Selective Hybrid RSS/AOA Approximate Maximum ... - Semantic Scholar
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Selective Hybrid RSS/AOA Approximate Maximum Likelihood Mobile intra cell Localization Leila Gazzah, Leila Najjar and Hichem Besbes Carthage University, Lab. COSIM, Higher School of Communication of Tunis, Tunisia Email: {gazzah.leila, leila.najjar, hichem.besbes}@supcom.rnu.tn
Abstract—We present in this paper two selective combined schemes for mobile location. These schemes are based respectively on the Linear Least Squares (LLS) and the Approximate Maximum Likelihood (AML) approaches. The proposed positioning techniques apply to intra cell localization in which the serving base station (BS) Angle Of Arrival (AOA) and the best two Received Signal Strength (RSS) measurements taken at some reference points (with known positions) are combined for mobile user localization. Such nodes are either localized users or anchors. The proposed approach achieves better performance compared to the selective RSS LLS method. The use of only the two best RSS measures reduces the location bias effect induced by propagation. It also decreases the computational burden implied by multiple RSS combining. Simulation results show that, for certain scenarios, the proposed selective hybrid RSS/AOA LLS (respectively AML) with two nodes outperforms the selective RSS LLS with up to seven nodes (respectively to ten nodes)and achieves equivalent performances to AOA-based approaches. Keywords: Radio Signal Strength, Angle-Of-Arrival, intra-cell, selective, hybrid, Linear Least Squares, Approximate Maximum Likelihood.
I. I NTRODUCTION By offering multiple innovative applications, radiolocation has gained an increasing interest in wireless communication research field. Fleet management, interactive applications and location-based billing are some examples of new location based services triggered by mobile positioning technology [1]. Radiolocation can be derived from measurements of Time of Arrival (TOA), Angle of Arrival (AOA), Time Difference of Arrival (TDOA) and Received Signal Strength (RSS) [2]. In this work, we investigate in the intra cell localization framework the potential enhancement in geolocation that can be obtained by incorporating the serving BS AOA measure and the two higher RSS measurements obtained at known location points within the cell. Our motivation is that one would expect to improve the accuracy of the localization when discarding the weaker signals resulting from multiple reflections (NLOS) and by adding the AOA information. Because of the noisy measurements, the mobile station (MS) position is typically determined by some statistical estimation which processes a set of nonlinear equations constructed from the RSS and/or AOA statistics with knowledge of the BSs and anchors geometry. Among the different proposed techniques, one particularly interesting approach, known as Approximate Maximum Likelihood (AML), was developed in [3] and shown to achieve near-optimal performance without the complexity of full maximum likelihood estimation. One limitation of the
original AML method in [3] is that it relies on RSS data only and needs measurements from several BSs (all assumed to have the same measurement error variance). Furthermore, the use of a large number of nodes results in a network overload by requiring complex messaging. A recently proposed work [4] presents a weighted RSS/AOA combined scheme where multiple AOA and RSS measurements are used for emitter location. The Least Squares and Maximum Likelihood criteria are considered. In a cellular network framework, this scheme would require the cooperation between multiple BSs. We rather consider the intra cell localization propose to process a localization from the unique AOA and a subset of RSS intra cell measures. This approach is selective as it considers a few but the most reliable information and avoids highly attenuated and multiply reflected signals. In this paper, selective hybrid versions of the LLS and AML algorithms, where the two higher RSS of the cell are combined with the serving BS AOA, are studied. We show through simulations that this approach leads to superior performance than exclusively RSS-based schemes. The paper is organized as follows. In Section II, the LLS approach is developed. The proposed Selective Hybrid LLS (SHLLS) and Selective Hybrid AML (SHAML) approaches are derived in Section III. Section IV is devoted to simulations and numerical results. Finally, Section V concludes the paper. II. T HE LLS APPROACH A. RSS technique 1) RSS-Based Distance Estimation: Various techniques have been proposed in the literature for mapping RSS measurements to distance estimates. The basic model used in localization literature [5] is given by the formula RSSi = P L(d0 ) − 10γ log10 (
di ) + XσRSSi , d0
(1)
where RSSi is the received signal strength in dB at the ith among N nodes. In (1), di is the true distance between the sender and the ith receiver, γ is the path-loss exponent, P L(d0 ) is the power loss in dB at a reference distance d0 and XσRSSi in dBm is a random variable representing the noise in the measured RSS. Contrarily to time varying noise sources, the errors induced by shadowing can not be averaged out by taking multiple measurements and can be modeled in logarithmic scale by a zero mean Gaussian distribution with variance σRSSi [5]. We collect N RSS measurements from known positions anchors or localized users where N ≥ 3.
2
This measured value RSSi is mapped to a distance estimate dbi . This mapping from RSSi to dbi requires an expression for the distance dbi as a function of RSSi and can be obtained by solving (1) for d, yielding (−RSSi +XσRSS )/10γ
di = k10
i
1 T (A A1 )−1 AT1 P. (10) 2 1 An LLS selective approach can be envisaged where rather than using the N measured distances, only a subset, corresponding to the higher RSS measurements are considered. θ=
,
(2)
where k is a constant incorporating both P L(d0 ) and γ log10 (d0 ). Given the empirical values of k and γ, the resulting model in (2) can be used to compute a distance estimate dbi from a measured RSSi as
B. AOA LLS technique This scheme supposes multi-cell cooperation. Denoting by αi the AOA estimated at the ith BS, then
dbi = k10(−RSSi )/10γ . The measured distance dbi are assumed to be corrupted by independent zero-mean Gaussian noise with covariance given by n o Q = diag vardb , vardb , ..., vardbN , 1
RSS), kr,i = kr − ki and r is the reference point that is used to obtain the linear model. From the above expressions, the LLS solution for θ can be written as [6]
2
y − yi = (x − xi )tg(αi ), i = 1, ..., N.
