Bayesian Time Delay Estimation of GNSS Signals in Dynamic Multipath Environments

Abstract

A sequential Bayesian estimation algorithm for multipath mitigation is presented, with an underlying movement model that is especially designed for dynamic channel scenarios. In order to facilitate efficient integration into receiver tracking loops, it builds upon complexity reduction concepts that previously have been applied within maximum likelihood (ML) estimators. To demonstrate its capabilities under different GNSS signal conditions, simulation results are presented for both BPSK modulated and BOC(1,1) modulated navigation signals.

Full text

Hindawi Publishing Corporation
International Journal of Navigation and Observation
Volume 2008, Article ID 372651, 11 pages
doi:10.1155/2008/372651
Research Article
Bayesian Time Delay Estimation of GNSS Signals in
Dynamic Multipath Environments
Michael Lentmaier,1Bernhard Krach,2and Patrick Robertson2
1Vodafone Chair Communications Systems, Dresden University of Technology, TU Dresden, 01062 Dresden, Germany
2German Aerospace Center DLR, Institute of Communications and Navigation, Oberpfaffenhofen, 82234 Wessling, Germany
Correspondence should be addressed to Michael Lentmaier, michael.lentmaier@ifn.et.tu-dresden.de
Received 1 August 2007; Revised 7 December 2007; Accepted 17 March 2008
Recommended by Letizia Presti
A sequential Bayesian estimation algorithm for multipath mitigation is presented, with an underlying movement model that
is especially designed for dynamic channel scenarios. In order to facilitate efficient integration into receiver tracking loops, it
builds upon complexity reduction concepts that previously have been applied within maximum likelihood (ML) estimators. To
demonstrate its capabilities under different GNSS signal conditions, simulation results are presented for both BPSK-modulated
and BOC-(1,1) modulated navigation signals.
Copyright © 2008 Michael Lentmaier et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
1. INTRODUCTION
A major error source within global navigation satellite
systems (GNSSs) comes from multipath, the reception of
additional signal replica due to reflections, which introduce
a bias into the estimate of the delay lock loop (DLL) of
a conventional navigation receiver. For efficient removal of
this bias, it is possible to formulate advanced maximum
likelihood (ML) estimators that incorporate the echoes into
the signal model and are capable of achieving the theoretical
limits given by the Cram´
er Rao bound. The drawback of
ML estimator techniques is that the parameters are assumed
to be constant during the time of observation. Independent
estimates are obtained for successive observation intervals,
whose length has to be adapted to the dynamics of the
channel.
In this paper, we consider the important practical case
of dynamic channel scenarios and assess how the time-
delay estimation can be improved if information is available
about the temporal evolution of the channel parameters. Our
approach is based on Bayesian filtering, the optimal and well-
known framework to address such dynamic state estimation
problems [1]. Sequential Monte Carlo (SMC) methods [2,3]
are used for computing the posterior probability density
functions (PDFs) of the signal parameters.
2. A COMPARISON OF VARIOUS MULTIPATH
MITIGATION APPROACHES
Figure 1 gives an overview of the relationships between
different multipath mitigation and estimation approaches. In
fact, we have chosen to discriminate approaches according
to their primary objective. The left column represents the
class of techniques that attempt to mitigate the effect of
multipath in different ways. This can for example be achieved
by modifications of the antenna response, either by means of
hardware design or with signal processing techniques (e.g.,
beamforming). The majority of the remaining mitigation
techniques are in some way aligning the more or less
traditional receiver components (e.g., the early/late corre-
lator) to the signal received in the multipath environment.
The probably most simple technique is the adjustment of
the correlator spacing applied in the Narrow Correlator
[4]. Other well-known examples of this category are the
Strobe Correlator [5], the Gated Correlator [6], or the Pulse
Aperture Correlator [7]. To incorporate new signal forms
(such as BOC), these methods need “tuning” in order to
suffer as little as possible from multipath. On the other
hand, multipath estimation techniques (right column) treat
multipath (in particular the delay of the paths) as something
to be estimated from the channel observations, so that its

