Approximate Conditional Mean Particle Filtering for Linear/Nonlinear Dynamic State Space Models

Abstract

We consider linear systems whose state parameters are separable into linear and nonlinear sets, and evolve according to some known transition distribution, and whose measurement noise is distributed according to a mixture of Gaussians. In doing so, we propose a novel particle filter that addresses the optimal state estimation problem for the aforementioned class of systems. The proposed filter, referred to as the approximate conditional mean particle filter (ACM-PF), is a combination of the approximate conditional mean filter and the sequential importance sampling particle filter. The algorithm development depends on approximating a mixture of Gaussians distribution with a moment-matched Gaussian in the weight update recursion. A condition indicating when this approximation is valid is given. In order to evaluate the performance of the proposed algorithm, we address the blind signal detection problem for an impulsive flat fading channel and the tracking of a maneuvering target in the presence of glint noise. Extensive computer simulations were carried out. For computationally intensive implementations (large number of particles), the proposed algorithm offers performance that is comparable to other state-of-the-art particle filtering algorithms. In the scenario where computational horsepower is heavily constrained, it is shown that the proposed algorithm offers the best performance amongst the considered algorithms for these specific examples.

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Approximate Conditional Mean Particle
Filtering for Linear/Nonlinear Dynamic State
Space Models
Derek Yee, James P. Reilly*, T. Kirubarajan, and K. Punithakumar
Department of Electrical and Computer Engineering,
McMaster University, 1280 Main St. W.,
Hamilton, Ontario, Canada L8S 4K1
Abstract
We consider linear systems whose state parameters are separable into linear and nonlinear sets, and
evolve according to some known transition distribution, and whose measurement noise is distributed
according to a mixture of Gaussians. In doing so, we propose a novel particle filter that addresses the
optimal state estimation problem for the aforementioned class of systems. The proposed filter, referred
to as the ACM-PF, is a combination of the approximate conditional mean filter and the sequential
importance sampling particle filter. The algorithm development depends on approximating a mixture of
Gaussians distribution with a moment-matched Gaussian in the weight update recursion. A condition
indicating when this approximation is valid is given.
In order to evaluate the performance of the proposed algorithm, we address the blind signal
detection problem for an impulsive flat fading channel and the tracking of a maneuvering target in
the presence of glint noise. Extensive computer simulations were carried out. For computationally
intensive implementations (large number of particles), the proposed algorithm offers performance that is
comparable to other state–of–the–art particle filtering algorithms. In the scenario where computational
horsepower is heavily constrained, it is shown that the proposed algorithm offers the best performance
amongst the considered algorithms for these specific examples.
J. Reilly, corresponding author: ph: 905 525 9140 x22895, fax: 905 521 2922, email: reillyj@mcmaster.ca

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I. INTRODUCTION
In many real–world applications, unknown quantities are to be estimated given a set of
noisy observations. Common examples include target tracking [4], [15], channel tracking in
wireless communications [18], [21], and the extraction of speech signals from contaminating
audio environments [6], [13], [28]. In many cases, the unknowns can be characterized by a
process equation and the observations by a measurement equation, which together, form the so-
called dynamic state space model (DSSM). With this DSSM, we can adopt a Bayesian filtering
approach, and thereby, recast the problem to one of tracking a hidden process given a set of
noisy observations.
Often, observations arrive sequentially in time. Therefore, it is more appropriate to consider
filters that recursively estimate the unknowns of interest. More specifically, we aim to recursively
compute the exact posterior probability density function (pdf) of interest. Indeed, within the
Bayesian framework, the posterior pdf captures all the information about the unknowns. Thus,
if an algorithm can recursively deduce the exact posterior pdf of interest, we can refer to it as
the optimal solution of the aforementioned Bayesian filtering problem.
For a linear Gaussian DSSM, the celebrated Kalman filter (KF) provides the optimal solution
[17]. In general, however, the optimal filter is analytically intractable for the unyielding nonlinear,
non–Gaussian DSSM. Thus, a number of researchers have introduced ingenious approximations
that have resulted in mathematically tractable suboptimal filters.
Historically, the Extended Kalman filter (EKF) is the first of such filters [2]. The main idea is
to invoke linearization and thereby form an approximation of the original nonlinear model that
is amenable to an application of the KF. The resulting recursive formulas constitute the EKF. In
a number of applications, the EKF has performed adequately. However, there also exist many
scenarios where the EKF has performed poorly, in particular for non–Gaussian distributed noise
disturbances. Thus, it was suggested to consider filters that involve a collection of EKF’s.
These filters are known as Gaussian Sum filters (GSFs) [2], [26], and the underlying as-
sumption is to approximate the true posterior pdf by a Gaussian mixture approximation. Each
component in the Gaussian mixture approximation, also called a mixand, is computed by a EKF
or a KF. Thus, depending on the nature of the non-Gaussianity, these filters utilize a bank of
EKF’s or KF’s to construct an approximation of the true posterior pdf. As such, these methods are

