IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 1, JANUARY 2008
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Modeling Wireless Fading Channels via Stochastic Differential Equations: Identification and Estimation Based on Noisy Measurements C. D. Charalambous, R. J. C. Bultitude, X. Li, and J. Zhan
Abstract— This paper is concerned with modeling and identification of wireless channels using noisy measurements. The models employed are governed by Stochastic Differential Equations (SDEs) in state space form, while the identification method is based on the Expectation-Maximization (EM) algorithm and Kalman Filtering. The algorithm is tested against real channel measurements. The results presented include state space models for the channels, estimates of inphase and quadrature components, and estimates of the corresponding Doppler Power Spectral Densities (DPSDs), from sample noisy measurements. Based on the available measurements, it is concluded that state space models of order two are sufficient for wireless flat fading channel characterization. Index Terms— Stochastic differential equations, wireless fading channels, identification, estimation, kalman filter.
I. I NTRODUCTION
T
HIS paper is concerned with developing wireless channel models based on system identification algorithms, and Kalman filtering from noisy measurement data. These models are important for developing a novel channel simulator that replicates wireless channel data, and produces outputs that vary in a similar manner to the variations encountered in practice. The approach considered is new and consists of the following two steps. • Development of mathematical models in state space form using SDEs. • Application of the Expectation-Maximization (EM) algorithm to identify the channel model parameters of SDEs. Unlike work found in the literature which employ the models described in [2], [16] (see also [11] - [14]), and are concerned with matching second order statistics (see [20]), in this paper the emphasis is to develop models and estimation techniques which match wireless channel realizations. Specifically, time domain methods are considered by postulating that the inphase and quadrature components of the channel are solutions of SDEs, while noisy measurements are employed Manuscript received July 17, 2006; revised April 27, 2007, accepted July 23, 2007. The associate editor coordinating the review of this paper and approving it for publication was I. Collings. This work was supported by the Cyprus Research Promotion Foundation and the Communications Research Center (CRC) Canada, Ottawa, Canada. C. D. Charalambous is with the Department of Electrical and Computer Engineering, University of Cyprus, Nicosia, Cyprus, and an Adjunct Professor with the School of Information Technology and Engineering, University of Ottawa, Ottawa, Canada (e-mail:
[email protected]). R. J. C. Bultitude is with the Communications Research Centre (CRC) Canada, Ottawa, Canada (e-mail:
[email protected]). X. Li is with the Department of Electrical and Communications Engineering, Helsinki University of Technology, Espoo, Finland (e-mail:
[email protected]). J. Zhan is with the Wuhan Strong Automation Control Co. Ltd, Wuhan, China. Digital Object Identifier 10.1109/TWC.2008.060482.
to extract dynamic channel models using estimation and identification techniques. Since the coefficients of the SDEs are unknown, they are estimated using identification techniques from noisy observed data. Thus, the proposed channel simulator consists of an estimation part and an identification part as follows. The inphase and quadrature components are governed by SDEs, the estimation part is employed to filter out the noise and extract least-squares estimates of the inphase and quadrature components, while the identification part is employed to extract the model parameters from noisy observed data. This is a non-linear estimation problem, hence it is dealt with using Kalman Filtering and the EM algorithm [7], [9], [15], [18]. Several simulations are performed to illustrate the concepts. Specifically, given a realization of the noisy measurements, the Kalman-Filter and the EM algorithm are run in parallel to identify the model parameters, filter the noise, and generate least-squares estimates of realizations of the channel inphase and quadrature components. Moreover, from the estimates of the channel realizations of inphase and quadrature components, estimates of their corresponding DPSDs are also obtained. In addition, the EM algorithm is used to estimate the noise variance associated with additive Gaussian noise channels. The proposed methodology is complimentary to previous work found in the literature, which is concerned with matching second order statistics to measurement data [2], [11], [12], [13], [14], [16], [17], [20]. In these