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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
Stochastic Differential Equations for
Modeling, Estimation and Identification of
Mobile-to-Mobile Communication Channels
Mohammed M. Olama, Member, IEEE, Seddik M. Djouadi, Member, IEEE,
and Charalambos D. Charalambous, Senior Member, IEEE
Abstract—Mobile-to-mobile networks are characterized by
node mobility that makes the propagation environment time
varying and subject to fading. As a consequence, the statistical
characteristics of the received signal vary continuously, giving
rise to a Doppler power spectral density (DPSD) which varies
from one observation instant to the next. The current models
do not capture and track the time varying characteristics. This
paper is concerned with dynamical modeling of time varying
mobile-to-mobile channels, parameter estimation and identification from received signal measurements. The evolution of the
propagation environment is described by stochastic differential
equations, whose parameters can be determined by approximating the band-limited DPSD using the Gauss-Newton method.
However, since the DPSD is not available online, we propose
to use a filter-based expectation maximization algorithm and
Kalman filter to estimate the channel parameters and states, respectively. The scheme results in a finite dimensional filter which
only uses the first and second order statistics. The algorithm
is recursive allowing the inphase and quadrature components
and parameters to be estimated online from received signal
measurements. The algorithms are tested using experimental
data collected from moving sensor nodes in indoor and outdoor
environments demonstrating the method’s viability.
Index Terms—Mulipath fading channels, stochastic differential
equations, Doppler spectral density, Kalman filter, expectation
maximization, estimation and identification.
I. I NTRODUCTION
M
OBILE-TO-MOBILE (or ad hoc) wireless networks
comprise nodes that freely and dynamically selforganize into arbitrary and/or temporary network topology
without any fixed infrastructure support [1]. They require
direct communication between a mobile transmitter and a
mobile receiver over a wireless medium. Such mobile-tomobile communication systems differ from the conventional
cellular systems, where one terminal, the base station, is
stationary and only the mobile station is moving. As a consequence, the statistical properties of mobile-to-mobile links are
Manuscript received September 26, 2007; revised June 5, 2008 and August
17, 2008; accepted October 8, 2008. The associate editor coordinating the
review of this paper and approving it for publication was H. Xu.
M. M. Olama is with the Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 (e-mail:
olamahussemm@ornl.gov).
S. M. Djouadi is with the Electrical Engineering & Computer Science
Department, University of Tennessee, 414 Ferris Hall, Knoxville, TN 379962100 (e-mail: djouadi@eecs.utk.edu).
C. D. Charalambous is with the Electrical & Computer Engineering
Department, University of Cyprus, 75 Kallipoleos Street, P.O. Box 20537
1678, Nicosia, Cyprus (e-mail: chadcha@ucy.ac.cy).
Digital Object Identifier 10.1109/TWC.2009.071068
different from cellular ones [2][3]. Copious ad hoc networking
research exists on layers in the open system interconnection
(OSI) model above the physical layer. However, neglecting
the physical layer while modeling wireless environment is
error prone and should be considered more carefully [4]. The
experimental results in [5] show that the factors at the physical
layer not only affect the absolute performance of a protocol,
but because their impact on different protocols is non-uniform,
it can even change the relative ranking among protocols for
the same scenario. The importance of the physical layer is
demonstrated in [6] by evaluating the Medium Access Control
(MAC) performance.
Most of the research on mobile-to-mobile channel modeling, such as [2][3][7][8][9], deals mainly with deterministic
wireless channel models. In these models the speed of the
nodes are assumed to be constant and the statistical characteristics of the received signal are assumed to be fixed in time.
The Doppler power spectral density (DPSD) is then fixed
from one observation instant to the next. But in reality, the
propagation environment varies continuously due to mobility
of the nodes at variable speeds causing network topology to
dynamically change, the angle of arrival of the wave upon
the receiver can vary continuously, and objects or scatters
move in between the transmitter and the receiver resulting
in appearance or disappearance of existing paths from one
instant to the next. As a result, the current models that assume
fixed statistics can no longer capture and track complex
time variations in the propagation environment. These time
variations compel us to introduce more advanced dynamical
models based on stochastic differential equations (SDEs), in
order to capture higher order dynamics of mobile-to-mobile
channels.
Recently, there have been several papers on the application
of SDEs to modeling propagation phenomena in radar scattering and wireless communications. SDEs have been successfully used to analyze K-distributed noise in electromagnetic
scattering in [11]. Autoregressive stochastic models for the
computer simulation of correlated Rayleigh fading processes
are investigated in [12]. A first-order stochastic autoregressive
model for a flat stationary wireless channel is introduced in
[13]. Stochastic channel models based on SDEs for cellular
networks have been presented in [14][15][31]. Some preliminary results using SDEs to model ad hoc channels were presented initially in [10]. The advantage of using SDE methods
is based on the computational simplicity of the algorithm
c 2009 IEEE
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OLAMA et al.: STOCHASTIC DIFFERENTIAL EQUATIONS FOR MODELING, ESTIMATION AND IDENTIFICATION OF MOBILE-TO-MOBILE
simply because estimation is done recursively. This means
that there is no need to store and process all measurements;
rather, at each time step the estimator is updated using the
previous estimator values and the new innovations. In our
case, since we are also dealing with identification of timevarying parameters, in addition to estimating the SDE models,
this offers a considerable advantage both in the simplicity of
presentation as well as in the computation complexity.
In this paper, the deterministic DPSD derived in [2][3] is
used to develop dynamical stochastic state space models for
mobile-to-mobile channel, which consider the inphase and
quadrature components as stochastic processes. The random
variables characterizing the instantaneous power in static
channel models are generalized to dynamical models including
random processes with time varying (TV) statistics. Inphase
and quadrature components of the TV mobile-to-mobile channel and their statistics are derived from the stochastic state
space models. Since these models are based on state space
representations, we propose to estimate the channel parameters
as well as the inphase and quadrature components directly
from received signal level measurements, which are usually
available or easy to obtain in any wireless ad hoc or sensor
network. A filter-based expectation maximization (EM) algorithm [16][26][27] and Kalman filter [17] are employed in the
estimation process. These filters use only the first and second
order statistics and recursive and therefore can be implemented
online. The standard EM algorithm [26] has a wide range
of applications, such as in the estimation of speech signals
[28], in localization of narrowband sources [29] and in speech
coding [30] to cite a few.
