VT-2005-00647.R2
1
Position and Velocity Tracking in Mobile
Networks Using Particle and Kalman Filtering
with Comparison
Mohammed M. Olama, Seddik M. Djouadi, Ioannis G. Papageorgiou, and Charalambos D.
Charalambous
Abstract—This paper presents several methods based on signal
strength and wave scattering models for tracking a user. The
received signal level method is first used in combination with
Maximum Likelihood (ML) estimation and triangulation to
obtain an estimate of the location of the mobile. Due to non-lineof-sight conditions and multipath propagation environments this
estimate lacks acceptable accuracy for demanding services, as
numerical results reveal. The 3D wave scattering multipath
channel model of Aulin is employed together with recursive
nonlinear Bayesian estimation algorithms to obtain improved
location estimates with high accuracy. Several Bayesian
estimation algorithms are considered such as the extended
Kalman filter (EKF), the particle filter (PF), and the unscented
particle filter (UPF). These algorithms cope with nonlinearities in
order to estimate the mobile location and velocity. Since the EKF
is very sensitive to the initial state, we propose to use the ML
estimate as an initial state to the EKF. In contrast to the EKF
tracking approach, the PF and the UPF approaches do not rely
on linearized motion models, measurement relations, and
Gaussian assumptions. Numerical results are presented to
evaluate the performance of the proposed algorithms when
measurement data do not correspond to the ones generated by
the model. This shows the robustness of the algorithm based on
modeling inaccuracies.
Index Terms—Kalman filtering, location tracking, maximum
likelihood estimation, multipath fading channels, and particle
filtering.
T
I. INTRODUCTION
HE need for an efficient and accurate mobile station (MS)
positioning system is growing day by day. This has been
stressed by a federal order issued by the federal
communications commission (FCC), which mandates all
wireless service providers to provide public safety answering
points with information to locate an emergency 911 caller
with an accuracy of 100 meters for 67% of the cases [1]. It is
M. M. Olama and S. M. Djouadi are with the Electrical and Computer
Engineering Department, University of Tennessee, Knoxville, TN 37996
USA, (phone: 865-974-5447; fax: 865-974-5483; e-mail: {molama,
mdjouadi}@utk.edu).
I. G. Papageorgiou and C. D. Charalambous are with the Electrical and
Computer Engineering Department, University of Cyprus, Nicosia, 1678
Cuprus, (e-mail: {ioannisp, chadcha}@ucy.ac.cy).
also expected that the FCC will tighten its requirements in the
near future [2]. Many other applications, such as vehicle fleet
management, location sensitive billing, intelligent transport
systems, fraud protection, and mobile yellow pages have
driven the cellular industry to research new and promising
technologies for MS positioning.
The problem of determining the location and velocity of a
MS has been studied extensively in the last few years. The
current standards in estimating the location and velocity are
based mostly on time signal information, such as time
difference of arrival (TDOA), enhanced observed time
differences (E-OTD), observed time difference of arrival
(OTDOA), global positioning system (GPS), etc., [3]-[7].
However, not all of these methods meet the necessary needs
imposed by certain services. In addition, most of them require
new hardware since localization is not inherent in the current
wireless systems, for instance, GPS demands a new receiver
and TDOA, E-OTD, OTDOA requires additional location
measurement units in the network [8]. Adding extra hardware
means extra cost for implementation, which can be reflected
on both consumers and operators. Researchers have also
suggested several MS location methods based on signal power
measurements such as in [9] and [10], where a certain
minimization problem is solved numerically to get an initial
estimate of the MS position, and then a smoothing procedure
such as linear regression [9], or the Kalman filter [10] are
applied to obtain a more accurate estimate.
In this paper, several MS tracking methods based on
maximum likelihood estimation (MLE) [11], and recursive
nonlinear Bayesian estimation (RNBE) algorithms such as the
extended Kalman filter (EKF) [12], the particle filter (PF)
[13], and the unscented particle filter (UPF) [14] are proposed.
The MLE algorithm employs the average received power
measurements based on the lognormal propagation channel
model to obtain an initial MS location estimate [3, 16]. These
measurements are readily available through network
measurement reports or radio measurements, in idle or active
mode, for any MS unit in 2G and 3G cellular networks. The
RNBE algorithms employ the instantaneous electric field
measurements based on the 3D multipath channel model of
Aulin [15] to account for multipath and non-line-of-sight
(NLOS) characteristics of the wireless channel as well as the
VT-2005-00647.R2
dynamicity of the MS. The received instantaneous electric
field in this model is a nonlinear function of the position and
velocity of the MS. The EKF approach is based on linearizing
the nonlinear system model around the previous estimate, and
therefore is very sensitive to the initial state. This motivates
the use of the ML estimate of the MS location as an initial
state to the EKF. Particle filtering approaches approximate the
optimal solution numerically based on the physical model,
rather than applying an optimal filter to an approximate model
such as in the EKF. They provide general solutions to many
problems where linearization and Gaussian approximations
are intractable or yield low performance. The more nonlinear
the model is or the more non-Gaussian the noise is, the more
potential PFs have, especially in applications where
computational power is rather cheap and the sampling rate is
moderate. In this paper, particle filtering is implemented for
the generic PF and the more recent UPF.
