Received Signal Strength-Based Wireless Localization via ...

Received Signal Strength-Based Wireless Localization via Semidefinite Programming Robin Wentao Ouyang, Albert Kai-Sun Wong, Chin-Tau Lea and Victoria Ying Zhang Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Kowloon, Hong Kong [email protected], [email protected], [email protected], [email protected]

Abstract—Wireless localization has drawn significant attention over the past decade and the received signal strength (RSS) based localization scheme provides a low-cost, low-complexity and easy-implementation solution. When the statistics of the RSS measurement error is known, the Maximum Likelihood (ML) estimator is asymptotically optimal. However, due to the nature of the localization problem itself, the formed ML estimator is nonconvex, causing the search for the global minimum very difficult. In addition, its performance highly depends on the initial point provided if a local optimization method is applied to find the solution. To circumvent this problem, we apply the Semidefinite Programming (SDP) relaxation technique to the RSS-based localization problem. After reformulation and relaxation, we finally form a convex SDP estimator. A superior property of a convex estimator is that the solution is not affected by the initial point provided since any local minimum is also its global minimum. The Cramer-Rao Lower Bound (CRLB) is then derived as a benchmark for the performance comparison. Simulation results show that the proposed SDP estimator exhibit excellent performance in the RSS-based localization system and it is very suitable for the case when there are only very limited base stations hearable. Index Terms—Wireless localization, Received Signal Strength (RSS), Maximum Likelihood (ML), Semidefinite Programming (SDP), relaxation, Cramer-Rao Lower Bound (CRLB).

I. I NTRODUCTION Wireless localization has gained considerable attention over the past decade [1], [2]. The capability of accurately positioning a Mobile Station (MS) in the cellular networks enables many innovative applications, for example, emergency services, friends finding, elderly tracking, location-based services and intelligent transportation. Most of the current localization techniques [3]–[7] for wireless networks are based on the measurements of one or several (hybrid) physical parameters of the radio signal transmitted between the Base Stations (BSs) and the MS. These parameters include time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA) and received signal strength (RSS). There exist inherent tradeoffs between the location accuracy and the implementation complexity among these techniques, and the RSS-based localization scheme provides a low-cost, low-complexity and easy-implementation solution. Determining the location of an MS given the measurements of one or several aforementioned parameters is actually an estimation problem. The commonly used estimators are mainly two categories, the Linearized Least Square (LLS) estimator

[3]–[5] and the Maximum Likelihood (ML) estimator [6], [7]. Although the ML estimator is asymptotically optimal, due to the property of the wireless localization problem itself, the formed ML estimator is nonconvex, causing the search for the global minimum difficult. In addition, the solution of the ML estimator by local optimization methods, for example, the gradient search method, highly depends on the starting point provided. A poor initialization often leads to very bad estimation. To overcome this problem, several researchers propose the LLS estimator. Though the LLS estimator is much easier and has explicit solution, its accuracy is not as good as the ML estimator, especially when the variance of the measurement noise is large. Therefore, the result by the LLS estimator is then proposed to provide a starting point for the corresponding ML estimator, and often, such a concatenation produces a satisfying solution. Several other methods have been proposed in [8]–[10]. Though these methods are either convex or less sensitive to the initial point provided, they cannot compete the corresponding ML estimator provided the global minimum of the ML estimator has been found. Recently, applying Semidefinite Programming (SDP) [11] relaxation technique to wireless localization problems has been studied in [12], [13]. The basic idea is to relax the original nonconvex problem via SDP to be a convex problem. Taking advantage of the convex optimization technique, we can find the global minimum of the convex optimization problem fast and efficiently. [12], [13] state that the estimators formed via SDP relaxation technique are highly satisfactory compared to other techniques; however, they do not compare the performance of their SDP estimators with other estimators through simulations. Moreover, the SDP estimators in [12], [13] are proposed for general wireless localization problems given pairwise distance information. It is expected that directly applying these general SDP estimators to the RSS-based localization system may result in large errors due to the extra errors introduced when first estimating pairwise distances from the RSS measurements. Motivated by this, we design an SDP estimator specially for the RSS-based wireless localization system in this paper. The estimator directly relates to the RSS measurements rather than pairwise distance information. We first reformulate the

