Random Variables and Power Spectrum Density

[rsingh628]

Hello all, I having issues wrapping my head around how to approach this question I have from a study guide. Not quite sure how to find the PSD using the given parameters. Any help appreciated.


Approach: So, the PSD is the Fourier transform of the autocorrelation function R(m) = E(X_k*X_(k+m)), where m is the lag. I guess my issue is finding the autocorrelation function.

 

[rsingh628]

Update: think I figured something out: after finding the autocorrelation function and doing some rearranging, I get
R(m) = 2*delta(m) + a*delta(m+d) + a*delta(m-d)
S(e^j*omega) = 2 + 2a(cos(d*omega)) using inverse Euler's to get a real function instead of a sum of complex exponentials

So, plugging in values of a and d I get the following results. Can anyone kindly confirm?

 

[blamocur]

Not sure I agree with your formula for R(m) -- can you show how you got it?

But I do get the same answers

 

[rsingh628]

Not sure I agree with your formula for R(m) -- can you show how you got it?

But I do get the same answers

Here's my work to arrive at the answers:

 

[blamocur]

Shouldn't it be [imath](1+a^2) \delta(m)[/imath] instead of [imath]2\delta(m)[/imath]?

 

[rsingh628]

Shouldn't it be [imath](1+a^2) \delta(m)[/imath] instead of [imath]2\delta(m)[/imath]?

Oh yes you are correct, wonder why I missed that. Thanks for pointing that out!
Is that where our discrepancy is?

 

[blamocur]

Oh yes you are correct, wonder why I missed that. Thanks for pointing that out!
Is that where our discrepancy is?

This is the only issue I've found, but, as I mentioned earlier, this did not effect the answer because while effecting the scales that error has no effect on the shapes.

 

[rsingh628]

This is the only issue I've found, but, as I mentioned earlier, this did not effect the answer because while effecting the scales that error has no effect on the shapes.

Thank your for your help.