AdaptiveFiltering.cs

Kalman filter that will linearize non-linear process around the covariance and current average value is called the extended Kalman filter. The equations are defined as following: 

Time update: 

x(k) = f(x(k-1), u(k-1), 0 )  

P(k) = A(k)*P(k-1)*A(k)^T + W(k)*Q*W(k)^T  

Measurement update: 

K(k) = P(k)*H(k)^T *( H(k)*P(k)*H(k)^T + V(k)* R*V(k)^T )^-1  

x(k) = x(k) + K(k) * (z(k) - h(x(k), 0))  

P(k) = (I - K(k)*H(k))*P(k)  

The process x(k) has a known mathemathical model. When we have an application where we need to measure x(k) as accurately as possible the Kalman filter can help us reduce the noise. We use the added information from the mathemathical model to more effectively filter the signal. 

Description of symbols: 

s - number of parallel inputs (columns in x and z).  

f - non-linear function relating x(k-1) to x(k)  

x(k) - size: n x s. vector state (value) of the process in each column  

u(k) - control vector input. size: l x s  

B - size: n x l maps control input u(k-1) to x(k)  

Q - process noise covariance  

R - measurement noise covariance  

z - size: m x s. Measurement vector  

h - Non-linear function relates x(k) to the measurement z(k)  

w(k) - process noise  

v(k) - measurement noise  

A(k) - size: n x n. matrix of partial derivates of f with respect to x  

H(k) - matrix of partial derivates of h with respect to x  

W(k) - matrix of partial derivates of f with respect to w  

V(k) - matrix of partial derivates of h with respect to v  

All parameters can be modified by the user before each iteration of the filter. One iteration of the filter is achieved by calling the Update method. The process noise and measurement noise are assumed to be independent and gaussian.[1] An Introduction to the Kalman Filter, Greg Welch and Gary Bishop

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