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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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7.3. Kalman filtering

Vector or multivariate approach given:

XK : state multivector (n×1)

xK : state vector of results

YK : multivector of observations (m×1)

yK : vector of observations and results

7.3.1. State equation

images

with A(K)= state matrix (n×n), deterministic matrix

and NK = process noise vector (l×1)

that we will choose centered, white and of correlation matrix (covariance matrix in the general case).

images :correlation matrix of the process noise vector NK

images : deterministic matrix

7.3.2. Observation equation

images

with

H(K): matrix of measurements or of observations (m×n), deterministic matrix;

WK : measurement noise vector of observations vector (p×1) that we choose, like NK, centered, white and of correlation matrix(covariance matrix in the general case);

images correlation matrix of the measurement noise vector WK

G(K) : (m × p) : deterministic matrix

The noises NK and WK are independent, and, as they are centered:

K and j

We will suppose, in what follows, that WKX0.

By iteration ...

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