Aulin’s model postulates knowledge of the instantaneous
received field at the MS, which is obtained through the
circuitry of the mobile unit. The proposed RNBE algorithms
take into account NLOS condition as well as multipath
propagation environments. They require only one base station
(BS) to estimate the MS location instead of at least three BSs
as found in the literature [10], [16]. However, an initial MS
location estimate that requires at least three BSs, such as the
MLE and triangulation method, will improve the convergence
of the RNBE filter. Particle filtering has been used in several
tracking wireless applications [17]-[20], but the channel
models used do not take into account the multipath properties
of the wireless channel. To the best of our knowledge, the
utilization of the PF and/or the UPF together with the classical
wireless channel model to extract the MS location and
velocity is new. The performance of the proposed algorithms
is computed numerically and in the presence of parameters
uncertainty. Numerical results indicate that the proposed UPF
algorithm is highly accurate and superior to other approaches.
The main contribution of this paper is to develop MS
location and velocity estimation algorithms, which use Aulin’s
3D wave scattering model and are based on particle filtering.
The performance is compared with that of the EKF. The
choice of this channel model is to account for the multipath
properties and NLOS of wireless networks. These methods
support existing network infrastructure and channel signaling
and only use one BS. Moreover, simulations are performed
under uncertainties in the model parameters of the channel to
evaluate the robustness of the algorithm.
The paper is structured as follows. In Section II, we
describe the mathematical models for the location and velocity
estimation algorithms. The MLE and the EKF approaches for
MS location estimation are presented in Sections III and IV,
respectively. The PF and the UPF approaches for MS location
and velocity estimation, which are the main contribution of
this paper, are presented in Sections V and VI, respectively. In
Section VII we present numerical results, and evaluate the
robustness of the proposed algorithms to uncertainties due to
random variations in the channel parameters. Section VIII
2
provides concluding remarks.
II. SYSTEM MATHEMATICAL MODELS
A. The Lognormal Propagation Channel Model
Here we consider a 2D geometry with the MS located at
( x0 , y0 )
and
the
BSs
located
at
((x
BS1
)
, yBS1 ), ( xBS2 , yBS2 ),..., ( xBSB , yBS B ) .
The
general
lognormal propagation channel model is described by [21]:
⎛d
PLsb (db ) = PL(d0b ) + 10ε b log ⎜ b
⎜ d0
⎝ b
⎞
⎟ + X bs
⎟
⎠
(1)
where db ≥ d 0b , s ∈ {1, 2,.., S } , b ∈ {1, 2,.., B} , PLsb (db ) is the
path loss from the bth BS at distance db for the sth sample,
d 0b is the reference distance, εb is the path loss exponent and
X bs ∼ N (0; σ b2 ) is a Gaussian random variable (RV)
represents the shadowing variance due to gross variations in
the terrain profile and changes in the local topography. In
cellular networks, the MS preserves and frequently updates, in
idle and active mode, the received power of the strongest nonserving BSs (e.g., in GSM the 6 strongest [22]) in addition to
the one of the serving cell. Exploiting these measurements
from surrounding BSs lead to estimate the location of the MS.
The MLE approach described in Section III that employs this
channel model is used to estimate the MS location.
B. Aulin’s Scattering Model
The basic 3D wireless scattering channel model described
in [15], which assumes that the electric field, denoted by E(t),
at any receiving point ( x0 , y0 , z0 ) is the resultant of P plane
waves (see Fig. 1), in which the receiver moves in the X-Y
plane having velocity v in a direction making an angle γ
with the X-axis, is given by:
E (t ) =
P
P
∑ E (t ) = ∑ r cos (ω t + ω t + θ ) + e(t )
n =1
n
n =1
n
c
n
(2)
n
where
Z
nth multipath
component
(x0, y0, z0)
MS
O
Y
βn
αn
γ
X
v
VT-2005-00647.R2
3
Fig.1. Aulin’s 3D multipath channel model.
ωn =
θn = −
2πυ
λ
( cos(γ − α n ) cos β n )
(3)
The measurement equation can be found from Aulin’s
scattering model (2), (3), and (4), which can be written in
discrete form as:
P
2π ⎛ x0 cos α n cos β n + y0 sin α n cos β n ⎞
⎜
⎟ + φn
λ ⎝ + z0 sin β n
⎠
n =1
(4)
of the nth component, λ is the wavelength, e(t ) is a white
Gaussian noise, and P is the total number of paths. It can be
seen from (3) and (4) that the Doppler and phase shifts depend
on the velocity and location of the receiver, respectively.
Clearly, (2) assumes transmission of a narrowband signal.
This assumption is valid only when the signal bandwidth is
smaller than the coherence bandwidth of the channel.
Nevertheless, the above model is not restrictive since it can be
modified to represent a wideband transmission by including
multiple time-delayed echoes. In this case, the delay spread
has to be estimated. A sounding device is usually dedicated to
estimating the time delay of each discrete path such as the
Rake receiver [23].