The proposed models and estimation algorithms are tested
using received signal level measurement data collected from
two moving Crossbow’s TelosB wireless sensor nodes [18],
in indoor and outdoor environments. The experimental results,
presented in this paper, demonstrate the modeling, estimation
and identification algorithms viability. The proposed models can be used in the development of a practical channel
simulator that replicates wireless channel characteristics, and
produces outputs that vary in a similar manner to the variations
encountered in a real-world channel environment.
The remainder of this paper is organized as follows. Section II presents the deterministic DPSD of mobile-to-mobile
channels as described in [2]. Section III discusses the proposed
stochastic mobile-to-mobile channel models. Section IV introduces the filter-based EM algorithm together with the Kalman
filter, to estimate recursively the channel parameters and states,
respectively, from received signal measurements. Section V
discusses the experimental setup, numerical results and link
performance. Section VI provides concluding remarks.
II. D ETERMINISTIC DPSD OF M OBILE - TO -M OBILE
C HANNELS
Dependent on mobile speed, wavelength, and angle of
incidence, the Doppler frequency shifts on the multipath rays
give rise to a DPSD. The cellular DPSD for a received fading
carrier of frequency fc is given by [9]
⎧
⎨ 1 2 , |f − fc | < f1
S(f )
c
1− f −f
=
(1)
f1
⎩
pG/πf1
0,
otherwise
8
3.5
6
2.5
1755
3
2
4
1.5
1
2
0.5
0
−1
−0.5
0
0.5
0
−6 −5 −4 −3 −2 −1
1
alpha = 0
0
1
2
3
4
5
6
alpha = 0.25, f2 = 4*f1
5
12
4
10
8
3
6
2
4
1
0
−4
2
−2
0
2
4
0
−3
alpha = 0.5, f2 = 2*f1
−2
−1
0
1
2
3
alpha = 1
Fig. 1. Mobile-to-mobile deterministic DPSDs for different values of α’s,
with parameters fc = 0, f1 = 1, and pG = π.
where f1 is the maximum Doppler frequency of the mobile, p
is the average power received by an isotropic antenna, and G is
the gain of the receiving antenna. For a mobile-to-mobile link,
with f1 and f2 as the sender and receiver’s maximum Doppler
frequencies, respectively, the degree of double mobility, denoted by α is defined by α = [min(f1 , f2 )/ max(f1 , f2 )], so
0 ≤ α ≤ 1, where α = 1 corresponds to a full double mobility
and α = 0 to a single mobility like the cellular link, implying
that cellular channels are a special case of mobile-to-mobile
channels. The corresponding deterministic mobile-to-mobile
DPSD is for |f − fc | < (1 + α)fm [2][7],
f − f 2
S(f )
1+α
c
√
√
1−
= K
(1 + α)fm
(pG)2 /π 2 fm α
2 α
=
0, otherwise
(2)
where K(·) is the complete elliptic integral of the first
kind, and fm = max(f1 , f2 ). Fig. 1 shows deterministic
mobile-to-mobile DPSDs for different values of α’s. Thus,
a generalized DPSD has been found where the U-shaped
spectrum of cellular channels is a special case.
The deterministic mobile-to-mobile DPSD is used in the
next section to derive a method based on the SDEs for
the inphase and quadrature components of the channel via
approximations by rational functions.
III. S TOCHASTIC M OBILE - TO -M OBILE C HANNEL
M ODELS
A. General Representation of Time Varying Channels
The general TV model of a wireless channel is typically
represented by the following multipath band-pass impulse
response [19]
C(t; τ )
=
J(t)
Ij (t, τ ) cos(2πfc t) − Qj (t, τ ) sin(2πfc t)
j=1
δ τ − τj (t)
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
where C(t; τ ) is the band-pass response of the channel at
time t, due to an impulse applied at time t − τ , J(t) is
the random number of multipath components, and the set
J(t)
{Ij (t, τ ), Qj (t, τ ), τj (t)}j=1 describes the random TV
inphase component, quadrature component, and arrival time
of the different paths, respectively. Let sl (t) be the transmitted
signal, then the band pass representation of the received signal
is given by
y(t) =
J(t)
Ij (t, τ ) cos(2πfc t) − Qj (t, τ ) sin(2πfc t)
j=1
sl t − τj (t) +νI (t) cos(2πfc t) − νQ (t) sin(2πfc t) (4)
where {νI (t)}t≥0 and {νQ (t)}t≥0 are two independent and
identically distributed (iid) white Gaussian noise processes.
The DPSD is the fundamental channel characteristic on
which dynamical channel models are based on. The approach
presented here is based on traditional system theory using
the state space approach [20] while capturing the spectral
characteristics of the channel. The main idea in constructing the dynamical models for mobile-to-mobile channels is
to factorize the DPSD into an approximate even transfer
function, and then use a stochastic realization [21] to obtain
a state space representation for the inphase and quadrature
components. The dynamical models, introduced here, are
based on the fundamental assumption that the inphase and
quadrature components of the fading channel are assumed to
be conditionally uncorrelated Gaussian random variables [22],
and thus conditionally independent.
The mobile-to-mobile channel is considered as a dynamical
system for which the input-output map is described by (3).
In order to identify the random process associated with S(f )
in (2) in the form of an SDE, we need to find a transfer
function, H(f ) whose magnitude square equals S(f ), i.e.,
S(f ) = |H(f )|2 . This is√equivalent to S(s) = H(s)H(−s),
where s = i2πf and i = −1. However, since S(f ) in bandlimited, it does not satisfy the Paley and Wiener condition
[23] and therefore is not factorizable. In order to factorize it,
the DPSD has to be first approximated by a rational transfer
function, denoted S̃(f ), and is discussed next.
B. Approximating the Deterministic Mobile-to-Mobile DPSD
A number of rational approximation methods can be used to
approximate the mobile-to-mobile DPSD [24], the choice of
which depends on the complexity and the required accuracy.