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localization problem and then relax it via SDP to be a convex problem. The proposed SDP estimator can be used in the cellular networks and also the wireless sensor networks as long as the path loss model [14] for wireless transmission is valid. Simulation results demonstrate that our proposed SDP estimator exhibits excellent performance in the RSSbased localization system which is almost the same as (and sometimes even better than) the asymptotically optimal ML estimator and much better than the other estimators. The remainder of this paper is organized as follows. Section II briefly describes the RSS measurement model. Section III first introduces the ML estimator and then reformulates the problem using Chebyshev norm. Section IV relaxes the reformulated estimator via SDP to form a convex estimator. The Cramer-Rao Lower Bound (CRLB) is derived in Section V as a benchmark for the performance comparison. Simulation results are shown in Section VI and Section VII draws the conclusions. II. M EASUREMENT M ODEL For the convenience of later use, we start by introducing the notations used throughout this paper. R, Rn and Sn denote the set of real numbers, n-vectors and symmetric n by n matrices respectively. tr(A) represents the trace of the matrix A. (·)T is the transpose operator. The identity n by n matrix is noted as In . u denotes the Euclidean norm on the vector u. For two symmetric matrices A and B, A B means A − B is positive semidefinite. We use [u]i and [A]i,j to denote the ith element of the vector u and the element at the ith row jth column of the matrix A respectively. Denote the unknown coordinates of the MS as θ = [x, y]T (θ ∈ R2 ) and the coordinates of the ith BS (known) as θi = [xi , yi ]T (θi ∈ R2 ), with i = 1, 2, . . . , N (where N is the total number of BSs that the MS can hear). The received signal strength (RSS) (from the ith BS and received by the MS or vice versa), denoted as Pi , can be related to the distance between the MS and the ith BS through the path loss model in wireless transmission [14] PT − Pi = L0 + 10γ log10

θ − θi + mi , d0

i = 1, 2, . . . , N

(1) where PT is the transmission power, L0 denotes the path loss value at the reference distance d0 , γ is the path loss exponent and mi is a Gaussian random variable representing the lognormal shadow fading effects in multipath environments. PT − Pi is also denoted as the path loss Li . III. C HEBYSHEV N ORM F ORMULATION In this section, we first introduce the Cheyshev Norm formulation of the RSS-based localization problem and then derive its equivalent form without logarithm in the expression, which is to facilitate the application of the followed semidefinite relaxation. In (1), mi ’s are often modeled as independent and identically distributed (IID) Gaussian random variables with zero mean and standard deviation σ. Then the joint conditional pdf

of the observation vector L = [L1 , . . . , LN ]T (L ∈ RN ) given θ is p(L|θ)

⎧ ⎪ ⎨ Li − L0 − 10γ log10 1 √ exp − = ⎪ 2σ 2 2πσ ⎩ i=1 N

θ−θi d0

2 ⎫ ⎪ ⎬ ⎪ ⎭

. (2)

The corresponding Maximum Likelihood (ML) estimator is therefore 2 N θ − θi ˆ 10γ log10 − (Li − L0 ) . (3) θ = arg min θ d0 i=1 (3) happens to be a Nonlinear Least Square (NLS) estimator. This NLS estimator can be used even when the distribution of mi is not Gaussian or totally unknown, while the corresponding ML estimator changes or cannot be formed. i − (Li − L0 ) as the residual If we consider 10γ log10 θ−θ d0 ri , which is the difference between the true value and the measurement, then (3) can be considered as to minimize a penalty function on the residual vector r = [r1 , . . . , rN ]T (r ∈ RN ) θˆ = arg min f (r) (4) θ

with f (r) = r2 . Now we introduce another penalty function (Chebyshev norm [11]) f (r) = r∞ = maxi [r]i . Under such a penalty function, (4) becomes θ − θi ˆ − (Li − L0 ) . (5) θ = arg min max 10γ log10 i θ d0 An advantage to use the Chebyshev norm as the penalty function is that after certain manipulations [11], we can form an equivalent problem without logarithm as shown below. Since θ − θi > 0 and positive scaling of the objective function will not influence the minimizer, therefore, (5) is equivalent to θ − θi 2 ˆ (6) θ = arg min max log10 i θ βi2 Li −L0

where βi2 = d20 10 5γ . By noting that 2 log10 θ − θi βi2

θ − θi 2 βi2 , log = max log10 10 β2 θ − θi 2

i θ − θi 2 βi2 = log10 max , , βi2 θ − θi 2

(7)