It can be seen that the noisy instantaneous received field in
(2) depends parametrically on the location and velocity of the
receiver. Consequently, this expression is used to estimate the
MS location and velocity by using the EKF, the PF, and/or the
UPF. Next, we formulate the location estimation as a filtering
problem in state-space form [24]. The general form, once
discretized, is given by:
(5)
where f (.,.) and h(.,.) are known vector functions, k is the
estimation step, z k are the output measurements at time step
k, and x k is the system state at time step k and must not be
confused with location coordinates. Further, w k and v k are
the discrete zero-mean, independent state and measurement
noise processes, with covariance matrices Q and R ,
respectively.
Now let x k = [ xk , xk , yk , yk ] denote the state of the MS at
T
time k, where xk and yk are the Cartesian coordinates of the
MS, xk and yk are the velocities of the MS in the X and Y
directions, respectively. We choose the case where the
velocity of the MS is not known and is subject to unknown
accelerations. The dynamics of the MS can be written as [17]:
2
0 ⎤
⎡ xk ⎤ ⎡1 Δ k 0 0 ⎤ ⎡ xk −1 ⎤ ⎡ Δ k / 2
⎢
⎥
⎢ ⎥ ⎢
⎢
⎥
⎥
xk ⎥ ⎢0 1 0 0 ⎥ ⎢ xk −1 ⎥ ⎢ Δ k
0 ⎥ ⎡ wk −1,1 ⎤
⎢
xk =
=
+
⎢
⎥ (6)
⎢ yk ⎥ ⎢0 0 1 Δ k ⎥ ⎢ yk −1 ⎥ ⎢ 0
Δ k2 / 2 ⎥ ⎢⎣ wk −1,2 ⎥⎦
⎥
⎢ ⎥ ⎢
⎥ ⎢
⎥⎢
Δ k ⎥⎦
⎣⎢ yk ⎦⎥ ⎣⎢0 0 0 1 ⎦⎥ ⎣⎢ yk −1 ⎦⎥ ⎢⎣ 0
where Δ k is a (possibly non-uniform) measurement interval
between time k –1 and k.
2π xk2 + yk2
ωn =
shift, θ n is the phase shift, rn is the amplitude, φn is the phase
z k = h( x k , v k )
)
(7)
where
and α n , β n are spatial angles of arrival, ωn is the Doppler
x k = f (x k −1 , w k −1 )
(
zk = h(x k , vk ) = ∑ rnk cos ωc tk + ωnk tk + θ nk + v(tk )
λ
k
θn =
k
( cos(γ
− α nk ) cos β nk
k
−2π ⎛ xk cos α nk cos β nk + yk sin α nk cos β nk
⎜
λ ⎜⎝ + z0 sin β nk
)
(8
)
⎞
⎟⎟ + φnk
⎠
(9
)
Clearly, the measurement equation h(.,.) is a nonlinear
function of the state-space vector, as observed in (7), (8), and
(9). If we assume approximate knowledge of the channel,
which is attainable either through channel estimation at the
receiver (e.g., GSM receiver), or through various estimation
techniques (e.g., least-squares, ML), then this problem falls
under the broad area of nonlinear parameter estimation from
noisy data which can be solved using the RNBE algorithms.
These algorithms will be discussed in Sections IV, V, and VI.
The MLE algorithm that employs the lognormal propagation
channel model is discussed in the next section.
III. THE MLE APPROACH FOR MS LOCATION ESTIMATION
In this section, The MLE method that employs the
lognormal propagation channel model described in section IIA is considered for the MS location estimation. This method
exploits the received power measurements at the MS which
are available from network measurement reports (NMR).
Thus, we write the likelihood function and then maximize it
with respect to the distances θ = d = ( d1 , d 2 ,..d B ) from each
BS, where θ is the parameter to be estimated. The ML
estimator, denoted by θˆ = dˆ = dˆ , dˆ ,.., dˆ , represents the
(
1
2
B
)
most possible MS/BS distances based on the measurements
available at the MS.
Consider the measurement vector for the sth sample from
all BSs, denoted by PLs (d ) = ( PL1s ( d1 ), PLs2 ( d 2 ),.., PLsB ( d B ) ) .
The distribution function for this vector is the B-variate
normal distribution given by:
p ( PLs (d) | θ ) = ( 2π )
−B / 2
( det(Σs ) )
−1 2
) (
(
T
s
s
⎛ 1⎛
exp ⎜ − ⎜ PLs (d) − PL (d) Σ−s 1 PLs (d) − PL (d)
⎝ 2⎝
s
(10)
)
PLs (d) ∼ N B PL (d); Σ s ,
where
s
(
) ⎞⎟⎠ ⎟⎞⎠
(
s
s
s
PL (d) = PL1 (d1 ), PL 2 (d 2 ),.., PL B (d B )
)
is the mean path
loss from each BS, and Σ s is the covariance matrix.