In this paper, we consider a numeric approach based on the
Gauss-Newton method for iterative search [24], which is used
to generate a stable, minimum phase, real rational transfer
function, denoted by H̃(s), to identify the best model from
the data of H(2πf ) as
min
a,b
l
2
w(2πfk )H(2πfk ) − H̃(2πfk )
sn
bn−1 sn−1 + · · · + b1 s + b0
+ an−1 sn−1 + · · · + a1 s + a0
S̃(s)
=
H̃(s)
=
and a := {an−1 , · · · , a1 , a0 }, b := {bn−1 , · · · , b1 , b0 },
w(2πf ) is the weight function and l is the number of frequency points. Several variants have been suggested in the
(7)
Fig. 2(c) shows S(f ) and S̃(f ) according to (7) and (8) for
different values of α’s. It can be noticed that this simple
approximation method is less accurate than the Gauss-Newton
method, but is easier to implement. In the next section,
the approximated deterministic DPSD is used to develop
stochastic mobile-to-mobile channel models.
C. Stochastic Mobile-to-Mobile Channel Models
Several stochastic realizations can be used to obtain a
state space representation for the inphase and quadrature
components of mobile-to-mobile channel, the choice of which
depends on the application. The stochastic observable canonical form (OCF) [20][32] is used to realize (6) for the inphase
and quadrature components as
dXI,j (t)
=
AI XI,j (t)dt + BI dWjI (t)
Ij (t)
=
CI XI,j (t) + fjI (t)
dXQ,j (t)
=
AQ XQ,j (t)dt + BQ dWjQ (t)
Qj (t)
=
CQ XQ,j (t) +
fjQ (t)
where
1
n
XI,j (t) = [XI,j
(t), · · · , XI,j
(t)]T
⎡
(6)
K2
s4 + 2ωd2 (1 − 2ζ 2 )s2 + ωd4
K
s2 + 2ζωd s + ωd2
and if the approximate density S̃(f ) coincides with the exact
density S(f ) at f = 0 and f = fmax , then the arbitrary
parameters {ζ, ωd , K} can be computed explicitly as
1
S(0)
2πfmax
ζ =
1− 1−
, ωd = 2
S(fmax )
1 − 2ζ 2
K = ωd2 S(0)
(8)
(5)
k=1
where H̃(s) =
literature, where the weighting function gives less attention
to high frequencies [24]. This approach is chosen since it is
highly accurate; however it requires moderate computational
cost and is sensitive to the initial state.
Fig. 2(a) shows the DPSD, S(f ), and its approximation
S̃(f ) via different orders using the Gauss-Newton method.
The higher the order of S̃(f ) the better the approximation
obtained. It can be seen that approximation with 4th order
transfer function gives a very good approximation. Fig. 2(b)
shows S(f ) and S̃(f ) for different values of α’s via 4th order
even function. It can be noticed that S̃(f ) approximates S(f )
with high accuracy. A higher order model would add more
complexity. Now we consider a special case of (6) where
the coefficients of the approximate DPSD, a and b, can be
computed explicitly with reasonable accuracy. Let
⎢
⎢
⎢
AI = AQ = ⎢
⎢
⎣
1
n
XQ,j (t) = [XQ,j
(t), · · · , XQ,j
(t)]T
⎤
0
1
0
···
0
⎥
0
0
1
···
0
⎥
⎥
..
..
..
..
..
⎥
.
.
.
.
.
⎥
⎦
0
0
0
···
1
−a0 −a1 −a2 · · · −an−1
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OLAMA et al.: STOCHASTIC DIFFERENTIAL EQUATIONS FOR MODELING, ESTIMATION AND IDENTIFICATION OF MOBILE-TO-MOBILE
bn−1
bn−2
..
.
⎥
⎥
⎥
⎥ , CI = CQ = [1, 0, · · · , 0]
⎥
⎦
4.5
where
X(t) =
A(t)
=
B(t)
=
C(t)
=
alpha = 0.25
3.5
S(f)
and XI,j (t), XQ,j (t) are state vectors of the inphase and
quadrature components. Ij (t) and Qj (t) correspond to the
inphase and quadrature components, respectively, {WjI (t)}t≥0
and {WjQ (t)}t≥0 are independent standard Brownian motions, which correspond to the inphase and quadrature
components of the jth path respectively, the parameters
{an−1 , · · · , a0 , bn−1 , · · · , b0 } are obtained from the approximation of the deterministic DPSD, and fjI (t) and fjQ (t) are
arbitrary functions representing the line-of-sight (LOS) of the
inphase and quadrature components respectively, characterizing further dynamic variations in the environment. The LOS
functions can be defined as f I (t) = r0 cos(ω0 t + ϕ0 ) and
f Q (t) = r0 sin(ω0 t + ϕ0 ) where the parameters {r0 , ω0 , ϕ0 }
correspond to the LOS component [22].
Time-domain simulation of mobile-to-mobile channels can
be performed by passing two independent white noise processes through two identical filters, H̃(s) , obtained from the
factorization of the deterministic DPSD, one for the inphase
and the other for the quadrature component, and realized in
their state space form as described in (9) and (10). Fig. 3
shows time domain simulation of the inphase and quadrature
components, and the attenuation coefficient for a mobile-tomobile channel with parameters ν1 = 36 km/hr (10 m/s) and
ν2 = 24 km/hr (6.6 m/s) in which α = 0.66. In Fig. 3 GaussNewton method is used to approximate the deterministic
DPSD with 4th order transfer function. The simulation is
performed using Simulink in Matlab.