2 International Journal of Navigation and Observation
Mitigation Estimation
Modification of standard DLL
detector
•Narrow correlator
•Double-delta correlator
•Strobe correlator
•Pulse aperture correlator
.
.
.
Antenna characteristics
•Choke ring
•Beamforming
.
.
.
Static Dynamic
Maximum likelihood
•MEDLL
•Vision correlator &
MMT
•SAGE
•Newton-ty pe
•Antenna array signal
processing techniques
.
.
.Practical ML integration:
•Signal compression
•Loop-aided ML
•ML-in-the-loop
.
.
.
MAP with static prior
Sequential estimation
•Bayesian filtering
◦Kalman filter
variants
◦Sequential Monte
Carlo methods
.
.
.
Figure 1: Classification of multipath mitigation approaches.
effects can be trivially removed at a later processing stage. For
the estimation techniques, we have differentiated between
static and dynamic approaches, according to the underlying
assumption of the channel dynamics. Examples for static
multipath estimation are those belonging to the family of
ML estimators, often using different efficient maximization
strategies over the likelihood function [8–13]. For static
channels without availability of prior information, the ML
approach is optimal and performs significantly better than
other techniques, especially if the echoes have short delay.
An estimator based on sequential importance sampling (SIS)
methods (particle filtering) for static multipath scenarios has
been considered in [14], which has the advantage that prior
channel knowledge can be incorporated.
As a first step towards addressing dynamic channels, one
can incorporate ML estimators in receiver loops or formulate
quasisequential estimators [15,16]. Finally, dynamic esti-
mators that target the computation of the posterior PDF
conditioned on the received channel output sequence at the
receiver can be applied on a per single range basis or operate
in the position domain. In this paper, we concentrate on
dynamic estimators applied per each range. The sequential
Monte Carlo approach has also been suggested in the com-
munications field for estimation of time-varying channel
responses in spread spectrum systems [17,18].
3. SIGNAL MODEL
Assume that the complex valued baseband-equivalent
received signal is equal to
z(t)=
Nm

i=1
ei(t)·ai(t)·c(t)∗gt−τi(t)+n(t), (1)
where c(t) is a delta-train code sequence that is modulated on
a pulse g(t), Nmis the maximum number of allowed paths
reaching the receiver (to restrict the modeling complexity),
ei(t) is a binary function that controls the activity of the
ith path, and ai(t)andτi(t) are their individual complex
amplitudes and time delays, respectively. The signal is
disturbed by additive white Gaussian noise n(t). Grouping
blocks of Lsamples at times (m+kL)Ts,m=0, ...,L−1,
together into vectors zk,k=0, 1, ...,nwhilst assuming
the parameter functions ei(t), ai(t), and τi(t) being constant
within the corresponding time interval and equal to ei,k,ai,k,
and τi,k, this can be rewritten as
zk=CGτkEkak+nk=sk+nk.(2)
In the compact form on the right hand side, the samples of
the delayed pulses g(τi,k) are stacked together as columns
of the matrix G(τk)=[g(τ1,k), ...,g(τNm,k)], Cis a
matrix representing the convolution with the code, and the
delays and amplitudes are collected in the vectors τk=
[τ1,k,...,τNm,k]Tand ak=[a1,k,...,aNm,k]T,respectively.
Furthermore, for concise notation we use Ek=diag [ek]
whilst the elements of the vector ek=[e1,k,...,eNm,k]T,
ei,k∈[0, 1], determine whether the ith path is active or not
by being either ei,k=1 corresponding to an active path or
ei,k=0 for a path that is currently not active. The term sk
denotes the signal hypothesis and is completely determined
by the channel parameters τk,ak,and ek. Using (2), we can
write the associated likelihood function as
pzk|sk=1
(2π)Lσ2L·exp −1
2σ2zk−skHzk−sk.
(3)
The likelihood function will play a central role in the
algorithms discussed in this paper; its purpose is to quantify
the conditional probability of the received signal conditioned
on the unknown signal (specifically the channel parameters).

