A ROBUST KALMAN FILTER FOR ESTIMATION AND TRACKING OF A
CLASS OF PERIODIC DISCRETE EVENT PROCESSES
Stephen D. Elton1 and Benjamin J. Slocumb2
1 Electronics and
Surveillance Research Laboratory
Defence Science and Technology Organisation and
Cooperative Research Centre for Robust and Adaptive Systems
P.O. Box 1500, Salisbury, SA 5108, Australia
2Electronic Systems Laboratory, Georgia Tech Research Institute
Georgia Institute of Technology, Atlanta, GA 30332-0840, U.S.A.
ABSTRACT
This paper discusses a Kalman lter approach to parameter
estimation and tracking for a class of discrete event processes. The proposed estimation techniques operate on the
recorded event arrival time sequence of a pulse train signal
with pulse occurrence times corrupted by timing noise. In
adopting a state space approach to signal modelling, a number of real-world conditions are considered and this leads to
the formulation of a Kalman lter estimator that is robust
to missing and false data, and to signal model mismatch.
1. INTRODUCTION
In this article we demonstrate the merits of a state space formulation to parameter estimation and tracking for a class of
discrete event signals known as point processes. This work
is motivated by an estimation problem for pulsed radar signals collected by a passive electronic surveillance sensor in
which the observed sequence of pulses is replaced by a series of delta functions. The characterisation and identication of a radar source for electronic defence purposes is
made possible by the estimation of parameters associated
with the intercepted radar emissions, e.g., the estimation of
pulse train periodicity. In this paper we take a mean square
estimation approach and use a Kalman lter to compute recursive parameter estimates based on a state space model
for the pulse train signal of interest (SoI).
Previous research that relates to the current paper includes the development of parameter estimation techniques
for constant period signals [1]. We extend and improve on
this earlier work by addressing the following issues: (i) initialisation of the Kalman lter, (ii) robustness to signal
model mismatch, (iii) robustness to missing and false data,
and (iv) the estimation and tracking of the time-varying
period of a radar pulse train signal. The application of the
Kalman lter is discussed here in the context of the analysis
of an isolated SoI. That is not to say, however, that it could
not be used for multiple, time interleaved pulse train signals
S.D. Elton acknowledges the funding of the activities of the
Cooperative Research Centre for Robust and Adaptive Systems
by the Australian Commonwealth Government under the Cooperative Research Centres Programme.
associated with a complex electromagnetic signal environment, either through the addition of pre-ltering techniques
or by extending the single Kalman lter to a Kalman lter
bank that processes multiple pulse train signals in parallel
as described in [2,3].
2. POINT PROCESS SIGNAL MODEL
Although we allow for signal pre-ltering, by processing signals recorded by a sensor using the observed event time of
arrival (TOA) sequence of a pulse stream, this admits a particularly simple type of discrete time signal model known
as a point process. Pulses arrival time measurements are
modelled by a series of delta functions, and in this representation, pulse observations are distinguishable from each
other only by their occurrence times. Consider a point process signal, p(t) say, with N arrival times ftn g, where p(t)
is dened by
p(t) =
NX
?1
n=0
(t ? tn ):
(1)
The ftn g form a time ordered sequence with tn+1 > tn ; 8n,
measured by a surveillance sensor using leading edge timing circuitry. The observed pulse train arrival time sequence
may be nominally periodic, or exhibit a time dependency
corresponding to some form of temporal modulation. In either case, the pulse TOAs will be subject to random timing
perturbations, termed jitter. In this paper we will consider
two types of jitter model known as non-cumulative jitter
(NCJ) and cumulative jitter (CJ) as described in [1].
3. STATE SPACE CONCEPTS
In formulating the pulse train parameter estimation problem of interest, and to allow the use of \standard" Kalman
lter notation, we will assume that a sequence of discrete
pulse arrival time measurements fzn g is available such that
zn = tn + vn ; n = 0; 1; : : : ;
(2)
where the ftn g now represent scheduled event arrival times
with
tn+1 = tn + Tn + un+1 ; n = 0; 1; : : : ; (3)
t0 = t + u 0 ;
(4)
such that Tn denotes the pulse train period or pulse repetition interval (PRI), at time tn , and t the pulse TOA phase
which denes the start of the event arrival time sequence
relative to an arbitrary time origin. The sequence fvn g corresponds to a NCJ component with distribution N (0; v2 ),
and fun g a CJ component with distribution N (0; u2 ). To
allow for a possible time-varying periodicity in the observed
pulse arrival time sequence we adopt a random walk process
as our model for the time evolution of Tn , thus
Tn+1 = Tn + n ;
(5)
where Tn is assumed to be a slowly varying parameter and
where n is a Gaussian random variable with distribution
N (0; 2 ). A stochastic model such as this proves useful in
tracking what may really be a deterministic time-varying
signal periodicity, e.g., a linear ramp, step and dwell, or
sinusoidal PRI modulation. A value of equivalent to
a few percent of the mean PRI was found to be sucient
for tracking. Alternatively, one could use exponential data
weighting to provide the Kalman lter with a similar capability for tracking slowly varying periodicities by assigning
more weight to the most recently acquired TOA data.
