A Joint TOA/AOA Constrained Minimization Method ...

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[3] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [4] J. G. Foschini and M. G. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wirel. Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [5] J. G. Foschini, “Layered space–time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J., vol. 1, no. 2, pp. 41–59, Autumn 1996. [6] M. Sellathurai and J. G. Foschini, “Stratified diagonal layered space–time architectures: Signal processing and information theoretic aspects,” IEEE Trans. Signal Process., vol. 51, no. 11, pp. 2943–2954, Nov. 2003. [7] A. Lozano and C. Papadias, “Layered space–time receivers for frequency-selective wireless channels,” IEEE Trans. Commun., vol. 50, no. 1, pp. 65–73, Jan. 2002. [8] N. Al-Dhahir and A. H. Sayed, “The finite-length multi-input multioutput MMSE-DFE,” IEEE Trans. Signal Process., vol. 48, no. 10, pp. 2921–2936, Oct. 2000. [9] S. Qureshi, “Adaptive equalization,” Proc. IEEE, vol. 73, no. 9, pp. 1349– 1387, Sep. 1985. [10] S. Liu and Z. Tian, “Near-optimum soft decision equalization for frequency selective MIMO channels,” IEEE Trans. Signal Process., vol. 52, no. 3, pp. 721–733, Mar. 2004. [11] X. Wautelet, A. Dejonghe, and L. Vandendorpe, “MMSE-based fractional turbo receiver space–time BICM over frequency-selective MIMO fading channels,” IEEE Trans. Signal Process., vol. 52, no. 6, pp. 1804–1809, Jun. 2004. [12] J. L. J. Cimini, “Analysis and simulation of a digital mobile channel using orthogonal frequency division multiplexing,” IEEE Trans. Commun., vol. COM-33, no. 7, pp. 665–675, Jul. 1985. [13] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., vol. 33, no. 2, pp. 100–109, Feb. 1995. [14] R. Dinis, R. Kalbasi, and D. Falconer, “Iterative layered space– time receivers for single-carrier transmission over severe time dispersive channels,” IEEE Commun. Lett., vol. 8, no. 9, pp. 579–581, Sep. 2004. [15] B. Lu, G. Yue, and X. Wang, “Performance analysis and design optimization of LDPC-coded MIMO OFDM systems,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 348–361, Feb. 2004. [16] S. Ahmed, M. Sellathurai, T. Ratnarajah, and C. Cowan, “Low complexity iterative equalization for severe time dispersive MIMO channels,” in Proc. IEEE 40th Annu. Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, Oct. 2006, pp. 2102–2106. [17] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1989. [18] S. ten Brink, “Convergence of iterative decoding,” Electron. Lett., vol. 35, no. 13, pp. 806–808, May 1999. [19] S. ten Brink, “Convergence behaviour of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1727–1737, Oct. 2001. [20] A. Ashikhmin, G. Kramer, and S. ten Brink, “Extrinsic information transfer functions: Model and erasure channel properties,” IEEE Trans. Inf. Theory, vol. 50, no. 11, pp. 2657–2673, Nov. 2004. [21] M. Tüchler, R. Koetter, and A. C. Singer, “Turbo equalization: Principles and new results,” IEEE Trans. Commun., vol. 50, no. 5, pp. 754–766, May 2002. [22] P. Schniter, “Low-complexity equalization of OFDM in doubly selective channels,” IEEE Trans. Signal Process., vol. 52, no. 4, pp. 1002–1010, Apr. 2004. [23] D. Reynolds and X. Wang, “Low-complexity turbo-equalization for diversity channels,” Signal Process., vol. 81, no. 5, pp. 989–995, May 2001. [24] R. Koetter, A. C. Singer, and M. Tuchler, “Turbo equalization,” IEEE Signal Process. Mag., vol. 21, no. 1, pp. 67–80, Jan. 2004. [25] B. M. Hochwald and S. ten Brink, “Achieving near-capacity on a multipleantenna channel,” IEEE Trans. Commun., vol. 51, no. 3, pp. 389–398, Mar. 2003. [26] C. Schlegel and L. C. Perez, Trellis and Turbo Coding. New York: Wiley, 2004. [27] B. Lu, X. Wang, and Y. Li, “Iterative receivers for space–time block coded OFDM system in dispersive fading channels,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 213–225, Apr. 2002. [28] T. M. Cover and J. A. Thomas, Elements of Information Theory. Englewood Cliffs, NJ: Prentice–Hall, 1991.