Michael Lentmaier et al. 3
3.1. Efficient likelihood computation
In [11], a general concept for the efficient representation
of the likelihood (3) was presented, which is applicable to
many of the existing ML multipath mitigation methods.
The key idea of this concept is to formulate (3) through
avectorzc,kresulting from an orthonormal projection of
the observed signal zkonto a smaller vector space, so that
zc,kis a sufficient statistic according to the Neyman-Fisher
factorization [19] and hence suitable for estimating sk.In
other words, the reduced signal comprises the same infor-
mation as the original signal itself. In practice, this concept
becomesrelevantastheprojectioncanbeachievedby
processing the received signal (2) with a bank of correlators
and a subsequent decorrelation of the correlator outputs.
A variant of this very general concept, applied in [13], has
also been referred to as the Signal Compression Theorem
in [20] for a set of special projections that do not require
the step of decorrelation due to the structure of the used
correlators. For instance, unlike the correlation technique
used in [8], the one suggested in [13] already projects onto
an orthogonal and thus uncorrelated subspace, similar to the
code matched correlator technique proposed in [11]. Due to
complexity reasons, all practically relevant realizations of ML
estimators [8,13] operate in a projected space, namely after
correlation. The corresponding mathematical background
will be discussed below, including also interpolation of the
likelihood and elimination of complex amplitudes as further
methods for complexity reduction.
3.1.1. Data compression
As explained above, the large vector containing the received
signal samples zkis linearly transformed into a vector zc,kof
much smaller size. Following this approach, the likelihood
according to (2)canberewrittenas
pzk|sk=1
(2π)Lσ2Lexp −zH
kzk
2σ2
·exp RzH
kQcQH
csk
σ2−sH
kQcQH
csk
2σ2
=1
(2π)Lσ2Lexp −zH
kzk
2σ2
·exp RzH
c,ksc,k
σ2−sH
c,ksc,k
2σ2,
(4)
with the compressed received vector zc,kand the compressed
signal hypothesis sc,k:
zc,k=QH
czk,sc,k=QH
csk,(5)
and the orthonormal compression matrix Qc, which needs to
fulfill
QcQH
c≈I,QH
cQc≈I,(6)
to minimize the compression loss. According to [11], the
compression can be two-fold so that we can factorize
Qc=QccQpc (7)
into a canonical component decomposition,givenbyanL×Ncc
matrix Qcc,andaprincipal component decomposition,given
by an Ncc ×Npc matrix Qpc.In[11], two choices for Qcc are
proposed
Qcc =⎧
⎨
⎩
CG(τb)R−1
cc Signal matched,
C(τb)R−1
cc Code matched, (8)
where the elements of the vector τbdefine the positions
of the individual correlators. The noise-free outputs of the
corresponding correlator banks are illustrated in Figure 2.To
decorrelate the bank outputs (CG(τb))Hyand C(τb)Hy,as
mentioned above, the whitening matrix Rcc can be obtained
from a QR decomposition of CG(τb)andC(τb), respectively.
Apart from practical implementation issues, both correlation
methods given by (8) are equivalent from a conceptual point
of view. For details on the compression through Qpc,the
reader is referred to [11].
3.1.2. Interpolation
In order to compute (4) independently of the sampling grid,
advantage can be made of interpolation techniques. Using
the discrete Fourier transformation (DFT), with Ψbeing the
DFT matrix and Ψ−1being its inverse counterpart (IDFT),
we get
sc,k=QH
cCΨ−1diag Ψg(0)ΩτkEkak
=MscΩτkEkak,(9)
with Ω(τk) being a matrix of column-wise stacked vectors
with Vandermonde structure [10,11], such that the element
at row pand column qcomputes with
RΩτkp,q=cos2π(p−1)τq,k
NgTs,
IΩτkp,q=−sin 2π(p−1)τq,k
NgTs.
(10)
Ngis the length of the pulse gin samples. The advantage
of the interpolation is that it can take place in the reduced
space. The most costly computations in (9) can be carried out
in precalculations as the matrix Msc, whose row dimension
corresponds to the dimension of the reduced space and
whose column dimension is Ng, is constant.
3.1.3. Amplitude elimination
In a further step, we reduce the number of parameters by
optimizing (4) for a given set of τkand ekwith respect to
the complex amplitudes ak, which can be achieved through a
closed-form solution. Using
Sc,k=MscΩ(τk)Ek(11)
and obtaining S+
c,kby removing zero columns from Sc,k,one
yields the corresponding amplitude values of the active paths:
a+
k=(S+H
c,kS+
c,k)−1S+H
c,kzc,k.(12)

4 International Journal of Navigation and Observation
21.510.50−0.5−1−1.5−2
Delay (μs)
−0.2
0
0.2
0.4
0.6
0.8
1
(a) BPSK signal matched
21.510.50−0.5−1−1.5−2
Delay (μs)
−0.2
0
0.2
0.4
0.6
0.8
1
(b) BPSK code matched
21.510.50−0.5−1−1.5−2
Delay (μs)
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
(c) BOC(1,1) signal matched
21.510.50−0.5−1−1.5−2
Delay (μs)
−1.5
−1
−0.5
0
0.5
1
1.5
(d) BOC(1,1) code matched
Figure 2: Output of the canonical component type correlator banks CG(τb)andC(τb) for BPSK and BOC(1,1).
As we have introduced a potential source of performance
loss by eliminating the amplitudes and thus practically
are disregarding their temporal correlation, we propose to
optimize (4) using
zc,k=1
Q·
Q−1