Assuming the noise is independent identically distributed
(iid), then the logarithm likelihood function is the log product
of the sample likelihood functions given by:
VT-2005-00647.R2
4
⎛
1
L ( θ | PLs ( d ) ) = log ⎜
⎜ (2π ) SB / 2 ( det( Σ ) ) S / 2
s
⎝
(
s
⎛
− ∑ ⎜ PLs ( d ) − PL (d )
s =1 ⎝
S
)
⎞
⎟
⎟
⎠
)
(
s
⎞
Σ −s 1
PLs (d ) − PL ( d ) ⎟
2
⎠
T
(11)
k
where S is the total number of samples. Maximizing (11) first
s
with respect to PL (d) , the score function yields:
^
PLb (db ) =
1 S
∑ PLsb (db ),
S s =1
∀b ∈ {1, 2,.., B}
(12)
Solving for d̂ using the invariance property of the MLE [11],
it can be shown that:
dˆb = 10
⎧⎪ 1
⎨
⎪⎩10ε b
⎡1
⎢
⎢⎣ S
S
⎤ ⎫⎪
s =1
⎦⎭
∑ PLsb ( db ) − PL ( d0b )⎥⎥ ⎬⎪
(13)
is the MLE for the distance of the bth BS from the MS. Next,
we perform triangulation using the least squares error method
[25] to estimate the MS location ( x0 , y0 ) , by solving:
(
)
2⎫
⎧B
arg min ⎨∑ db − dˆb ⎬
x0 , y0
⎩ b =1
⎭
K k = Pk HTk ⎡⎣ H k Pk HTk + Vk R k VkT ⎤⎦
xˆ k = x k + K k ( zk − h(x k , 0))
Pˆ = (I − K H )P
(14)
k
In the next section, the EKF approach that employs the
channel model of Aulin to estimate the MS location and
velocity is discussed.
IV. THE EKF APPROACH FOR MS LOCATION AND VELOCITY
ESTIMATION
Consider the general discrete-time dynamical system model
described in (5). Let the known probability density functions
(PDFs) of the process noise w k and the measurement noise
v k be p (w k ) and p (v k ) , respectively. As usual, w k and v k
are assumed to be mutually independent. The set of entire
measurements from the initial time step to time step k is
denoted by Z k = {z i }i =1 . The distribution of the initial
k
condition x0 is assumed to be given by p (x0 | Z 0 ) = p (x0 ) .
The EKF is based on linearizing the nonlinear system
models around the previous estimate. The general algorithm
for the discrete EKF can be described by the time-update
equations given as [12]:
x k = f (xˆ k −1 , 0)
(15)
P = A Pˆ AT + W Q WT
k
k
k −1
k
k
k −1
k
and the measurement-update equations given as:
(16)
k
where
⎡1
⎢
0
∂f
Ak =
(xˆ k −1 , 0) = ⎢
⎢
0
∂x
⎢
⎣0
Δk 0 0 ⎤
⎥
1 0 0 ⎥
,
0 1 Δk ⎥
⎥
0 0 1⎦
⎡ Δ k2 / 2
0 ⎤
⎢
⎥
Δ
0 ⎥
∂f
k
Wk =
( xˆ k −1 , 0) = ⎢
,
⎢ 0
∂w
Δ k2 / 2 ⎥
⎢
⎥
⎢⎣ 0
Δ k ⎥⎦
∂h
Vk =
( xˆ k , 0) = 1,
∂v
∂h
Hk =
(xˆ k , 0) = [ H 1k H 2k H 3k H 4k ] ,
∂x
P
⎡
⎛ 2π
⎞⎤
H 1k = ∑ ⎢ rnk sin ωc tk + ωnk tk + θ nk ⎜
cos α nk cos β nk ⎟ ⎥
⎝ λ
⎠⎦
n =1 ⎣
(
)
( ) ( )
⎡
⎛
2π t
⎢ − rn sin ωc tk + ωn tk + θ n ⎜
cos β nk
k
k
k
⎜
λ xk2 + yk2
H 2k = ∑ ⎢
⎝
n =1 ⎢
⎢ x cos γ − α + y sin γ − α
k
nk
k
k
nk
⎣ k
P
⎡
⎛ 2π
H 3k = ∑ ⎢ rnk sin ωc tk + ωnk tk + θ nk ⎜
sin α nk cos β nk
⎝ λ
n =1 ⎣
(
P
The performance of this location estimation algorithm is
discussed through numerical results and compared to the
following algorithms in Section VII.
k
−1
(
(
)
)
(
(
))
)
(
(
)
( ) ( )
⎠⎥
⎥
⎦
⎞⎤
⎟⎥
⎠⎦
⎞⎤
cos β nk ⎟ ⎥
2
2
⎟⎥
⎝ λ xk + yk
⎠⎥
⎥
γ k − α nk
⎦
⎛
2π t
) ⎜⎜
(
⎞⎤
( ) ⎟⎟ ⎥⎥
(
⎡
⎢ − rn sin ωc tk + ωn tk + θ n
k
k
k
H 4k = ∑ ⎢
⎢
n =1
⎢ y cos γ − α − x sin
k
nk
k
⎣ k
P
(17)
( )
))
K is the gain matrix, P̂ is the estimation error covariance,
and γ k = arctan ( yk / xk ) . The notation x k denotes the a priori
state estimate at time step k and xˆ k the a posteriori state
estimate given measurement zk. P and Pˆ are defined
k
k
similarly.