As the DPSD varies from one instant to the next, the channel
parameters {an−1 , · · · , a0 , bn−1 , · · · , b0 } also vary in time,
and have to be estimated online from time domain measurements. In Section IV, we propose to recursively estimate the
channel parameters as well as the inphase and quadrature
components directly from received signal measurements, using
the EM algorithm together with the Kalman filter. Without loss
of generality, we consider the case of flat fading, in which the
ad hoc channel has purely a multiplicative effect on the signal
and the multipath components are not resolvable, and can be
considered as a single path [19]. Following the state space
representation in (9) and the band pass representation of the
received signal in (4), the TV fading channel can be represented using a general stochastic state space representation of
the form
dX(t) =
y(t) =
4
(10)
A(t)X(t)dt + B(t) dW (t)
C(t)X(t) + D(t) ν(t)
T
XI (t)T XQ (t)T
AI (t)
0
0
AQ (t)
BI (t)
0
0
BQ (t)
cos(ωc t)CI − sin(ωc t)CQ
(11)
3
2.5
Original S(f)
Appr. with order = 2
Appr. with order = 4
Appr. with order = 6
2
1.5
−50
−40
−30
−20
−10
0
10
Frequency (Hz)
20
30
40
50
(a)
S(w)
Appr. S(w)
alpha = 0.5
0.25
alpha = 0.33
0.2
alpha = 0.25
alpha = 0.2
Magnitude (W)
⎢
⎢
⎢
BI = BQ = ⎢
⎢
⎣ b1
b0
⎤
0.15
0.1
0.05
0
−60
−40
−20
0
20
40
60
Frequency (Hz)
(b)
S(w)
Appr. S(w)
alpha = 0.5
0.25
alpha = 0.33
alpha = 0.25
0.2
Magnitude (W)
⎡
1757
alpha = 0.2
0.15
0.1
0.05
0
−60
−40
−20
0
20
40
60
Frequency (Hz)
(c)
Fig. 2. DPSD, S(f ), and its approximation, S̃(f ), (a) using the GaussNewton method for different orders of S̃(f ) (b) using the Gauss-Newton
method via 4th order function for different values of α’s (c) using the special
case approximation method via 4th order function for different values of α’s.
Note that f1 = 10 Hz.
D(t) =
ν(t)
=
cos(ωc t) − sin(ωc t)
(12)
T
T
νI (t) νQ (t) , dW (t) = dW I (t) dW Q (t)
In this case, y(t) represents the received signal measurements,
X(t) is the state variable of the inphase and quadrature
components, and ν(t) is a continuous-time measurement noise.
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1
1
0
−1
−2
0
−1
0
0.5
1
1.5
−2
2
Attenuation Coefficient
2
0.5
1
1.5
2
IV. M OBILE - TO -M OBILE C HANNEL E STIMATION V IA THE
EM A LGORITHM AND K ALMAN F ILTERING
Attenuation Coefficient
r(t) [dB]
0
1.5
1
−10
−20
0.5
0
0
10
2.5
r(t)
are
obtained
from
{an−1 , · · · , a0 , bn−1 , · · · , b0 }
approximating the deterministic DPSD. However, in reality
one can not have access to the DPSD online and at all
times during the estimation process. In the next section, we
estimate the channel parameters, the inphase and quadrature
components directly from received signal measurements.
Quadrature
2
Q(t)
I(t)
Inphase
2
0
0.5
1
Time (sec)
1.5
2
−30
0
0.5
1
Time (sec)
1.5
2
This section describes the procedure employed to estimate
the mobile-to-mobile channel model parameters and states
associated with (11), using the EM algorithm [16][26] together
with Kalman filtering [17]. We consider the discrete-time
(sampled) version of (11) given as [21]
xk+1
yk
Fig. 3.
Inphase and
quadrature components {I(t), Q(t)}, and the
attenuation coefficient I 2 (t) + Q2 (t), for a mobile-to-mobile channel with
α = 0.66.
= Ak xk + Bk wk
= Ck xk + Dk vk
(18)
where subscript k belongs to the index set {0, 1, · · · }, xk ∈
R2n is a discrete-time state vector, yk ∈ R1 is a discretetime measurement vector, wk ∈ R2 is a discrete-time state
D. Solution to the Stochastic State Space Model
noise, vk ∈ R2 is a discrete-time measurement noise, Ak =
The stochastic TV state space model described in (11) and Φ(tk+1 , tk ) where Φ(t, t0 ) is the fundamental matrix of
t
(11), Bk2 = tkk+1 Φ(tk+1 , s)B(s)B(s)T ΦT (tk+1 , s) ds,
(12) has a solution given by [21][32]
t
yk = y(tk ), Ck = C(tk ), Dk = D(tk ), and vk = v(tk ). The
ΦL (t, u)BL (u)dWL (u) (13) noise processes wk and vk are assumed to be independent zero
XL (t) = ΦL (t, t0 )XL (t0 ) +
t0
mean and unit variance Gaussian processes.
The system parameters θk = {Ak , Bk , Ck , Dk } and states
where L = I or Q, and ΦL (t, t0 ) is the fundamental matrix,
which satisfies Φ̇L (t, t0 ) = AL (t)ΦL (t, t0 ) and ΦL (t0 , t0 ) are unknown and can be estimated through received signal
measurement data, YN = {y1 , y2 , · · · , yN }. The parameters
is the identity matrix.
A simple computation shows that the mean of XL (t) is given are identified using a filter-based EM algorithm and the
channel states are estimated using the Kalman filter. The
by [21]
Kalman filter is introduced next.
E[XL (t)] = ΦL (t, t0 ) E[XL (t0 )]
(14)
and the covariance matrix of XL (t) is given by
ΣL (t) = ΦL (t, t0 )V ar XL (t0 ) ΦTL (t, t0 )
(15)
t
T
+
ΦL (t, u)BL (u)BL
(u)ΦTL (t, u) du
t0
Differentiating (15) shows that ΣL (t) satisfies the Riccati
equation
Σ̇L (t) = A(t)ΣL (t) + ΣL (t)AT (t) + B(t)B T (t)
(16)
For the time invariant case, AL (t) = AL and BL (t) = BL ,
(13), (14), and (15) simplify to
t
eAL (t−u) BL dWL (u)
XL (t) = eAL (t−t0 ) XL (t0 ) +
t0
E[XL (t)] =
ΣL (t) =
+
eAL (t−t0 ) E[XL (t0 )]
T
eAL (t−t0 ) V ar XL (t0 ) eAL (t−t0 )
t
T AT
eAL (t−u) BL BL
e L (t−u) du
(17)
t0
It can be seen in (14) and (15) that the mean and variance
of the inphase and quadrature components are functions of
time. Note that the statistics of the inphase and quadrature
components, and therefore the statistics of the mobile-tomobile channel, are time varying.