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more powerful, and more complex to implement. Indeed, if we do not introduce any alleviating
procedure, the number of mixands exponentially increases over time. Moreover, if the GSF
employs a bank of EKF’s, it will still suffer from its inaccuracies. Therefore, it is of interest to
consider alternative methods.
One such method is known as the approximate conditional mean (ACM) filter [22], [31]. As
shown in [22], for a linear DSSM with non-Gaussian observation noise distributed in accordance
with a Gaussian mixture distribution, the ACM filter yields near optimal performance. Thus, for
this scenario, the ACM filter provides an efficient alternative over the computationally expensive
GSF.
Up to this point, it is apparent that we are without a universally effective approach for online
signal processing of difficult nonlinear non–Gaussian DSSM’s. Therefore, it is necessary to
consider alternative solutions. Recently, particle filtering has emerged as a promising solution
to the general nonlinear non-Gaussian filtering problem [9], [11]. The underlying idea is to
use a randomly weighted set of samples or particles to recursively build in time, a point-mass
approximation of the true posterior pdf. Unlike traditional methods described earlier, particle
filters do not make any type of approximating assumptions; rather, they build an approximation
of the entire posterior pdf itself. In fact, particle filters are applicable to almost any DSSM
where signal variations are present. This is even true for nonlinear dynamics and noise distributed
according to non-Gaussian distributions. Consequently, particle filters are expected to outperform
popular, traditional EKF type algorithms. Indeed, in the last few years, there have been an
abundant number of papers on particle filtering and their applications.
A particularly significant contribution is the development of an efficient particle filter (PF)
called the mixture Kalman filter (MKF) [20]. The MKF exploits the conditional linear Gaussian
sub–structure of the considered DSSM. Since this approach reduces the dimension of the space
in which the particles are sampled from, it is more efficient than a standard implementation of
the PF. Although the assumption of a linear Gaussian sub–structure limits the amenable class
of DSSM’s, a few important classes of DSSM’s satisfy this assumption. These include jump
Markov linear systems (JMLS) [12], partially observed Gaussian (POG) DSSM [3] and the
class of DSSM’s corresponding to the transmission of data over a wireless channel in mobile
communications [5], [16].
Another important class of models that has received relatively little attention are DSSM’s in