(11)
If the angles of arrival from N ≥ 2 BSs are known, the MS position can be estimated using a standard LLS approach. The appropriate system of linear equations can be written as [7]
where vardb = E(dbi − di )2 .
A2 θ = B,
(3)
(12)
i
2) The selective RSS LLS approach: This section gives a brief description of the original LLS approach in [6]. Assuming N ≥ 3 known position nodes (anchors) involved in the positioning process, the vector of distances to be recovered from RSS measurements is T
∆ = [d1 , ..., dN ] ,
(4)
where di is the true distance from sender to the ith receiver. We consider a 2-D coordinate system. These distances are related to the MS unknown position θ = [x y]T and the N anchors positions (xi , yi ) as d2i = s + ki − 2xxi − 2yyi ,
(5)
where A2 =
tg(α1 ) tg(α2 ) .. .
−1 −1 .. .
,B =
tg(αN ) −1
x1 tg(α1 ) − y1 x2 tg(α2 ) − y2 .. .
.
xN tg(αN ) − yN
(13) The estimated vector of coordinates can then be calculated as θ = (AT2 A2 )−1 AT2 B.
(14)
Note that contrarily to the work in [4], in our framework of intra cell localization, only one AOA from the serving BS is available. III. T HE PROPOSED SHLLS AND SHAML A PPROACHES
with 2
2
s=x +y ,
(6)
ki = x2i + yi2 .
(7)
The non-linear model discussed in (5) contains the parameter s which is quadratic in x and y. In order to obtain a linear model, an alternative technique was proposed in [7] for canceling out these non-linear terms. By fixing expressions for the rth reference point or anchor in (5), subtracting it from the remaining equations for i = 1, 2, ..., N (i 6= r), and rearranging the terms, we then obtain the following linear model A1 θ =
1 P, 2
(8)
with A1 =
x1 − xr x2 − xr .. . xN − xr
b2 b2 dr − d1 − kr,1 db2 − db2 − kr,2 2 r ,P = .. . 2 2 b b yN − yr dr − dN − kr,N y1 − yr y2 − yr .. .
,
(9) where the distances di are replaced by their measures (from
In the proposed approaches, the serving BS, referenced by node 1, provides the measure of the unique AOA α and the RSS measurements are taken as the two higher RSS measurements in the cell, referenced by nodes 2 and 3. A. The SHLLS approach Combining (11) and (5), and rewriting in matrix form, we obtain Dθ = C, where
2 (x3 − x2 ) 2 (y3 − y2 ) D= sin α − cos α db22 − db23 − k2,3 . C= x1 sin α − y1 cos α
(15) , (16)
The location of the MS can then be determined by solving (15) using LLS method as θ = (DT D)−1 DT C.
(17)
3
B. The SHAML approach In the proposed approach, the vector of measured parameters becomes h
∆H = db2 , db3 , α
iT
,
(18)
3 X i=2
QH
i=2
(26)
where
where α is the AOA at the home BS corrupted by an additive zero-mean Gaussian noise with variance varAOA independent from RSS noise, modeling the possible NLOS effect. We consider ∆H the covariance matrix n o = diag vardb2 , vardb3 , varAOA .
3 X qi s + ki − db2i = qi (2xi x + 2yi y) + 2w(x − x1 ).
pi
=
qi
=
(x − xi )/(dbi + di )di vardbi , (y − yi )/(dbi + di )di var b ,
w
=
1/2 (α − β) /d21 varAOA .
di
Then in matrix notation
(19)
2Aθ = b1 s + b2 ,
(27)
where
The location parameters of ∆H is denoted
P3 P3 px i=2 pi yi − w P3 i=2 i i P , 3 i=2 qi xi + w i=2 qi yi P3 pi i=2 P3 = , qi # " Pi=2 3 b2 ) − 2wy1 p (k − d i i i P3i=2 . = b2 i=2 qi (ki − di ) + 2wx1
T
RH (θ) = [d2 , d3 , β] .
A
(20)
Then, the probability density function (pdf) [7] of ∆H given θ is JH −3/2 −1/2 , (21) f (∆H | θ) = (2π) det (QH ) exp − 2
b1 b2
where JH is expressed as JH
= =
vardbi
T
Q−1 H (∆H
to
− RH (θ)) 2
θ=
(α − β) + . varAOA
(22)
∂RH (θ) ∂θ
T
Q−1 H (∆H − RH (θ)) = 0.
(23)
Computing the gradient of RH (θ) gives the following quantity x−x y−y 2
∂RH (θ) = ∂θ
d2 x−x3 d3 1 − y−y d21
2
d2 y−y3 d3 x−x1 d21
.
(24)
Then (23) is equivalent to the following system of equations bi + di 3 (x − x ) −d X i di vardbi i=2 bi + di 3 (y − y ) −d X i i=2
di vardbi
+
(y − y1 ) (α − β) d21 varAOA
=
0,(25)
−
(x − x1 ) (α − β) d21 varAOA
=
0.
Substituting the quantity di − dbi = d2i − db2i / dbi + di , the system of equations above becomes 3 X i=2
3 X pi s + ki − db2i = pi (2xi x + 2yi y) − 2w(y − y1 ), i=2
1 T −1 T (A A) A (b1 s + b2 ) = v1 s + v2 , 2
(29)
with
Thus, the ML estimate is the vector θ that minimizes JH . Setting the gradient of JH with respect to θ to zero yields ∂JH = ∂θ
(28)
Performing the Least Squares (LS) estimator [6], [7] leads
(∆H − RH (θ)) 2 3 dbi − di X i=2
=
1 T −1 T 1 (A A) A b1 and v2 = (AT A)−1 AT b2 . 2 2 Therefore, v1 =
x = v1 (1)s + v2 (1) and y = v1 (2)s + v2 (2).
