[Retracted] Modeling, Real-Time Estimation, and Identification of UWB Indoor Wireless Channels

Abstract

Stochastic differential equations (SDEs) are used to model ultrawideband (UWB) indoor wireless channels. We show that the impulse responses for time-varying indoor wireless channels can be approximated in a mean-square sense as close as desired by impulse responses that can be realized by SDEs. The state variables represent the inphase and quadrature components of the UWB channel. The expected maximization and extended Kalman filter are employed to recursively identify and estimate the channel parameters and states, respectively, from online received signal strength measured data. Both resolvable and nonresolvable multipath received signals are considered and represented as small-scaled Nakagami fading. The proposed models together with the estimation algorithm are tested using UWB indoor measurement data demonstrating the method’s viability and the results are presented.

1. Introduction

Ultrawideband (UWB) communication systems have recently attracted significant interest from both the research community and industry since the Federal Communications Commission (FCC) allowed limited unlicensed operation of UWB devices in the USA [1]. They are commonly defined as systems that have either more than 20% relative bandwidth or more than 500 MHz absolute bandwidth. UWB technology has many benefits, including high data rate, low interference, less sensitivity to multipath fading, low transmit power, and availability of low cost transceivers [2]. Industrial standards such as IEEE 802.15.3a and 802.15.4a have been established in recognizing these developments.

The ultimate performance limits of a communication system are determined by the channel it operates in [3]. Realistic channel models are thus of utmost importance for system design and testing. UWB propagation channels show fundamental differences from conventional (narrowband) ones in many respects [4, 5], and therefore the established (narrowband) channel models cannot be used. A number of UWB channel models have been proposed in the literature. A model for frequency range below 1 GHz is suggested in [6]. A statistical model that is valid for a frequency range from 3 to 10 GHz is proposed in [7] and is accepted by the IEEE 802.15.4a task group as a standard model for evaluation of UWB system proposals. Significant experimental work in office, residential, and industrial environments has been reported in this field such as in [8, 9]. Most of the proposed channel models are based on characterizing the discrete multipath components. Although these models are able to capture the statistics of the channel, they cannot be specified by a finite number of parameters since their impulse responses are general functions of time and space and therefore are not easy to estimate directly from measurements.

A necessary and sufficient condition for representing any time-varying (TV) impulse response (IR) in stochastic state-space form is that it is factorizable into the product of two separate functions of time and space [10]. However, in general this is not the case for the IR of wireless channels. We show that the IR of indoor wireless channels can be approximated in the mean-square sense as close as desired by factorizable impulse responses that can be realized by stochastic differential equations (SDEs) in state-space form. In particular, the SDEs are used to model UWB indoor channels and are combined with system identification algorithms to extract various parameters of the channel from received signal measurement data. The expected maximization (EM) and the extended Kalman filter (EKF) are employed in estimating channel parameters as well as the inphase and quadrature components, respectively. The EM and EKF are chosen since they are recursive and therefore can be implemented online. These algorithms have been recently utilized in [11–13] to estimate the channel parameters and states in narrowband environments, and therefore the formulations of these algorithms are not presented in this paper. Experiments are conducted in our UWB laboratory to collect received signal strength measured data, which are used to determine the applicability of the proposed models. These 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 UWB channel environment.

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 analyze𝐾-distributed noise in electromagnetic scattering in [14]. Autoregressive stochastic models for the computer simulation of correlated Rayleigh fading processes are investigated in [15]. A first-order stochastic autoregressive model for a flat stationary wireless channel is introduced in [16]. Stochastic channel models based on SDEs for cellular and ad hoc networks have been presented in [12, 17, 18]. Some preliminary results using SDEs to model UWB channels were presented initially in [19]. The advantage of using SDE methods is based on the computational simplicity of the algorithm 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.

The paper is organized as follows. In Section 2, the general TV narrowband and UWB indoor wireless channel impulse responses are introduced. In Section 3, we show that the impulse responses for TV indoor wireless channels can be approximated in a mean-square sense as close as desired by impulse responses that can be realized by SDEs. The stochastic UWB channel models are developed in Section 4. In Section 5, experimental setup and numerical results are presented. Finally, Section 6 provides concluding remarks.