As in any nonlinear estimation problem, the convergence of
the EKF to the true value of the location depends on the initial
parameter value; therefore we first develop the MLE method
to obtain an initial estimator of adequate accuracy for the
EKF. This hybrid algorithm, as numerical results indicate, has
improved accuracy for the final MS location estimate.
The EKF described above utilizes the first term in a Taylor
expansion of the nonlinear measurement model in (7). It
always approximates p (x k | Z k ) by a Gaussian distribution.
However, if the true density is non-Gaussian, then a Gaussian
model may not describe it precisely. In such cases PFs yield
an improvement in performance in comparison to that of an
EKF. The design of the PF is discussed in the next section.
VT-2005-00647.R2
5
V. THE PF APPROACH FOR MS LOCATION AND VELOCITY
ESTIMATION
The PF is a technique for implementing a recursive
Bayesian filter by Monte Carlo simulations. The key idea is to
represent the required posterior density function by a set of
random
samples
{ω ( j )}
k
N
j =1
{xˆ ( j )}
k
N
j =1
with
associated
weights
N
p (x k | Z k ) ≈ ∑ ωk ( j ) δ ( x k − xˆ k ( j ) )
(18)
j =1
We therefore have a discrete weighted approximation to the
true posterior p (x k | Z k ) . The weights are chosen using the
principle of importance sampling [26]:
ωk ( j ) ∝
p ( z k | x k ( j ) ) p ( x k ( j ) | x k −1 ( j ) )
(19)
q ( x k ( j ) | x k −1 ( j ) , z k )
where q ( x k ( j ) | x k −1 ( j ) , z k ) is the importance proposal
distribution function that generates the samples
{xˆ ( j )}
k
N
j =1
.
The choice of this distribution function is one of the most
critical design issues and determines the type of the PF. The
optimal proposal distribution function that minimizes the
variance of the weights conditioned on x k −1 ( j ) and z k is
q ( x k | x k −1 ( j ) , z k )opt = p ( x k | x k −1 ( j ) , z k ) [26].
However, analytical evaluation of the optimal proposal
function is not possible for many models, and thus has to be
approximated using local linearization [26] or the unscented
transformation [14]. In this paper, the unscented
transformation method is considered and the resulting filter is
called the unscented particle filter (UPF) that is described in
Section VI.
Nonetheless, the most popular choice of proposal function
is the transition prior q ( x k | x k −1 ( j ) , z k ) = p ( x k | x k −1 ( j ) ) .
This filter is called the generic PF and is discussed herein.
Although this choice of proposal function results in higher
Monte Carlo variations than the optimal, it is usually simple to
implement.
The time-update stage of the generic PF [27] is performed
by passing the random samples
{xˆ ( j )}
k −1
N
j =1
through the
system model (6) to obtain the time-updated samples
k
by:
N
j =1
which yields:
ωk ( j ) =
and to compute estimates based on these samples
and weights. In this case the posterior density at time k can be
approximated as:
{x ( j )}
the time updated PDF p (x k | Z k −1 ) .
The measurement-update stage can be described by
substituting
the
choice
of
proposal
distribution
q ( x k | x k −1 ( j ) , zk ) = p ( x k | x k −1 ( j ) ) into (19) and normalizing
. Namely, the time-updated samples are obtained
x k ( j ) = f ( xˆ k −1 ( j ) , w k −1 ( j ) )
(20)
p ( zk | x k ( j ) )
∑
N
j =1
(21)
p ( zk | x k ( j ) )
We define a discrete density over
{x ( j )}
k
N
j =1
with
probability mass ωk ( j ) associated with each sample x k ( j ) .
Then we get the measurement-update samples
{xˆ ( j )}
k
N
j =1
through
a
resampling
process,
such
that
Pr {xˆ k ( i ) = x k ( j )} = ωk ( j ) for any i. Several resampling
schemes are presented in the literature such as: systematic
[28], stratified, and residual resampling [29]. However, the
specific choice of resampling scheme does not significantly
affect the performance of the PF. Therefore, systematic
resampling is used in all of the experiments in Section VII
since it is simple to implement. The estimate of the PF at time
k is chosen to be the mean of the samples {xˆ k ( j )} j =1 .
N
In the next section, an approximate version of the optimal
proposal distribution is considered in order to have a more
accurate MS location estimate.
VI. THE UPF APPROACH FOR MS LOCATION AND VELOCITY
ESTIMATION S
The UPF results from using a scaled unscented
transformation (SUT) method to approximate the optimal
proposal distribution within a particle filter framework. The
SUT provides more accurate approximation than linearization
methods [14]. In particular, the SUT calculates the posterior
covariance accurately to the 3rd order, whereas linearization
methods such as the EKF rely on a first order biased
approximation. The SUT method is introduced next.
A. The SUT Method
The SUT method still approximates the proposal
distribution by a Gaussian distribution, but it is specified using
a minimal set of deterministically chosen sample points. These
sample points completely capture the true mean and
covariance of the Gaussian distribution, and when propagated
through the true nonlinear system, captures the posterior mean
and covariance accurately to the 3rd order for any nonlinearity.
