
OK so I should prove
 
Augustin
Augustin
a process is WSS iff these two conditions are true
 
sepideh
sepideh
1. E(X_t) does not depend on t.
2. The autocovariance depends only on the lag.
 
Augustin
Augustin
8:34 PM
yes
 
sepideh
sepideh
3.autocorrelation depends only on the lag
 
Augustin
Augustin
3 is a consequence of 2
because autocorrelation is covariance/sigma^2
 
sepideh
sepideh
OK
so when we say R(\tau) it means that autocovariance=f(\TAU)
and there's no difference whether to showIt's just two things.
1. E(X_t) does not depend on t.
2. The autocovariance depends only on the lag.
or
1.E(X_t) dows not depend on t
2.autocorrelation depends only on the lag
?????
 
Augustin
Augustin
yes it's the same thing because autocorrelation is just a renormalization of autocovariance
 
sepideh
sepideh
OK thanks
but about my second question
here in my question we have
which is f(A,B,\Phi;t)
so the autocovariance matrix should be 3x3
right?
 
Augustin
Augustin
8:42 PM
$A$ and $B$ are constant scalars, vectors, random variables, random vectors ?
 
sepideh
sepideh
the exercise says that
B, A, and Ф are
independent random variables. A and B and Ф are uniformly distributed
over (0,1] and (0,2] and (0, 2π] respectively
 
Augustin
Augustin
so X_t is a 1D random variable, for each t
 
sepideh
sepideh
so why?
 
Augustin
Augustin
then the autocovariance function is a real valued function
 
sepideh
sepideh
autocovariance matrix is 1x1
????
why?
 
Augustin
Augustin
8:45 PM
because it's the covariance of 1D random variables
 
sepideh
sepideh
X(t) a function of three random variables!!!
 
Augustin
Augustin
yes but X(t) is 1D
the autocovariance is not between A, B and phi, it's between X(t) and X(t+h)
 
sepideh
sepideh
OK
there's an example in the tutorial
X(T) = A sin(w0 t + Ф)
A and Ф are independent
A is uniformly distributed over (0,1)
Ф is uniformly distributed over (0, 2π)
and is solved like this:
so this is true!
I think I understood now
thanks for your help
 
Augustin
Augustin
yes, so X is WSS
 
sepideh
sepideh
yeah
very nice
thanks alot
!!!
 
Augustin
Augustin
8:50 PM
you're welcome
 
sepideh
sepideh
bye
 
Augustin
Augustin
bye
 

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