2. The General Time-Varying Impulse Response for Indoor Wireless Channels

The general TV impulse response (in complex baseband) of an indoor wireless fading channel is typically represented by Saleh-Valenzuela (SV) model given as [20] 𝐶(𝑡;𝜏)=^𝐿𝐾_𝑙=1^𝑙_𝑘=1𝑎_𝑘𝑙(𝑡,𝜏)exp𝑗𝜙_𝑘𝑙(𝑡,𝜏)×𝛿𝑡−𝑇_𝑙(𝑡)−𝜏_𝑘𝑙(𝑡), (1) where𝐶(𝑡;𝜏)is the impulse response of the channel at time𝑡, due to an impulse applied at time𝑡−𝜏,𝑎_𝑘𝑙(𝑡,𝜏)and𝜙_𝑘𝑙(𝑡,𝜏)are, respectively, the random TV tap weight and phase of the𝑘th component in the𝑙th cluster,𝑇_𝑙(𝑡)is the delay of the𝑙th cluster,𝜏_𝑘𝑙(𝑡)is the delay of the𝑘th multipath component (MPC) relative to the𝑙th cluster arrival𝑇_𝑙(𝑡),𝛿(⋅)is the Dirac delta function,𝐾_𝑙is the total number of MPCs within the𝑙th cluster, and𝐿is the total number of clusters that can either be assumed fixed [21] or considered to be a random variable [7]. Let𝑠(𝑡)be the transmitted signal; the received signal is then given by 𝑦(𝑡)=^𝐿𝐾_𝑙=1^𝑙_𝑘=1𝑎_𝑘𝑙(𝑡,𝜏)exp𝑗𝜙_𝑘𝑙(𝑡,𝜏)×𝑠𝑡−𝑇_𝑙(𝑡)−𝜏_𝑘𝑙(𝑡)+𝑣(𝑡), (2) where𝑣(𝑡)is the measurement noise process.

For narrowband systems, complex Gaussian fading is conventionally used to describe the small-scale fading. More precisely, the equivalent complex baseband representation consists of Rayleigh-distributed amplitude and uniformly distributed phase. This can be related theoretically to the fact that a large number of multipath components fall into each resolvable delay bin, so that the central limit theorem is valid [3]. Therefore,𝑎_𝑘𝑙(𝑡,𝜏)and𝜙_𝑘𝑙(𝑡,𝜏)are statistically independent Rayleigh and uniform (over[0,2𝜋]) random processes, respectively [20].

In UWB systems, the central limit theorem is not valid, and a number of alternative amplitude distributions have been proposed in the literature. The most common empirically determined amplitude distribution in many UWB environments is Nakagami distribution, which is observed in [6, 8] and considered in the IEEE 802.15.4a standard [7]. Therefore, in UWB systems,𝑎_𝑘𝑙(𝑡,𝜏)and𝜙_𝑘𝑙(𝑡,𝜏)are statistically independent Nakagami and uniform random processes, respectively. In the next section, the corresponding impulse response with𝑎_𝑘𝑙(𝑡,𝜏)and𝜙_𝑘𝑙(𝑡,𝜏)for the time-varying indoor wireless channels in (1) is approximated in a mean-square sense as close as desired by SDEs.

3. Approximating the Time-Varying Impulse Response for Indoor Wireless Channels by SDEs

Now, we want to represent the TV IR in (1) with a stochastic state-space form in order to allow well-developed tools of estimation and identification to be applied to this class of problems. The following theorem states a necessary and sufficient condition for the realization of the TV IR.

Theorem 1 (see [10]). The impulse response𝐶(𝑡;𝜏)of a TV system has a stochastic state-space realization if and only if it is factorizable; that is, there exist functions𝑔(⋅)and𝑓(⋅)such that for all𝑡and𝜏, one has 𝐶(𝑡;𝜏)=𝑔(𝑡)𝑓(𝜏). (3)

It is readily seen from the expression of the IR𝐶(𝑡;𝜏)of the indoor wireless channels in (1) that in general it is not factorizable in the form (3) since𝑎_𝑘𝑙(𝑡,𝜏)and𝜙_𝑘𝑙(𝑡,𝜏)are arbitrary functions of𝑡and𝜏. However, one will show that in general𝐶(𝑡;𝜏)can be approximated as close as desired by a factorizable IR function.

Theorem 2. In general, the IR𝐶(𝑡;𝜏)of the indoor wireless channel in (1) can be approximated as close as desired by a factorizable function.