A Joint TOA/AOA Constrained Minimization Method for Locating Wireless Devices in Non-Line-of-Sight Environment Saleh Al-Jazzar, Member, IEEE, Mounir Ghogho, Member, IEEE, and Desmond McLernon, Member, IEEE Abstract—Non-line-of-sight (NLOS) propagation degrades the performance of wireless location systems. Thus, developing algorithms that are robust to NLOS considerations is of great importance. Based on time-ofarrival (TOA) and angle-of-arrival (AOA) measurements, this paper introduces a new approach, which consists of incorporating the coordinates of dominant scatterers as unknowns in the location algorithm. It is assumed that the first arriving path signal at each base station (BS) experiences a single dominant scatterer, but the BSs are allowed to have different dominant scatterers. Locating the mobile station is accomplished by means of a nonlinear optimization procedure under nonlinear constraints. Two algorithms are proposed: The first algorithm assumes that hybrid TOA/AOA measurements are available at three BSs. In the second algorithm, the AOA is assumed to be available at the serving BS only. The performance of the proposed algorithms is assessed and compared with that of existing algorithms through extensive simulations. Index Terms—Dominant scatterer, joint time of arrival and angle of arrival, non-line-of-sight, wireless location.

I. I NTRODUCTION In emergency call systems, it is important to physically locate the caller’s phone as accurately as possible. In the landline phone case, it is easy to locate the caller through the switches and the already known connections in the network. In the wireless mobile phone case, the serving base station (BS) could easily be identified, but locating the caller within this BS is impossible without some location techniques, the most common of which uses either direction finding, ranging, or differential ranging. Direction finding, or angle-of-arrival (AOA) measurement, involves the use of directional antennas to determine the direction from which a signal was received at one or more receivers [1]–[3]. Location systems based on ranging are discussed in [3]–[6]. Methods for measuring the range include time-of-arrival (TOA), signal strength, and phase estimation. In this paper, we consider the use of hybrid TOA/AOA at three BSs. We choose measurements at three BSs only because of hearability limitations [7]–[9]. We also address the case where AOA measurement is available at the serving BS only. The primary wireless location error is caused by the non-line-ofsight (NLOS) effect. Due to the lack of a line-of-sight (LOS) path between the mobile station (MS) and a BS, the measured signal path length (or, equivalently, TOA) is generally much larger than the true or LOS path; thus, the TOA measurement will face a deviation error from its true value. This happens due to the reflection and/or diffraction in the channel. Unfortunately, traditional TOA algorithms have been developed based on LOS assumptions between the transmitter and receivers, and consequently, they perform very poorly when the TOAs/AOAs are corrupted by an NLOS error. Several approaches for mitigating NLOS error have been proposed in the literature [7]–[15]. In [7]–[9], the authors apply a statistical solution to a scattering model NLOS problem. The methods developed Manuscript received July 29, 2007; revised January 13, 2008 and February 23, 2008. First published April 18, 2008; current version published January 16, 2009. The review of this paper was coordinated by Dr. Y. Gao. S. Al-Jazzar is with the Department of Electrical and Computer Engineering, The Hashemite University, Zerqa 13115, Jordan. M. Ghogho and D. McLernon are with the School of Electronic and Electrical Engineering, University of Leeds, LS2 9JT Leeds, U.K. Digital Object Identifier 10.1109/TVT.2008.923670

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in [12]–[15] use TOA measurements to perform positioning, whereas, in [10] and [11], AOA measurements were used. Algorithms that use hybrid TOA/AOA measurements were presented in [16] and [17]. Most of the algorithms in the literature assume that either there is at least one BS with LOS [10], [11], [14], [15] or there are more than three BSs that can detect the signal from the MS [10]–[13], [15]. In many practical cases, measurements are available only from three BSs for a given MS, and usually, they are all NLOS. In this paper, a new approach is proposed to tackle the NLOS problem, as will be described next. The signal received from the MS at a BS consists of multiple delayed and attenuated versions of the transmitted signal. For location purposes, only the first arriving path is of interest. In the proposed approach, the following assumption is made: The first arriving path experiences a single dominant scatterer. Such an assumption was used in the literature in [18], with the difference that the approach in [18] uses multiple NLOS signals at one BS, whereas our approach uses single NLOS signals at multiple BSs. In the simulation section, different scattering models will be used to characterize the position of these dominant scatterers. The justification for assuming a single dominant scatterer follows from the fact that the BS in macrocells is installed at a high level compared to the MS. Thus, there are no significant scatterers surrounding the BS. In addition, a single dominant reflector scatterer may be considered realistic. The main idea behind the proposed approach is to incorporate the coordinates of the dominant scatterers as unknown parameters that need to be estimated to improve the estimation of the MS’s coordinates. This is accomplished by means of a nonlinear optimization procedure utilizing TOA and AOA measurements, under nonlinear geometrical constraints. Two algorithms are proposed: The first algorithm assumes that the hybrid TOA/AOA measurements are available at three BSs. In the second algorithm, AOA is assumed to be available only at the serving BS. Utilizing the dominant scatterers’ coordinates in obtaining the MS location is the major difference between the two algorithms proposed here and the algorithm proposed in [16]. In addition, compared with the algorithm in [17], the two proposed algorithms do not require a storage database of the scatterers’ coordinates around the BS. The proposed algorithms do not require the BSs to be within the LOS of the MS, and they can be applied to the case where the measurements are available at three BSs only. The rest of this paper is organized as follows: Section II presents the problem formulation. Algorithms based on hybrid TOA/AOA measurements are developed in Section III. Discussion on identifiability and convexity is presented in Section IV. Simulation results are given in Section V, followed by concluding remarks in Section VI.