(30)
Substituting the last expressions in (6), a quadratic in s is obtained 2
2
(v1 (1)s + v2 (1)) + (v1 (2)s + v2 (2)) = s.
(31)
Solving this quadratic yields three possible cases upon which the root selection routine (RSR) selects the adequate solution as follows • Case 1. The roots are real and there is only one positive root. Then s takes the value of this positive root and θ is estimated by applying (30). • Case 2. The roots are real and are both positive. Then θ is calculated by applying (30) for each root and the corresponding value of JH is determined by performing (22). The selected solution is the root that leads to the minimum value of JH and the estimated (x, y) is the corresponding θ. • Case 3. The roots are negative (or complex). In this case we take the absolute value (of the real part) and process as in the Case 2. However, the design matrix A and the noise vectors b1 and b2 contain the quantities pi , qi and w which in their turn include the unknown true mobile coordinates. Thus, an iterative procedure can be used (in numerical simulations, the
4
number of iterations taken n = 5 is usually sufficient for convergence). First, the SHAML algorithm approximates the parameters pi , qi and w starting with an initial mobile position and then updates iteratively this estimate using the LS solution given in (31). In order to determine an initial position to start with, consider s + ki − db2i = 2(xxi + yyi ), (x − x1 ) sin(α) − (y − y1 ) cos(α) = 0.
(32)
In matrix notation, this is equivalent to Bθ =
1 c1 s + c2 , 2
(33)
Fig. 1. BS and nodes configuration. Small circles indicate the 13 nodes location and small stars indicate possible user location.
where
x2 y2 , y3 B = x3 sin(α) − cos(α) 1 b2 1 2 (k2 − d2 ) 1 . b2 c1 = 1 , c2 = 2 (k3 − d3 ) 0 x1 sin(α) − y1 cos(α)
(34)
An initial position can be obtained from the LS estimate as 1 θ = (B T B)−1 B T ( c1 s + c2 ). (35) 2 Since the solution given in (35) is expressed again in terms of s, we combine (35) and (6) and we obtain a quadratic in s. Then, the adequate root is chosen similarly as discussed for (31). IV. N UMERICAL R ESULTS This section presents simulation results that illustrate the performance of the selective approaches SHLLS and SHAML using the home BS AOA measure and the two higher RSS measurements of the cell. The obtained performance is compared to the selective RSS LLS method using a larger number (N ≥ 3) of nodes and to the AOA LLS method using 2 BSs and implying a multi-cell cooperation which is not our framework as intra cell localization is considered. In our simulation study, we assume that a network of a BS and 12 beacons and sensor nodes is deployed over a region A of size (200 × 200m2 ) (Refer to figure 1). The region A is discretized into an evenly-spaced grid with spacing of 10m as shown in figure 1. Our simulation results represent an average over 1000 trials for each MS position on the discretized grid. RSS and AOA measurements are assumed to be corrupted by independent Gaussian noise with variances σRSS (related to vardb) and varAOA , respectively. Two scenarios are envisaged, in the first, all the RSS sensors are assumed to have the same standard deviation σRSS = 6dB. In the second, the shadowing is more severe for more distant receivers and grows as a function of the distance. The path loss exponent in the model (1) is fixed to γ = 2. Unless otherwise mentioned, the AOA standard deviation is fixed to 3◦ .
Fig. 2. Performance of the selective RSS LLS as a function of the number of the best selected nodes.
In the first simulation, we study the performance of the selective RSS LLS as a function of the number of the higher RSS selected nodes involved in the localization. Figure 2 displays the experimental Cumulative Distribution Function (CDFs), which is evaluated as the rate of trials where the estimation range error is lower than abscissa value, this curve is evaluated for different number of best selected nodes. It shows that the location error decreases as a function of the number of nodes. By selective RSS LLS we refer to the choice of the higher RSS among the 13 nodes measurements. In the second simulation, we compare the performance of the proposed SHLLS and SHAML techniques and the original AOA LLS used with 2 BSs to the selective RSS LLS approach with 3 or more nodes. Table 1 lists the Root Mean Squares Error (RMSE) on the MS coordinates (in meters), averaged over the MS positions (xi , y i ) on the grid. For the position (xi , y i ), it is evaluated as v u M u 1 X RM SEi = t [(b xi − xi )2 + (b y i − y i )2 ] M i =1
where M is the number of Monte Carlo trials. Table 1: RMSE performance of the SHLLS and SHAML, AOA LLS and selective RSS LLS with varAOA = 30 .