Proof. The IR𝐶(𝑡;𝜏)of the indoor wireless channel has finite energy; that is, 𝐶(𝑡;𝜏)∈𝐿²([0,∞)×[0,∞)), (4) where𝐿²([0,∞)×[0,∞))is the Hilbert space of square integrable complex valued functions defined on[0,∞)×[0,∞)with the norm 𝑓²₂∶=_{[)0,∞)[0,∞}𝑓(𝑡;𝜏)²𝑓()𝑑𝜏𝑑𝑡<∞,𝑡;𝜏∈𝐿²([0,∞)×[0,∞)). (5) Likewise define𝐿²([0,∞))as the standard Hilbert space of square integrable complex valued functions defined on[0,∞)under the norm ‖𝑥‖²₂∶=^∞₀|𝑥(𝑡)|²𝑑𝑡,𝑥∈𝐿²([0,∞)). (6)
The space𝐿²([0,∞))contains all finite energy signals defined on[0,∞). The IR𝐶(𝑡;𝜏)of the channel has a finite energy and belongs to𝐿²([0,∞)×[0,∞)); that is, ‖𝐶‖²₂∶=_{[)0,∞)[0,∞}|𝐶(𝑡;𝜏)|²𝑑𝜏𝑑𝑡<∞. (7) Since the transmitted and received signals are finite energy signals, the IR can be viewed as an integral operator mapping transmitted signals in𝐿²([0,∞))into𝐿²([0,∞)); that is, if𝑠∈𝐿²([0,∞)), then 𝑦(𝑡)=^∞₀𝐶(𝑡;𝜏)𝑠(𝑡−𝜏)𝑑𝜏∈𝐿²([0,∞)). (8) The tensor space 𝐿²([0,∞))⊗𝐿²𝐹()([0,∞))∶=𝑡,𝜏∈𝐿²)×[∶𝐹()=([0,∞0,∞))𝑡,𝜏^𝑛_𝑘=1𝛼_𝑘(𝑡)𝜙_𝑘(𝜏),𝛼_𝑘(𝑡)∈𝐿²,𝜙([0,∞))_𝑘(𝜏)∈𝐿²([0,∞)),∀𝑛integer (9) is dense in𝐿²([0,∞)×[0,∞)); that is, for any𝜀>0, there exist{𝛼_𝑘}^𝑛₁,{𝜙_𝑘}^𝑛₁⊂𝐿²([0,∞))such that [22] 𝜇_𝑛∶=𝐶(𝑡;𝜏)−^𝑛_𝑘=1𝛼_𝑘(𝑡)𝜙_𝑘(𝜏)₂≤𝜀 (10) and𝜇_𝑛→_𝑛→∞0. This implies that in the‖⋅‖₂-norm,𝐶(𝑡;𝜏)can be approximated to any desired accuracy by an IR of the form∑^𝑛_𝑘=1𝛼_𝑘(𝑡)𝜙_𝑘(𝜏), which is factorizable by putting𝑔(𝑡)∶=𝛼₁(𝑡)𝛼₂(𝑡)⋯𝛼_𝑛(𝑡)and𝑓(𝜏)∶=𝜙₁(𝜏)𝜙₂(𝜏)⋯𝜙_𝑛(𝜏)^𝑇, where𝑇denotes vector or matrix transpose.
The optimal approximation of𝐶(𝑡;𝜏)by functions in𝐿²([0,∞))⊗𝐿²([0,∞))corresponding to (10) can be written as 𝜇^∗_𝑛∶=inf_{𝛼𝑘(𝑡)∈𝐿2,𝜙([0,∞))𝑘(𝜏)∈𝐿2([0,∞))}𝐶(𝑡;𝜏)−^𝑛_𝑘=1𝛼_𝑘(𝑡)𝜙_𝑘(𝜏)₂. (11) For arbitrary𝑛, expression (11) is nothing but the shortest distance between the impulse function𝐶(𝑡;𝜏)and the space𝐿²([0,∞))⊗𝐿²([0,∞)); that is,𝜇^∗_𝑛=dist(𝐶(𝑡;𝜏),𝐿²([0,∞))⊗𝐿²([0,∞))). The problem is to find the minimizing functions{𝛼_𝑘}^𝑛₁,{𝜙_𝑘}^𝑛₁. This problem has been solved in [22] for arbitrary positive𝑛. However, for fixed positive𝑛, the problem becomes the shortest distance, denoted by𝜇^𝑜_𝑛∶=dist(𝐶(𝑡;𝜏),𝑆), from𝐶(𝑡;𝜏)to the set 𝑆∶=𝐹(𝑡,𝜏)=^𝑛_𝑘=1𝛼_𝑘(𝑡)𝜙_𝑘(𝜏),𝛼_𝑘(𝑡)∈𝐿²,𝜙([0,∞))_𝑘(𝜏)∈𝐿²([0,∞)),𝑛>0fixed. (12) Note that𝑆is not a subspace