Consider the state equation described in (5). For simplicity,
let x k = f (x k −1 ) , where x k −1 an nx dimensional random
vector and assume x k −1 has mean xk −1 and covariance Pk −1 .
Then, a set of 2nx + 1 weighted samples or sigma points
where w k −1 ( j ) is a sample drawn from the PDF p (w k −1 ) of
Si = {Wi , X i }
the system noise. The samples {x k ( j )} j =1 are distributed as
completely capture the true mean and covariance of the prior
N
are deterministically chosen so that they
VT-2005-00647.R2
6
x0 ( j ) = E ⎡⎣ x 0 ( j ) ⎤⎦
random vector x k −1 . A selection scheme that satisfies this
requirement is [14]:
X k0−1 = xk −1
X
i
k −1
X ki−1 = xk −1
( m)
W0
(
−(
= xk −1 +
( nx + λ ) Pk −1 )i ,
)
T
P0 ( j ) = E ⎡( x 0 ( j ) − x0 ( j ) ) ( x 0 ( j ) − x0 ( j ) ) ⎤
⎣
⎦
x0a ( j ) = E ⎡⎣ x 0a ( j ) ⎤⎦ = ⎡( x0 ( j ) )
⎣
T
i = 1,… , nx
( nx + λ ) Pk −1 , i = nx + 1,… , 2nx
= λ / ( nx + λ )
W0( c ) = λ / ( nx + λ ) + (1 − α 2 + β )
Wi ( m ) = Wi ( c ) = 1/ {2 ( nx + λ )} ,
where
λ =α
parameters,
(
2
( nx + κ ) − nx ,
( nx + λ ) Pk −1 )i
i = 1,… , 2nx
α , β , and κ
are
scaling
is the ith row or column of the
propagated
through
the
nonlinear
function
X ki = f (X ki−1 ), i = 0,… , 2nx . And the estimated mean and
covariance of x k are computed as follows:
i =0
i =0
{X
i
k
(23)
− xk }{X ki − xk }
T
( na + λ ) Pka−1 ( j ) ⎤⎦
(26)
• Performing the time update stage as:
i =0
2 na
{
B. The UPF Design
The UPF uses the same framework as the regular PF,
except that it approximates the optimal proposal distribution
by a Gaussian distribution using the SUT method. In
particular, the SUT is used to generate and propagate a
Gaussian proposal distribution for each particle to get:
q ( x k ( j ) | x k −1 ( j ) , z k )opt ≈ N ( xk ( j ) , Pk ( j ) )
(24)
and j = 1,… , N . That is, at time k − 1 the SUT is used with
the new data, to compute the mean and covariance of the
importance distribution for each particle. Next, the jth particle
is sampled from this distribution.
The description of the UPF approach in this section is
mainly based on [14]. In the implementation of the UPF, the
augmented state vector is defined as the concatenation of the
T
original state and noise variables as x ak = ⎡⎣ xTk wTk vk ⎤⎦ . Then
the SUT sigma point selection scheme is applied to this new
augmented state vector to calculate the corresponding sigma
matrix, X ka . The complete UPF is described as follows [14]:
1. Initialization ( k = 0 ) : Draw the particles {x0 ( j )} j =1 from
N
}{
}
Pk ( j ) = ∑ Wi ( c ) X i ,xk ( j ) − xk ( j ) X i ,xk ( j ) − xk ( j )
i =0
These estimates of the mean and covariance are accurate to
the 3rd order for any nonlinear function. In comparison, the
EKF only calculates the posterior mean and covariance
accurately to the first order with all higher order moments
truncated
the prior p (x0 ) and set:
• Calculating sigma points:
X ka−1 ( j ) = ⎡ xka−1 ( j ) xka−1 ( j ) ±
⎣
X kx ( j ) = f (X kx−1 ( j ) ,X kv−1 ( j ) ) , xk ( j ) = ∑ Wi ( m )X i ,xk ( j )
xk = ∑ Wi ( m )X ki
c)
2. Now for k = 1, 2,… , the importance sampling step is
performed by the following steps:
2 na
2 nx
Pk = ∑ Wi (
(25)
where E[.] is the expectation operator.
matrix square root of ( nx + λ ) Pk −1 , Each sigma point is now
2 nx
T
T
P0a ( j ) = E ⎡⎢( x 0a ( j ) − x0a ( j ) ) ( x 0a ( j ) − x0a ( j ) ) ⎤⎥
⎣
⎦
0⎤
⎡ P0 ( j ) 0
⎢
⎥
0⎥
=⎢ 0
Q
⎢ 0
0
R ⎥⎦
⎣
(22)
i
0 0⎤
⎦
T
(27)
2 na
Zk ( j ) = h (X kx ( j ) ,X kn−1 ( j ) ) , zk ( j ) = ∑ Wi ( m ) Zi , k ( j )
i =0
• Performing the measurement update stage as:
2 na
Pz k zk = ∑ Wi ( c ) {Zi , k ( j ) − zk ( j )} {Zi , k ( j ) − zk ( j )}
T
i =0
2 na
Pxk z k = ∑ Wi (
c)
i =0
K k = Pxk z k Pz−k1z k
{X ( j ) − x ( j )}{Z ( j ) − z ( j )}
, x ( j ) = x ( j ) + K ( z − z ( j ))
T
x
i, k
k
k
k
Pk ( j ) = Pk ( j ) − K k Pz k z k K
and
then
i, k
k
k
k
k
T
k
sampling
(
(28)
)
xk ( j )
from
q ( x k ( j ) | x k −1 ( j ) , zk ) = N xk ( j ) , Pk ( j ) .
