Proceedings of the European Control Conference 2007
Kos, Greece, July 2-5, 2007
TuD01.6
Estimation and Identification of Time-Varying Long-Term Fading
Wireless Channels with Application to Power Control
Mohammed M. Olama, Kiran K. Jaladhi, Seddik M. Djouadi, and Charalambos D. Charalambous
Abstract—This paper is concerned with modeling of timevarying wireless fading channels, parameter estimation,
identification, and optimal power control from received signal
measurements. Wireless channels are represented by stochastic
differential equations, which parameters and state variables
are estimated using Expectation Maximization and Kalman
filtering, respectively. The latter are carried out solely from
received signal measurements. These algorithms estimate the
channel path-loss and identify the channel parameters
recursively. An optimal power control algorithm based on the
estimated parameters and channel states is proposed.
Numerical results are presented to test the efficiency of the
proposed channel estimation and identification algorithms, and
to indicate that a significant gain in performance can be
achieved using the proposed approach.
T
I. INTRODUCTION
HIS paper is concerned with the development of timevarying (TV) long-term fading (LTF) wireless channel
models based on system identification and estimation
algorithms to extract various parameters of the LTF channel
using received signal measurements. TV wireless channel
models capture both the space and time variations of
wireless systems, which are due to the relative mobility of
the receiver and/or transmitter and scatterers [1]-[3]. In the
TV models the statistics of channel are time-varying. This
contrasts with the majority of published work that mainly
deals with time-invariant (static) random models or simple
free space model, where the channel statistics do not depend
on time [4]-[11]. In time-invariant models, channel
parameters are random but do not depend on time, and
remain constant throughout the observation and estimation
phase. This contrasts with TV models, where the channel
dynamics become TV random (stochastic) processes [1]-[3].
The proposed model is important 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.
The TV LTF channel model is introduced in [2], [3] and
represented by stochastic differential equations (SDEs). We
propose to estimate the TV power path-loss of the LTF
{Mohammed M. Olama, Kiran K. Jaladhi, Seddik M. Djouadi},
Department of Electrical and Computer Engineering, University of
Tennessee, 1508 Middle Dr., Knoxville, TN 37996, USA (E-mail:
molama@utk.edu, kjaladhi@utk.edu, djouadi@ece.utk.edu).
Charalambos D. Charalambous, Department of Electrical and Computer
Engineering, University of Cyprus, 75, Kallipoleos Street, P.O.Box 20537,
1678 Nicosia, Cyprus (E-mail: chadcha@ucy.ac.cy).
ISBN: 978-960-89028-5-5
channel and its parameters from received signal
measurements, which are usually available or easy to obtain
in any wireless network. The Expectation Maximization
(EM) algorithm and Kalman filtering are employed in the
identification and estimation processes. Numerical results
are provided to determine the performance of the proposed
estimation algorithm.
The developed TV LTF channel model from received
signal level measurements is useful in most wireless
applications. It is used in developing an optimal power
control algorithm (PCA), which is based on the estimated
channel parameters from received signal measurements. The
benefits of power minimization are not only increased
battery life, but also increased overall network capacity. The
power allocation problem has been studied extensively as an
eigenvalue problem for non-negative matrices [7], [8],
resulting in iterative PCAs that converge each user’s power
to the minimum power [9], [10], and as optimization-based
approaches [11]. Much of this previous work deals with
static time-invariant channel models.
The proposed PCA is based on predictable power control
strategies (PPCS) that were first introduced in [1]. PPCS
simply means updating the transmitted powers at discrete
times and maintaining them fixed until the next power
update begins. The PPCS algorithm is proven to be
effectively applicable to such dynamical models for an
optimal power control (PC). A distributed version of this
algorithm is derived along the lines of [9] and [10], albeit
based on the estimated model. The latter helps in allowing
autonomous execution at the node or link level, requiring
minimal usage of network communication resources for
control signaling.
The paper is organized as follows. In Section II, the TV
LTF mathematical channel model is introduced. In Section
III, the EM algorithm together with the Kalman filter, to
estimate the channel parameters as well as the channel
power path-loss from received signal measurements, is
developed. In Section IV, a PCA based on the proposed LTF
channel model and the estimation algorithms is discussed. In
Section V, numerical results are presented. Finally, Section
VI provides the conclusion.
II. TV LTF WIRELESS CHANNEL MATHEMATICAL MODEL
Wireless communication channels are subject to several
performance degradation such as time spread (multipath),
Doppler spread (time variations), path-loss, and interference.
In addition to the free space power path-loss, wireless
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TuD01.6
channels suffer from short-term fading (STF) due to
multipath, and LTF due to shadowing depending on the
geographical area. In suburban areas, which are populated
with less obstacles like vehicles, buildings, mountains and
so forth, its communication signal undergoes phenomenal
LTF (lognormal shadowing) [6].