methodologies, the main concept is either based on Jakes sum of sinusoids time method or is based on generating the inphase and quadrature components of the channel by passing independent white Gaussian noise through a specific filter. That is, the power spectral densities of I, Q, denoted by SI (ω), SQ (ω), respectively, are obtained via SI (ω) = |H(ω)|2 SvI (ω), SQ (ω) = |H(ω)|2 SvQ (ω), where SvI (ω), SvQ (ω) are the power spectral densities of inphase and quadrature noises, vI , vQ , respectively, and H(ω) is the filter. For white noise with unit variance, SvI (ω) = 1, SvQ (ω) = 1. Given that one knows the channel then the DPSDs SvI (ω), SvQ (ω) or the autocorrelation of inphase and quadrature components are obtained from [2] or [11] - [14], [16], [20], for a specific transmitting and receiving antenna. However, there are many different channels and propagation environments, hence the corresponding DPSDs SvI (ω), SvQ (ω) are very difficult to classify and determine prior to receiving a specific realization of the channel from measurements. On the other hand, the receiver is often corrupted by Gaussian noise, hence filtering in required before the channel is estimated. In this paper, we formulate the channel estimation and identification problem using inverse problem techniques. Specif-
c 2008 IEEE 1536-1276/08$25.00 °
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 1, JANUARY 2008
ically, we do not assume any knowledge of the DPSDs or autocorrelations of the inphase and quadratute components. Rather, we construct the model as well as the parameters and filter out the noise from measurements of a noisy channel. The approach compliments existing methods in the sense that both estimation and identification techniques are invoked, and hence the method of extracting models adapts to the measurements available. The mathematical basis for the methodology presented is the following. In general, any filter H(ω) can be realized using SDEs provided, one can find an H(ω), whose magnitude square equals |H(ω)|2 . This realization problem is studied by Paley-Wiener [4], and states the following. Given a non-negative integrable function S(ω), such that the R∞ S(ω)| Paley-Wiener condition −∞ | log 1+ω 2 dω < ∞ holds, then there exists a causal, stable, minimum phase H(ω), such that |H(ω)|2 = S(ω). This implies that S(ω) is factorizable by S(ω) = H(ω)H(−ω). The transfer function H(ω) represents the causal, stable, and minimum phase system, which is driven by white noise. Clearly, using realization theory, one can postulate that the inphase and quadrature components, I, Q, respectively, of the channel are generated by solutions of stochastic differential equations with corresponding power spectral densities SI (ω), SQ (ω), respectively. In wireless applications the degree and coefficients of the inphase and quadrature DPSDs SI (ω), SQ (ω), respectively, are unknown, and hence they must be estimated from noisy channel measurements (realizations). Clearly, the only possible approach to extract model parameters and estimates of inphase and quadrature components is via filtering and identification algorithms as proposed above. Once the model parameters are identified and the noise is filtered out, then the simulation of wireless channels can be done by solving specific SDEs, to generate estimates of channel realizations. In this paper, we use the above methodology to find the appropriate SDEs (e.g. order and model parameters) which generate measurement data for the inphase and quadrature components of a noisy channel. In order to identify these SDEs, we first assume a model, then determine the parameters of the model using the EM algorithm from noisy measurements, and then extract a least-square estimate of the model, and compare it with the experimental results. Our results show that SDEs of dimension two, for inphase and quadrature components have a very good fit to experimental data. The same holds when we compare the estimated DPSDs of the determined models with the experimental DPSDs. The paper is organized as follows. Section II introduces the mathematical models. Section III discusses the EM algorithm, and Section IV presents various simulations. The simulator is found at http://www.eng.ucy.ac.cy/chadcha/systemidentification.html [19], and it is designed to allow users to input their measurements and extract SDEs which best fit the noisy data. II. M ATHEMATICAL M ODELS FOR FADING C HANNELS A. Input-Output Fading Channel First, we describe the measurement model used to collect the real-world data at the receiver. Consider the transmission of signal s(t) at a carrier frequency ωc , through an additive Gaussian noise flat fading, Rayleigh or Ricean channel. Then,
the received signal is y(t)
= (I(t) cos(ωc t) − Q(t) sin(ωc t))s(t) + DI vI (t) cos(ωc t) − DQ vQ (t) sin(ωc t)
(1)