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the form of
x1
k
= A1(x2
k−n:k)x1
k−1+ F1(x2
k−n:k) + w1
k
(1)
x2
k
∼ p(x2
= M(x2
k|x2
k)x1
k−n:k−1)
(2)
yk
k+ H(x2
k) + ek
(3)
where A1(·), F1(·), M(·), H(·) are known functions with appropriate dimensions, n is some
arbitrary integer, w1
k∼ N(w1
k;0,Q1
k), p(x2
k|x2
k−n:k−1) is a known possibly non–Gaussian transi-
tion distribution and ekis observation noise distributed according to a Gaussian Mixture Model
(GMM) of the form
p(ek) =
N
?
j=1
pjN(ek;e(j)
k,R(j)
k)
(4)
where?N
Like most filtering problems, the estimates of interest are the maximum a posteriori (MAP)
estimate of xk= [x1T
j=1pj= 1.1
k,x2T
k]T, the minimum mean square error (MMSE) estimate of xkand its
associated conditional covariance covp(xk|y1:k)[xk], i.e.,
? xMAP
k
= argmax
?
?
xk
?
p(xk|y1:k)
?
(5)
Ep(xk|y1:k)[xk] =xkp(xk|y1:k)dxk
(6)
covp(xk|y1:k)[xk] =
˜ xk˜ xT
kp(xk|y1:k)dxk
(7)
where ˜ xk= xk− Ep(xk|y1:k)[xk]. However, these estimates are generally intractable, thus, it is
necessary to resort to some sort of approximate solution. One of the more popular suboptimal
solutions is the MKF [20], which in the literature is also known as the Rao–Blackwellized
PF [3] or the Marginalized PF [25]. The MKF enjoys asymptotic convergence in the posterior
distribution and the typical estimates associated with this distribution. In fact, as the number
of particles approaches infinity, the estimates converge toward the true Bayesian estimators of
interest. In terms of variance in the estimates, the MKF also bests the standard PF. Thus, it
1A mixture of Gaussians effectively models a wide variety of pdf’s. This is a useful property as it increases the utility of
the DSSM. There are a number of techniques to fit a GMM to an arbitrary pdf. The most popular approaches include the
expectation–maximization (EM) algorithm [8] and its variants, e.g., [23]. Other techniques also exist; for a sample see [24],
[27].

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has received considerable attention in the literature and has found success in many challenging
signal processing problems, e.g., see [3], [5], [12]. However, for finite number of particles, it
may be possible to outperform the former. To this end, we introduce a novel filter known as the
approximate condition mean PF (ACM–PF), and show through simulations, that the proposed
algorithm renders an appealing alternative to the sequential importance sampling PF (SIS–PF)
and the well known MKF.
The remainder of this paper is organized as follows. In Section 2, we review the SIS–PF,
the MKF and introduce the proposed ACM–PF. Section 3 presents some simulation results, and
Section 4 concludes this paper.
Notation: We use (·)1:m to indicate all the elements from time 1 to time m. The notation
diag(x) denotes the diagonal matrix with vector x on its diagonal. Lastly, a N–dimensional
complex Gaussian random vector x has a pdf of the form
CN(x;µ,Σ) =
1
πN|Σ|e−(x−µ)HΣ−1(x−µ).
II. PARTICLE FILTER BASED ALGORITHMS
A. Sequential Importance Sampling Particle Filter
A standard PF such as the sequential importance sampling PF (SIS–PF) employs a weighted
set of samples to approximate the joint posterior pdf p(x1
if a set of particles {x1,(i)
x2
k
,x2,(i)
k
) ∼ q(x1
(x1,(i)
k
,x2,(i)
k
) is assigned a so–called importance weight
k,x2
k|y1:k). According to this method,
k
,x2,(i)
k
}Np
i=1is drawn from an importance distribution q(x1
k|x1,(i)
k,x2
k|x1
1:k−1,
1:k−1,yk), i.e., (x1,(i)
k,x2
1:k−1,x2,(i)
1:k−1,yk) for i = 1,...,Np, and each particle
w(i)
k∝w(i)
k−1
p(yk|x1,(i)
k
,x2,(i)
k
,x2,(i)
)p(x1,(i)
|x1,(i)
k
,x2,(i)
k
1:k−1,x2,(i)
|x1,(i)
1:k−1,y1:k)
1:k−1,x2,(i)
1:k−1)
q(x1,(i)
kk
,
(8)
(6) and (7) can be approximated by [10], [11]:
?Ep(xk|y1:k)[xk] =
?
Np
?
Np
?
j=1w(j)
i=1
˜ w(i)
kx(i)
k
(9)
covp(xk|y1:k)[xk] =
i=1
˜ w(i)
k˜ x(i)
k˜ x(i)T
k
(10)
where ˜ x(i)
k= x(i)
k−?Ep(xk|y1:k)[xk] and ˜ w(i)
k= [?Np
k]−1w(i)
k
is the normalized importance
weight.