Wiener Hopf filter in time series.

• nikki92
In summary: Your name]In summary, to obtain the cross correlation vector p, you will need to use the formula E[U(n)d*(n)], where U(n) is a column vector of the signal u(n) and its previous values up to n-8, and d*(n) is the complex conjugate of the optimal delay value. The expected values of u(n) and d(n) can be calculated using the mean of the signal and the optimal delay value, respectively. You will need to shift the column vector U(n) to calculate the expected values for each element of p. Once you have all the values, they can be arranged in a 9x1 column vector for use in calculations. Examples and more detailed explanations
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.

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.

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!

1. What is a Wiener Hopf filter in time series?

A Wiener Hopf filter is a mathematical technique used to filter out noise from a time series data. It is based on the Wiener-Hopf equation, which is used to find the optimal filter coefficients that minimize the mean-square error between the filtered signal and the original signal.

2. How does a Wiener Hopf filter work?

A Wiener Hopf filter works by using a weighted average of the previous values of a time series to estimate the current value. It calculates the weights based on the autocovariance and cross-covariance functions of the time series. These weights are then used to filter out noise and improve the accuracy of the time series data.

3. What are the advantages of using a Wiener Hopf filter?

One of the main advantages of using a Wiener Hopf filter is its ability to effectively remove noise from a time series data. It also takes into account the correlation between different time points, making it a more accurate filtering method compared to other techniques. Additionally, the filter can be easily adapted to different types of time series data.

4. Are there any limitations of using a Wiener Hopf filter?

One limitation of using a Wiener Hopf filter is that it requires knowledge of the autocovariance and cross-covariance functions of the time series, which may not be available in some cases. Additionally, the filter assumes that the noise in the time series is uncorrelated with the signal, which may not always be true.

5. How is a Wiener Hopf filter applied in real-world scenarios?

A Wiener Hopf filter is commonly used in signal processing applications, such as in telecommunications and audio processing. It is also used in fields such as geophysics, finance, and weather forecasting to filter out noise from time series data and improve the accuracy of predictions and analysis.

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