Instrumentation & Measurement Magazine 25-1 - 38

clutter densities and multipath. Optimal information selection
under sensor uncertainties using a predicted performance
bound is also discussed.
Target Tracking
Target tracking is the procedure to estimate the current
states, including the position and velocity of one or more targets
based on uncertain measurements from one or multiple
sensors. In general, the number of targets is unknown and
time-varying. Hence, in addition to estimating the states, we
need to estimate the number of active targets at each time step.
If the measurements from the sensors do not have any false
alarms and miss detections, estimating the number of targets
will be fairly simple. However, almost all the sensors used for
different applications are not perfect; thus they provide false
alarms and miss a few targets. An additional challenge is deciding
which measurement comes from which target in order
to use that measurement information to update the state estimates.
Due to these uncertainties, estimates will have errors
and a confidence region which is the covariance of the estimates.
This information should be provided to the user so that
any decision the user might make would incorporate the estimation
uncertainties.
To perform target tracking, the most important pieces of
information needed are the target kinematic model and the
measurement model. Target kinematic model, which provides
a better prediction at any given time, is used to improve our
estimates over time even with the same measurement uncertainties.
The kinematic model of a target is given as:
x Fx v
k 1 kk k
  
where xk is the state vector of the target at time k, Fk
(1)
is the state
transition matrix, and vk is the process noise. An example state
for tracking a target in a two-dimensional plane is [x-position,
y-position, x-velocity, y-velocity]. The state transition matrix
Fk
sumed to be known.
The measurement model depends on the sensor type, and
it can be written as:

zk
where hk


hx w kk k
uk

if originated from a target
if false alarm
(2)
is the measurement matrix, usually nonlinear, wk
is
the measurement error and uk is a false measurement. A target
might be detected with a probability, which is called the probability
of detection (Pd
cases, the Pd
), and it is mostly less than zero. In some
could even be less than 0.5. The number of false
alarms reported by a sensor in a scan depends on many factors,
and the spatial distribution of the false alarms could be uniform
or non-uniform. Knowledge of the false alarm density is
needed in most of the tracking algorithms.
In addition to measurement accuracy, the following challenges
must also be handled with measurement uncertainties
in order to get better tracking performance.
38
is usually known. The estimation becomes harder due to the
process noise vk, which is unknown, but its distribution is as◗
Measurement bias: unknown measurement biases
must be estimated and incorporated into the tracking
algorithms.
◗ Sensor location uncertainties: location of the sensors are
needed to estimate the target location from a measurement.
If a wrong sensor location is assumed in a tracking
algorithm, that will lead to multiple issues, mainly information
from multiple sensors may be fused.
◗ Multipath measurements: with a given measurement, we
cannot easily say whether that the measurement comes
directly from the targets or it is reflected on a surface
before it reaches the sensor. Ignoring multipath possibilities
will result in ghost tracks, i.e., we would think that
there is a target in a wrong direction or range.
◗ Multiple measurements from a target: due to high-resolution
sensors, we may get multiple measurements from
a target. With false alarms and closely spaced targets, it
is a challenge to associate the correct measurements with
the targets.
Information Prediction
In this section, we discuss how to predict the information that
we can get with the available sensor resources. This information
prediction can be used to optimally select the sensor
resources. In addition, the evaluation of a target tracking algorithm's
performance is a key factor in designing a tracking
system, tuning the parameters and most importantly comparing
the performance with other algorithms.
Posterior Cramér-Rao Lower Bounds (PCRLB)
The PCRLB calculates the theoretical lower bound on the accuracy
of any tracker with given sensor characteristics [8], which
helps in the design of the tracking system and sensor management.
The PCRLB is defined as the inverse of the Fisher
Information Matrix (FIM). Let Xk
kk
state vector, and 
the sensor measurements Zk
C k X ZX X ZX J k 

  
kk k kk k

XZ be an unbiased estimate of Xk
bound of the error covariance matrix as:

be an unknown and random
with
. Then the PCRLB gives the lower
 1
  
where J(k) is the FIM and E is the expectation over (Xk
,Zk
).
The FIM at time step k can be considered as the sum of prior
information and the information obtained from the sensors at
this time. The prior information is calculated using the target
kinematic model and the information from the last time step.
The measurement information will be calculated as a function
of the measurement model and sensor location. A measurement
information reduction factor will be used to incorporate
the probability of false alarms and the probability of missed
detections.
Several PCRLB derivations are proposed for handling various
measurement uncertainties. Fig. 1 shows the predicted
performance of two different approaches in handling sensor
location uncertainties [4]. The PCRLB plots help to understand
the amount of improvement possible with one approach
IEEE Instrumentation & Measurement Magazine
February 2022
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Instrumentation & Measurement Magazine 25-1

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