l=0
zc,k−l,(13)
with the adjustable averaging coefficient Q. When evaluating
(4), we substitute sc,kby
sc,k=Sc,kak, (14)
where the elements of the vector akthat are indicated to
have an active path (ak,i:i→ek,i=1)aresetequalto
the corresponding elements of a+
k. All other elements (ak,i:
i→ek,i=0) can be set arbitrarily as their influence is masked
by the zero elements of ek.
3.2. Review of the ML Concept
The concept of ML multipath estimation has drawn substan-
tial research interest since the first approach was proposed in
[8]. Despite being treated differently in various publications,
the objective is the same for all ML approaches, namely to
find the signal parameters that maximize (3)foragiven
observation zk:
sk=arg max
skpzk|sk.(15)
The signal parameters are thereby assumed as being constant
throughout the observation period k.Different maximiza-
tion strategies exist, which basically characterize the different
approaches. Despite offering great advantages for theoretical
analysis, the practical advantage of the generic ML concept is
questionable due to a number of serious drawbacks.
(i) The ML estimator assumes that the channel is static
for the observation period and is not able to exploit

Michael Lentmaier et al. 5
its temporal correlation throughout the sequence
k=1, ...,n. Measured channel scenarios have shown
significant temporal correlation [21].
(ii) Despite being of great interest in practice, the esti-
mation of the number of received paths is often not
addressed. The crucial problem here is to correctly
estimate the current number of paths to avoid over
determination, since an overdetermined estimator
will tend to use the additional degrees of freedom
to match the noise by introducing erroneous paths.
There exist various techniques based on model selec-
tion that can be employed to estimate the number
of paths [22] but they suffer from the problem of
having to dynamically adjust the decision thresholds.
Typically, only a single hypothesis is tracked, which
in practice causes error event propagation.
(iii) The ML estimator does only provide the most likely
parameter set for the given observation. No relia-
bility information about the estimates is provided.
Consequently, ambiguities and multiple modes of the
likelihood are not preserved by the estimator.
ML estimators require that the estimated parameters
remain constant during the observation period. Due to data
modulation and phase variations in practice, this period,
which is often referred to as the coherent integration time,
is limited to a range of 1 millisecond–20 milliseconds. To
reach a sufficient noise performance with a ML estimator
in practice, it is required to extend its observation interval
approximately to the equivalent averaging time of a con-
ventional tracking loop, which is commonly in the order of
several hundred coherent integration periods. To overcome
this problem, the observations are forced to be quasicoherent
by aiding the ML estimator with a phase locked loop (PLL)
and a data removal mechanism [8].
4. SEQUENTIAL ESTIMATION
4.1. Optimal solution
In Section 3, we have established the models of the under-
lying time-variant processes. The problem of multipath
mitigation now becomes one of sequential estimation of a
hidden Markov process. We want to estimate the unknown
channel parameters based on an evolving sequence of
received noisy channel outputs zk. The channel process for
each range of a satellite navigation system can be modeled
as a first-order Markov process if future channel parameters
given the present state of the channel and all its past states
depend only on the present channel state (and not on any
past states). We also assume that the noise affecting successive
channel outputs is independent of the past noise values; so
each channel observation depends only on the present channel
state.
Intuitively, we are exploiting not only the channel
observations to estimate the hidden channel parameters (via
the likelihood function), but we are also exploiting our
prior knowledge about the statistical dependencies between
successive sets of channel parameters. We know from channel
Measurements (observed)
zk−2zk−1zk
Time
k−2
Time
k−1
Time
k
Likelihood
p(zk|xk)
xk−2xk−1xk
p(xk|xk−1)
State transition PDF
Hidden states
Figure 3: Illustration of the hidden Markov estimation process for
three time instances. Our channel measurements are the sequence
{zi,i=1, ...,k}, and the channel parameters to be estimated are
{xi,i=1, ...,k}.
measurements that channel parameters are time varying but
not independent from one time instance to the next; for
example, an echo usually experiences a “life-cycle” from its
first occurrence, then a more or less gradual change in its
delay and phase over time, until it disappears [21]. These
measurements also show that the common channel models
considered in communication systems [17,18]arenot
adequately reflecting the properties that are crucial for high-
resolution signal delay estimation as required in navigation
systems.
Now that our major assumptions have been established,
we may apply the concept of sequential Bayesian estimation.
The reader is referred to [23] which gives a derivation of
the general framework for optimal estimation of temporally
evolving (Markovian) parameters by means of inference,
and we have chosen similar notation. The entire history of
observations (over the temporal index k)canbewrittenas
Zk={zk,k=1, ...,k}.(16)
Similarly, we denote the sequence of parameters of our
hidden Markovian process by
Xk={xk,k=1, ...,k}.(17)
Here, xkrepresents the characterization of the hidden
channel state, including the parameters that specify the signal
hypothesis skgiven in (2). Our goal is to determine the
posterior PDF of every possible channel characterization
given all channel observations: p(xk|Zk) (see Figure 3).
Once we have evaluated this posterior PDF, we can either
determine that channel configuration that maximizes it—the
so-called maximum a posteriori (MAP) estimate; or we can
choose the expectation—equivalent to the minimum mean
square error (MMSE) estimate. In addition, the posterior
distribution itself contains all uncertainty about the current
range and is thus the optimal measure to perform sensor data
fusion in an overall positioning system.