As
described
above,
the
channel
parameters
A. Channel State Estimation: The Kalman Filter
The Kalman filter estimates the channel states xk for given
system parameter θk and measurements Yk . It is described by
the following equations [17][32]
T
−2
x̂k/k = Ak−1 x̂k−1/k−1 + Pk/k Ck−1
Dk−1
(yk − Ck−1 Ak−1
x̂k−1/k−1 ), x̂k/k−1 = Ak−1 x̂k−1/k−1 , x̂0/0 = m0
(19)
where k = 1, 2, · · · , N , and Pk/k is given by
−1
P̄k/k
=
−1
−2
Pk−1/k−1
+ ATk−1 Bk−1
Ak−1
−1
Pk/k
=
T
−2
−2
−2
−2
Ck−1
Dk−1
Ck−1 + Bk−1
− Bk−1
P̄k/k ATk−1 Bk−1
Pk/k−1
=
2
Ak−1 Pk−1/k−1 ATk−1 + Bk−1
(20)
2
T
2
T
where Bk−1
= Bk−1 Bk−1
and Dk−1
= Dk−1 Dk−1
. The
channel parameters θk = {Ak , Bk , Ck , Dk } are estimated
using the EM algorithm which is introduced next.
B. Channel Parameter Identification: The EM Algorithm
The filter-based EM algorithm uses a bank of Kalman filters
to yield a maximum likelihood (ML) parameter estimate of
the Gaussian state space model [26]. The EM algorithm is an
iterative numerical algorithm for computing the ML estimate.
Each iteration consists of two steps: the expectation and the
maximization steps [16][26][27]. The filtered expectation step
only uses filters for the first and second order statistics. The
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OLAMA et al.: STOCHASTIC DIFFERENTIAL EQUATIONS FOR MODELING, ESTIMATION AND IDENTIFICATION OF MOBILE-TO-MOBILE
memory cost is modest and the filters are decoupled and hence
easy to implement in parallel on a multi-processor system [26].
The algorithm yields parameter estimates with nondecreasing
values of the likelihood function, and converges under mild
assumptions [27].
Let θk = {Ak , Bk , Ck , Dk } denotes the system parameters in (18) and {Pθk ; θk ∈ Θ} denotes a family of probability
measures induced by the system parameters θk , where Θ is
the parameter space R2n×2n × R2n×2 × R2n × R2 in which
θk lives. The EM algorithm computes the ML estimate of
the system parameters θk , given the data Yk . The expectation
step evaluates the conditional expectation of the log-likelihood
function given the complete data as
dPθk
(21)
Λ(θk , θ̂k ) = Eθ̂k log
| Yk
dPθ̂k
where θ̂k denotes the estimated system parameters at time step
k. The maximization step finds
θ̂k+1 ∈ arg max Λ(θk , θ̂k )
(22)
θk ∈Θ
The expectation and maximization steps are repeated until the
sequence of model parameters converge to the real parameters.
The EM algorithm is described by [16][26]
Âk
=
E
k
xi xTi−1
| Yk
!
i=1
B̂k2
E
k
xi xTi
| Yk
"−1
(4)
Lk
+
Ĉk
=
=
1 E
(xi − Ai−1 xi−1 )(xi − Ai−1 xi−1 )T |Yk
k
i=1
i=1
D̂k2
=
+
1 T
(yi yiT ) − (yi xTi )Ci−1
− Ci−1 (yi xTi )T
E
k
i=1
T
|Yk
Ci−1 (xi xTi )Ci−1
where E(·) denotes the expectation operator.
The system (23) gives the EM parameter estimates at each
iteration for the model (18). Furthermore, since Λ(θk , θ̂k ) is
continuous in both θk and θ̂k the EM algorithm converges to a
stationary point in the likelihood surface [26][27]. The system
parameters {Âk , B̂k2 , Ĉk , D̂k2 } can be computed from the
conditional expectations [16]
(1)
Lk
= E
k
#
xTi Qxi
|Yk
$
(2)
Lk
(3)
Lk
= E
= E
i=1
k
i=1
xTi−1 Qxi−1
(26)
=
1
1
(1)
(1)
− T r(Nk Pk|k ) −
T r(Ni−1 P̄i|i )
2
2 i=1
−
1 −1 (1)
−1
ri + 2xTi|i−1 Pi|i−1
−2xTi|i Pi|i
2 i=1
k
(1)
(1)
−2
ri|i−1 − xTi|i Ni xi|i + xTi|i−1 Bi−1
Ai−1 P̄i|i
(1)
−2
Ni−1 P̄i|i ATi−1 Bi−1
xi|i−1
where T r(·) denotes the matrix trace. In (26), ri and
(1)
Ni satisfy the following recursions
⎧ (1) (1)
−2
T
ri = Ai−1 − Pi|i Ci−1
Di−1
Ci−1 Ai−1 ri−1
⎪
⎪
⎪
(1)
⎪
−2
T
⎪
⎨ +2P
i/i Qxi|i−1 −Pi|i Ni Pi|i Ci−1 Di−1
yi − Ci−1 xi|i−1
(27)
⎪
(1)
(1)
⎪
⎪ ri|i−1
= Ai−1 ri
⎪
⎪
⎩ (1)
r = 02n×1
% 0
(1)
(1)
−2
−2
Ni = Bi−1
Ai−1 P̄i|i Ni−1 P̄i|i ATi−1 Bi−1
− 2Q
(1)
N0 = 02n×2n
(2)
2) Filter estimate of Lk ,
(2)
Lk
= E
k
#
xTi−1 Qxi−1 |Yk
$
(28)
i=1
i=1
k
#
$
(1)
k
=
xTi Qxi |Yk
k
i=1
k
E
k
#
i=1
1 E
(xi xTi ) − Ai−1 (xi xTi−1 )T − (xi xTi−1 )ATi−1
k
i=1
Ai−1 (xi−1 xTi−1 )ATi−1 |Yk
!
