Hello,
I am currently studying the "Fundamentals of Adaptive Filtering" from Ali
H. Sayed and I have a problem in the first chapter. I am not really
looking
for the answer but the logical reasoning to get the answer.
If someone has the book I am referring to the Example 1.4.3 as to how to
get the answer tanh(y1+y2).
The problem is as follows:
I want to measure a value 'x'. The received signal is y=x+v. 'v' is a
zero-mean Gaussian random variable with unit-variance.
The optimal estimator for x is E(x|y). The conditional expectation when y
is given. This is fundamentals in filtering theory so I think everybody
knows it.
So when we have 'one' sample of the received sample 'y', the optimal
estimator is 'tanh(y)'.
My problem is. how to solve this when I have 'two' received samples?
The value of 'x' remains the same in both samples. For example the
received samples are 'y1' and 'y2';
y1 = x + v1;
y2 = x + v2;
The book says the derivation is trivial and the answer is 'tanh(y1+y2)',
but I cant seem to get this.
I would really appreciate is someone give me a lead.
Basically, if the estimator of x is indicated as x', then;
x' = E(x|y)
But when I have two samples, my confusion is, is it:
x' = E(x|y=y1 AND y=y2) or E(x|y=y1 OR y=y2)?
I am pretty sure it should be AND, then in that case;
x' = Integration[x.pdf(x|y=y1 and y=y2)]
Now the problem is to get pdf(x|y1 and y2) ?
Can we take y1 and y2 as independent events and write;
pdf(x|y1 and y2) = pdf(x|y1) x pdf(x|y2) ?
Any help is greatly appreciated.
Thank you very much.
Best Regards,
Maduranga
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