and is not a convex set since it is not closed under addition. Therefore, the argument presented in [22] does not hold anymore since the orthogonal projection onto the set𝑆is not linear. Following [22], the impulse response is viewed as an integral operator𝑇mapping transmitted signals from𝐿²([0,∞))into𝐿²([0,∞)); that is, if𝑠_𝑙∈𝐿²([0,∞)), then𝑦_𝑙(𝑡)∈𝐿²([0,∞)), where 𝑦_𝑙(𝑡)=𝑇𝑠_𝑙(𝑡)∶=^∞₀𝐶(𝑡;𝜏)𝑠_𝑙(𝑡−𝜏)𝑑𝜏. (13) Since the impulse response is finite energy, the operator𝑇is a Hilbert-Schmidt or a trace class 2 operator [22, 23]. Let us denote the class of Hilbert-Schmidt operators acting from𝐿²([0,∞))into𝐿²([0,∞))by𝐶₂and the Hilbert-Schmidt norm‖⋅‖_HSis defined by ‖𝑇‖_HS=_{[0,∞)[0,∞)}|𝐶(𝑡;𝜏)|²𝑑𝜏𝑑𝑡,𝑇∈𝐶₂. (14) The operator𝑇admits a spectral factorization of the form [22, 23] 𝑇=^∞_𝑖=1𝜆_𝑖𝜈_𝑖⊗𝜓_𝑖, (15) where⊗is the tensor product,𝜆_𝑖>0with𝜆_𝑖≥𝜆_𝑖+1,𝑖=1,2,…,and both{𝜈_𝑖}^∞₁and{𝜓_𝑖}^∞_𝑖=1are orthonormal sequences in𝐿²([0,∞))and are given by 𝜈_𝑖(𝑡)=^∞₀𝐶(𝑡;𝜏)𝜓_𝑖(𝜏)𝑑𝜏,𝜓_𝑖(𝜏)=^∞₀𝐶(𝑡;𝜏)𝜈_𝑖(𝑡)𝑑𝑡. (16) The sum (15) has either a finite or countably infinite number of terms. The above representation is unique. The Hilbert-Schmidt norm of𝑇is also given by ‖𝑇‖_HS=_𝑖𝜆²_𝑖<∞. (17) The spectral factorization (15) yields the following representation for the impulse response𝐶(𝑡;𝜏)[23]: 𝐶(𝑡;𝜏)=^∞_𝑖=1𝜆_𝑖𝜈_𝑖(𝑡)𝜓_𝑖(𝜏). (18) It follows that the minimum in (11) is given by taking. 𝛼_𝑘(𝑡)=𝜆_𝑘𝜈_𝑘(𝑡),𝜙_𝑘(𝜏)=𝜓_𝑘(𝜏),𝑘=1,2,…,𝑛. (19) To further illustrate this result, note that{𝜈_𝑖}^∞_𝑖=1and{𝜓_𝑖}^∞_𝑖=1are orthonormal systems, and simple computations yield 𝐶(𝑡;𝜏)−^𝑛_𝑘=1𝜆_𝑘𝜈_𝑘(𝑡)𝜓_𝑘(𝜏)²₂=‖𝐶(𝑡;𝜏)‖²₂−^𝑛_𝑘=1𝜆²_𝑘. (20) Schmidt in [24] showed that for any other functions𝑓_𝑖(𝑡),𝑔_𝑖(𝜏)∈𝐿²([0,∞)),𝑖=1,2,…,𝑛, the following inequality holds: 𝐶(𝑡;𝜏)−^𝑛_𝑘=1𝑓_𝑘(𝑡)𝑔_𝑘(𝜏)²₂≥‖𝐶(𝑡;𝜏)‖²₂−^𝑛_𝑘=1𝜆²_𝑘, (21) and therefore the minimum in (11) is given by (19). The optimal approximation follows as 𝜇^𝑜_𝑛=𝐶(𝑡;𝜏)−^𝑛_𝑘=1𝜆_𝑘𝜈_𝑘(𝑡)𝜓_𝑘(𝜏)₂=^∞_𝑘=𝑛+1𝜆²_𝑘 (22) and𝜇^𝑜_𝑛→_𝑛→∞0. That is, by increasing𝑛, the RHS of (22) can be made arbitrarily small. In other words, for large enough𝑛, the following approximation is optimal in a mean-square sense: 𝐶(𝑡;𝜏)≈^𝑛_𝑖=1𝜆_𝑖𝜈_𝑖(𝑡)𝜓_𝑖(𝜏) (23) and is factorizable.