"
k
k
−1
T
T
E
E
yi x i | Y k
xi xi |Yk
(23)
1 (yi − Ci−1 xi )(yi − Ci−1 xi )T |Yk
E
k
i=1
xTi Syi + yiT S T xi |Yk
where Rij = {ei eTj /2; i, j = 1, 2}. The other terms in
(23) can be computed similarly from (24). The conditional
(1)
(2)
(3)
(4)
expectations {Lk , Lk , Lk , Lk } are estimated from measurements Yk as follows [26]
(1)
1) Filter estimate of Lk ,
k
=
k
where Q, R and S are given by
%
&
ei eTj + ej eTi
Q =
2
&
%
T
ei ej
R =
2
e
i
; i, j = 1, 2, · · · , 2n
(25)
S =
2
in which ei is the unit vector in the Euclidean space; that
is ei = 1 in the ith position, and 0 elsewhere. For instance,
consider the case 2n = 2, then
"
!
k
(3)
(3)
Lk (R11 ) Lk (R21 )
T
xi xi−1 |Yk =
E
(3)
(3)
Lk (R12 ) Lk (R22 )
i=1
k
=
E
i=1
(1)
Lk
i=1
=
1759
|Yk
$
T
xi Rxi−1 + xTi−1 RT xi |Yk
(24)
k
#
$
#
$
= Eθ xT0 Qx0 |Yk +Eθ
xTi Qxi |Yk
#
$
− Eθ xTk Qxk |Yk
(2)
i=1
(1)
Therefore, Lk can be obtained from Lk .
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
(3)
3) Filter estimate of Lk ,
(3)
=
Lk
E
k
T
xi Rxi−1 + xTi−1 RT xi |Yk
(29)
i=1
=
1 (3)
1 (3)
T r Ni−1 P̄i|i
− T r Nk Pk|k −
2
2 i=1
−
1 (3)
−1 (3)
−1
−2xTi|i Pi|i
ri + 2xTi|i−1 Pi|i−1
ri|i−1
2 i=1
k
k
−
(3)
−2
xTi|i Ni xi|i + xTi|i−1 Bi−1
Ai−1 P̄i|i
(3)
−2
T
Ni−1 P̄i|i Ai−1 Bi xi|i−1
(3)
(3)
In this case, ri and Ni satisfy the following recursions
⎧ (3) (3)
−2
T
ri = Ai−1 − Pi|i Ci−1
Di−1
Ci−1 Ai−1 ri−1
⎪
⎪
⎪
⎪
−2
T
⎪ −Pi|i Ni(3) Pi|i Ci−1
yi − Ci−1 xi|i−1
Di−1
⎪
⎪
⎨ + 2P R + 2P B −2 A P̄ RT A
i−1
i|i
i|i i−1 i−1 i|i
x
⎪
i−1|i−1
⎪
⎪
(3)
(3)
⎪
⎪
ri|i−1 = Ai−1 ri
⎪
⎪
⎩ (3)
r = 02n×2n
⎧ 0
(3)
(3)
−2
−2
⎪
⎨ Ni = Bi−1 Ai−1 P̄i|i Ni−1 P̄i|i ATi−1 Bi−1
−2
−2
T
−2RP̄i|i Ai−1 Bi−1 − 2Bi−1 Ai−1 P̄i|i RT
⎪
⎩ N (3) = 0
2n×2n
0
(4)
4) Filter Estimate of Lk ,
(4)
Lk
= E
k
T
xi Syi + yiT S T xi |Yk
(30)
i=1
=
k
T −1 (4)
(4)
−1
xi|i Pi|i ri − xTi|i−1 Pi|i−1
ri|i−1
i=1
(4)
ri
where
satisfies the following recursions (see Eq. 31)
(i)
Using the filters for Lk (i = 1, 2, 3, 4) and the
Kalman filter described earlier, the system parameters θk =
{Ak , Bk , Ck , Dk } can be estimated through the EM algorithm
described in (23). Experimental results that show the viability
of the above algorithm in estimating the channel parameters as
well as the inphase and quadrature components are discussed
in the next section.
V. E XPERIMENTAL S ETUP AND L INK P ERFORMANCE
In this section, we carry out an experiment to measure the
received signal strength of moving sensors in a wireless sensor
platform. Then the EM algorithm and the Kalman filter are
used to estimate the mobile-to-mobile channel parameters,
the inphase and quadrature components from the measured
received signal. The wireless sensors used in our experiment
are Crossbow’s TelosB sensor nodes, which have the following
specifications: IEEE 802.15.4 compliant, data rate is 250
kbps, carrier frequency is 2.4 GHz, and has USB interface.
These sensors are implemented with a Chipcon CC2420 RF
transceiver chip which provides a built-in received signal
strength indicator (RSSI) [18].
Our experimental setup consists of two moving transceivers
(sensors 1 and 2) and one passive receiver (sensor 3) connected to a workstation. At each time step, sensors 1 and 2
broadcast a packet containing a source address and the RSSI
of the most recently received packet from the other sensor.
Sensor 3 never transmits; rather, it forwards packets from
sensors 1 and 2 to a workstation for analysis. The mobileto-mobile channel between sensor 1 and 2 is time varying
since both sensors move with different (variable) velocities and
directions. Indoor and outdoor environments are considered. In
the estimation and identification process, a 4th order mobileto-mobile channel model as described in (18) is considered.
The system parameters θk = {Ak , Bk , Ck , Dk } can then be
represented as
⎡
⎡
⎤
⎤
0 1 0 0
b1 δ12
⎢ a1 a2 0 0 ⎥
⎢ b2 δ22 ⎥
⎢
⎥
⎥
Ak = ⎢
⎣ 0 0 0 1 ⎦ , Bk = ⎣ δ31 b3 ⎦
δ41 b4
0 0 a3 a4
Ck = [cos(ωc tk ) 0 − sin(ωc tk ) 0]
(32)
Dk = [d1 cos(ωc tk ) − d2 sin(ωc tk )]
Note that Bk in (32) is not block Diagonal as in (12). We
have included several variables, denoted δij , that have very
small values (close to zero) for the other entries to make Bk2
nonsingular which is required for the estimation algorithm.