469

Fig. 1. Layout of the MS and three-BS geometry.

The estimated distance at BSi (between BSi and MS) would be ˆ li = Ri + ηi + μi

(3)

where ηi is the effective NLOS component, and μi is the measurement noise, which can be considered as additive white Gaussian noise. Since NLOS causes the signal to arrive from a path that is longer than the true distance, then ηi ≥ 0. A layout of the MS and multiple-BS geometry is shown in Fig. 1. Next, we will propose a method for accurately locating the MS in the presence of this NLOS error. III. TOA/AOA C ONSTRAINED A LGORITHM (TA-CA) Two versions of this algorithm will be derived, given the available measurements: The first version, which is referred to as the TA-CA6 algorithm, uses six measurements (three TOA and three AOA measurements) at all the three BSs that can hear the MS. The other version, which is referred to as the TA-CA4 algorithm, assumes that four measurements are available (three TOA measurements are available at the three BSs, and only one AOA measurement is available at the serving BS), due to the fact that the AOA measurement may not be reliable at the nonserving BSs [16]. A. TA-CA Using TOA/AOA Measurements From All BSs (TA-CA6)

II. P ROBLEM F ORMULATION Assume that we have a transmitting MS (with location ψ m = (xms , yms )) and TOA measurements are taken at a receiving BSi (with location ψ i = (xi , yi )). The range is derived from the TOA and then projected onto the (x, y) plane, because the elevation of the antennas of the BSs increases the measured range. Then, the true distance between the MS and the BS is Ri = ψ m − ψ i  =



(xms − xi )2 + (yms − yi )2 .

If we assume that the scatterer coordinates are ψ s,i = (xs,i , ys,i ) and it is the point at which the signal is reflected as it propagates from the MS to the ith BS, then the propagation distance li due to the reflection off the scatterer (without considering the measurement noise) consists of the sum of the distance between the ith BS and the scatterer position ψ s,i and the distance between the scatterer position and the MS position ψ m , i.e., li = Ri + ηi = ψ m − ψ s,i  + ψ s,i − ψ i  = dms,i + ds,i

(1) where

The true AOA is

y

− yi φi = arctan xms − xi ms

 .

(2)

dms,i = ds,i =

 (x − xs,i )2 + (yms − ys,i )2  ms (xs,i − xi )2 + (ys,i − yi )2 .

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(4)

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This is shown in Fig. 1. This is true for any scattering model. NLOS error ηi is considered as the major source of error, and its effect is much more severe than measurement error μi . A typical value for ηi has been measured, and it can average between 500 and 700 m [19]. Thus, for the three BSs, there will be three equations from (4) with eight unknowns, six of which are the scatterers’ coordinates for the three BSs, i.e., ψ s,i for i = 1, 2, 3, and two of which are the MS coordinates ψ m . As for the AOA measurements, the angle of the received signal will actually be the bear angle of the scatterer with respect to the BS, as shown in Fig. 1. For the three BSs, the AOA at each BS is

 θi = arctan

ys,i − yi xs,i − xi

 .

(5)

For the three BSs, (5) provides three equations. Another set of equations can be formulated by looking at the triangle (ψ i , ψ m , ψ s,i ) in Fig. 1. The following equation can be deduced:

the distance between the estimated MS position (xms , yms ) and the BSi position should be less than or equal to li , i.e.,



(xms − xi )2 + (yms − yi )2 ≤ li .