6 International Journal of Navigation and Observation
It can be shown that the sequential estimation algorithm
is recursive, as it uses the posterior PDF computed for time
instance k−1 to compute the posterior PDF for instance
k(see Figure 4). For a given posterior PDF at time instance
k−1, p(xk−1|Zk−1), the prior PDF p(xk|Zk−1) is calculated
in the so-called prediction step by applying the Chapman-
Kolmogorov equation:
pxk|Zk−1=pxk|xk−1pxk−1|Zk−1dxk−1, (18)
with p(xk|xk−1) being the state transition PDF of the
Markov process. In the update step, the new posterior PDF for
step kis obtained by applying Bayes’ rule to p(xk|zk,Zk−1)
yielding the normalized product of the likelihood p(zk|xk)
and the prior PDF:
pxk|Zk=pxk|zk,Zk−1
=pzk|xk,Zk−1pxk|Zk−1
pzk|Zk−1
=pzk|xkpxk|Zk−1
pzk|Zk−1.
(19)
The term p(zk|xk)=p(zk|sc,k=sc,k) follows from (4)and
represents the probability of the measured channel output
(often referred to as the likelihood value), conditioned on a
certain configuration of channel parameters at the same time
step k. The denominator of (19)doesnotdependonxk,and
so it can be computed by integrating the numerator of (19)
over the entire range of xk(normalization).
To summarize so far, the entire process of prediction
and update can be carried out recursively to calculate the
posterior PDF (19) sequentially, based on an initial value of
p(x0|z0)=p(x0). The evaluation of the likelihood function
p(zk|xk) is the essence of the update step. Similarly,
maximizing this likelihood function (i.e., ML estimation)
would be equivalent to maximizing p(xk|Zk) only in the
case that the prior PDF p(xk|Zk−1)doesnotdependon
Zk−1, and when all values of xkare a priori equally likely.
Since these conditions are not met, evaluation of p(xk|Zk)
entails all the above steps.
4.2. Sequential estimation using particle filters
The optimal estimation algorithm relies on evaluating the
integral (18),whichisusuallyaverydifficult task, except for
certain additional restrictions imposed on the model and the
noise process. So, very often a suboptimal realization of a
Bayesian estimator has to be chosen for implementation. In
this paper, we use a sequential Monte Carlo (SMC) filter, in
particular a sampling importance resampling particle filter
(SIR-PF) according to [23]. In this algorithm, the posterior
density at step kis represented as a sum and is specified by a
set of Npparticles:
pxk|Zk≈
Np

j=1
wj
k·δxk−xj
k, (20)
Movement model
p(xk|xk−1)Prediction
stage
Prior
p(xk|Zk−1)
Measurements
zkLikelihood
p(zk|xk)Update
stage
k=k+1
Posterior
p(xk|Zk)
Figure 4: Illustration of the recursive Bayesian estimator.
where each particle with index jhas a state xj
kand has a
weight wj
k. The sum over all particles’ weights is one. In SIR-
PF, the weights are computed according to the principle of
importance sampling where the so-called proposal density
is chosen to be p(xk|xk−1=xj
k−1), and with resampling
at every time step. For Np→∞, the approximate posterior
approaches the true PDF.
The key step, in which the measurement for instance k
is incorporated, is in the calculation of the weight wj
kwhich
for the SIR-PF can be shown to be the likelihood function:
p(zk|xj
k). The characterization of the channel process enters
in the algorithm, when at each time instance k, the state
of each particle xi
kis drawn randomly from the proposal
distribution, that is, from p(xk|xj
k−1).
4.3. Exploiting linear substructures
If there exist linear substructures in the model, it is possible
to reduce the computational complexity of the filter by
means of marginalization over the linear state variables [24],
also known as Rao-Blackwellization [25]. In a marginalized
filter, particles are still used to estimate the nonlinear states,
while for each of the particles the linear states can be
estimated analytically. In our case, since the measurement
zkis a linear function of the complex amplitudes ak, the
likelihood function can be factorized as
pzk|sc,k=pzk|τk,ek,ak=fzk,τk,ek·gzk,τk,ek,ak,
(21)
where the function g(zk,τk,ek,ak) is Gaussian with respect
to ak,and f(zk,τk,ek) is proportional to p(zk|sc,k).
Marginalization of (21) over the linear state variables gives
ak
pzk|τk,ek,akdak=fzk,τk,ek∝pzk|sc,k.
(22)
Assuming that the amplitudes are block-wise independent
(Q=1), it follows that the weights of the SIR particle filter
are equal to p(zk|xj
k)=p(zk|sc,k=sj
c,k),whichisgivenby
(4).