5
selective RSS LLS
Approach
SHLLS
SHAML
nb of nodes
2
2
3
σRSS = 3dB
7.87
5.02
11.23 9.02 7.83 7.44 6.43 4.97
σRSS = 6dB
8.69
6.014
12.06 9.31 8.53 8.37 7.45 5.93
σRSS = 9dB
9.31
6.93
12.85 9.73 9.29 8.57 7.67 6.91
5
7
8
9
10
The AOA LLS technique leads to an RMSE of 5.94. As can be seen from the results given in table 1, although the selective RSS technique uses a larger number of nodes, the SHAML and SHLLS methods show better performance. Comparing the RMSE position errors obtained using the novel SHLLS (respectively SHAML) algorithm involving at most only two nodes apart from the serving BS to the selective RSS LLS technique with three nodes, an enhancement of 33.6%, 33.7%, 35.4% (respectively 62%, 60.5%, 59.2%) for σRSS = 3dB , 6dB and 9dB respectively is perceived on the accuracy of the MS estimate. Even for high values of shadowing, the accuracy of selective RSS LLS surpasses that of SHLLS (respectively SHAML) only if seven nodes (respectively ten nodes) are involved in the positioning process which may not necessarily be available in a real-world scenario. In addition, for low values of shadowing (σRSS = 6dB), comparing the RMSE position errors obtained using the original AOA LLS technique with two BSs to the novel SHLLS (respectively SHAML) algorithm involving only the cell BS and two additional nodes, an enhancement of 27.5% (respectively 0.74%) is perceived on the accuracy of the mobile position estimate. Then, the performance of the AOA LLS and the SHAML techniques remains nearly the same. Eventhough, assuming that such a difficult scenario is available, the accuracy enhancement towards the proposed SHLLS algorithm is of just 0.4%, 1.6%, 0, 2% after involving seven more nodes when using the selective RSS LLS approach. Moreover, when we increase the shadowing variance σRSS , corresponding to a more dense environment, the selective RSS LLS performance degrades. However, the SHLLS (respectively SHAML) is less influenced by the increase of this feature because of the contribution of the added AOA information and the choice of the two best RSS measurements. Figures 3 (a) and 3 (b) illustrate CDF of the average location error for two scenarios (shadowing fixed or variable) for the SHLLS, SHAML, AOA LLS and selective RSS LLS approaches. As shown in these two figures although the number of used nodes with the SHLLS and SHAML techniques is lower than that of the selective RSS LLS approach, they outperform it in both fixed and distance-dependant shadowing variance cases. Figure 3 (a) shows that the average location error of SHLLS below 10m occurs in more than 66% (respectively of SHAML at 90%) of the measurements. On the other hand, in the selective RSS LLS approach an average location error below 10m occurs in less than 35, 54% of the measurements. However, for the AOA LLS approach an average location error remains nearly the same of the SHAML approach. Figure 4 compares the performance of SHAML and selective RSS LLS methods in both fixed and distance-dependant shadowing. It shows a better robustness of the SHAML
(a)
(b) Fig. 3. Comparison between 2 nodes SHAML, SHLLS, AOA LLS and 3 nodes selective RSS LLS techniques with (a) σRSS = 6dB and (b) distancedependent shadowing where for d in [0, 20m[, σRSS = 1dB, and an increase of 1dB per additional distance of 20m.
approach in the case of severe shadowing. For example an average location error of 7.5m is obtained with a CDF 8% and 71.6% for selective RSS LLS and SHAML respectively for a fixed shadowing, it corresponds to CDF of 21% and 79% in the case of variable shadowing.
Fig. 4. Comparison between 2 cases of shadowing: fixed (σRSS = 6dB)and distance-dependent shadowing where for d in [0, 20m[, σRSS = 1dB, and an increase of 1dB per additional distance of 20m.
We then study the performance of the AOA LLS, the selective RSS LLS, the SHLLS and the SHAML techniques against the AOA standard deviation. As shown in figure 5, the accuracy of the MS estimate calculated with the SHLLS, the SHAML and the AOA LLS algorithms is enhanced when more accurate AOA measurements are used. As expected, no
6
effect is introduced on the position estimate calculated with the selective RSS LLS technique since it does not use the AOA information. It is worth noting that the performance of the proposed SHAML algorithm is still better than that of the selective RSS LLS method using any AOA standard deviation. A slight advantage of SHAML is obtained w.r.t. the AOA LLS for high values of AOA standard deviation.
Fig. 5. Influence of the AOA variances on the position accuracy with 2 nodes SHAML, SHLLS and AOA LLS and 3 nodes selective RSS LLS techniques.
V. C ONCLUSION In this paper, we proposed a two selective hybrid RSS/AOA schemes: the Linear Least Squares and the Approximate Maximum Likelihood approaches for intra-cell localization in wireless networks. The main contribution of our combining scheme is based on using the serving BS AOA measure and the two best RSS measurements of the cell localized nodes (users with known positions or anchors). Simulation results showed that the SHAML outperforms the SHLLS technique. Moreover, in some geometric cases and even for high values of shadowing, SHLLS (respectively SHAML) ranging obtained from 2 nodes are as accurate as selective RSS LLS results obtained from 7 nodes (respectively 10 nodes). For high AOA standard deviation, corresponding to severe NLOS effects, the SHAML outperforms the AOA LLS approach envisageable only in multi cell localization frame. R EFERENCES [1] C. Drane, M. Macnaughtan, and C. Scott, “Positioning GSM telephones”, IEEE Communications Magazine, vol. 36, no. 4, pp.46-54, 1998. [2] J. J. Caffery Jr, “Wireless Location in CDMA Cellular Radio Systems”, Boston, MA: Kluwer Academic, 2000. [3] Mohamed Zhaounia, Mohamed Adnan Landolsi, and Ridha Bouallegue, “Hybrid TOA/AOA Approximate Maximum Likelihood Mobile Localization”, Hindawi Publishing Corporation Journal of Electrical and Computer Engineering, 2010. [4] Sichun Wang, Brad R.Jackson, Robert Inkol, “Hybrid RSS/AOA emitter location estimation based on least squares and maximum likelihood criteria”, In Proc. IEEE Communications (QBSC), Kingston, pp. 24-29, Juin 2012. [5] A. LaMarca, June. Hightower, I. Smith, and S. Consolvo, “Self-mapping in 802.11 location systems”, In Proc. 7th International Conference on Ubiquitous Computing (Ubicomp05), pp. 87-104, Tokyo, Japan, September 2005.
[6] J. J. Caffery, “A new approach to the geometry of TOA location”, in Proc. IEEE Vehic. Technol. Conf. (VTC), vol. 4, Boston, MA, Sep. 2000, pp. 1943-1949. [7] S. M. Kay, “Fundamentals of Statistical Signal Processing: Estimation Theory”, Prentice-Hall PTR, Upper Saddle River, NJ, USA, 1993.