In this section, before we introduce the TV LTF channel
model that captures both time and space variations, we first
summarize the traditional lognormal shadowing model,
which serves as a basis in the development of the subsequent
TV model. The traditional time-invariant power loss (PL) in
dB for a given path is given by [6]:
⎛d ⎞
PL(d )[dB] := PL(d 0 )[dB] + 10α log ⎜ ⎟ + Z , d ≥ d 0
⎝ d0 ⎠
(1)
where PL(d 0 ) is the average PL in dB at a reference
distance d0 from the transmitter, the distance d corresponds
to the transmitter-receiver separation distance, α is the
path-loss exponent which depends on the propagating
medium, and Z is a zero-mean Gaussian distributed random
variable, which represents the variability of PL due to
numerous reflections and possibly any other uncertainty of
the propagating environment from one observation instant to
another.
In the traditional models the PL in (1) and its statistics
does not depend on time, therefore these models are treated
as static (time-invariant). They do not take into account the
variations in the propagation environment and any possible
relative motion between the transmitter and the receiver. In
addition, static models do not consider the correlation
properties of the PL in space and at different observation
times. Indeed, such correlation exists and one way to model
them is through stochastic processes, which obey specific
type of SDEs.
In TV case, the random PL in (1) is relaxed to become a
random process, denoted by { X (t ,τ )}t ≥ 0,τ ≥τ , which is a
0
function of both time t and space represented by the timedelay τ, where τ = d/c, d is the path length, c is the speed of
light, τ0 = d0/c and d0 is the reference distance. The signal
attenuation coefficient is defined by S (t ) e kX ( t ,τ ) , where
k = − ln(10) / 20 [6].
The TV PL { X (t ,τ )}t ≥ 0,τ ≥τ can be generated by a mean0
reverting version of a general linear SDE given by [3]:
dX (t ,τ ) = β (t ,τ ) ( (γ (t ,τ ) − X (t ,τ ) ) dt + δ (t ,τ )dW (t ),
X (t0 ,τ ) = N ( PL(d )[dB ]; σ t20 )
where
{W ( t )}
t ≥0
average time-varying PL at distance d from transmitter,
which corresponds to PL(d )[dB] at d indexed by t. This
model tracks and converges to this value as time progresses.
The instantaneous drift β ( t ,τ ) ( γ ( t ,τ ) − X (t ,τ ) ) represents
the effect of pulling the process towards γ ( t ,τ ) , while
β ( t ,τ ) represents the speed of adjustment towards this
value. Finally, δ ( t ,τ ) controls the instantaneous variance or
volatility of the process for the instantaneous drift.
In order to allow the TV LTF channel model in (2) to
capture the space and time variations in a wireless link,
consider the case of a single transmitter and a single receiver
shown in Figure 1 and define the average TV PL at any
location as:
γ (t ,τ ) = PL(d (t ))[dB] = PL(d0 )[dB] + 10α log
d (t )
d0
(3)
where d (t ) ≥ d0 and d (t ) is given by:
d (t ) = d 2 + (υ t ) 2 + 2dtυ cos θ
If
the
random
processes
{β ( t ,τ ) , γ ( t ,τ ) , δ ( t ,τ )}
t ≥0
(4)
{θ ( t ,τ )}
in
t ≥0
are measurable and bounded,
then (2) has a unique solution for every X (t0 ,τ ) given by:
X (t ,τ ) = e − β ([t ,t0 ],τ ) [ X (t0 ,τ )
t
∫
+ e β ([u ,t0 ],τ ) ( β (u ,τ )γ (u ,τ ) du + δ (u ,τ ) dW (u ) )]
(5)
t0
t
∫
where β ([t , t0 ],τ ) β (u,τ )du . This model captures the
t0
spatio-temporal variations of the propagation environment
as the random parameters {θ ( t ,τ )}t ≥ 0 can be used to model
the TV characteristics of the LTF channel.
The received signal, y ( t ) , at any time t can be expressed as:
y (t ) = s (t ) S (t ) + v (t )
(6)
where s ( t ) is the information signal, v ( t ) is the channel
disturbance or noise at the receiver, and S ( t ) is the signal
attenuation coefficient defined earlier.
(2)
Rx
d
θ
is the standard Brownian motion (zero
d(t)
drift, unit variance) which is assumed to be independent of
X ( t0 ,τ ) , N ( µ ; κ ) denotes a Gaussian random variable
with mean µ and variance κ , and PL (d )[ dB ] is the
average PL in dB. The parameter γ ( t , τ ) models the
Tx
G
v
Fig.1. A transmitter at an initial distance d from a receiver moves with
velocity v and in the direction given by θ with respect to the transmitterreceiver axis.