Michael Lentmaier et al. 7
If there exists a temporal correlation of the amplitudes,
an optimal marginalized filter requires the implementation
of a separate Kalman filter for each of the particles. As we do
not want to increase the complexity per particle, we take the
correlation into account by adjustment of Qin (13).
4.4. Choice of appropriate channel process
To exploit the advantages of sequential estimation for
our task of multipath mitigation/estimation, we must be
able to describe the actual channel characteristics (channel
parameters) so that these are captured by p(xk|xk−1).
In other words, the model must be a first-order Markov
model, and all transition probabilities must be known. In our
approach, we approximate the channel as follows.
(i) The channel is totally characterized by a direct path
(index i=1) and at most Nm−1echoes,
(ii) each path has complex amplitude ai,kand relative
delay and Δτi,k=τi,k−τ1,k; where echoes are
constrained to have delay τi,k≥τ1,k, that is, Δτi,k≥0,
(iii) the different path delays follow the process:
τ1,k=τ1,k−1+α1,k−1·Δt+nτ,
Δτi,k=Δτi,k−1+αi,k−1·Δt+nτ,(23)
(iv) each parameter αi,kthat specifies the speed of the
change of the path delay follows its own process:
αi,k=1−1
K·αi,k−1+nα, (24)
(v) the magnitudes and phases of the individual paths,
represented by the complex amplitudes ai,k, are elimi-
nated according to (12)and(14) for the computation
of the likelihood (4),
(vi) each path is either “on” or “off,” as defined by channel
parameter ei,k∈{1≡“on,0≡“off },
(vii) where ei,kfollows a simple two-state Markov process
with a-symmetric crossover and same-state probabil-
ities:
pei,k=0|ei,k−1=1=ponoff, (25)
pei,k=1|ei,k−1=0=poffon.(26)
The model implicitly incorporates three i.i.d noise sources:
Gaussian nτand nαas well as the noise process driving the
state changes for ei,k. These sources provide the randomness
of the model. The parameter Kdefines how quickly the
speed of path delays can change (for a given variance of nα).
Finally, Δtis the time between instances k−1andk.We
assume all model parameters (i.e., K,Δt, noise variances, and
the “on”/“off” Markov model) to be independent of k(see
Figure 5). The hidden channel state vector xkcan therefore
be represented as
τ1,k,Δτ2,k,...,ΔτNm,k,α1,k,...,αNm,k,ei,k,...,eNm,kT.
(27)
Note that the model implicitly represents the number of
paths
Nm,k=
Nm

i=1
ei,k(28)
as a time-variant parameter.
When applied to our particle filtering algorithm, drawing
from the proposal density is simple. Each particle stores
the above-channel parameters of the model, and then the
new state of each particle is drawn randomly from p(xk|
xj
k−1) which corresponds to drawing values for nαand nτas
well as propagating the “on”/“off” Markov model, and then
updating the channel parameters for kaccording to (23)–
(26).
4.5. Practical issues
4.5.1. Model matching
It is important to point out that a sequential estimator is
only as good as its state transition model matches the real
world situation. The state model needs to capture all relevant
hidden states with memory and needs to correctly model
their dependencies, while adhering to the first order Markov
condition. Furthermore, any memory of the measurement
noise affecting the likelihood function p(zk|xk)mustbe
explicitly contained as additional states of the model x,so
that the measurement noise is i.i.d.
The channel state model is motivated by channel
modeling work for multipath prone environments such as
the urban satellite navigation channel [21,26]. In fact,
the process of constructing a channel model in order to
characterize the channel for signal level simulations and
receiver evaluation comes close to our task of building a
first-order Markov process for sequential estimation. For
particle filtering, the model needs to satisfy the condition
that one can draw states with relatively low computational
complexity. Adapting the model structure and the model
parameters to the real channel environment is a task for
current and future work. It may even be possible to envisage
hierarchical models in which the selection of the current
model itself follows a process. In this case, a sequential
estimator will automatically choose the correct weighting of
these models according to their ability to fit the received
signal.
4.5.2. Integration into a receiver
For receiver integration, the computational complexity of
the filtering algorithm is crucial. From a theoretical point
of view, it is desirable to run the sequential filter clocked
corresponding to the coherent integration period of the
receiver and with a very large number of particles. From the
practical point of view, however, it is desirable to reduce the
sequential filter rate to the navigation rate and to minimize
the number of particles. Existing ML approaches can help
here to achieve a flexible complexity/performance tradeoff,
as strategies already developed to extend the observation