The estimation includes the channel parameters, inphase
and quadrature components, and the received signal, which
are then compared to the ones obtained from measurement
data. It is assumed that the received signal measurement
data are corrupted by white noise sequences. Fig. 4 and 5
show respectively indoor and outdoor measured and estimated
received signals using the EM algorithm together with Kalman
filter for 500 sampled data taken from measurements between
sensor 1 and 2. At a certain time instant (k = 400), indoor
system parameters are estimated as
⎡
⎤
0
1
0
0
⎢ −0.3066 0.0016
⎥
0
0
⎥(33)
 = ⎢
⎣
⎦
0
0
0
1
0
0
−0.5940 0.0059
⎡
0.8531
−0.0369 −0.0284
⎢
−0.3962
0.0649
0.0032
2
B̂ = ⎢
⎣
−0.0742
0.0074
0.0753
3.1804 · 10−4 −0.0532 0.0064
⎤
2.8531 · 10−4
−0.0193 ⎥
⎥
(34)
⎦
0.0021
0.0853
Ĉ
=
[0.958 0 − 0.2868 0] , D̂2 = [2.0262] ,
while outdoor system parameters are estimated as
⎡
0
1
0
0
⎢ −0.7151 0.0037
0
0
 = ⎢
⎣
0
0
0
1
0
0
−0.1500 0.0515
⎡
0.5824
−0.0735 −0.0735
⎢
−0.7452
0.0846
0.0083
B̂ 2 = ⎢
⎣
−0.0864
0.0365
0.0454
1.8643 · 10−4 −0.0643 0.0820
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⎤
⎥
⎥(35)
⎦
OLAMA et al.: STOCHASTIC DIFFERENTIAL EQUATIONS FOR MODELING, ESTIMATION AND IDENTIFICATION OF MOBILE-TO-MOBILE
1761
⎧ (4) (4)
−2
T
⎪
⎨ ri = Ai−1 − Pi|i Ci−1 Di−1 Ci−1 Ai−1 ri−1 + 2Pi|i Syi
(4)
(4)
ri/i−1 = Ai−1 ri
⎪
⎩ (4)
r0 = 02n×1
0
(31)
0
−20
Est.
Meas.
−20
−10
−10
−40
−20
−60
Est.
Meas.
−40
−60
0
10
20
30
40
50
Received Signal (dB)
Received Signal (dB)
−20
−30
−40
−50
−80
−30
0
10
20
100
150
200
30
40
50
250
300
Samples
350
−40
−50
−60
−70
−60
−80
−70
0
50
100
150
200
250
300
Samples
350
400
450
−90
0
500
Fig. 4. Indoor measured and estimated received signals from sensor 2 by
using a 4th order ad hoc channel model.
450
500
30
Indoor
Outdoor
25
20
MSE
= [0.624 0 − 0.7815 0] , D̂2 = [1.1725].
From Fig. 4 and 5, it can be noticed that the received signals
from indoor and outdoor environments have been estimated
with very high accuracy. It takes a few iterations (about
5 iterations) for the estimation algorithm to converge. The
root mean square errors (RMSE) for indoor and outdoor
environments are shown in Fig. 6. It can be seen that indoor
RMSE is higher than the one for outdoor because of reflections
from walls and objects in indoor environment.
Now, we want to compare the performance of the stochastic mobile-to-mobile link in (11) with the cellular one. We
consider BPSK is the modulation technique and the carrier
frequency is fc = 900 MHz. We test 10000 frames of P = 100
bits each. We assume mobile nodes are vehicles, with the
constraint that the average speed over the mobile nodes is 30
km/hr. This implies ν1 + ν2 = 60 km/hr, thus for a mobile-tomobile link with α = 0 we get ν1 = 60 km/hr and ν2 = 0. The
cellular case is defined as the scenario where a link connects a
mobile node with speed 30 km/hr to a permanently stationary
node, which is the base station. Thus, there is only one mobile
node, and the constraint is satisfied. We consider the non-lineof-sight case (fI = fQ = 0), which represents an environment
with large obstructions.
Fig. 7 shows the attenuation coefficient, r(t) =
I 2 (t) + Q2 (t), for both the cellular case and the worstcase mobile-to-mobile case (α = 0). It can be observed that
a mobile-to-mobile link suffers from faster fading by noting
the higher frequency components in the worst-case mobile-to-
400
Fig. 5. Outdoor measured and estimated received signals from sensor 2 by
using a 4th order ad hoc channel model.
⎤
1.5395 · 10−4
−0.0375 ⎥
⎥
⎦
0.0264
0.0753
Ĉ
50
15
10
5
0
0
50
100
150
200
250
300
Samples
350
400
450
500
Fig. 6. Received signal estimates RMSE for indoor and outdoor environments
using the EM algorithm together with the Kalman filter.
mobile link. Also it can be noticed that deep fading (envelope
less than −12 dB) on the mobile-to-mobile link occurs more
frequently and less bursty (48% of the time for the mobileto-mobile link and 32% for the cellular link). Therefore, the
increased Doppler spread due to double mobility tends to
smear the errors out, causing higher frame error rates.
Consider the data rate given by Rb = P/Tc = 5 Kbps
which is chosen such that the coherence time Tc equals
the time it takes to send exactly one frame of length P
bits, a condition where variation in Doppler spread greatly
impacts the frame error rate (FER). Fig. 8 shows the link
performance for 10000 frames of 100 bits each. It is clear
that the mobile-to-mobile link is worse than the cellular link,
but the performance gap decreases as α −→ 1. This agrees
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 4, APRIL 2009
0
r(t) [dB]
−10
−20
32%
Cellular
−30
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
r(t) [dB]
0
−10
−20
48%
ACKNOWLEDGEMENT
Ad hoc
−30
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
Fig. 7. Rayleigh attenuation coefficient for cellular link and worst-case
mobile-to-mobile link.