Next, we establish a state space representation for the
pulse train period estimation and tracking problem by considering a linear stochastic time-invariant dynamic system
driven by additive white Gaussian noise. In this context,
let us dene the state vector, xn , by xn = [tn Tn ]T , where
tn and Tn are as above. The dynamics of the system are
described by the state equation
xn+1 = F xn + Gwn ; n = 0; 1; : : :
(6)
where F is the state transition matrix and G the disturbance transition matrix. The initial state of the system,
x0 , is assumed to be a Gaussian random variable with
known mean, x 0 = E fx0 g, and covariance, P 0 = E f(x0 ?
x 0 )(x0 ? x 0 )T g. The vectors fw n g correspond to a zeromean Gaussian white driving noise sequence, termed process noise, , where wn = [wn(1) wn(2)]T , and with known covariance, E fw n wTm g = Qnm , where nm is the Kronecker
delta function and
2
2
0
0
u
w
(1)
(7)
Q=
0 w2 (2) = 0 2 ;
such that n wn(2) . The inclusion of process noise in the
state equation allows the Kalman lter a degree of robustness to modelling errors in the assumed state space model.
We will make use of it for instance, to track time-varying
periodicities with minimal a priori information via the random walk model. Also, process noise may be associated
with a cumulative TOA jitter model for pulse train signals.
The state variables, fxn g, are related to the scalar TOA
measurements, fzn g, according to the measurement equation
zn = H T xn + vn ; n = 0; 1; : : : ;
(8)
T
where H denotes the measurement matrix, and fvn g is
a zero-mean Gaussian white measurement noise sequence,
corresponding to a NCJ component with known covariance
E fvn vm g = Rnm , where R = v2 . We further assume that
E fwn(l) vm g = 0; l = 1; 2; 8n; m, i.e., the process and mea-
surement noise sequences are uncorrelated. In the above
state space representation, the matrices F , G and H T are
as follows
F = 10 11 ; G = 10 01 ; H T = 1 0 :
(9)
4. ESTIMATION AND SMOOTHING
The Kalman lter recursions (e.g., [4]) may be used to
compute conditional mean state estimates fx^ njn g, based
on measurements up to and including the nth pulse TOA,
which we denote by Z n , i.e., x^ njn = E fxn jZ n g. The lter also provides a recursive update of the associated state
covariance matrix, njn , conditioned on Z n . Whether or
not this covariance accurately reects the true performance
of the lter, rests on the validity of modelling assumptions
used in the state space formulation. In addition to estimating and tracking the pulse train period, Tn , the Kalman
lter lends itself naturally to the problem of time series or
pulse TOA prediction for radar pulse train signals and has
application for example, to deceptive jamming as a form of
electronic countermeasure. Finally, xed-point smoothing
may be used to signicantly improve on the Kalman lter
estimate of TOA phase, t , contained in the initial state
vector, x0 , and this may be taken advantage of in several
pulse train signal analysis applications.
4.1. Consistent Initialisation of the Kalman Filter
To implement the Kalman lter, we must rst initialise the
lter recursions in terms of x^ 0j0 and 0j0 . Naturally, the
value we give to 0j0 should be commensurate with the
uncertainty in the value we assign to x^ 0j0 , otherwise the
potential exists for poor convergence of the lter parameter estimates to their true values. Based on the two-point
dierencing method outlined in [4], and the assumed NCJ
and CJ timing noise variances v2 and u2 , values for x^ 0j0 and
0j0 can be obtained from the data as follows and leads to
consistent lter initialisation1 . Using the rst two noisy
event TOA observations, which for convenience we now denote by z?1 and z0 , with T?1 a constant, we compute the
initial ltered state estimate, x^ 0j0 , viz
t^0j0 = z0 and T^0j0 = z0 ? z?1 :
(10)
The associated initial state covariance matrix is then
2
2
2
2
+
2
+
v
u
v
u
0j0 = v2 + u2 2v2 + u2 :
(11)
4.2. PRI Tracking Example
In Figure 1 we demonstrate the excellent PRI tracking capability of the Kalman lter for a TOA sequence with a
linear ramp PRI modulation. In this example, v2 = 6:25 10?10 s2 , u2 = 0 and 2 = 3:60 10?9 s2 .