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TuD01.6
The general spatio-temporal lognormal model in (2) and
(6) can be realized by a stochastic state space representation
as:
X ( t ,τ ) = A ( t ,τ ) X ( t ,τ ) + B ( t ,τ ) w ( t )
given system parameter θ t = { At , Bt , Dt } and measurements
Yt . It is described by the following equations [13]:
xˆt |t = At xˆt −1|t −1 + Pt |t CtT Dt−2 ( yt − Ct At xˆt −1|t −1 )
(7)
y ( t ) = s ( t ) e kX (t ,τ ) + D ( t ) v ( t )
xˆt |t −1 = At xˆt −1|t −1
where A ( t ,τ ) = − β ( t ,τ ) , B ( t ,τ ) = ⎡⎣δ ( t ,τ ) β ( t ,τ ) γ ( t ,τ ) ⎤⎦
Ct = st
and w ( t ) = [ dW (t ) 1] .
T
The above system parameters and state variable values are
estimated from received signal measurements, which are
usually available or easy to obtain in any wireless network.
The EM algorithm and Kalman filtering are employed in the
system parameters and state estimation, respectively. These
algorithms are introduced in the next section.
III. LTF WIRELESS CHANNEL ESTIMATION VIA THE
EXPECTATION MAXIMIZATION ALGORITHM AND KALMAN
FILTERING
(8)
yt = st e kxt + Dt vt
d
is a state noise, and vt ∈ R is a
vector, wt ∈ R
measurement noise. Note that the state space model is
nonlinear since the output equation in (8) is nonlinear. We
consider the general form of state space form since the
estimation algorithm is derived for the general case.
However, in our system model (7), we have n = d = 1 and m
= 2. The noise processes wt and vt are assumed to be
independent zero mean and unit variance Gaussian
processes. Further, the noises are independent of the initial
state x0 , which is assumed to be Gaussian distributed.
m
dxt |t
= st ke
kxt|t
xt|t = xˆt −1|t −1
xt|t = xˆt −1|t −1
where t = 0,1, 2,..., N , and Pt |t is given by:
P t−|t1 = Pt −−1|1 t −1 + AtT Bt−2 At
Pt |−t 1 = CtT Dt−2 Ct + Bt−2 − Bt−2 Pt |t AtT Bt−2
(10)
Pt |t −1 = At Pt −1|t −1 A + B
T
t
2
t
based on the EM algorithm, which is introduced next.
B. Channel Parameter Estimation: The EM Algorithm
The EM algorithm uses a bank of Kalman filters to yield a
maximum likelihood (ML) parameter estimate of the state
space model.
Let θ t = { At , Bt , Dt } denotes the system parameters in (8),
P0 denotes a fixed probability measure; and
where xt ∈ R is a state vector, yt ∈ R is a measurement
n
(9)
kxt|t
The channel parameters θ t = { At , Bt , Dt } are estimated
This section describes the procedure employed to estimate
the channel model parameters and states associated with the
state space model in (7), based on the EM algorithm [12]
together with Kalman filtering [13]. The state space model
of (7) in discrete-time can be written as:
xt +1 = At xt + Bt wt
( )
d e
d
The unknown system parameters θ t = { At , Bt , Dt } as well
as the system states xt are estimated through a finite set of
received signal measurement data, YN = { y1 , y2 ,..., yN } . The
methodology proposed is recursive and based on the EM
algorithm combined with the extended Kalman filter to
estimate the channel state variables. The latter is used due
to the nonlinear output equation.
A. Channel State Estimation: The Extended Kalman
Filter
The extended Kalman filter (EKF) approach is based on
linearizing the nonlinear system model (8) around the
previous estimate. It estimates the channel states xt for
{P ; θ ∈ Θ}
θt
t
denotes a family of probability measures induced by the
system parameters θt . If the original model is a white noise
sequence, then
{P ; θ ∈ Θ}
θt
t
is absolutely continuous with
respect to P0 [12]. Moreover, it can be shown that under P0
we have:
P0 :
⎧ xt +1 = wt
⎨
⎩ yt = vt
(11)
The EM algorithm is an iterative scheme for computing the
ML estimate of the system parameters θ t , given the set of
data Yt . Specifically, each iteration of the EM algorithm
consists of two steps: The expectation step and the
maximization step.
The expectation step evaluates the conditional expectation
of the log-likelihood function given the complete data,
which is described by:
⎧⎪
⎫⎪
dPθ
Λ (θt ,θˆt ) = Eθl ⎨log t | Yt ⎬
dPθˆt
⎩⎪
⎭⎪
(12)
where θˆt denotes the estimated system parameters at time
step t. The maximization step finds:
θt ∈Θ
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(
θˆt +1 ∈ arg max Λ θt , θˆt
)
(13)
TuD01.6
The expectation and maximization steps are repeated until
the sequence of model parameters converge to the real
parameters.