• Evaluating the importance weights as:
p ( zk | x k ( j ) ) p ( x k ( j ) | x k −1 ( j ) )
ωk ( j ) ∝
q ( x k ( j ) | x k −1 ( j ) , zk )
(29)
and then normalizing the importance weights for j = 1,… , N .
3. Finally, a resampling process such as systematic
resampling is performed to obtain N random particles
xˆ k ( j ) , Pˆ k ( j ) , and the output is generated in the same
(
)
manner as for the generic PF.
In the next section, numerical examples are presented to
illustrate the accuracy of the proposed algorithms.
VT-2005-00647.R2
7
RMSE ( k ) =
1 MC i
∑ xˆ k − xktrue
MC i =1
(
) ( xˆ
T
i
k
− x ktrue
)
(30)
where MC is the number of Monte Carlo simulations
performed, and xˆ ik is the filter position estimate ( x, y )T (or
velocity estimate ( x, y )T ), at time k in Monte Carlo run i. The
overall RMSE is defined as:
RMSE =
1 L 1 MC i
∑ ∑ xˆ k − xktrue
L k =1 MC i =1
(
) ( xˆ
T
i
k
−x
true
k
)
(31)
where L is the total number of simulation time steps after the
convergence of the filter.
Figure (2a) and (2b) show one realization illustrating the
convergence of the proposed algorithms to the real position
and velocity of a moving MS, respectively. Figure (3) shows
X (meters)
3000
3185
3180
2800
3175
2600
3170
35
2400
2400
Y (meters)
[30], cell radii is 5000 m, number of samples S is 10, number
of BSs for triangulation is 5, radio-frequency is 900MHz, and
100 Monte Carlo simulations were performed.
Next, we consider the simulation setup for the EKF, the PF,
and the UPF approaches that employ Aulin’s channel model
for MS location and velocity estimation. The simulation setup
for the MLE approach remains the same, only now we are
trying to locate a single MS. The envelope of the received
signal for all paths, rn’s, are generated as Rayleigh iid RVs
with parameter 0.5. an , β n , and φn are generated as uniform
iid RVs in [0, 2π], [0, 0.2π], and [0, 2π], respectively. The
total number of paths P is 6 (represents urban environment).
The filters have the following parameters: Number of time
steps (measurements) is 50 with Δ k = 0.1 seconds, process
noise covariance Q and measurement noise variance R are
I 2× 2 and 0.01, respectively, where I 2× 2 is the twodimensional identity matrix, the initial PDF of the MS
position is assumed to be uniform over the entire cell size
which represents the worst-case as far as choosing an initial
PDF is considered, the initial PDF of the MS velocity is
Gaussian distributed with mean 65 meters/sec and variance
10, number of particles is 500. The SUT parameters are set to
α = 1, β = 0, and κ = 0, and finally the mean estimate of all
particles is used as the final estimate. The position (or
velocity) root mean square error (RMSE) is used as a
performance measure and is defined as:
3200
36
37
2200
2000
2140
1800
1400
Actual
PF
EKF
EKF/MLE
UPF
2130
1600
2120
25
0
26
10
27
20
28
29
30
40
50
Filter Steps
(a)
Vx (meters/second)
variance σ b2 is 8 dB, reference distance d 0b is 200 m for all b
3400
Vy (meters/second)
In this numerical example, the performance of the proposed
MS location and velocity estimation algorithms is determined.
We consider first the ML estimate of the MS location in which
we employ a typical, yet realistic, wireless communication
simulation setup. The service area consists of a 19-cell cluster.
The BSs are placed over a uniform hexagonal pattern of cells
which are centrally equipped with omni-directional antennas.
MSs are placed randomly in the central cell and the number of
arranged users is 1000. Path-loss exponent ε b is 3.5, path-loss
the position and velocity RMSE for each time according to
(30), respectively, and the overall position and velocity RMSE
for the convergent runs using (31) are shown in Table (1).
200
55
150
48
42
100
47
50
0
150
51
Actual
PF
EKF
EKF/MLE
UPF
100
47
33
34
35
50
0
0
10
20
30
40
50
Filter Steps
(b)
Fig. 2. (a) Location and (b) velocity estimates of a moving MS generated by
the different filters.
Velocity RMSE (m/s) Location RMSE (m)
VII. NUMERICAL RESULTS
300
EKF
PF
MLE
EKF/MLE
UPF
200
15
0
40
100
0
5
10
15
20
25
30
35
43
40
45
50
150
10
100
0
40
45
50
0
5
10
15
20
25
30
35
40
45
50
Filter Steps
Fig. 3. Location and velocity estimates RMSE (k) generated by the different
algorithms.
