Kalman, state-space model

In summary, the given problem involves a time discrete stocastic signal described by a specific formula and observed under the influence of white noise. The task is to find a space-state model using the given state. After some confusion and seeking help, the problem was successfully solved.
  • #1
mr.t
7
0

Homework Statement


A time discrete stocastic signal is described by
[tex]s(k) = w(k-1) + aw(k-2)[/tex], |a|<1
and w(n) is white gaussian noise with [tex]m_w = 0, \sigma_w^2 = 1[/tex]. It is observed under influence of white noise:
[tex]y(k) = s(k) + v(k)[/tex]
where v(n) is white gaussian noise with [tex]m_v = 0, \sigma_v^2=1[/tex]. v(n) and w(n) are independant.

Problem: Find the space-state model:
[tex]x(k+1) = Ax(k) + Bw(k)
y(k) = Cx(k) + v(k)[/tex]

By using the state:
[tex]x(k) = \bmatrix s(k) \\ w(k-1) \endbmatrix[/tex]

Homework Equations


(given above)

The Attempt at a Solution


I have only solved these problems when there is a AR-part. As this is an ARMA(0,2) I have no clue and need help. If its just an MA-part, then the whole A-matrix is zero? And how should I use the fact that I am suppose to use the specified states? How does that affect the state-space model?

Im confused, please help me!
Thanks!
 
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  • #2
Just want to let you guys know that I've solved it. (pretty sure at least :tongue2:)
 
  • #3


I would approach this problem by first understanding the concept of state-space models and how they are used in signal processing. A state-space model is a mathematical representation of a system that describes the behavior of a system over time. It is commonly used in control systems, signal processing, and other fields of engineering.

In this particular problem, we are given a stochastic signal described by an ARMA(0,2) model. This means that the signal is a combination of two moving average terms, with the first term being delayed by one time step. We are also given the observation equation, which is a combination of the signal and white noise.

To find the state-space model, we need to express the signal in terms of a state vector x(k) and then derive the state equations. In this case, we are given the state x(k) as a combination of the signal s(k) and the previous noise term w(k-1). Using this state vector, we can express the signal equation as x(k+1) = Ax(k) + Bw(k).

To find the A and B matrices, we can substitute the given ARMA(0,2) model into the state equation and solve for A and B. This will give us the state equations in terms of the state vector x(k). Similarly, we can express the observation equation as y(k) = Cx(k) + v(k) and solve for the C matrix.

In conclusion, the state-space model for this problem is:

x(k+1) = \bmatrix 0 & 1 \\ a & 0 \endbmatrix x(k) + \bmatrix 0 \\ 1 \endbmatrix w(k)

y(k) = \bmatrix 1 & 0 \endbmatrix x(k) + v(k)

where the state vector x(k) is defined as x(k) = \bmatrix s(k) \\ w(k-1) \endbmatrix.

I hope this explanation helps you understand how to approach and solve this problem. It is always important to have a strong understanding of concepts and techniques before attempting to solve problems.
 

1. What is a Kalman filter?

A Kalman filter is a mathematical algorithm used to estimate the state of a system based on noisy measurements. It is commonly used in control theory and signal processing to improve the accuracy of predictions and reduce the effects of measurement errors.

2. How does a Kalman filter work?

A Kalman filter uses a system model and measurements to recursively update its estimates of the true state of a system. It combines current measurements with a prediction based on past estimates to produce a more accurate estimate of the current state.

3. What is a state-space model?

A state-space model is a mathematical representation of a dynamic system. It describes how the state of the system changes over time based on its inputs and internal dynamics. Kalman filters are often used to estimate the state of a system based on a state-space model.

4. What are the applications of Kalman filters?

Kalman filters have a wide range of applications, including navigation systems, control systems, robotics, and signal processing. They are particularly useful in situations where there is uncertainty or noise in measurements and the true state of a system needs to be estimated.

5. Are there any limitations of Kalman filters?

While Kalman filters are effective in many applications, they do have some limitations. They are based on linear equations and assumptions, so they may not perform well in nonlinear systems. They also require knowledge of the system model and accurate measurements to produce reliable estimates.

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