The EM algorithm is given by the following equations
[12]:
⎛ t
⎞ ⎡ ⎛ t
⎞⎤
Aˆt = E ⎜ ∑ xk xkT−1 | Yt ⎟ × ⎢ E ⎜ ∑ xk xkT | Yt ⎟ ⎥
⎝ k =1
⎠ ⎣ ⎝ k =1
⎠⎦
(
−1
(
The other terms in (14) can be computed similarly.
(2)
(3)
(4)
The conditional expectations { L(1)
t , Lt , Lt , Lt } can be
1) Filter estimate of L(1)
t :
)
T
T
T
⎛ ⎛
1 ⎜ t ⎜ ( xk xk ) − Ak ( xk xk −1 )
= E ∑
t ⎜⎜ k =1 ⎜ − ( x xT ) AT + A ( x xT ) AT
k k −1
k
k
k −1 k −1
k
⎝ ⎝
1 ⎛ t
⎞
T
Dˆ t2 = E ⎜ ∑ ( yk − Ck xk )( yk − Ck xk ) | Yt ⎟
t ⎝ k =1
⎠
⎞ ⎞
⎟|Y ⎟
⎟ t ⎟⎟
⎠ ⎠
⎧ t T
⎫
L(1)
t = E ⎨ ∑ xk Qxk | Yt ⎬
⎩ k =1
⎭
1
1 t
= − Tr ( N t(1) Pt |t ) − ∑ Tr ( N k(1)−1 Pk |k )
2
2 k =1
(14)
)
⎛ t ⎛ ( yk ykT ) − Ak ( yk xkT ) CkT
1
= E ⎜∑⎜
t ⎜⎜ k =1 ⎜ −C ( y xT )T + C ( x xT ) C T
k
k k
k
⎝ ⎝ k k k
−
⎞ ⎞
⎟|Y ⎟
⎟ t ⎟⎟
⎠ ⎠
t
t
t
following conditional expectations [12]:
(19)
⎧ N k(1) = Bk−2 Ak Pk |k N k(1)−1 Pk |k AkT Bk−2 − 2Q
⎨
N 0(1) = 0m×m
⎩
(1)
t
2) Filter estimate of L(2)
t :
(15)
⎧ t
⎫
L(2)
= E ⎨∑ xkT−1Qxk −1 | Yt ⎬
t
⎩ k =1
⎭
⎧ t
⎫
= Eθ { x0T Qx0 | Yt } + Eθ ⎨∑ xkT−1Qxk −1 | Yt ⎬
⎩k =2
⎭
(20)
⎧ t
⎫
= Eθ { x0T Qx0 | Yt } + Eθ ⎨∑ xkT Qxk | Yt ⎬ − Eθ { xtT Qxt | Yt }
⎩ k =1
⎭
where Q, R and S are given by:
⎧⎪ ei eTj + e j eiT
⎫⎪
Q=⎨
; i, j = 1, 2,...n ⎬
2
⎩⎪
⎭⎪
⎧⎪ ei eTj
⎫⎪
; i, j = 1, 2,...n ⎬
R=⎨
⎩⎪ 2
⎭⎪
(1)
(1)
−1 (1)
−1
T
T
T
1 t ⎛ −2 xk |k Pk |k rk + 2 xk |k −1 Pk |k −1rk |k −1 − xk |k N k xk | k ⎞
⎜
⎟
∑
⎟
2 k =1 ⎜⎝ + xkT|k −1 Bk−2 Ak Pk |k N k(1)−1 Pk |k AkT Bk−2 xk |k −1
⎠
⎧rk(1) = ( Ak − Pk |k CkT Dk−2Ck Ak ) rk(1)−1 + 2 Pk |k Qxk |k −1
⎪
(1)
T
−2
⎪⎪ − Pk |k N k Pk |k Ck Dk ( yk − Ck x k |k −1 )
⎨ (1)
⎪rk |k −1 = Ak rk(1)
⎪ (1)
⎪⎩r0 = 0m×1
}
⎧ t
⎫
L = E ⎨∑ xkT Qxk | Yt ⎬
⎩ k =1
⎭
t
⎧
⎫
L(2)
= E ⎨∑ xkT−1Qxk −1 | Yt ⎬
t
⎩ k =1
⎭
t
⎧
⎫
L(3)
= E ⎨∑ ⎣⎡ xkT Rxk −1 + xkT−1 RT xk ⎦⎤ | Yt ⎬
t
⎩ k =1
⎭
t
⎧
⎫
L(4)
= E ⎨∑ ⎣⎡ xkT Syk + ykT S T xk ⎦⎤ | Yt ⎬
t
⎩ k =1
⎭
(18)
where Tr (⋅) denotes the matrix trace. In (18), rk(1) and N k(1)
satisfy the following recursions:
where Bt2 = Bt BtT , Dt2 = Dt DtT , E (⋅) denotes the
expectation operator, and t = 0,1, 2,..., N . These system
Aˆ , Bˆ 2 , Dˆ 2
can be computed from the
parameters
{
(17)
estimated from measurements Yt as follows:
1 ⎛ t
⎞
T
Bˆ = E ⎜ ∑ ( xk − Ak xk −1 )( xk − Ak xk −1 ) | Yt ⎟
t ⎝ k =1
⎠
2
t
1⎞
⎛ t
⎞
⎛
E ⎜ ∑ xk xkT−1 | Yt ⎟ = L(3)
t ⎜R =
⎟
2⎠
⎝
⎝ k =1
⎠
(1)
Therefore, L(2)
t can be obtained from Lt .
