TABLE OF CONTENTS It is shown how best to combine two independent estimates of a variable to form a weighted mean value, this operation being central to the process of Kalman filtering [1, 2]. The following development assumes only a knowledge by the reader of elementary statistical principles. A full mathematical derivation of the Kalman filter is beyond the intended scope of this book. Therefore, the reader interested in a mathematical treatise on the subject is referred to the excellent text by Jazwinski [3].
Consider the situation in which two independent estimates, x 1 and x 2, are provided of a quantity x, where 
and 
are their respective variances. It is required to combine the two estimates to form a weighted mean, corresponding to the 'best', or minimum variance, estimate, 
. In general, the weighted mean may be expressed as:
| (A.1) |  |
where w 1 and w 2 are the weighting factors and w 1 + w 2 = 1. The expected or mean value of 
, written as E( 
), is given by:
| (A.2) |  |
The variance of a quantity x is defined to be E[{ x - E( x)} 2]. Hence the variance of 
, denoted ? 2, may be written as:
| (A.3) |  |
Since x 1 and x 2 are independent, ( x 1 - E( x 1)) and ( x 2