Selective Hybrid RSS/AOA Approximate Maximum Likelihood Mobile intra cell Localization Leila Gazzah, Leila Najjar and Hichem Besbes Carthage University, Lab. COSIM, Higher School of Communication of Tunis, Tunisia Email: {gazzah.leila, leila.najjar, hichem.besbes}@supcom.rnu.tn
Abstract—We present in this paper two selective combined schemes for mobile location. These schemes are based respectively on the Linear Least Squares (LLS) and the Approximate Maximum Likelihood (AML) approaches. The proposed positioning techniques apply to intra cell localization in which the serving base station (BS) Angle Of Arrival (AOA) and the best two Received Signal Strength (RSS) measurements taken at some reference points (with known positions) are combined for mobile user localization. Such nodes are either localized users or anchors. The proposed approach achieves better performance compared to the selective RSS LLS method. The use of only the two best RSS measures reduces the location bias effect induced by propagation. It also decreases the computational burden implied by multiple RSS combining. Simulation results show that, for certain scenarios, the proposed selective hybrid RSS/AOA LLS (respectively AML) with two nodes outperforms the selective RSS LLS with up to seven nodes (respectively to ten nodes)and achieves equivalent performances to AOA-based approaches. Keywords: Radio Signal Strength, Angle-Of-Arrival, intra-cell, selective, hybrid, Linear Least Squares, Approximate Maximum Likelihood.
I. I NTRODUCTION By offering multiple innovative applications, radiolocation has gained an increasing interest in wireless communication research field. Fleet management, interactive applications and location-based billing are some examples of new location based services triggered by mobile positioning technology [1]. Radiolocation can be derived from measurements of Time of Arrival (TOA), Angle of Arrival (AOA), Time Difference of Arrival (TDOA) and Received Signal Strength (RSS) [2]. In this work, we investigate in the intra cell localization framework the potential enhancement in geolocation that can be obtained by incorporating the serving BS AOA measure and the two higher RSS measurements obtained at known location points within the cell. Our motivation is that one would expect to improve the accuracy of the localization when discarding the weaker signals resulting from multiple reflections (NLOS) and by adding the AOA information. Because of the noisy measurements, the mobile station (MS) position is typically determined by some statistical estimation which processes a set of nonlinear equations constructed from the RSS and/or AOA statistics with knowledge of the BSs and anchors geometry. Among the different proposed techniques, one particularly interesting approach, known as Approximate Maximum Likelihood (AML), was developed in [3] and shown to achieve near-optimal performance without the complexity of full maximum likelihood estimation. One limitation of the
original AML method in [3] is that it relies on RSS data only and needs measurements from several BSs (all assumed to have the same measurement error variance). Furthermore, the use of a large number of nodes results in a network overload by requiring complex messaging. A recently proposed work [4] presents a weighted RSS/AOA combined scheme where multiple AOA and RSS measurements are used for emitter location. The Least Squares and Maximum Likelihood criteria are considered. In a cellular network framework, this scheme would require the cooperation between multiple BSs. We rather consider the intra cell localization propose to process a localization from the unique AOA and a subset of RSS intra cell measures. This approach is selective as it considers a few but the most reliable information and avoids highly attenuated and multiply reflected signals. In this paper, selective hybrid versions of the LLS and AML algorithms, where the two higher RSS of the cell are combined with the serving BS AOA, are studied. We show through simulations that this approach leads to superior performance than exclusively RSS-based schemes. The paper is organized as follows. In Section II, the LLS approach is developed. The proposed Selective Hybrid LLS (SHLLS) and Selective Hybrid AML (SHAML) approaches are derived in Section III. Section IV is devoted to simulations and numerical results. Finally, Section V concludes the paper. II. T HE LLS APPROACH A. RSS technique 1) RSS-Based Distance Estimation: Various techniques have been proposed in the literature for mapping RSS measurements to distance estimates. The basic model used in localization literature [5] is given by the formula RSSi = P L(d0 ) − 10γ log10 (
di ) + XσRSSi , d0
(1)
where RSSi is the received signal strength in dB at the ith among N nodes. In (1), di is the true distance between the sender and the ith receiver, γ is the path-loss exponent, P L(d0 ) is the power loss in dB at a reference distance d0 and XσRSSi in dBm is a random variable representing the noise in the measured RSS. Contrarily to time varying noise sources, the errors induced by shadowing can not be averaged out by taking multiple measurements and can be modeled in logarithmic scale by a zero mean Gaussian distribution with variance σRSSi [5]. We collect N RSS measurements from known positions anchors or localized users where N ≥ 3.
2
This measured value RSSi is mapped to a distance estimate dbi . This mapping from RSSi to dbi requires an expression for the distance dbi as a function of RSSi and can be obtained by solving (1) for d, yielding (−RSSi +XσRSS )/10γ
di = k10
i
1 T (A A1 )−1 AT1 P. (10) 2 1 An LLS selective approach can be envisaged where rather than using the N measured distances, only a subset, corresponding to the higher RSS measurements are considered. θ=
,
(2)
where k is a constant incorporating both P L(d0 ) and γ log10 (d0 ). Given the empirical values of k and γ, the resulting model in (2) can be used to compute a distance estimate dbi from a measured RSSi as
B. AOA LLS technique This scheme supposes multi-cell cooperation. Denoting by αi the AOA estimated at the ith BS, then
dbi = k10(−RSSi )/10γ . The measured distance dbi are assumed to be corrupted by independent zero-mean Gaussian noise with covariance given by n o Q = diag vardb , vardb , ..., vardbN , 1
RSS), kr,i = kr − ki and r is the reference point that is used to obtain the linear model. From the above expressions, the LLS solution for θ can be written as [6]
2
y − yi = (x − xi )tg(αi ), i = 1, ..., N.