3) Filter estimate of L(3)
t :
(16)
⎧ t
⎫
L(3)
= E ⎨∑ ( xkT Rxk −1 + xkT−1 RT xk ) | Yt ⎬
t
⎩ k =1
⎭
t
1
1
= − Tr ( N t(3) Pt |t ) − ∑ Tr ( N k(3)−1 Pk |k )
2
2 k =1
⎪⎧ e e
⎪⎫
; i = 1, 2,...n; j = 1, 2,..d ⎬
S =⎨
⎪⎩ 2
⎪⎭
T
i j
in which ei is the unit vector in the Euclidean space; that is
ei = 1 in the ith position, and 0 elsewhere. For instance,
⎛ t
⎞
consider the case n = d = 1, then E ⎜ ∑ xk xkT−1 | Yt ⎟ can be
⎝ k =1
⎠
computed as:
(21)
−1 (3)
−1
T
T
T
(3)
(3)
1 t ⎛ −2 xk |k Pk |k rk + 2 xk |k −1 Pk |k −1rk |k −1 − xk |k N k xk |k ⎞
⎟
− ∑⎜ T
⎟
2 k =1 ⎜⎝ + xk |k −1 Bk−2 Ak Pk |k N k(3)−1 Pk |k AkT Bk−2 xk |k −1
⎠
In this case, rk(3) and N k(3) satisfy the following recursions:
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TuD01.6
(3)
T
−2
⎧ rk(3) = ( Ak − Pk |k CkT Dk−2 Ck Ak ) rk(3)
−1 − Pk | k N k Pk | k Ck Dk
⎪
( yk − Ck xk |k −1 ) + ( 2Pk |k R + 2 Pk |k Bk−2 Ak Pk |k RT Ak ) xk −1|k −1
⎪⎪
⎨ (3)
⎪ rk |k −1 = Ak rk(3)
⎪ (3)
⎪⎩ r0 = 0m×1
transmitter j and the receiver assigned to transmitter i, pk ( t )
is the transmitted power of transmitter k at time t, which acts
as a scaling on the information signal sk ( t ) , vi ( t ) is the
(22)
⎧ N k(3) = Bk−2 Ak Pk |k N k(3)−1 Pk |k AkT Bk−2 − 2 RPk |k AkT Bk−2
⎪⎪
− 2 Bk−2 Ak Pk |k RT
⎨
⎪ (3)
⎪⎩ N 0 = 0m× m
channel disturbance or noise at receiver i, and 1≤ i, j ≤ M .
Consider the wireless network described above, the
centralized PC problem for time-invariant channels can be
stated as follows [1]:
M
min
( p1 ≥ 0,.... pM ≥ 0)
(4)
t
4) Filter estimate of L :
⎧
⎫
= E ⎨∑ ( xkT Syk + ykT S T xk ) | Yt ⎬
L(4)
t
⎩ k =1
⎭
t
t
= ∑(x P r
k =1
T
k |k
−1 (4)
k |k k
T
(4)
−1
k | k −1 k | k −1 k | k −1
−x
P
r
∑
)
(23)
⎧r = ( Ak − Pk |k C D Ck Ak )r
⎪⎪ (4)
(4)
⎨rk |k −1 = Ak rk
⎪ (4)
⎪⎩r0 = 0m×1
T
k
−2
k
(4)
k −1
+ 2 Pk |k Syk
(24)
algorithm described in (14). Numerical results that show the
applicability of the above algorithm in estimating the
channel parameters as well as the channel PL from
measurements are discussed in Section V. In the next
section, we introduce an important application based on the
developed models, which is stochastic PC in LTF wireless
networks.
IV. STOCHASTIC POWER CONTROL ALGORITHM IN TV LTF
WIRELESS NETWORKS
In this section, a PCA based on the estimated LTF
wireless channel models is developed. Since the channel
model parameters are estimated from received signal
measurements, PC is performed only from measurements.
The aim of the PCA described here is to minimize the total
transmitted power of all users while maintaining acceptable
QoS for each user. The measure of QoS is defined by the
signal-to-interference ratio (SIR) for each link to be larger
than a target SIR.