(6) can be rewritten as

θ − θi 2 βi2 . θˆ = arg min max log10 max , i θ βi2 θ − θi 2 (8)


Since log10 (x) is a strictly monotonically increasing function in its domain (0, +∞) (there is a one-one mapping between log10 (x) and x, and when log10 (x) is maximized, x is also maximized), therefore, (8) is equivalent to

θ − θi 2 βi2 ˆ , (9) , θ = arg min max max i θ βi2 θ − θi 2 which can be further simplified as

θ − θi 2 βi2 ˆ θ = arg min max . , i θ βi2 θ − θi 2

(10)

(11)

where s.t. is short for subject to. Obviously, in the above formulation, t > 0. We then rewrite (11) as

s.t. θ − θi ≤ 2

θ − θi ≥

+ ki ≤ βi2 t

+ ki ≥ βi2 t−1

i = 1, . . . , N

(13)

In the above formulation, tr(X) − 2θiT θ + ki ≤ βi2 t are affine constraints, and tr(X) − 2θiT θ + ki ≥ βi2 t−1 are convex constraints since tr(X) is linear in X, −2θiT θ is linear in θ and t−1 is convex on t > 0. However, the equality constraint X = θθT is not affine, therefore, the above formulation is still nonconvex. To make it convex, we relax the equality constraint X = θθT to an inequality constraint X θθT (semidefinite relaxation), and then express it as a linear matrix inequality by using a Schur complement [11]: ˆ X, ˆ tˆ) = arg min t (θ, θ,X,t

s.t. tr(X) −

ˆ tˆ) = arg min t (θ,

θ,t βi2 t βi2 t−1

tr(X)

− 2θiT θ − 2θiT θ T

X = θθ .

θ,t

2

ˆ X, ˆ tˆ) = arg min t (θ, s.t. tr(X)

ˆ tˆ) = arg min t (θ, s.t.

In this section, we apply semidefinite relaxation technique to the original nonconvex problem (12) and form a convex problem. For simplicity, we define ki = θi 2 . Introducing an auxiliary variable X = θθT (X ∈ S2 ), we can rewrite (12) as θ,X,t

Introducing an auxiliary variable t ∈ R, (10) can then be cast as [11]

θ − θi 2 ≤t βi2 βi2 ≤ t i = 1, . . . , N, θ − θi 2

IV. S EMIDEFINITE P ROGRAMMING R ELAXATION

(12) i = 1, . . . , N.

Note that (12) is actually the same as (11) since the constraints in (12) already imply θ − θi 2 = 0 and t > 0. Therefore, we can derive (11) from (12). After the above algebraic manipulations, we now find an equivalent problem (12) of problem (5). We can observe that the objective function in (5) is nondifferentiable, but (12) has a differentiable objective function and is much simpler in the formulation than (5). We will refer to (12) as the original problem afterwards. We now analyze the convexity of the ML estimator (3), the Chebyshev norm formulation (5) and its equivalent form (12). It is clear the domain of (3) is {θ|θ = θi }. Since it is not continuous, (3) is not a convex optimization problem. Similarly, (5) is also nonconvex. In addition, (3) and (5) contain q(θ) = log10 θ − θi which is neither convex nor concave in the objective functions. As (12) is equivalent to (5), it is also nonconvex. Alternatively, it is easy to observe that θ − θi 2 ≥ βi2 t−1 in (12) are not convex constraints. An nonconvex optimization problem has several drawbacks. One is that it may contain several local minimums and saddle points, therefore by local optimization methods, the final solution highly depends on the starting point provided and there is no guarantee that the final solution converges to the global minimum. Another is that the search for the global minimum is very difficult and time-consuming.

