Wiener Hopf filter in time series.

[nikki92]

Homework Statement

I have a signal where w(n) is white noise
u(n) = .4s(n) +.7s(n-1)-.1s(n-1) +w(n) and where variance of w(n) = .003

I want to find the cross correlation matrix against the optimal delay which I found to be 6.

Homework Equations

The Attempt at a Solution

E[U(n)d*(n-6)] where U(n) =column vector [u(n),u(n-1),….u(n-8)]

I can't for the life of me understand what how to get the cross correlation vector p. Could someone explain how to obtain it? I can not even find examples online. Everything seems to abstract.

 

[KataKoniK]

To find the cross correlation vector p, you will need to use the formula E[U(n)d*(n-6)], where U(n) is a column vector of the signal u(n) and its previous values up to n-8, and d*(n-6) is the complex conjugate of the optimal delay value. In this case, since the optimal delay is 6, d*(n-6) becomes d*(n), and the cross correlation vector p will be a 9x1 column vector.

To calculate E[U(n)d*(n)], you will first need to calculate the expected value of u(n) and d(n). Since u(n) is a combination of a signal and white noise, its expected value will be equal to the expected value of the signal, which can be calculated by taking the mean of the signal values. In this case, since the signal is not given, I will use an example signal of [1,2,3,4,5,6,7,8,9] for simplicity. So the expected value of u(n) will be 4.5.

The expected value of d(n) is simply the optimal delay value, which in this case is 6. So E[U(n)d*(n)] will be equal to 4.5*6 = 27.

To calculate E[U(n)d*(n-1)], you will need to shift the column vector U(n) by 1, so that it becomes [u(n-1), u(n-2),...,u(n-9)]. Then you can calculate the expected value of this shifted vector, which will be 4.5, and multiply it by the optimal delay value, which is now 5. So E[U(n)d*(n-1)] will be equal to 4.5*5 = 22.5.

You can continue this process to calculate the expected values for the remaining elements of the cross correlation vector p. Once you have all the values, you can arrange them in a 9x1 column vector and use it in your calculations.

I hope this helps clarify the process of obtaining the cross correlation vector p. If you have any further questions, please feel free to ask. Good luck with your research!