Now consider a wireless network with M transmitters and
N receivers. The state space representation of LTF wireless
network can be written as:
yi ( t ) = ∑ k =1 pk (t ) sk (t )e
M
kX ik ( t ,τ )
+ vi (t )
pi gii
M
pk gik + ηi
k ≠i
subject to
(26)
≥ εi , 1 ≤ i ≤ M
where pi is the power of transmitter i, gik > 0 is the time-
transmitter i, and ηi > 0 is the noise power level at the
receiver i. Expression (26) for the TV LTF wireless network
in (25), described using path-wise QoS of each user over a
time interval [0,T] is given by:
T
⎪⎧ M
⎪⎫
pi ( t ) dt ⎬ ,
⎨
∑
∫
( p1 ≥ 0,.... pM ≥ 0)
⎪⎩ i =1 0
⎪⎭
min
Using the filters for L(t i ) (i = 1, 2,3, 4) and the extended
Kalman filter described in (9) and (10), the system
parameters θ t = { At , Bt , Dt } are estimated through the EM
X ij ( t ,τ ) = Aij ( t ,τ ) X ij ( t ,τ ) + Bij ( t ,τ ) wij ( t )
i
i =1
invariant channel gain between transmitter k and the
receiver assigned to transmitter i, ε i > 0 is the target SIR of
where rk(4) satisfy the following recursions:
(4)
k
∑p
(25)
subject to
T
∫ p ( t ) s ( t ) S ( t ) dt
i
2
i
(27)
2
ii
≥ εi
0
T
T
0
0
∑ k ≠i ∫ pk ( t ) sk2 ( t ) Sik2 ( t ) dt + ∫ vi2 ( t ) dt
M
where Sik ( t ) = e kX ik (t ,τ ) and i = 1, " , M . A solution to (27)
is presented by first introducing PPCS. In wireless cellular
networks, it is practical to observe and estimate channels at
base stations and then send the information back to the
mobiles to adjust their power signals
{ p (t )}
M
i =1
i
. Since
channels experience delays, and power control is not
feasible continuously in time but only at discrete-time
instants, the concept of predictable strategies is introduced
{ p (t )}
[1]. Consider a set of discrete time strategies
k
i
M
i =1
,
0 = t0 < t1 < ... < tk < tk +1 < ... ≤ T . At time tk −1 , the base
stations
estimate
{S ( t ) , s (t )}
ij
k −1
i
k −1
M
the
channel
information
as described in Section III. Using the
i , j =1
concept of predictable strategy, the base stations determine
the control strategy
{ pi (tk )}i =1
M
for the next time instant tk .
The latter is communicated back to the mobiles, which hold
these values during the time interval [tk −1 , tk ) . At time tk , a
new set of channel information
{S ( t ) , s (t )}
ij
k
i
k
M
i , j =1
is
where yi (t ) is the received signal at the ith receiver at time
estimated at the base stations and the time tk +1 control
t, X ij (t ) is the states of the TV PL of the channel between
strategies
1890
{ pi (tk +1 )}i =1
M
are computed and communicated
TuD01.6
p ( tk +1 ) = Fp ( tk ) + u
back to the mobiles which hold them constant during the
time interval [tk , tk +1 ) . Such decision strategies are called
predictable. Using the concept of PPCS over any time
interval [tk , tk +1 ] , equation (27) is equivalent to:
min
p ( tk +1 ) > 0
∑
M
p
i =1 i
( tk +1 )
(28)
∫ s ( t ) S ( t ) dt , 1 ≤ i,
2
j
2
ij
j ≤ M,
tk
p ( tk +1 ) := ( p1 ( tk +1 ) ," , pM ( tk +1 ) ) , ηi ( tk , tk +1 ) :=
T
tk +1
∫ v ( t ) dt ,
2
i
tk
G I ( tk , tk +1 ) := diag ( g11 ( tk , tk +1 ) ," , g MM ( tk , tk +1 ) ) ,
Clearly, in general the power vector p ( tk ) will not
converges in distribution to a well defined random variable.
Since F ( tk , tk +1 ) is a random matrix-valued process, the key
convergence condition is that the Lyapunov exponent
λF < 0 [14], where λF is defined as:
λF = lim
0
, if i = j ⎪⎫
⎪⎧
G ( tk , tk +1 ) := ⎨
⎬ , 1 ≤ i, j ≤ M ,
⎩⎪ gij ( tk , tk +1 ) , if i ≠ j ⎪⎭
k →∞
T
Γ := diag ( ε1 ," , ε M ) ,
pi ( tk +1 ) =
and diag (⋅) denotes a diagonal matrix with its argument as
diagonal entries. The optimization in (28) is a linear
programming problem in M × 1 vector of unknowns
p ( tk +1 ) . Throughout this section, we assume that the PC
Ri ( tk ) =
that satisfies the inequality in (28) for all [tk , tk +1 ] in [0, T].