2θiT θ 2θiT θ

+ ki ≤ βi2 t

tr(X) − + ki ≥ βi2 t−1 X θ 0. θT 1

i = 1, . . . , N

(14)

Now (14) is convex and the readily developed numerical tools (e.g., CVX [15]) for solving convex optimization problems can be used. An excellent property of a convex optimization problem is that any local minimum is also its global minimum. Therefore, we can guarantee the global minimum is achieved when a solution is obtained. The only difference between (14) and (13) is that we relax the equality constraint in (13) to an inequality constraint in (14). Therefore, after we solve (14), we need to check whether ˆ = θˆθˆT . If it does, we conclude that the solution satisfies X the minimizer of (14) is also the minimizer of (13). Moreover, since (13) is equivalent to (12), then the solution is also the global minimizer of (12). If not, at least we obtain a solution and a lower bound on the optimal value of the original problem (12) since we solve a relaxed problem on a larger set. Note that (12) is in fact a different expression of (10) which is an unconstrained optimization problem, therefore, any θˆ given by (14) is feasible for (12). A. Result Analysis ˆ Compared with the original problem (12), besides giving θ, ˆ which can provide us the SDP problem (14) also produces X, ˆ additional information about θ. Due to relaxation, θˆ given by (14) may not be the global ˆ = θˆθˆT is satisfied. However, we minimizer of (12) unless X can assume that θˆ given by (14) should lie around the global


minimizer of (12), and in consequence, the global minimizer of (12) should also lie around θˆ given by (14). If we view the unknown global minimizer of (12) as a ˜ we have random variable [12] and denote it as θ, ˜ = θ, ˆ E(θ˜θ˜T ) = X ˆ E(θ)

(15)

where E is the expectation operator. Then the covariance matrix of θ˜ is given by ˜ = E(θ˜θ˜T ) − E(θ)E( ˜ θ) ˜T =X ˆ − θˆθˆT . cov(θ)

(16)

ˆ and θˆ are the solutions of (14), they must satisfy Since X T ˆ ˆ ˆ − θˆθˆT 0. ˆ [X θ; θ 1] 0 which is equivalent to X T ˆ ˆ ˆ Therefore, we can observe that X − θθ can indeed serve as a covariance matrix. Such an interpretation coincides with ˆ − θˆθˆT = 0, then θ˜ should be our previous analysis, since if X ˜ ˜ = θ, ˆ i.e., θˆ given by (14) is deterministic and hence θ = E(θ) also the global minimizer of (12). Since (14) provides us the covariance matrix of the estimation, we can utilize it to trust an estimation with small variances of its elements while discard an estimation with large variances and require a new estimation. V. C RAMER -R AO L OWER B OUND A NALYSIS

where J is the Fisher information matrix (FIM) [16] with the element [J]i,j defined by 2 ∂ ln p(z|θ) , (18) [J]i,j = −Ez ∂[θ]i ∂[θ]j z denotes the observation vector and p(z|θ) is the joint conditional pdf of the observation vector given θ. In RSS-based wireless localization system, given the joint conditional pdf (2), we have [J]1,1 = [J]2,2

[J]1,2 = [J]2,1 =

(19)

N 1 ∂fi (θ) ∂fi (θ) σ 2 i=1 ∂x ∂y

A. Estimators Compared For the SDP estimator proposed in [12], it assumes that the distance information has been given without considering how such information is obtained. If the distance between the MS and the ith BS di = θ − θi is estimated through the RSS measurements, then by the corresponding ML estimator, dˆi is given by

2 di ˆ − (Li − L0 ) di = arg min 10γ log10 di d0 (22) Li −L0 10γ

.

After obtaining the distance information, the SDP estimator in [12] can then be applied. In addition, the LLS estimator for TOA in [3] can also be used since its linearized formulation has nothing to do with the statistics of the measurement noise. That means it can be applied to any scenario when distance information has been acquired no matter which distribution the associated error follows. In the subsequent simulations, besides the SDPRSS estimator (14) proposed in this paper, the following estimators are chosen for comparison: 1) SDP: the SDP estimator proposed in [12] with pairwise distance information given by (22). 2) LLS: the LLS estimator proposed in [3] with pairwise distance information given by (22). 3) ML: the ML estimator (3) with reasonable initialization (the true MS location is provided, which leads to the convergence to the global minimum). In addition, the CRLB on the RMSE for any unbiased estimator is presented as a benchmark. B. Simulation Scenario The unit used here is meter. We consider the scenario that there are N BSs evenly located on a circle centered at (0, 0) with radius rad = 200. The location of the ith BS is given by

where fi (θ) = 10γ log10 θ − θi 10γ x − xi ∂fi (θ) = ∂x ln 10 θ − θi 2 10γ y − yi ∂fi (θ) = . ∂y ln 10 θ − θi 2

