Target Tracking - MATLAB Simulations for Radar Systems Design - page 417

represent radar observations, while columns represent track files. In cases
where several observations correlate with more than one track file, a set of pre-
determined association rules can be utilized so that a single observation is
assigned to a single track file.

9.6. State Variable Representation of an LTI System

A linear time invariant system (continuous or discrete) can be described
mathematically using three variables. They are the input, output, and the state
variables. In this representation, any LTI system has observable or measurable
objects (abstracts). For example, in the case of a radar system, range may be an
object measured or observed by the radar tracking filter. States can be derived
in many different ways. For the scope of this topic, states of an object or an
abstract are the components of the vector that contains the object and its time
derivatives. For example, a third-order one-dimensional (in this case range)
state vector representing range can be given by

R
R ß
ßß

x =

(9.23)

R ß ßß where , , and are, respectively, the range measurement, range rate

(velocity), and acceleration. The state vector defined in Eq. (9.23) can be rep-
resentative of continuous or discrete states. In this topic, the emphasis is on
discrete time representation, since most radar signal processing is executed
using digital computers. For this purpose, an n-dimensional state vector has the
following form:

ß n … t

ß 1 … x 2 ß 2 … x n

x =

x 1

(9.24)

where the superscript indicates the transpose operation.

The LTI system of interest can be represented using the following state equa-
tions:

ß () Ax () Bw ()

= +

(9.25)

y () Cx () Dw () = + (9.26)

ß where: is the value of the n × 1 state vector; y is the value of the p × 1 out-

put vector; w is the value of the m × 1 input vector; A is an nn × matrix; B

is an nm × matrix; C is p × n matrix; and D is an p × m matrix. The