Recursive Estimation and Identification of Time-Varying Long- Term Fading Channels

This paper is concerned with modeling of time-varying wireless long-term fading channels, parameter estimation, and identification from received signal strength data. Wireless channels are represented by stochastic differential equations, whose parameters and state variables are estimated using the expectation maximization algorithm and Kalman filtering, respectively. The latter are carried out solely from received signal strength data. These algorithms estimate the channel path loss and identify the channel parameters recursively. Numerical results showing the viability of the proposed channel estimation and identification algorithms are presented.


INTRODUCTION
This 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][2][3].In the TV models, the statistics of channel are time-varying.This contrasts with the majority of published work that mainly deals with timeinvariant (static) random models or simple free space model, where the channel statistics do not depend on time [4][5][6].In time-invariant models, channel parameters are random but do not depend on time, and they remain constant throughout the observation and estimation phase.This contrasts with TV models, where the channel dynamics become TV random (stochastic) processes [1][2][3].
The TV LTF channel model is discussed in [2] and represented by stochastic differential equations (SDEs).We propose to estimate the TV power path loss of the LTF channel and its parameters from received signal strength data, which are usually available or easy to obtain in any wireless network.The expectation maximization (EM) algorithm [7] and Kalman filtering [8] are employed in the identification and estimation of the channel parameters and path loss.The proposed identification and estimation algorithms are recursive and only based on received signal measurements.Numerical results are provided to determine the performance of the proposed estimation algorithm.
The paper is organized as follows.In Section 2, the TV LTF mathematical channel model is introduced.In Section 3, the EM algorithm together with the Kalman filter, to estimate the channel parameters as well as the channel power loss from signal measurements, is developed.In Section 4, numerical results are presented.Finally, Section 5 provides the conclusion.

TV LTF WIRELESS CHANNEL MATHEMATICAL MODEL
Wireless 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, their communication signal undergoes phenomenal LTF (lognormal shadowing) [5].For such propagation environments, the random process power loss (PL) in dB, {X(t, τ)} t≥0,τ≥τ0 , which is a function of both time t and space represented by the time delay τ, is generated by a meanreverting version of a general linear time-varying SDE given by [2,3] where {W (t)} t≥0 is the standard Brownian motion (zero drift, unit variance) which is assumed to be independent of In [2,3], this model is shown to capture the spatiotemporal variations of the propagation environment as the random parameters {β(t, τ), γ(t, τ), δ(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 where s(t) is the information signal, v(t) is the channel disturbance at the receiver, and H(t) is the signal attenuation coefficient defined by H(t) e kX (t,τ) , where k = − ln (10)/20 [5].
The general spatiotemporal lognormal model in ( 1) and ( 2) can be realized by a stochastic state space system given by where and The above system parameters and state variable values are estimated from received signal measurements.The EM algorithm and Kalman filtering are employed in the system parameters and state estimation, respectively.These algorithms are introduced next.

WIRELESS CHANNEL ESTIMATION VIA THE EM ALGORITHM AND KALMAN FILTERING
This section describes the procedure employed to estimate the channel model parameters and states associated with the state space model in (3), based on the EM algorithm [7] together with Kalman filtering [8].Since the estimation and identification processes are carried out in discrete instants, we consider a sampled version of the state space model (3) in discrete time as where x t ∈ R n is a state vector, y t ∈ R d is a measurement vector, w t ∈ R m is a state noise, and v t ∈ R d is a measurement noise.Note that the state space model is nonlinear since the output equation in ( 4) is nonlinear.The channel parameters θ t = {A t , B t , D t } are unknown and they are estimated together with the path loss represented by the system states x t from a finite set of received signal measurement data, Y N = {y 1 , y 2 , . . ., y N }.The proposed methodology is recursive and based on the EM algorithm combined with the extended Kalman filter (EKF).The latter is used due to the nonlinear output equation.

Channel state estimation-The EKF
The EKF approach is based on linearizing the nonlinear system model (4) around the previous estimate.It estimates the channel states x t for given system parameter θ t = {A t , B t , D t } and measurements Y t .It is described by the following equations [8]: where t = 0, 1, 2, . . ., N, and P t|t is given by where The channel parameters θ t = {A t , B t , D t } are estimated based on the EM algorithm, which is introduced next.

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.It is an iterative scheme for computing the ML estimate of the system parameters θ t , given the set of data Y t .Specifically, each iteration of the EM algorithm consists of two steps: the expectation step and the maximization step [9].The filtered expectation step only uses filters for the firstand second-order statistics.The algorithm yields parameter estimates with nondecreasing values of the likelihood function and converges under mild assumptions [10].The expectation step evaluates the conditional expectation of the loglikelihood function given the complete data as where {F θt ; θ t ∈ Θ} denotes a family of probability measures induced by the system parameters θ t , and θ t denotes the estimated system parameters at time step t.The maximization step finds that The expectation and maximization steps are repeated until the sequence of model parameters converges to the real parameters.The EM algorithm is given by [7, 9] where E(•) denotes the expectation operator, and t = 0, 1, 2, . . ., N. The system parameters { A t , B 2 t , D 2 t } are computed from the following conditional expectations [7]: where Q, R, and S are given by in which e i is the unit vector in the Euclidean space, that is, e i = 1 in the ith position and 0 elsewhere.For instance, consider the case The other terms in ( 9) can be computed similarly.

NUMERICAL RESULTS
In this section, 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.The measurement data are generated by the system parameters: where γ m (τ) is the average PL at a specific location τ and it is chosen to be 25 dB, T is the observation interval, and the variances of the state and measurement noises are 10 −2 and 10 −6 , respectively.Figure 1 shows the actual and estimated received signals using the EM algorithm together with the extended Kalman filter for 500 sampled data.From Figure 1, it can be noticed that the received signal has been estimated with very good accuracy.Figure 2   for the filter to converge, and the steady-state performance of the proposed channel estimation algorithm using the EM together with Kalman filtering is excellent.Since our stochastic model in (3) is first-order, the computational cost of the proposed estimation algorithm is very low, and thus it can be implemented online.Moreover, the filters of the expectation step are recursive and decoupled, and hence easy to implement in parallel on a multiprocessor system [9].

CONCLUSION
This paper develops a general scheme for extracting mathematical LTF channel models from noisy received signal measurements.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.Future work includes combining identification and estimation with other performance requirements, such as power control, admission control, and base station assignment.

Figure 1 : 6 RMSEFigure 2 :
Figure 1: Real and estimated received signals for the channel model.