Here, {I(t)}t≥0 and {Q(t)}t≥0 denote the inphase and quadrature components of the channel, respectively, {vI (t)}t≥0 and {vQ (t)}t≥0 are two independent and identically distributed (iid) white Gaussian noise processes with probability density N (0; 1), and DI , DQ are constants which scale the variance of the noises {vI (t)}t≥0 and {vQ (t)}t≥0 , respectively. Clearly, if one has at hand the specific model that generates {I(t)}t≥0 and {Q(t)}t≥0 then simple least-squares estimation techniques such as Kalman filtering will provide estimates for {vI (t)}t≥0 and {vQ (t)}t≥0 . However, this is not available; the most we know is that under certain scenarios the DPSDs or Autocorreletion functions of {I(t)}t≥0 and {Q(t)}t≥0 have a specific functional form [2], [11] - [14], [16]. On the other hand, different propagation conditions give different type of DPSDs or Autocorreletion functions, and thus an alternative approach would be to investigate the inverse problem of generating both the model for {I(t)}t≥0 and {Q(t)}t≥0 as well as their least-squares estimates from received noisy data {y(t)}t≥0 . In principle, this is a nonlinear estimation problem. Below, we formulate this problem using nonlinear estimation techniques via the EM algorithm and Kalman filtering. Finally, we invoke measurements to evaluate our estimation and identification algorithms. B. State Space Realization of Flat Fading Channels First, we describe how to model {I(t)}t≥0 and {Q(t)}t≥0 via SDEs. Using the Paley-Wiener realization approach discussed in the introduction, the DPSD of the inphase and quadrature components can be approximated by a general nth order rational transfer function H(ω), having coefficients which are unknown. The unknown coefficients can be determined from received noisy measured data {y(t)}t≥0 via the EM identification algorithm. Since we favor time-domain methods which are computationally efficient, the nth order rational transfer function H(ω) is realized via SDEs using state space methods (see [1], [3], [10] for details). The stochastic differential equations can be represented in state space form, and hence the inphase and quadrature components are generated by n-dimensional SDEs: d dt XI (t) = AI XI (t) + BI wI (t); XI (0) ∼ N (0; Σ0 ), I(t) = MI XI (t)
(2)
d dt XQ (t) = AQ XQ (t) + BQ wQ (t); XQ (0) ∼ N (0; Σ0 ), Q(t) = MQ XQ (t)
(3)
In (2), (3), {wI (t)}t≥0 , {wQ (t)}t≥0 are m-dimensional iid white noise processes with probability density N (0; I), XI (0), XQ (0) are independent of {wI (t)}t≥0 , {wQ (t)}t≥0 , MI , MQ are 1 × n vectors, and I, Q are scalar processes, which are generated via solutions of (2), (3). Thus, the measurement equation (1) is represented by y(t)
=
(MI XI (t) cos(ωc t) − MQ XQ (t) sin(ωc t))s(t) + DI vI (t) cos(ωc t) − DQ vQ (t) sin(ωc t) (4)
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 1, JANUARY 2008
Since (2), (3) generate solutions pwhich are zero mean Gaussian processes, then the envelop I(t)2 + Q(t)2 , is Rayleigh distributed. The Ricean distribution can be treated similarly by adding a specular component [3]. Consequently, a fading channel can be represented via a general linear SDE as follows: d dt X(t)
y(t)
= AX(t) + Bw(t) = C(t)X(t) + D(t)v(t)
(5)
Here, y(t) represents the measurement at time t, T [XIT (t) XQ (t)]T denotes the state variables of the inphase and T quadrature components, w(t) = [wIT (t) wQ (t)]T denotes the process noise, which is assumed to be independent Gaussian, T and v(t) = [vIT (t) vQ (t)]T represents the measurement noise, which is also assumed to be independent Gaussian. In the above formulation the system parameters {A, B, C, D} are given by · ¸ · ¸ AI 0 BI 0 A= , B= 0 AQ 0 BQ (6) C(t) = [MI s(t) cos(ωc t) − MQ s(t) sin(ωc t)] D(t) = [DI cos(ωc t) − DQ sin(ωc t)] Most of the parameters in (6) are unknown and hence they can be estimated in order to build least-squares estimates of the inphase and quadrature components. In subsequent sections, we shall show that SDEs of the form (5) can be used to generate measurement data, provided the identification problem (inverse problem) associated with determining the parameters in (6) from measured noisy data {y(t)}t≥0 is solved. Thus, next we shall introduce an algorithm to determine {A, B, C, D}. III. S YSTEM I DENTIFICATION VIA THE E XPECTATION M AXIMIZATION A LGORITHM T OGETHER WITH K ALMAN F ILTER This section describes the EM algorithm [7], [9], [18] which is used to estimate the state space model parameters {A, B, C, D} of (5). For simplicity we discretize (5) and consider the discrete version. One can also work with the continuous time version of the EM algorithm by invoking the algorithm in [7]. Without loss of generality we discuss the application of EM algorithm to the following GaussianMarkov model xt+1 yt
= Axt + Bwt ; = Cxt + Dvt
x0 ∼ N (m0 ; Σ0 ),
(7)
Here, xt ∈