The corresponding SDE is then given by [10] 𝑑𝑋(𝑡)=𝑓(𝑡)𝑑𝑊(𝑡),𝑦(𝑡)=𝑔(𝑡)𝑋(𝑡), (24) where𝑋(𝑡)is the state of the channel and𝑊(𝑡)is the standard Brownian motion. Since state-space realizations of impulse responses are not unique [10], a realization of the following form in terms of the inphase and quadrature components for the𝑘th path within the𝑙th cluster can be used as [12, 17] 𝑑𝑋^𝐼_𝑘𝑙(𝑡)=𝐴^𝐼_𝑘𝑙(𝑡)𝑋^𝐼_𝑘𝑙(𝑡)𝑑𝑡+𝐵^𝐼_𝑘𝑙(𝑡)𝑑𝑊^𝐼_𝑘𝑙(𝑡),𝐼_𝑘𝑙(𝑡)=𝐻^𝐼_𝑘𝑙(𝑡)𝑋^𝐼_𝑘𝑙(𝑡),𝑑𝑋^𝑄_𝑘𝑙(𝑡)=𝐴^𝑄_𝑘𝑙(𝑡)𝑋^𝑄_𝑘𝑙(𝑡)𝑑𝑡+𝐵^𝑄_𝑘𝑙(𝑡)𝑑𝑊^𝑄_𝑘𝑙(𝑡),𝑄_𝑘𝑙(𝑡)=𝐻^𝑄_𝑘𝑙(𝑡)𝑋^𝑄_𝑘𝑙(𝑡), (25) where𝐼_𝑘𝑙(𝑡)and𝑄_𝑘𝑙(𝑡)are, respectively, the inphase and quadrature component processes,𝑋^𝐼_𝑘𝑙(𝑡)and𝑋^𝑄_𝑘𝑙(𝑡)are, respectively, the state vectors of the inphase and quadrature components,{𝑊^𝐼_𝑘𝑙(𝑡)}_𝑡≥0and{𝑊^𝑄_𝑘𝑙(𝑡)}_𝑡≥0are two independent standard Brownian motions which correspond to the inphase and quadrature components, respectively, and𝐴^𝐼_𝑘𝑙(𝑡),𝐴^𝑄_𝑘𝑙(𝑡),𝐵^𝐼_𝑘𝑙(𝑡),𝐵^𝑄_𝑘𝑙(𝑡),𝐻^𝐼_𝑘𝑙(𝑡), and𝐻^𝑄_𝑘𝑙(𝑡)are matrices of appropriate dimensions. Note that𝐼_𝑘𝑙(𝑡)and𝑄_𝑘𝑙(𝑡)are two independent Gaussian processes with zero-mean and equal variances; that is,Var(𝐼_𝑘𝑙(𝑡))=Var(𝑄_𝑘𝑙(𝑡))=Var(𝑁_𝑘𝑙(𝑡)), where𝑁_𝑘𝑙(𝑡)is either𝐼_𝑘𝑙(𝑡)or𝑄_𝑘𝑙(𝑡)[13]. They are related to the tap weight𝑎_𝑘𝑙(𝑡,𝜏)and phase𝜙_𝑘𝑙(𝑡,𝜏)by the expressions𝑎_𝑘𝑙(𝑡,𝜏)=𝐼²_𝑘𝑙(𝑡,𝜏)+𝑄²_𝑘𝑙(𝑡,𝜏)and𝜙_𝑘𝑙(𝑡,𝜏)=arctan(𝑄_𝑘𝑙(𝑡,𝜏)/𝐼_𝑘𝑙(𝑡,𝜏)), respectively. In the next section, the stochastic narrowband and UWB indoor channel models are developed.