8 International Journal of Navigation and Observation
τ1,k−2
α1,k−2
e1,k−2
τ1,k−1
α1,k−1
e1,k−1
τ1,k
α1,k
e1,k
Δτ2,k−2
α2,k−2
e2,k−2
Δτ2,k−1
α2,k−1
e2,k−1
Δτ2,k
α2,k
e2,k
ΔτNm,k−2
αNm,k−2
eNm,k−2
ΔτNm,k−1
αNm,k−1
eNm,k−1
ΔτNm,k
αNm,k
eNm,k
Direct path
First echo
Nm−1’th echo
.
.
..
.
..
.
.
··· ···
State xk−2State xk−1State xk
Figure 5: The Markov process chosen in this paper to model the channel with Nmpaths. Dotted arrows—shown only for a small subset of
the transitions—indicate the constraint that Δτi,k≥0. For an explanation of the terms, see the text leading up to (27).
periods of ML estimators can be used directly to reduce the
rate of the sequential filtering algorithm.
5. PERFORMANCE EVALUATION
To demonstrate the capabilities of the SMC-based Bayesian
time delay estimator proposed in Section 4,wehavecarried
out computer simulations for both static and dynamic
channel environments. The signal-to-multipath ratio (SMR)
was chosen constant and equal to 6dB, while the relative
phases are changing according to Δϕi,k=2πΔτi,kfcwith
fc=1575.42 MHz being the frequency of the L1 carrier. The
Bayesian estimator uses a time interval of Δt=1 millisecond
corresponding to the duration of a codeword. The amplitude
averaging coefficient is set to Q=10, and signal compression
is applied with Ncc =41 (code-matched correlators) Npc =
25.Thechannelparametersσ2
i,τ,σ2
i,α,K,and p(ei,k|ei,k−1)are
selected to fit the statistics of a real channel according to [21].
The SIR-PF uses the minimum mean square error (MMSE)
criterion to estimate the parameters xkfrom the posterior.
5.1. Static multipath scenario
The capability of multipath mitigation techniques is com-
monly assessed by showing the systematic error due to
a single multipath replica plotted as a function of the
relative multipath delay in a static channel scenario. In
Figure 6, the root mean square error (RMSE) is shown
for the proposed sequential estimator, implemented as a
SIR-PF with Ns=2000 particles. For comparison, the
performance of conventional DLLs with Narrow Correlator
706050403020100
Relative delay (m)
0
2
4
6
8
10
12
RMSE (m)
SIR-PF, path activitiy tracking
SIR-PF, single path model
SIR-PF, two path model
DLL narrow correlator
DLL strobe correlator
Figure 6: Static scenario: performance of SIR-PF for BPSK
modulation as function of relative multipath delay for different path
models.
and with Strobe Correlator is also shown. The simulated
signal corresponds to a GPS L1 signal with c(t) being a Gold
code of length 1023 that is modulated on a bandlimited
rectangular pulse. The chip rate is 1.023 Mchips/s so that
the duration of the codeword is 1millisecond. The one-
sided bandwidth of the resulting navigation signal is 5 MHz.
Estimators with fixed two-path model or fixed single path
model are also shown for comparison with the implicit path