In fact, the single equality constraint on dms,1 from (6) is equivalent to dms,1 ≤ l1 . Therefore, the equality constraint is not explicitly used in the optimization algorithm. Looking at (l1 , l2 , l3 , θ1 ), the number of equations is four, and the number of unknowns is eight (the MS’s and scatterers’ coordinates). Furthermore, the inequality constraints in (8) help find this solution. Again, the identifiability issue is discussed in Section IV. Since the measurements are noisy, we use the following cost function: FTA−CA4 (xms , yms , Ψs ) =

3 



3 

αi (dms,i + ds,i − ˆ li )2

i=1

+

3 

 βi

 arctan

i=1

ys,i − yi xs,i − xi



2

FTA−CA6 (xms , yms , Ψs )

Subject to

Ceq,i (xms , yms , Ψs )

ys,1 − y1 xs,1 − x1



2 − θˆ1

(9)

where αi , for (i = 1, 2, 3), and β are the weight factors that depend on the measurement accuracy. Therefore, the TA-CA4 algorithm is formulated as follows: Minimize

FTA−CA4 (xms , yms , Ψs )

Subject to

Cneq,i (xms , yms , Ψs ) for i = 1, 2, 3

(10)

where Cneq,i is the inequality constraint in (8). The TA-CA4 algorithm here requires the same measurements assumed by the HTA algorithm in [16] (three TOA measurements and an AOA measurement at the serving BS), but, in our TA-CA4 algorithm, we make use of the assumption that the signal is reflected off a single dominant reflector scatterer [which is implicitly shown in (4)]. A comparison between the algorithms will be given in Section VI.

− θˆi IV. D ISCUSSIONS ON I DENTIFIABILITY AND C ONVEXITY

subject to the equality constraints in (6), where Ψs is the set of all the scatterers’ coordinates for the different BSs, and αi and βi , for i = 1, 2, 3, are the weighting parameters that reflect the accuracy of the measurements. The identifiability issue is discussed in Section IV. The TA-CA6 algorithm (the first proposed algorithm) can therefore be summarized as follows: Minimize

 arctan

(6)

with θi and φi indicated in Fig. 1. Equation (6) provides three independent equations but introduces three unknowns φi for the three BSs. However, φi can be written in terms of the unknown MS as (2). Therefore, the number of unknowns is back to eight, and for the three BSs, we have six nonlinear equations (three from (4) and three from (5) relating the TOA/AOA parameters to the unknowns) and three equality constraints from (6). Since the TAO/AOA measurements are noisy, we propose the following nonlinear least-squares method. The unknown parameters (xms , yms ) are estimated by minimizing the following objective function:

FTA−CA6 (xms , yms , Ψs ) =

αi (dms,i + ds,i − ˆ li )2

i=1

+β d2ms,i = d2s,i + Ri2 − 2ds,i Ri cos (θi − φi )

(8)

for i = 1, 2, 3

(7)

where Ceq,i is the equality constraint in (6). Since this is a nonlinear optimization problem, we resort to numerical techniques. Here, we use fmincon.m in Matlab, which finds the least-squares error solution for nonlinear equations with nonlinear constraints. B. TA-CA Using the TOA Measurements at All BSs and the AOA Measurement at the Serving BS (TA-CA4) Here, we assume that AOA measurements are available at the serving BS only (as in [16]). In this case, the available equations are (l1 , l2 , l3 ) from (4), θ1 from (5), and dms,1 from (6). In addition, constraints on the MS position can be deduced by using the fact that

For identifiability, we consider the noise-free case. Since the scatterers’ positions are unknown, each range/distance measurement li implies that the scatterer and the MS are inside a disc with radius li and centered around the ith BS (see Fig. 2). This is unlike the case of LOS, where the possible positions of the MS are on a circle. Thus, given three range measurements, the possible positions of the MS are confined within the area that is common to the three discs. This area becomes smaller as the distance between the scatterers and the MS decreases and becomes a single point when all measurements are within LOS. If one of the range measurements is within LOS but the others are not, then the common area, or solution area, becomes an arc of a circle. When, in addition to the range measurements, AOA measurements are available, the solution area becomes even smaller. Indeed, each AOA measurement constrains the associated scatterer to be on a straight line, and this reduces the area of the possible positions of the MS. This is the reason our TA-CA6 algorithm, which uses three AOA measurements, gives a better performance than the TA-CA4 algorithm, which uses only one AOA measurement. Further constraints could be used to reduce the solution area. One of such constraints would be to assume that the scatterers are much closer to the MS than to the BSs, which is often the case in practice. For example, one can assume that the distance between each scatterer and the MS is less than t (t  1) times the distance between the scatterer and the corresponding BS. In this case, each range measurement li

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Fig. 2.

Geometry of the common area layout.