4. Stochastic State-Space Models for UWB Indoor Wireless Channels

In narrowband systems, the equivalent indoor complex baseband representation consists of Rayleigh-distributed amplitude and uniformly distributed phase [20]. Thus, the stochastic narrowband indoor state-space channel model can be represented as 𝑑𝑋_𝑘𝑙(𝑡)=𝐴_𝑘𝑙(𝑡)𝑋_𝑘𝑙(𝑡)𝑑𝑡+𝐵_𝑘𝑙(𝑡)𝑑𝑊_𝑘𝑙(𝑡),𝑦(𝑡)=^𝐿𝐾_𝑙=1^𝑙_𝑘=1𝐶^𝐼_𝑘𝑙(𝑡)𝑋_𝑘𝑙(𝑡)²+𝐶^𝑄_𝑘𝑙(𝑡)𝑋_𝑘𝑙(𝑡)²×𝑒^{𝑗arctan(𝐶𝑄𝑘𝑙(𝑡)𝑋𝑘𝑙(𝑡)/𝐶𝐼𝑘𝑙(𝑡)𝑋𝑘𝑙(𝑡))}×𝑠𝑡−𝑇_𝑙(𝑡)−𝜏_𝑘𝑙(𝑡)+𝐷(𝑡)𝑣(𝑡), (26) where 𝑋_𝑘𝑙(𝑡)=𝑋^𝐼_𝑘𝑙(𝑡)^𝑇𝑋^𝑄_𝑘𝑙(𝑡)^𝑇𝑇,𝑊_𝑘𝑙(𝑡)=𝑊^𝐼_𝑘𝑙(𝑡)^𝑇𝑊^𝑄_𝑘𝑙(𝑡)^𝑇𝑇,𝐴_𝑘𝑙(𝑡)=𝐴^𝐼_𝑘𝑙(𝑡)00𝐴^𝑄_𝑘𝑙(𝑡),𝐵_𝑘𝑙(𝑡)=𝐵^𝐼_𝑘𝑙(𝑡)00𝐵^𝑄_𝑘𝑙(𝑡),𝐶^𝐼_𝑘𝑙(𝑡)=𝐻^𝐼_𝑘𝑙(𝑡)0,𝐶^𝑄_𝑘𝑙(𝑡)=0𝐻^𝑄_𝑘𝑙(𝑡,𝐼_𝑘𝑙)(𝑡,𝜏)=𝐶^𝐼_𝑘𝑙(𝑡)𝑋_𝑘𝑙(𝑡),𝑄_𝑘𝑙(𝑡,𝜏)=𝐶^𝑄_𝑘𝑙(𝑡)𝑋_𝑘𝑙(𝑡), (27) where𝑣(𝑡)is the measurement noise which is assumed to be Gaussian with zero-mean and unit variance and the tap weight process𝑎_𝑘𝑙(𝑡,𝜏)=(𝐶^𝐼_𝑘𝑙(𝑡)𝑋_𝑘𝑙(𝑡))²+(𝐶^𝑄_𝑘𝑙(𝑡)𝑋_𝑘𝑙(𝑡))²and the phase process𝜙_𝑘𝑙(𝑡,𝜏)=arctan(𝐶^𝑄_𝑘𝑙(𝑡)𝑋_𝑘𝑙(𝑡)/𝐶^𝐼_𝑘𝑙(𝑡)𝑋_𝑘𝑙(𝑡))are independent Rayleigh- and uniform-distributed random processes, respectively. In this case, the tap weight process𝑎_𝑘𝑙(𝑡,𝜏)has the following statistics [25]: 𝐸𝑎_𝑘𝑙(𝑡,𝜏)=𝜋2Var𝑁_𝑘𝑙(𝑡),𝐸𝑎_𝑘𝑙(𝑡,𝜏)^𝑝=2Var𝑁_𝑘𝑙(𝑡)^𝑝/2Γ11+2𝑝,Var𝑎_𝑘𝑙(=𝜋𝑡,𝜏)2−2Var𝑁_𝑘𝑙(𝑡), (28) whereΓ(𝑝)is the gamma function.

In UWB systems, as mentioned earlier, the most common amplitude distribution of the received signal is Nakagami distribution [6–8]. Its probability density function is given by [26] 𝑓(𝑥)=2Γ(𝑚)𝑚Ω^𝑚𝑥^2𝑚−1−𝑚expΩ𝑥², (29) where𝑚≥0.5is the shape parameter,Γ(𝑚)is the Gamma function, andΩcontrols the spread of distribution. The𝑚-parameter is often modeled as a random variable [6]. For integer value of𝑚, the distribution describes𝑚orthogonal independent Rayleigh-distributed random variables. That is, for𝑀Rayleigh-distributed random variables𝑍_𝑖, the probability density function of random variable𝑌, defined as𝑌=∑^𝑀_𝑖=1𝑍²_𝑖, is given by a Nakagami distribution with parameter𝑚=𝑀[27].

