Joint Optimization of Detection and Tracking in Adaptive Radar Systems - Advances in Heuristic Signal Processing and Applications - page 117

bounds is usually not known and questionable. In this aspect, they can provide at
most semi-quantitative measures for offline performance evaluation of a tracking
filter.

• Analytic Model Approach: The second class of tools is referred to as the analytic

model approach [ 49 ]. In this methodology, the aim is to establish some (possibly
approximate) analytic relationships between the performance measure and some
“key” parameters of the algorithm (see, e.g., [ 8 , 51 , 59 ]). Although these tech-
niques provide analytically useful expressions, they are obtained under several
assumptions and approximations due to complexity of the big picture. Therefore,
their accuracy is still not as good as of the performance prediction approach.

• Performance Prediction Approach: This is an algorithmic approach. It aims at

obtaining an offline (or a deterministic) algorithm for calculation of one of the
performance measures of the tracking filter, usually the estimation error covari-
ance. Developing such a deterministic algorithm for the covariance propagation
is in general a hard task. However, this methodology is proven to produce much
more accurate results compared to the previous two techniques mentioned above
(see, for example, [ 20 , 46 ], and [ 47 ]).

In this section (and also in the entire chapter), our focus will be on the perfor-
mance prediction category which we refer to as non-simulation performance pre-
diction (NSPP) techniques. The key point in NSPP techniques is to obtain a de-
terministic recursion for the estimation error covariance , which then can be used
to quantify the filter's performance offline. In the simplest case, when there is no
clutter and no variation in target dynamics (i.e., no “target maneuver” in tracking
terminology), the Kalman filter [ 37 ] already has a deterministic covariance recur-
sion in the form of a (matrix) Riccati equation [ 1 ]. However, for the more complex
situations in which there is clutter or the target dynamics is time-varying, the error
covariance calculation of the filter under concern is no longer deterministic. This is
due to the presence of discrete type uncertainties introduced into the problem, which
makes the covariance calculation dependent on the measurements received, hence
stochastic.
To be able to make an NSPP for the filters in these situations, there are two main
methodologies proposed so far. The first methodology, which also pioneered the
NSPP topic, is the work of Fortmann et al. [ 20 ] where both types of uncertainties
(discrete and continuous) in the problem are globally averaged out. In this pioneer-
ing work [ 20 ], the authors applied this methodology for the Probabilistic Data As-
sociation Filter (PDAF) [ 7 ] and obtained a Riccati-like recursion for the determin-
istic calculation of its covariance. This recursion was named as the Modified Riccati
Equation (MRE) [ 20 ]. The MRE approach is further extended to multi-sensor case
(Multisensor PDAF—MSPDAF) by Frei [ 21 ], and recently studied in the context of
NSPP for Kalman filtering with intermittent observations [ 14 , 15 , 61 ].
Inspired by the work of Fortmann et al., the second methodology was proposed
by Li et al. in [ 46 ] where only the continuous uncertainties are averaged, while the
discrete uncertainties are retained in the propagation of the covariance. Similar to
Fortmann et al, in the proposal paper of their algorithm [ 46 ], they first derive it for
the PDAF and name it as the Hybrid Conditional Averaging (HYCA) algorithm.