7.4.3 Introduction to the Kalman Filter-Smoother

NOTATION

As far as possible, the notation is in line with that used in most textbooks on Least Squares procedures. However, due to the larger number of estimation steps in filter/smoother techniques than in classical Least Squares, a clear distinction between various quantities, such as the "residuals", is therefore necessary. This is accomplished by a consistent use of superscripts, for both parameters and residuals. The following rules for style will apply, unless explicitly stated otherwise.

General:

lowercase normal:	s	scalar
lowercase bold:	s	vector
uppercase normal:	S	matrix element(s)
uppercase bold:	S	matrix

Some important symbols used in the following developments are:

l	vector of measurements
x	state vector, that is vector of parameters
	state vector, with time subscript
	increment of the state vector
v	vector of residuals
,	approximate state vector, respective residuals
,	predicted state vector, respective residuals
,	adjusted (filtered) state vector, respective residuals
,	smoothed state vector, respective residuals
QII	covariance matrix of the measurements
P	weight matrix of the measurements (subscript l is omitted)
Q	covariance matrix of the predicted state vector
P	weight matrix of the predicted state vector
Ql	covariance between the measurements and the filtered state vector
A	design matrix for parametric adjustment
B	design matrix for condition adjustment
	transition matrix
W	vector of miscloses, or system noise (depending on context)
K	Kalman gain matrix

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© Chris Rizos, SNAP-UNSW, 1999