Michael Lentmaier et al. 9
706050403020100
Relative delay (m)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability of two path model
Figure 7: Static scenario: average probability of a two-path model
for the estimator with path activity tracking.
30002500200015001000500
Number of particles
3
4
5
6
7
8
9
RMSE (m)
Figure 8: Static scenario: RMSE performance as function of
number of particles Nsfor BPSK modulation.
activity tracking. The performance of a single path estimator
is comparable to that of a DLL with infinitesimal correlator
spacing and shows a considerable bias over a large delay
range. The estimator with fixed two-path model successfully
mitigates the multipath bias for delays greater than 30m.
However, for smaller delays it shows an increasing variance
and is outperformed by the single path estimator. The
estimator with path activity tracking is capable of combining
the advantages of both models. From the posterior, it
is possible to calculate the estimated average probability
P(Nm,k=2|Zk) of a two-path model, which is shown in
Figure 7, and indicates the transition between the models:
for small delays the two paths essentially merge to a single
one. Note that in these simulations, the model parameters
of the sequential estimator are still the ones designed for the
dynamic channel and not optimal for this static scenario. The
RMSE as function of the number of particles Nsis shown
in Figure 8 for a relative multipath delay of 14 m, which
corresponds to the worst case in Figure 6.
5.2. Dynamic scenario
The dynamic multipath channel scenario with up to Nm=3
paths, used in the following simulations and depicted in
Figure 9, has been generated according to the movement
model, whereby the parameters K=25000, σnα=10−10,
120100806040200
Time (s)
0
50
100
150
200
Pseudorange (m)
Figure 9: Considered dynamic multipath scenario: pseudoranges
over time of the direct path (continuous line) and the temporarily
present echoes.
12010080604020
Time (s)
−5
0
5
10
Error (m)
Figure 10: Dynamic scenario: performance of the DLL with narrow
correlator for BPSK modulation (upper/grey) and BOC(1,1) modu-
lation (lower/black). The segment marked with a dashed box shows
a situation where BOC(1,1) is clearly superior to BPSK.
σnτ=10−8,andponoff=poffon =0.0001 were chosen
to resemble a typical urban satellite navigation channel
environment [26].
Figure 10 shows for this multipath scenario at a carrier-
to-noise ratio of C/N0=45dB-Hz the BPSK and the
BOC(1,1) performance of a DLL with a narrow correlator
spacing of 10−7seconds, corresponding to a tenth of a
code chip. A DLL loop bandwidth of 1Hz was selected,
which appeared to deliver the best DLL performance for this
channel. For a fair comparison of the different modulation
techniques, both signals were generated with the same C/A
code sequence of length 1023, the same receiver bandwidth
of 5 MHz (one-sided), a sampling rate of 1/Ts=10.23 MHz,
and a coherent integration time of LTs=1 millisecond.
Although the results show an improvement for the BOC(1,1)
modulation, there still remains a considerable error due to
the multipath. This behavior is confirmed by a comparison
with the multipath error envelope, depicted in Figure 11.
While in some multipath delay regions, the error is reduced
by the BOC(1,1) modulation (see, e.g, the segment marked
by the dashed box in Figure 10), more sophisticated multi-
path mitigating techniques are required to reduce the errors
due to short range multipath.
The simulation results for the particle filter-based estima-
tor with Ns=20000 particles are given in Figures 12(a) and

10 International Journal of Navigation and Observation
350300250200150100500
Relative delay (m)
−10
−5
0
5
10
Error envelope (m)
Figure 11: Multipath error envelope of the DLL with narrow cor-
relator for BPSK modulation (dashed) and BOC(1,1) modulation
(inner/solid).
120100806040200
Time (s)
−10
−5
0
5
10
15
Error (m)
(a) BPSK modulation
120100806040200
Time (s)
−5
0
5
10
Error (m)
(b) BOC(1,1) modulation
Figure 12: Dynamic scenario: Performance of the sequential
estimator with particle filtering (black) compared to the DLL
(grey). The sequential estimator is clearly superior for multipath
mitigation.
12(b) for the same channel as in Figure 10. They show that
the performance could be improved substantially, resulting
in a reduction of the root mean square (RMS) error from
3.77 m to 0.769 m for BPSK modulation and from 2.61 m to
0.694 m for BOC(1,1) modulation, respectively.
The RMSE as function of the number of particles Nsis
shown in Figure 13.
32.521.510.5
×104
Number of particles
0
2
4
6
8
10
RMSE (m)
Figure 13: Dynamic scenario: RMSE performance as function of
number of particles Nsfor BPSK modulation.
6. CONCLUSIONS
We have demonstrated how sequential Bayesian estimation
techniques can be applied to the multipath mitigation
problem in a navigation receiver. The proposed approach
is characterized by code-matched, correlator-based signal
compression together with interpolation techniques for effi-
cient likelihood computation in combination with a particle
filter realization of the prediction and update recursion. The
considered movement model has been adapted to dynamic
multipath scenarios and incorporates the number of echoes
as a time-variant hidden channel state variable that is tracked
together with the other parameters in a probabilistic fashion.
A further advantage compared to ML estimation is that
the posterior PDF at the output of the estimator represents
reliability information about the desired parameters and
preserves the ambiguities and multiple modes that may occur
within the likelihood function. Simulation results for BPSK-
and BOC-(1,1) modulated signals show that in both cases
significant improvements can be achieved compared to a
DLL with narrow correlator. In this work, we have employed
two methods to reduce complexity: the signal compression to
facilitate the computation of the likelihood function as well
as a simple form of Rao-Blackwellization to eliminate the
complex amplitudes from the state space. Further work will
concentrate on additional complexity reduction techniques
such as more suitable proposal functions or particle filtering
algorithms such as the auxiliary particle filter that are
possibly more efficient with respect to the number of
particles when applied to our problem domain.
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