Since multiple orthogonal independent Rayleigh-distributed random variables can generate Nakagami distribution, the stochastic UWB indoor state-space channel model can be represented by 𝑑𝑋_𝑖𝑘𝑙(𝑡)=𝐴_𝑖𝑘𝑙(𝑡)𝑋_𝑖𝑘𝑙(𝑡)𝑑𝑡+𝐵_𝑖𝑘𝑙(𝑡)𝑑𝑊_𝑖𝑘𝑙(𝑡),𝑦(𝑡)=^𝐿𝐾_𝑙=1^𝑙_𝑘=1^𝑀𝑘𝑙_𝑖=1𝐶^𝐼_𝑖𝑘𝑙(𝑡)𝑋_𝑖𝑘𝑙(𝑡)²+𝐶^𝑄_𝑖𝑘𝑙(𝑡)𝑋_𝑖𝑘𝑙(𝑡)²×𝑒^{𝑗arctan(𝐶𝑄𝑖𝑘𝑙(𝑡)𝑋𝑖𝑘𝑙(𝑡)/𝐶𝐼𝑖𝑘𝑙(𝑡)𝑋𝑖𝑘𝑙(𝑡))}×𝑠𝑡−𝑇_𝑙(𝑡)−𝜏_𝑘𝑙(𝑡)+𝐷(𝑡)𝑣(𝑡), (30) where 𝑋_𝑖𝑘𝑙(𝑡)=𝑋^𝐼_𝑖𝑘𝑙(𝑡)^𝑇𝑋^𝑄_𝑖𝑘𝑙(𝑡)^𝑇𝑇,𝑊_𝑖𝑘𝑙(𝑡)=𝑊^𝐼_𝑖𝑘𝑙(𝑡)^𝑇𝑊^𝑄_𝑖𝑘𝑙(𝑡)^𝑇𝑇,𝐴_𝑖𝑘𝑙(𝑡)=𝐴^𝐼_𝑖𝑘𝑙(𝑡)00𝐴^𝑄_𝑖𝑘𝑙(𝑡),𝐵_𝑖𝑘𝑙(𝑡)=𝐵^𝐼_𝑖𝑘𝑙(𝑡)00𝐵^𝑄_𝑖𝑘𝑙(𝑡),𝐶^𝐼_𝑖𝑘𝑙(𝑡)=𝐻^𝐼_𝑖𝑘𝑙(𝑡)0,𝐶^𝑄_𝑖𝑘𝑙(𝑡)=0𝐻^𝑄_𝑖𝑘𝑙(𝑡,𝐼_𝑖𝑘𝑙)(𝑡,𝜏)=𝐶^𝐼_𝑖𝑘𝑙(𝑡)𝑋_𝑖𝑘𝑙(𝑡),𝑄_𝑖𝑘𝑙(𝑡,𝜏)=𝐶^𝑄_𝑖𝑘𝑙(𝑡)𝑋_𝑖𝑘𝑙(𝑡), (31) where𝐿is the total number of clusters,𝐾_𝑙is the total number of resolvable delay bins (paths), and𝑀_𝑘𝑙is the number of nonresolvable delay bins within the resolvable delay bin𝑘in cluster𝑙. In this case, the tap weight process𝑎_𝑘𝑙(𝑡,𝜏)has the following statistics [25]: 𝐸𝑎_𝑘𝑙(𝑡,𝜏)=2Var𝑁_𝑘𝑙(𝑡)Γ(1/2)2𝑀_𝑘𝑙+1Γ𝑀_𝑘𝑙,𝐸𝑎_𝑘𝑙(𝑡,𝜏)^𝑝=2Var𝑁_𝑘𝑙(𝑡)^𝑝/2Γ(1/2)2𝑀_𝑘𝑙+𝑝Γ𝑀_𝑘𝑙,Ω𝑝≥0,_𝑘𝑙=2Var𝑁_𝑘𝑙(𝑡Γ𝑀)^𝑘𝑙+1Γ𝑀_𝑘𝑙. (32)

It can be noticed in (30) that the received signal measurement,𝑦(𝑡), is a nonlinear function of the state variables of the model. The UWB channel parameters are estimated using the EM algorithm, and the inphase and quadrature components are estimated directly from received signal measurements using the EKF. A filter-based EM algorithm together with the EKF is employed to estimate the channel model parameters and states in (30). These filters use only the first- and second-order statistics and are also recursive and therefore can be implemented online. These algorithms have been recently utilized in [11–13] to estimate the channel parameters and states in narrowband environments, and therefore the formulations of these algorithms are not presented in this paper. Experimental results demonstrating the applicability of these algorithms in UWB indoor environments are discussed in the next section.