1 A Kalman lter is said to be consistent if the state estimation errors are compatible with the covariances computed by the
lter [4].
Kalman Filter PRI Tracking
Kalman Filter State Covariance: PRI (NCJ)
80
CRLB
2.5
70
2
True PRI
Estimated PRI
1.5
1
0
10
20
30
40
50
60
70
Pulse Index Number
Kalman Filter PRI Estimation Error
80
90
100
0.1
PRI Error (ms)
Performance Variance
60
10 log(1 / Variance)
PRI Estimate (ms)
3
50
40
Design Variance
30
0.05
20
0
10
-0.05
-0.1
0
0
0
10
20
30
40
50
60
Pulse Index Number
70
80
90
100
10
20
30
40
50
60
Pulse Index Number
70
80
90
100
Figure 1: Kalman lter tracking for a pulse train with a
PRI that is modulated by a linear ramp function.
Figure 2: Performance characterisation of the Kalman lter
PRI estimator under signal model mismatch.
5. ROBUSTNESS ISSUES
5.1. Robustness to Model Uncertainties
2 = 0, and use the results of [6] to perform an error analy-
For a Kalman lter to yield state estimates that are optimal
in the minimum mean square error sense, the true values of
the measurement and process noise covariances used in the
lter recursions must be known a priori. In many practical
estimation problems however, including the one of interest
here, these values are seldom known beforehand.
5.1.1. Optimal Estimation Under Model Mismatch
In [5], it has been shown that in certain situations, optimal
state estimates may be obtained despite the use of a Kalman
lter with incorrect noise covariances. By implementing the
lter in an appropriate form, the results obtained in [5] lend
themselves to the estimation problem that we address here.
Specically, by using the two-point dierencing method described earlier to initialise the Kalman lter recursions, one
can ensure that optimal state estimates are obtained for
pulse train signals under the following mismatch conditions:
1. Rd = R, Qd = Q = 0 and d0j0 = 0j0 , > 0,
2. Rd = R = 0, Qd = Q and d0j0 = 0j0 , > 0,
3. Rd = R, Qd = Q and d0j0 = 0j0 , > 0,
where is a scalar, and Rd and Qd denote the respective
design values of the measurement and process noise covariances used by the Kalman lter, rather than the true values
R and Q. Similarly, d0j0 denotes the lter design initial
state covariance matrix, as opposed to the true state covariance matrix 0j0 . While the lter will return optimal
state estimates under the above conditions of signal mismatch, the design state covariance, dnjn , computed by the
Kalman lter, will be consistent with the use of scaled versions of the noise covariances, such that dnjn = njn [5],
where njn characterises optimal performance.
To illustrate this result for constant PRI signals, we set
sis by comparing the actual
performance of the Kalman lter as characterised by pnjn = E f(xn ? x^ njn )(xn ? x^ njn )T g,
against the predicted performance, dnjn , based on lter design equations. In other words, the computation of dnjn
is then just part of the Kalman lter \machinery" used to
generate the state estimates fx^ njn g, rather than a true performance measure. In Figure 2 we plot dnjn and pnjn for a
constant period pulse train TOA sequence with NCJ alone
and a nominal jitter variance of R = v2 and u2 = 0 (Condition 1 above). Comparison is also made with the CramerRao lower bound (CRLB) for the PRI estimator obtained
in [1]. Using the two-point dierencing method and a design
measurement noise covariance of Rd = 10R, we conclude
from Figure 2 that the lter performs optimally in that
pnjn meets the CRLB; inspection of the design state covariance, dnjn , however, would actually suggest much poorer
performance. In the case of optimal xed-point smoothed
estimates under conditions of noise covariance mismatch, it
is a simple exercise to extend the work of [5] and identify
that the same conditions (1{3) above apply. As a point
of interest, we note also that the Kalman lter initialised
in the above manner provided identical PRI and smoothed
TOA phase estimates as a deterministic least squares analysis based on the NCJ pulse arrival time model, but which
does not require prior knowledge of v2 (e.g., see [1]).