(11)
If the angles of arrival from N ≥ 2 BSs are known, the MS position can be estimated using a standard LLS approach. The appropriate system of linear equations can be written as [7]
where vardb = E(dbi − di )2 .
A2 θ = B,
(3)
(12)
i
2) The selective RSS LLS approach: This section gives a brief description of the original LLS approach in [6]. Assuming N ≥ 3 known position nodes (anchors) involved in the positioning process, the vector of distances to be recovered from RSS measurements is T
∆ = [d1 , ..., dN ] ,
(4)
where di is the true distance from sender to the ith receiver. We consider a 2-D coordinate system. These distances are related to the MS unknown position θ = [x y]T and the N anchors positions (xi , yi ) as d2i = s + ki − 2xxi − 2yyi ,
(5)
where A2 =
tg(α1 ) tg(α2 ) .. .
−1 −1 .. .
,B =
tg(αN ) −1
x1 tg(α1 ) − y1 x2 tg(α2 ) − y2 .. .
.
xN tg(αN ) − yN
(13) The estimated vector of coordinates can then be calculated as θ = (AT2 A2 )−1 AT2 B.
(14)
Note that contrarily to the work in [4], in our framework of intra cell localization, only one AOA from the serving BS is available. III. T HE PROPOSED SHLLS AND SHAML A PPROACHES
with 2
2
s=x +y ,
(6)
ki = x2i + yi2 .
(7)
The non-linear model discussed in (5) contains the parameter s which is quadratic in x and y. In order to obtain a linear model, an alternative technique was proposed in [7] for canceling out these non-linear terms. By fixing expressions for the rth reference point or anchor in (5), subtracting it from the remaining equations for i = 1, 2, ..., N (i 6= r), and rearranging the terms, we then obtain the following linear model A1 θ =
1 P, 2
(8)
with A1 =
x1 − xr x2 − xr .. . xN − xr
b2 b2 dr − d1 − kr,1 db2 − db2 − kr,2 2 r ,P = .. . 2 2 b b yN − yr dr − dN − kr,N y1 − yr y2 − yr .. .
,
(9) where the distances di are replaced by their measures (from
In the proposed approaches, the serving BS, referenced by node 1, provides the measure of the unique AOA α and the RSS measurements are taken as the two higher RSS measurements in the cell, referenced by nodes 2 and 3. A. The SHLLS approach Combining (11) and (5), and rewriting in matrix form, we obtain Dθ = C, where
2 (x3 − x2 ) 2 (y3 − y2 ) D= sin α − cos α db22 − db23 − k2,3 . C= x1 sin α − y1 cos α
(15) , (16)
The location of the MS can then be determined by solving (15) using LLS method as θ = (DT D)−1 DT C.
(17)
3
B. The SHAML approach In the proposed approach, the vector of measured parameters becomes h
∆H = db2 , db3 , α
iT
,
(18)
3 X i=2
QH
i=2
(26)
where
where α is the AOA at the home BS corrupted by an additive zero-mean Gaussian noise with variance varAOA independent from RSS noise, modeling the possible NLOS effect. We consider ∆H the covariance matrix n o = diag vardb2 , vardb3 , varAOA .
3 X qi s + ki − db2i = qi (2xi x + 2yi y) + 2w(x − x1 ).
pi
=
qi
=
(x − xi )/(dbi + di )di vardbi , (y − yi )/(dbi + di )di var b ,
w
=
1/2 (α − β) /d21 varAOA .
di
Then in matrix notation
(19)
2Aθ = b1 s + b2 ,
(27)
where
The location parameters of ∆H is denoted
P3 P3 px i=2 pi yi − w P3 i=2 i i P , 3 i=2 qi xi + w i=2 qi yi P3 pi i=2 P3 = , qi # " Pi=2 3 b2 ) − 2wy1 p (k − d i i i P3i=2 . = b2 i=2 qi (ki − di ) + 2wx1
T
RH (θ) = [d2 , d3 , β] .
A
(20)
Then, the probability density function (pdf) [7] of ∆H given θ is JH −3/2 −1/2 , (21) f (∆H | θ) = (2π) det (QH ) exp − 2
b1 b2
where JH is expressed as JH
= =
vardbi
T
Q−1 H (∆H
to
− RH (θ)) 2
θ=
(α − β) + . varAOA
(22)
∂RH (θ) ∂θ
T
Q−1 H (∆H − RH (θ)) = 0.
(23)
Computing the gradient of RH (θ) gives the following quantity x−x y−y 2
∂RH (θ) = ∂θ
d2 x−x3 d3 1 − y−y d21
2
d2 y−y3 d3 x−x1 d21
.
(24)
Then (23) is equivalent to the following system of equations bi + di 3 (x − x ) −d X i di vardbi i=2 bi + di 3 (y − y ) −d X i i=2
di vardbi
+
(y − y1 ) (α − β) d21 varAOA
=
0,(25)
−
(x − x1 ) (α − β) d21 varAOA
=
0.
Substituting the quantity di − dbi = d2i − db2i / dbi + di , the system of equations above becomes 3 X i=2
3 X pi s + ki − db2i = pi (2xi x + 2yi y) − 2w(y − y1 ), i=2
1 T −1 T (A A) A (b1 s + b2 ) = v1 s + v2 , 2
(29)
with
Thus, the ML estimate is the vector θ that minimizes JH . Setting the gradient of JH with respect to θ to zero yields ∂JH = ∂θ
(28)
Performing the Least Squares (LS) estimator [6], [7] leads
(∆H − RH (θ)) 2 3 dbi − di X i=2
=
1 T −1 T 1 (A A) A b1 and v2 = (AT A)−1 AT b2 . 2 2 Therefore, v1 =
x = v1 (1)s + v2 (1) and y = v1 (2)s + v2 (2).