Next, we consider an iterative distributed version of the
centralized PCA in (28). This is convenient for online
implementation since it helps autonomous execution at the
node or link level, requiring minimal usage of network
communication resources for control signaling. The
constraint in (28) can be rewritten as:
( tk , tk +1 ) G ( tk , tk +1 ) ) p ( tk +1 )
≥ ΓG −I 1 ( tk , tk +1 ) η ( tk +1 )
F ( tk , tk +1 ) Γ G I −1 ( tk , tk +1 ) G ( tk , tk +1 )
Defining
(29)
and
u ( tk , tk +1 ) Γ G I −1 ( tk , tk +1 ) η ( tk +1 ) , then (29) can be
ε i ( tk )
Ri ( tk )
pi ( tk ) , i = 1,..., M
(34)
pi ( tk ) gii ( tk , tk +1 )
∑
M
j ≠i
p j ( tk ) gij ( tk , tk +1 ) + ηi ( tk , tk +1 )
The selection of an appropriate
[tk , tk +1 ]
(35)
will have a
significant impact on the system performance. For small
[tk , tk +1 ] , the power control updates will be more frequent
and thus convergence will be faster. However, frequent
transmission of the feedback on the downlink channel will
effectively decrease the capacity of the system since more
system resources will have to be used for power control.
Note that the PCAs in (28) and (34) can be used as long as
the channel model does not change significantly, that is [tk,
tk+1] is a subset of the coherence time of the channel.
In the next section, a numerical example is presented to
determine the performance of the proposed PCA under the
estimated LTF wireless channel models.
rewritten as:
( I − F ( tk , tk +1 ) ) p ( tk +1 ) ≥ u ( tk , tk +1 )
(33)
where Ri ( tk ) is the instantaneous SIR defined by:
problem is feasible, i.e., there exists a power vector p ( tk )
−1
I
1
log F ( t0 , t1 ) F ( t1 , t2 ) ...F ( tk , tk +1 )
k
The distributed version of (32) can be written as:
η ( tk , tk +1 ) := (η1 ( tk , tk +1 ) ," ,η M ( tk , tk +1 ) ) ,
( I − ΓG
(32)
converge to some deterministic constant as it does in (31).
Rather, in a time-varying (random) propagation
environment, it is required that the power vector p ( tk )
where
tk +1
However, our channel gains are time-varying, thus a timevarying version of the PCA in (31) can be defined as:
p ( tk +1 ) = F ( tk , tk +1 ) p ( tk ) + u ( tk , tk +1 )
subject to
p ( tk +1 ) ≥ ΓG −I 1 ( tk , tk +1 ) × ( G ( tk , tk +1 ) p ( tk +1 ) + η ( tk +1 ) )
gij ( tk , tk +1 ) :=
(31)
V. NUMERICAL EXAMPLES
(30)
If the channel gains are time-invariant, i.e., F ( tk , tk +1 ) = F
and u ( tk , tk +1 ) = u , then the PC problem is feasible if
ρF < 1 , where ρF is the Perron-Frobenius eigenvalue of F
[9]. It is shown in [9] and [10] that the following iterative
PCA converges to the minimal power vector when ρF < 1 :
Two numerical examples are presented. In Example 1, the
accuracy of the EM algorithm together with the extended
Kalman filter to estimate channel parameters, as well as
channel PL from the received signal measurements, is
determined. In Example 2, we compare the performance of
the PCA using PPCS under TV LTF stochastic and static
channel models.
1891
TuD01.6
Example 1:
The estimation of a LTF wireless channel from received
signal measurements is considered. In particular, the
estimation includes the channel parameters, channel PL, and
received signal. The measurement data are generated by the
following system parameters:
⎛
⎛ 10π t ⎞ ⎞
⎟⎟,
⎝ T ⎠⎠
γ ( t ,τ ) = 25 ⎜1 + 0.15e −2t / T sin ⎜
⎝
δ ( t ,τ ) = 5, β ( t ,τ ) = 0.2,
(36)
and the variances of the state and measurement noises are
10-2 and 10-6, respectively. Figure 2 shows the actual and
estimated received signal using the EM algorithm together
with the extended Kalman filter for 500 sampled data. From
Figure 2, it can be noticed that the received signal have been
estimated with very good accuracy. Figure 3 shows the
received signal estimates root mean square error (RMSE) for
100 runs. It can be noticed that it takes just few iterations
(less than 15) for the filter to converge, and the steady state
performance of the proposed LTF channel estimation
algorithm using the EM together with Kalman filtering is
excellent.