0
10
FER
alpha = 0
alpha = 0.5
alpha = 1
Cellular
−1
10
−2
10
15
16
17
18
19
20
Eb/N0 (dB)
quadrature components are estimated recursively with high
accuracy from received signal measurements. The proposed
algorithm consists of filtering based on the Kalman filter
to remove noise from data, and identification based on the
filter-based EM algorithm to determine the parameters of
the model which best describe the measurements. Indoor
and outdoor experimental setups are considered. Experimental
results indicate that the measured data can be regenerated
through a simple 4th order discrete-time stochastic differential
equation with excellent accuracy, and therefore demonstrate
the validity of the method.
21
22
23
24
25
Fig. 8. FER results for Rayleigh mobile-to-mobile link for different α’s and
compared with cellular link.
with the main conclusion of [7], that an increase in degree
of double mobility mitigates fading by lowering the Doppler
spread. The gain in performance is nonlinear with α, as the
majority of gain is from α = 0 to α = 0.5. Intuitively, it
makes sense that link performance improves as the degree
of double mobility increases, since mobility in the network
becomes distributed uniformly over the nodes in a kind of
equilibrium.
VI. C ONCLUSION
Stochastic models based on SDEs for mobile-to-mobile
wireless channels have been derived. These models take into
account the statistical and time variations in mobile-to-mobile
communication environments. The dynamics are captured by
a stochastic state space model, whose parameters are determined by approximating the deterministic DPSD. Inphase and
quadrature components of the channel and their statistics are
derived from the proposed model. The state space models have
been used to verify the effect of fading on a transmitted signal
in ad hoc networks. The channel parameters, the inphase and
The authors would like to thank Dr. Y. Li, Mr. T. Kuruganti,
and Mr. T. Goodspeed for providing the experimental data and
for their helpful comments and discussions.
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Mohammed M. Olama is currently a Research
Associate in the Computational Sciences and Engineering Division at Oak Ridge National Laboratory.
He received his Ph.D. degree from the Electrical and
Computer Engineering Department at the University
of Tennessee, Knoxville, in 2007, and his B.S. and
M.S. (with first class honors) degrees in electrical
engineering from the University of Jordan, in 1998
and 2001, respectively. From 2001 to 2003, he
completed 33 credit hours towards his Ph.D. degree
in the Applied Science Department, University of
Arkansas at Little Rock (UALR). He held an internship position in Oak
Ridge National Laboratory in the summer of 2007. He received the Scholarly
Activities Research Incentive Fund (SARIF) Summer Graduate Research
Assistantship for two consecutive years (2006 and 2007). From 1999 to
2001, he served as a full-time control engineer at the National Electric Power
Company (NEPCO) in Amman, Jordan. Dr. Olama received the best regular
paper award in the 1st Mediterranean Conference on Intelligent Systems
and Automation (CISA) in 2008. He also received a 2007 Significant Event
Award from the Computational Sciences and Engineering Division, Oak Ridge
National Laboratory. He is a member of the Phi Kappa Phi honor society. His
research interests include modeling, power control and location services for
wireless networks, estimation and identification, control over communication
networks, wide area measurement systems (WAMS), SCADA systems, and
discrete event systems.
1763
Seddik M. Djouadi received his Ph.D. degree from
McGill University, his M.Sc. degree from University
of Montreal, both in Montreal, his B.S. (with first
class honors) from Ecole Nationale Polytechnique,
Algiers, all in electrical engineering, respectively,
in 1999, 1992, and 1989. He is currently an Assistant Professor in the Electrical Engineering and
Computer Science Department at the University of
Tennessee, Knoxville. He was an assistant Professor
in University of Arkansas at Little Rock, and held
postdoctoral positions in the Air Force Research
Laboratory and Georgia Institute of Technology, where he was also a Design
Engineer with American Flywheel Systems Inc. He received five US Air
Force Summer Faculty Fellowships, and an Oak Ridge National Laboratory
Summer Fellowship.
Dr. Djouadi is a member of IEEE, the American Mathematical Society and
la Societe Mathematique de France. He received the Best paper award in the
1st Conference on Intelligent Systems and Automation 2008, the Ralph E.
Powe Junior Faculty Enhancement Award in 2005, the Tibbet Award with
AFS Inc. in 1999 and the American Control Conference Best Student Paper
Certificate (best five in competition) in 1998. He was selected by Automatica
as an outstanding reviewer for 2003-2004 and 2007-2007. His research interests include filtering and control of systems under communication constraints,
modeling and control of wireless networks, model reduction and control of
fluid flows, active vision and identification.
Charalambos D. Charalambous (SM’2005) received the Electrical Engineering B.S. degree in
1987, the M.E. degree in 1988, and the Ph.D. in
1992, all from Department of Electrical Engineering
from Old Dominion University, Virginia, USA.
In 2003 he joined the Department of Electrical
and Computer Engineering, University of Cyprus,
where he is currently Associate Professor and Acting
Dean of the School of Engineering. He was an
Associate Professor at University of Ottawa, School
of Information Technology and Engineering from
1999 to 2003. He has served on the faculty of McGill University, Department
of Electrical and Computer Engineering, as a non-tenure faculty member, from
1995 to 1999. From 1993 to 1995 he was a Post-Doctoral Fellow at Idaho
State University, Engineering Department. His research group ICCCSystemS,
Information, Communication and Control of Complex Systems is interested in
theoretical and technological developments concerning large scale distributed
communication and control systems and networks in science and engineering.
These include theory and applications of stochastic processes and systems
subject to uncertainty, communication and control systems and networks, large
deviations, information theory, robustness and their connections to statistical
mechanics.
Dr. Charalambous is currently an associate editor of the IEEE C OMMUNI CATIONS L ETTERS , and from 2002 to 2004 he served as an Associate Editor
of the IEEE T RANSACTIONS ON AUTOMATIC C ONTROL. He was a member
of the Canadian Centers of Excellence through MITACS (the mathematics of
information technology and complex systems), from 1998 to 2001. In 2001
he received the Premier’s Research Excellence Award of the Ontario Province
of Canada.
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