(30)
Substituting the last expressions in (6), a quadratic in s is obtained 2
2
(v1 (1)s + v2 (1)) + (v1 (2)s + v2 (2)) = s.
(31)
Solving this quadratic yields three possible cases upon which the root selection routine (RSR) selects the adequate solution as follows • Case 1. The roots are real and there is only one positive root. Then s takes the value of this positive root and θ is estimated by applying (30). • Case 2. The roots are real and are both positive. Then θ is calculated by applying (30) for each root and the corresponding value of JH is determined by performing (22). The selected solution is the root that leads to the minimum value of JH and the estimated (x, y) is the corresponding θ. • Case 3. The roots are negative (or complex). In this case we take the absolute value (of the real part) and process as in the Case 2. However, the design matrix A and the noise vectors b1 and b2 contain the quantities pi , qi and w which in their turn include the unknown true mobile coordinates. Thus, an iterative procedure can be used (in numerical simulations, the
4
number of iterations taken n = 5 is usually sufficient for convergence). First, the SHAML algorithm approximates the parameters pi , qi and w starting with an initial mobile position and then updates iteratively this estimate using the LS solution given in (31). In order to determine an initial position to start with, consider s + ki − db2i = 2(xxi + yyi ), (x − x1 ) sin(α) − (y − y1 ) cos(α) = 0.
(32)
In matrix notation, this is equivalent to Bθ =
1 c1 s + c2 , 2
(33)
Fig. 1. BS and nodes configuration. Small circles indicate the 13 nodes location and small stars indicate possible user location.
where
x2 y2 , y3 B = x3 sin(α) − cos(α) 1 b2 1 2 (k2 − d2 ) 1 . b2 c1 = 1 , c2 = 2 (k3 − d3 ) 0 x1 sin(α) − y1 cos(α)
(34)
An initial position can be obtained from the LS estimate as 1 θ = (B T B)−1 B T ( c1 s + c2 ). (35) 2 Since the solution given in (35) is expressed again in terms of s, we combine (35) and (6) and we obtain a quadratic in s. Then, the adequate root is chosen similarly as discussed for (31). IV. N UMERICAL R ESULTS This section presents simulation results that illustrate the performance of the selective approaches SHLLS and SHAML using the home BS AOA measure and the two higher RSS measurements of the cell. The obtained performance is compared to the selective RSS LLS method using a larger number (N ≥ 3) of nodes and to the AOA LLS method using 2 BSs and implying a multi-cell cooperation which is not our framework as intra cell localization is considered. In our simulation study, we assume that a network of a BS and 12 beacons and sensor nodes is deployed over a region A of size (200 × 200m2 ) (Refer to figure 1). The region A is discretized into an evenly-spaced grid with spacing of 10m as shown in figure 1. Our simulation results represent an average over 1000 trials for each MS position on the discretized grid. RSS and AOA measurements are assumed to be corrupted by independent Gaussian noise with variances σRSS (related to vardb) and varAOA , respectively. Two scenarios are envisaged, in the first, all the RSS sensors are assumed to have the same standard deviation σRSS = 6dB. In the second, the shadowing is more severe for more distant receivers and grows as a function of the distance. The path loss exponent in the model (1) is fixed to γ = 2. Unless otherwise mentioned, the AOA standard deviation is fixed to 3◦ .
Fig. 2. Performance of the selective RSS LLS as a function of the number of the best selected nodes.
In the first simulation, we study the performance of the selective RSS LLS as a function of the number of the higher RSS selected nodes involved in the localization. Figure 2 displays the experimental Cumulative Distribution Function (CDFs), which is evaluated as the rate of trials where the estimation range error is lower than abscissa value, this curve is evaluated for different number of best selected nodes. It shows that the location error decreases as a function of the number of nodes. By selective RSS LLS we refer to the choice of the higher RSS among the 13 nodes measurements. In the second simulation, we compare the performance of the proposed SHLLS and SHAML techniques and the original AOA LLS used with 2 BSs to the selective RSS LLS approach with 3 or more nodes. Table 1 lists the Root Mean Squares Error (RMSE) on the MS coordinates (in meters), averaged over the MS positions (xi , y i ) on the grid. For the position (xi , y i ), it is evaluated as v u M u 1 X RM SEi = t [(b xi − xi )2 + (b y i − y i )2 ] M i =1
where M is the number of Monte Carlo trials. Table 1: RMSE performance of the SHLLS and SHAML, AOA LLS and selective RSS LLS with varAOA = 30 .