5.1.2. Almost Optimal Estimation Under Model Mismatch
Extensive numerical calculations were used to characterise
the performance degradation that results from model uncertainty in terms of the noise covariances R and Q, for
constant period pulse train signals with 2 = 0. For the
case of a pulse arrival time sequence perturbed by NCJ
alone, the results of the previous section indicate that optimal state estimates will be obtained under Condition 1. For
a TOA sequence perturbed by either CJ or a combination of
NCJ and CJ, our numerical calculations based on [6], suggest that very little degradation in estimator performance
will actually be experienced if one continues to use the NCJ
model with w2 (1) set to zero, and with R chosen conservatively to match the largest jitter variance expected.
5.2. Robustness to Missing and False Observations
In the radar pulse train signal processing problem that motivates our work, the problem of missing and false data is
frequently encountered. In addition to providing a capability for tracking time-varying periodicities, the stochastic
period evolution model described in Section 3 has the desirable eect of making the Kalman lter estimator robust
to missing and false data; essentially by allowing for some
variability in the observed PRI sequence.
6. CONCLUDING REMARKS
While the addition of process noise as modelled by Equation (5), is useful for period tracking purposes, this tech-
PRI Estimate (ms)
TDOA
Magill Filter Estimate
2
1.5
1
0
10
20
30
40
50
60
70
Pulse Index Number
Magill Filter PRI Estimation Error
10
20
30
80
90
100
80
90
100
0.15
PRI Error (ms)
5.1.3. Multiple Signal Model Approach
A multiple signal model or Magill lter approach may also
be used to account for signal model uncertainties, e.g., in
jitter variance identication/estimation, and leads to parallel processing strategies that adapt to an incoming data
stream (e.g., see [4]). Consider an unknown signal parameter , that is assumed to be included in the nite discrete
set f1 ; 2 ; : : : ; M g. A sequence of noisy observations is
used to feed a bank of M Kalman lters, each lter tuned
to a specic signal model that incorporates the unknown
parameter m , and yields a set of M outputs or conditional
ltered state estimates, fx^ (m )njn g. These estimates are
combined to form a conditional mean ltered state estimate
x^ njn , from a weighted sum of the fx^ (m )njn g. The contribution to this sum from the mth Kalman lter is weighted
according to the a posteriori probability, Pr(m jZ n ), that
= m , where Pr(m jZ n ) is computed on-line as part of
the estimation process.
To illustrate this approach as a method for system model
identication, we consider a bank of ten Kalman lters designed to process a pulse train signal with an unknown
NCJ variance, but which is contained in the discrete set
f(0.62, 2.50,
5.62, 10.00, 15.62, 22.50, 30.62 40.00, 50.63,
62.50)10?9 s2 g. The Magill lter was initialised by assuming that prior to recording any data, each signal model is
equally likely to correspond to the true model. Interestingly,
from our previous observations, each of the ten Kalman lters will yield identical state estimates, however, only the
lter with the correct jitter variance will produce an innovations sequence that is consistent with the lter design onestep predicted measurement covariance, and this leads to
the correct jitter variance identication. The performance
of the Magill lter is shown in Figure 3 for a pulse train signal with a constant PRI of 1:5 ms and v2 = 1:0 10?8 s2 .
Comparison is also made with the time dierence of arrivals (TDOAs) fzn+1 ? zn g. After 100 pulses had been
processed, the correct jitter level had been identied with
a lter computed a posteriori probability of 0.99.
Magill Filter PRI Estimation: NCJ
2.5
0.1
0.05
0
-0.05
0
40
50
60
Pulse Index Number
70
Figure 3: Magill lter performance for a constant PRI pulse
train TOA sequence with unknown NCJ.
nique will result in suboptimal PRI estimates for constant
period sources. One means of improving the performance of
the Kalman lter estimator for this situation is to take an
adaptive estimation approach based on the multiple signal
model method of Section 5.1.3 or the interacting multiple
model discussed in [4]. The lter bank would consist of just
two lters, one with process noise as described above, and
one without. Alternatively, one could run two independent
Kalman lters and switch between the state space models
as appropriate by monitoring the normalised innovations
squared. This is referred to as an adjustable level process
noise lter [4]. To incorporate robustness to missing and
false data, one could use TOA gating with minimal additional computational complexity as described in [7].
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