Table 1. Performance comparison for MS location and velocity estimation
algorithms using the MLE, EKF, EKF/MLE, PF, and the UPF approaches.
MLE
EKF
EKF/
PF
UPF
MLE
_
39
6
2
2
Diverged runs
73.46
142.38
11.23
4.31
3.81
Position RMSE (m)
_
51.36
16.52
1.01
0.96
Velocity RMSE
(m/sec)
VT-2005-00647.R2
8
From Figure (3) and Table (1), it can be noticed that the
accuracy of the MLE approach is satisfactory. However, in
realistic NLOS and multipath conditions this method does not
perform well. Nevertheless, it can be used as an initial
condition for the EKF to find a more accurate estimator. It
has been also observed that the accuracy increases as the
number of samples, S, increases and σ b2 , ε b decrease, as
expected. For more accurate estimates Aulin’s channel model
is employed together with the EKF, PF, and UPF.
We observe in Figure (2) that the EKF/MLE, PF, and UPF
estimators converge to the actual location and velocity within
a few iterations (less than 5). While the EKF position and
velocity estimates oscillate with large deviation around the
actual position and velocity. This is because the EKF truncates
higher order series expansion terms and is sensitive to the
initial state. However, the latter can be improved by using the
ML estimate as an initial estimate for the EKF. Since it takes
less than 5 iterations for the filters to converge near the actual
value as shown in Figures (2), the RMSE (k) in (30) is
calculated starting from the iteration k = 5. Only convergent
runs are used in the RMSE calculations. Figure (3) shows that
the performance of the PF and the UPF approaches are about
the same and superior to other approaches. The superior
performance of the UPF is clearly evident. Table (1) shows
the number of runs that diverged and the performance for each
approach. The latter shows the appropriateness of choosing
the PF and the UPF for this kind of problems. We have
observed that using fewer particles does not affect
significantly the unscented particle filter, while the
performance of the particle filter deteriorates. The high
accuracy is due to the appropriateness of Aulin’s channel
model and the efficiency of the particle filtering in this
particular application.
Figure (4) shows how robust the particle filtering approach
is if we assume that we only know the channel parameters
{rn , α n , β n } within certain tolerances. Specifically,
rn = rn0 (1 + δ rn0 ),
δ rn ≤ 5%, 10%, 20% and 30%
0
α n = α n (1 + δα n ), δα n ≤ 5%, 10%, 20% and 30%
0
0
0
β n = β n (1 + δβ n ),
Velocity RMSE (m/sec) Location RMSE (m)
0
δβ n ≤ 5%, 10%, 20% and 30%
0
0
25
20
15
10
5
0
5
10
15
20
25
30
35
40
25
45
50
Exact
5% Error
10% Error
20% Error
30% Error
20
15
(32)
where rn0 , α n0 and β n0 are the nominal (actual) values of the
channel parameters.
Figure (4) is generated by assuming that the real channel
has parameters rn0 , α n0 and β n0 , while in the estimation stage
the channel model parameters used are uniformly distributed
about their nominal values as in the uncertainty model (32),
and varying the uncertainty percentage from 5% to 30%. It
can be noticed that the location and velocity RMSE still
converge even if the channel parameters have errors. The
higher the error is, the longer time it takes for the filter to
converge. It can also be seen that the final RMSE increases for
higher errors in channel parameters as expected.
The high accuracy, consistency and performance of the
proposed UPF approach, makes it suitable to be used in any
location and velocity estimation applications, particularly
those which require high accuracy such as emergency
services.
VIII. CONCLUSION
New estimation methods are proposed to track the position
and velocity of a MS in a cellular network. They are based on
Aulin’s scattering model combined with the EKF, PF, and
UPF estimation algorithms. Since the instantaneous electric
field is a nonlinear function of the MS location and velocity,
the EKF, PF, and UPF are appropriate for the estimation
process.
Numerical results for typical simulations including in the
presence of parameters uncertainty show that they are highly
accurate and consistent. The performance of the PF and the
UPF estimation methods are superior to the EKF. This is due
to the sensitivity of the EKF to the initial condition and
Gaussian assumptions. An alternative is to use the ML
estimate that employs the lognormal channel model, as the
initial EKF state. The use of nonlinear models and/or nonGaussian noise is the main explanation for the improvement in
accuracy over linear algorithms such as the EKF. These
methods also excel in using inherent features of the cellular
system, i.e., they support existing network infrastructure and
channel signaling. The assumptions are knowledge of the
channel and access to the instantaneous received field, which
are obtained through channel sounding samples from the
receiver circuitry. Future work will focus on generating
efficient channel estimation algorithms, to remove the
assumption on partial knowledge of the channel. Work on
building a pilot application to test the performance of the PF
and/or the UPF in realistic conditions is on-going together
with the incorporation of channel model parameters estimation
algorithms.
10
5
0
5
10
15
20
25
30
35
40
45
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50
Filter Steps
Fig. 4. The UPF location and velocity estimates RMSE (k) for imperfect
knowledge of channel parameters.
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ETSI TR 101 115 V8.2.0 (2000-04), Annex V.A.
9