5
selective RSS LLS
Approach
SHLLS
SHAML
nb of nodes
2
2
3
σRSS = 3dB
7.87
5.02
11.23 9.02 7.83 7.44 6.43 4.97
σRSS = 6dB
8.69
6.014
12.06 9.31 8.53 8.37 7.45 5.93
σRSS = 9dB
9.31
6.93
12.85 9.73 9.29 8.57 7.67 6.91
5
7
8
9
10
The AOA LLS technique leads to an RMSE of 5.94. As can be seen from the results given in table 1, although the selective RSS technique uses a larger number of nodes, the SHAML and SHLLS methods show better performance. Comparing the RMSE position errors obtained using the novel SHLLS (respectively SHAML) algorithm involving at most only two nodes apart from the serving BS to the selective RSS LLS technique with three nodes, an enhancement of 33.6%, 33.7%, 35.4% (respectively 62%, 60.5%, 59.2%) for σRSS = 3dB , 6dB and 9dB respectively is perceived on the accuracy of the MS estimate. Even for high values of shadowing, the accuracy of selective RSS LLS surpasses that of SHLLS (respectively SHAML) only if seven nodes (respectively ten nodes) are involved in the positioning process which may not necessarily be available in a real-world scenario. In addition, for low values of shadowing (σRSS = 6dB), comparing the RMSE position errors obtained using the original AOA LLS technique with two BSs to the novel SHLLS (respectively SHAML) algorithm involving only the cell BS and two additional nodes, an enhancement of 27.5% (respectively 0.74%) is perceived on the accuracy of the mobile position estimate. Then, the performance of the AOA LLS and the SHAML techniques remains nearly the same. Eventhough, assuming that such a difficult scenario is available, the accuracy enhancement towards the proposed SHLLS algorithm is of just 0.4%, 1.6%, 0, 2% after involving seven more nodes when using the selective RSS LLS approach. Moreover, when we increase the shadowing variance σRSS , corresponding to a more dense environment, the selective RSS LLS performance degrades. However, the SHLLS (respectively SHAML) is less influenced by the increase of this feature because of the contribution of the added AOA information and the choice of the two best RSS measurements. Figures 3 (a) and 3 (b) illustrate CDF of the average location error for two scenarios (shadowing fixed or variable) for the SHLLS, SHAML, AOA LLS and selective RSS LLS approaches. As shown in these two figures although the number of used nodes with the SHLLS and SHAML techniques is lower than that of the selective RSS LLS approach, they outperform it in both fixed and distance-dependant shadowing variance cases. Figure 3 (a) shows that the average location error of SHLLS below 10m occurs in more than 66% (respectively of SHAML at 90%) of the measurements. On the other hand, in the selective RSS LLS approach an average location error below 10m occurs in less than 35, 54% of the measurements. However, for the AOA LLS approach an average location error remains nearly the same of the SHAML approach. Figure 4 compares the performance of SHAML and selective RSS LLS methods in both fixed and distance-dependant shadowing. It shows a better robustness of the SHAML
(a)
(b) Fig. 3. Comparison between 2 nodes SHAML, SHLLS, AOA LLS and 3 nodes selective RSS LLS techniques with (a) σRSS = 6dB and (b) distancedependent shadowing where for d in [0, 20m[, σRSS = 1dB, and an increase of 1dB per additional distance of 20m.
approach in the case of severe shadowing. For example an average location error of 7.5m is obtained with a CDF 8% and 71.6% for selective RSS LLS and SHAML respectively for a fixed shadowing, it corresponds to CDF of 21% and 79% in the case of variable shadowing.
Fig. 4. Comparison between 2 cases of shadowing: fixed (σRSS = 6dB)and distance-dependent shadowing where for d in [0, 20m[, σRSS = 1dB, and an increase of 1dB per additional distance of 20m.
We then study the performance of the AOA LLS, the selective RSS LLS, the SHLLS and the SHAML techniques against the AOA standard deviation. As shown in figure 5, the accuracy of the MS estimate calculated with the SHLLS, the SHAML and the AOA LLS algorithms is enhanced when more accurate AOA measurements are used. As expected, no
6
effect is introduced on the position estimate calculated with the selective RSS LLS technique since it does not use the AOA information. It is worth noting that the performance of the proposed SHAML algorithm is still better than that of the selective RSS LLS method using any AOA standard deviation. A slight advantage of SHAML is obtained w.r.t. the AOA LLS for high values of AOA standard deviation.
Fig. 5. Influence of the AOA variances on the position accuracy with 2 nodes SHAML, SHLLS and AOA LLS and 3 nodes selective RSS LLS techniques.
V. C ONCLUSION In this paper, we proposed a two selective hybrid RSS/AOA schemes: the Linear Least Squares and the Approximate Maximum Likelihood approaches for intra-cell localization in wireless networks. The main contribution of our combining scheme is based on using the serving BS AOA measure and the two best RSS measurements of the cell localized nodes (users with known positions or anchors). Simulation results showed that the SHAML outperforms the SHLLS technique. Moreover, in some geometric cases and even for high values of shadowing, SHLLS (respectively SHAML) ranging obtained from 2 nodes are as accurate as selective RSS LLS results obtained from 7 nodes (respectively 10 nodes). For high AOA standard deviation, corresponding to severe NLOS effects, the SHAML outperforms the AOA LLS approach envisageable only in multi cell localization frame. R EFERENCES [1] C. Drane, M. Macnaughtan, and C. Scott, “Positioning GSM telephones”, IEEE Communications Magazine, vol. 36, no. 4, pp.46-54, 1998. [2] J. J. Caffery Jr, “Wireless Location in CDMA Cellular Radio Systems”, Boston, MA: Kluwer Academic, 2000. [3] Mohamed Zhaounia, Mohamed Adnan Landolsi, and Ridha Bouallegue, “Hybrid TOA/AOA Approximate Maximum Likelihood Mobile Localization”, Hindawi Publishing Corporation Journal of Electrical and Computer Engineering, 2010. [4] Sichun Wang, Brad R.Jackson, Robert Inkol, “Hybrid RSS/AOA emitter location estimation based on least squares and maximum likelihood criteria”, In Proc. IEEE Communications (QBSC), Kingston, pp. 24-29, Juin 2012. [5] A. LaMarca, June. Hightower, I. Smith, and S. Consolvo, “Self-mapping in 802.11 location systems”, In Proc. 7th International Conference on Ubiquitous Computing (Ubicomp05), pp. 87-104, Tokyo, Japan, September 2005.
[6] J. J. Caffery, “A new approach to the geometry of TOA location”, in Proc. IEEE Vehic. Technol. Conf. (VTC), vol. 4, Boston, MA, Sep. 2000, pp. 1943-1949. [7] S. M. Kay, “Fundamentals of Statistical Signal Processing: Estimation Theory”, Prentice-Hall PTR, Upper Saddle River, NJ, USA, 1993.