471

Fig. 3. Comparisons of the location estimation error using the HTA, TA-CA4, and TA-CA6 algorithms in DOS environment for various scatterer radii.

implies that the scatterer and the MS are inside an annular region, instead of a disc, and this annular region is defined as the common area between a disc with radius li and a disc with radius (1 − 2t)li . Since, in the general case, the position of the MS cannot uniquely be identified, it statistically makes sense to choose the “centroid” of the common area, which is defined by the measurements and the constraints, as a solution. Since the common area can be made small by exploiting many measurements/constraints, this solution appears to give a better performance than existing methods. In our methods, we use the Matlab minimization function fmincon.m to estimate the MS position. Since this may provide a local optimum, an initialization is required. Indeed, the minimized cost functions are generally not convex. Recall that, even in the case of LOS, the minimization problem is not convex [20]. It is worth pointing out that, in the case of high signal-to-noise ratio, the cost functions will be flat over the solution area. Extensive simulations show that the cost functions are convex over the region including the solution area. V. S IMULATION R ESULTS The results in this section are averaged over 1000 ensemble runs. The locations of the BSs are (0, 0), (8.66, 0), and (4.33, 7.5), with all units in kilometers. The MS location is (xms , yms ), where xms = 4.33 · (u + 0.5) (in kilometers), yms = 0.5 · (7.5 + u) (in kilometers), and u is a random variable uniformly distributed in the region [0, 1]. Ring of scatterers (ROS) and disk of scatterers (DOS) environments are assumed to model the NLOS effect [21]. In the ROS model, the scatterers are located on a ring centered around the MS with radius Rr . The angular distribution of the transmitted (or, equivalently, first arriving) signal from the MS is uniformly distributed over [0, 2π]. In addition, the scatterers are uniformly distributed around the ring. In the DOS model, the scatterers are located on a solid circular disk of fixed radius Rd , with the MS at the center. The distance to a scatterer from the MS rDOS is uniformly distributed over [0, Rd ], and the angle is uniformly distributed over [0, 2π]. The range measurement error is assumed to have 50-m2 variance, and the AOA error was modeled as a uniform error with a 2.5◦ standard deviation (unless stated otherwise). The weight factors (α1 , α2 , α3 , β, γ) in (9) are all set to 1. Both the proposed versions of TA-CA are compared with the HTA algorithm in [16]. Figs. 3 and 5 present the location estimation errors of the algorithms versus the different scatterer radii in DOS and ROS environments,

Fig. 4. Comparisons of the location estimation error using the HTA, TA-CA4, and TA-CA6 algorithms in ROS environment for various scatterer radii.

Fig. 5. Comparisons of the CDF of the estimation error using the HTA, TA-CA4, and TA-CA6 algorithms in DOS environment.

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algorithm uses four measurements and the TA-CA6 algorithm uses six measurements to locate the MS. Both TA-CA algorithms are based on the estimation of the MS location by utilizing the unknown dominant scatterers’ coordinates in the location estimation process. The TA-CA6 algorithm utilizes the dominant scatterer coordinates, together with TOA/AOA measurements at three BSs to form an optimization problem, where an objective function is minimized under equality constraint. The TA-CA4 algorithm functions similarly, with the difference of using one AOA measurement, taken at the serving BS. Simulation results showed that the new algorithms compare favorably against an existing localization algorithm. R EFERENCES

Fig. 6. Comparisons of the CDF of the estimation error using the HTA, TACA4, and TA-CA6 algorithms in ROS environment.

Fig. 7. Comparisons of the location estimation error using the HTA, TA-CA4, and TA-CA6 algorithms in DOS environment for various AOA noise.

respectively. Figs. 5 and 6 show the error cumulative distribution function (CDF) of the different algorithms, where the radius of scatterer was set to 150 m for the serving BS and 250 m for the other BSs in DOS and ROS environments, respectively. All figures indicate that the TA-CA6 algorithm gives a better performance than the TA-CA4 algorithm, and both were better than the HTA algorithm. In addition, Fig. 7 shows the CDF of the estimation error for the different algorithms with an AOA noise standard deviation of 2◦ . Both figures show that the TA-CA6 algorithm outperformed the TA-CA4 algorithm, which, in turn, was better than the HTA algorithm. The reason both of the TA-CA algorithms are performing better than the HTA algorithms is the fact that they use the unknown dominant scatterers’ coordinates in their localization algorithms. The reason the TA-CA6 algorithm is performing better than the TA-CA4 algorithm is because it uses more AOA measurements, which enables a more accurate solution to be obtained. VI. C ONCLUSION Two algorithms called TA-CA4 and TA-CA6 are proposed and investigated via simulation for mitigating the effect of NLOS error. The reason for calling them TA-CA4 and TA-CA6 is that the TA-CA4

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