5. Experimental Setup and Numerical Results

A simple Gaussian pulse with clean pulse shape and narrow pulse width is chosen as UWB source signal. The experimental setup is similar to the one in [28]. It comprises a 300-picosecond Gaussian pulse (see Figure 1) that modulates a carrier signal centered at 8 GHz and is transmitted through an omnidirectional UWB antenna. Multiple directional Vivaldi subarray receiving antennas are located at distinct locations in an indoor environment to receive the modulated pulse signal. Each received modulated Gaussian pulse is amplified through a low noise amplifier (LNA) and then stored in a multichannel Tektronix TDS8200 sampling oscilloscope.

Figure 1 

Transmitted signal of a 300-picosecond Gaussian pulse shape.

The transmitter is located next to a metal wall. The receiving antenna is placed 10 cm away from the transmitter and then moved along the same direction away from the transmitter for 20 cm, 50 cm, 1 m, and 2 m. This scenario is demonstrated in Figure 2. In this environment, the transmitted signal suffers from reflection from the metal wall, ceiling, or floor or is scattered by the corner of the metal table, and then it reaches the receiver. Figure 3 shows the measured received signal which consists of three clusters. The measurement data are amplified by 30 dB and it can be seen in Figure 4 that the dominant path in each cluster is best fit to Nakagami distribution with different parameters. For example, the dominant path in the first cluster is best fit to Nakagami distribution with parameters𝑚=3andΩ= 0.28. This indicates that there are three nonresolvable delay bins within this path (i.e.,𝑀₁₁= 3). Figure 5 shows the measured and estimated received signals using the EM algorithm together with the EKF for the three clusters using a 4th-order model. It can be noticed that the UWB received signal has been estimated with very high accuracy and it takes little iteration for the estimation algorithm to converge. At a certain time instant, the system parameters for the dominant path in the 1st cluster, which consists of three nonresolvable delay bins, are estimated as follows: 𝐴₁₁₁=,𝐴01000.19280.0556000001000.1484−0.0966₂₁₁=,𝐴01000.29170.0556000001000.3563−0.0682₃₁₁=,𝐵01000.2014−0.0586000001000.1881−0.2669₂₁₁₁=,𝐵0.16990.24510.01220.00090.00040.09610.0287−0.00040.01220.00870.16280.02130.00770.02540.2959−0.0066₂₂₁₁=,𝐵0.00060.01820.14290.00770.0043−0.02560.0076−0.25970.07950.00560.00430.0218−0.01770.01920.1541−0.1632₂₃₁₁=,𝐶0.25610.00130.33100.00690.01290.0213−0.01070.09210.00440.00150.08240.00760.15410.14290.1632−0.3334_𝐼111=𝐶^𝐼₂₁₁=𝐶^𝐼₃₁₁=,𝐶1000_𝑄111=𝐶^𝑄₂₁₁=𝐶^𝑄₃₁₁=,𝐷0010₂=[0.8765]. (33)

Figure 2 

The indoor environment considered in our experiment.

Figure 3 

The measured UWB received signal.

(a)

(b)

(c)

Figure 4 

Histogram of measurement data for the dominant paths within the three clusters that are best fit to Nakagami distributions.

(a)

(b)

(c)

Figure 5 

Measured and estimated received signals using the EM algorithm combined with the EKF for the 1st, 2nd, and 3rd clusters.

Notice that the EM algorithm estimates𝐵²and𝐷²instead of𝐵and𝐷. Also note that the parameters𝐿,𝐾_𝑙,𝑀_𝑘𝑙,𝑇_𝑙, and𝜏_𝑘𝑙are assumed known through the estimation process. In fact, the parameters𝐿,𝐾_𝑙, and𝑀_𝑘𝑙can be estimated by various estimation algorithms such as minimum description length [29], multiple hypothesis testing [30], exponential fitting test [31], and linear piecewise variation [32]. And a sounding device is usually dedicated to estimate the time delay of each discrete path (i.e.,𝑇_𝑙and𝜏_𝑘𝑙) such as the rake receiver [33].

6. Conclusion

This paper describes a general scheme for extracting mathematical UWB indoor channel models from noisy received signal measurements. The UWB channel models are represented in stochastic state-space form, in which its system output produces Nakagami-distributed received signal strength and it is shown to approximate the general TV IR of the channel as close as desired. Experimental results indicate that the measured data can be regenerated with high accuracy.

Acknowledgments

This paper has been authored by employees of UT-Battelle, LLC, under Contract no. DE-AC05-00OR22725 with the US Department of Energy. Also, this work was supported in part by NSF Grant no. CMMI-1334094.