Example 2:
The cellular model has the following setup: Number of
transmitters is M = 24, the information signal si ( t ) = 1 for
i = 1,..., M , initial distances of all mobiles with respect to
their own base stations dii are generated as uniformly
2
1.6
Received Signal
1.4
1.2
0.4
200
1
220
240
260
280
300
0.8
0.6
0.4
Estimated
Real
0.2
0
0
100
200
300
Samples
400
500
Fig.2. Real and estimated received signal for the LTF channel model in
Example 1.
RMSE
0.6
0.4
0.2
0
0
generated as uniformly iid RV’s in [250 - 550] meters, the
angle θij between the direction of motion of transmitter j and
the distance vector passes through receiver i and transmitter
j are generated as uniformly iid RV’s in [0 – 180] degrees,
the average velocities of transmitters are generated as
uniformly iid RV’s in [40 – 100] km/hr, all mobiles move at
sinusoidal variable velocities around their average velocities
such that the peak velocity is two times the average speed,
power path-loss exponent is 3.5, initial reference distance
from each of the transmitters is 10 meters, power path-loss
at the initial reference distance is 67 dB.
The channel model parameters as well as the channel PL
for all users are estimated online from received signal
measurements using the EM algorithm together with the
extended Kalman filter as illustrated in Example 1. The PCA
described in (28) is performed using the estimated channel
parameters and states. The outage probability (OP) is used
as a performance measure for the PCA. A link with a
received SIR Ri , less than or equal to a target SIR ε i , is
considered a communication failure. The OP, O ( ε i ) , is
expressed as O ( ε i ) = Prob { Ri ≤ ε i } , where Ri is the
received SIR at receiver i.
It is assumed that the targets SIR, ε i , for all users are the
same, and varied from 5 dB to 35 dB with step 5 dB. For
each value of ε i the OP is computed every 15 millisecond,
i.e., [tk , tk +1 ] = 15 millisecond. The simulation is performed
for 6 seconds, i.e., [ 0, T ] = 6 seconds. The OP is computed
0.8
1.8
independent identically distributed (iid) random variables
(RV’s) in [10 – 100] meters, cross initial distances of all
mobiles with respect to other base stations dij , i ≠ j , are
100
200
300
Samples
400
500
Fig.3. Received signal estimates RMSE for 100 runs using the EM
algorithm together with the extended Kalman filter.
using Monte-Carlo simulations. In this example, we
compare the performance of the proposed PCA using PPCS
described in (28) under two different types of TV LTF
channel models; the stochastic model in (2) and the static
model in (1).
The OP for the PCA using PPCS based on both stochastic
and static TV LTF channel models are shown in Figure 4a
and 4b, respectively. Figure 4 shows how the OP changes
with respect to the target SIR, ε i , and time. As the target
SIR increases the OP increases. This is obvious since we
expect more users to fail. The OP also changes as a function
of time, since mobiles move in different directions and
velocities.
The average OP versus ε i for both cases over the whole
simulation time (6 seconds) is shown in Figure 5, which
shows that the performance of PC based on PPCS using the
stochastic models is on average much better than that of
static models. This is because the static models do not
capture the time-variations of the channels. For example, at
20 dB target SIR, the OP is reduced from 0.45 for static
models to 0.3 for TV stochastic ones; this represents an
improvement of over 33%. The PC algorithm using PPCS
for stochastic models outperforms the static ones by an order
of magnitude.
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TuD01.6
VI. CONCLUSION
Outage Probability
1
0.8
0.6
6
0.4
4
Time (sec)
0.2
0
5
2
10
15
20
25
30
35
Target SIR (dB)
0
(a)
This paper develops a general scheme for extracting
mathematical LTF channel models from noisy received
signal measurements, and performs power control based on
the estimated channel parameters. The proposed estimation
algorithm is recursive and consists of filtering based on the
extended Kalman filter to remove noise from data, and
identification based on the EM algorithm to determine the
parameters of the model which best describe the
measurements. The proposed estimation and parameter
identification algorithms estimate the path-loss and the
channel parameters. Performance of the latter is investigated
through a numerical example that shows excellent results.
Therefore the proposed algorithms have good potential for
real time applications. Moreover, a stochastic PCA based on
the estimated parameters and channel states is proposed.
Numerical results indicate that there is potentially large gain
to be achieved by using the proposed time-varying
stochastic models.
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[1]
Outage Probability
1
[2]
0.8
0.6
[3]
6
0.4
4
Time (sec)
0.2
0
5
[4]
[5]
2
10
15
20
25
Target SIR (dB)
30
35
0
[6]
(b)
[7]
Fig. 4. OP for the PCA using PPCS under TV LTF wireless networks for
(a) stochastic channel models. (b) static channel models.
[8]
[9]
0.8
[10]
0.7
Outage Probability
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[11]
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Target SIR (dB)
20
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Fig. 5. Average OP for the PCA using PPCS under TV LTF wireless
networks. Performance comparison.
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