The Kalman Equation

Before defining the Kalman filter itself, it is necessary to define some notation that will make the formulation and interpretation of the results simpler. First, let the vector of available observations be defined as 

let the time point at which the prediction is to be made be denoted by t. If s t, then the problem to be solved is called a prediction or forecasting problem. If s = t, then the problem is called filtering. If s t, then the problem is called smoothing. Irrespective of the time horizon, a general term for this problem is (state) estimation. Finally, define the conditional estimate of the state given the information currently available as 

If fi = 2 = t, then only a single t will be written to simplify the notation. 

Theorem 5.5 Basic Kalman Filter. For the state-space model previously defined with initial conditions Xo o = Pq and ZQ Q,for f = 0. n, the Kalman filter can be defined as 

Proof The proof for these equations can be found in (Kalman 1960). Q.E.D. 

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