VI. S IMULATION R ESULTS

= d0 10

The Cramer-Rao Lower Bound (CRLB) [16] on the covariance matrix of any unbiased estimator is given by Ez (θˆ − θ)(θˆ − θ)T ≥ J −1 (17)

2 N 1 ∂fi (θ) σ 2 i=1 ∂x 2

N 1 ∂fi (θ) = 2 σ i=1 ∂y

Defining the location estimation error as e = θˆ − θ, then for any unbiased location estimator, its root mean square error (RMSE) E(e2 ) is lower bounded by E(e2 ) = E(ˆ x − x)2 + E(ˆ y − y)2 (21) ≥ [J −1 ]1,1 + [J −1 ]2,2 = tr(J −1 ) Therefore, we define tr(J −1 ) as the CRLB on the RMSE of any unbiased location estimator.

xi = rad cos (20)

2π(i − 1) 2π(i − 1) , yi = rad sin N N

(23)

N varies from 3 to 8, the path loss exponent γ is set to 3, and the standard deviation σ of mi in (1) varies from 1 dB to 6 dB. The simulations are done via MATLAB.


400 LLS SDP ML SDPRSS CRLB

350 300

RMSE (m)

250 200 150 100 50 0

1

2

3 4 Standard deviation (dB)

5

6

5

6

(a) 400 LLS SDP ML SDPRSS CRLB

350 300

RMSE (m)

250 200 150 100 50 0

1

2


D. Effect of the Standard Deviation σ

(b) 400 LLS SDP ML SDPRSS CRLB

350 300

RMSE (m)

250 200 150 100 50 0

1

2


typical geometric MS-BSs configurations: one is (0, 20) which is very close to the centroid of the triangle formed by the BSs, the second is (120, -20) which is close to one of the BSs while far away from the other two, and the third is (220, 80) which is outside the convex hull formed by the BSs. The RMSEs of different estimators for different MS locations are shown in Fig. 1 (each result is based on 500 independent runs). Through simulations, we find that all these estimators are more or less biased. Therefore, their performance cannot be well lower bounded by the CRLB. However, the CRLB can serve as a benchmark representing the minimum RMSE that any unbiased location estimator can achieve. Through Fig. 1, it is clear that the geometric configurations of the MS and the BSs significantly influence the location accuracy. When the MS is at (0, 20), the performance among all the estimators do not differ too much when σ is small. As σ becomes large, the performance of LLS degrades very quickly. In addition, SDPRSS performs better than ML. When the MS is at (120, -20), SDPRSS and ML exhibit very close performance and their RMSEs are even lower than the CRLB, while the other two estimators perform much worse. The performance of LLS degrades very quickly as σ increases. When the MS is at (220, 80), LLS shows very poor performance, almost exponentially degrading as σ increases. Though SDP is much better than LLS, its performance is still poor. Nevertheless, SDPRSS and ML exhibit steady and excellent performance and their performance is even better than the CRLB when σ becomes large. In addition, ML is slightly better than SDPRSS.

5

6

(c) Fig. 1. RMSE versus σ when the MS is at different locations with N =3. (a) MS is at (0, 20), (b) MS is at (120, -20), (c) MS is at (220, 80).

C. Effect of the Geometric Layout It is well known that the geometric layout of the MS and the BSs has significant impact on the location accuracy, which is known as geometric dilution of precision (GDOP). To investigate its effect, we keep the BS locations fixed (N = 3) and choose three different MS locations which represent three

The effect of the standard deviation σ of the log-normal shadow fading variable mi on the average location accuracy has also been shown in Fig. 1. We can observe that no matter where the MS is, the RMSE of any estimator shows degradation as σ increases. Compared with LLS and SDP, the performance degradation of SDPRSS and ML is much slower and steadier. To average out the effect of the geometric layout, we fix the locations of the BSs (N = 4) and uniformly sample 2000 random MS locations inside the convex hull formed by these 4 BSs. The RMSEs versus different standard deviations are depicted in Fig. 2. It can be observed that on average, the performance of SDPRSS is (slightly) better than ML, and there exists obvious performance gap between SDP and SDPRSS. Moreover, LLS exhibits very poor performance. E. Effect of the Number of Hearable BSs Besides σ, the number of hearable BSs N also has impact on the location accuracy. In the following simulation, we vary N from 3 to 8 while keeping the MS location inside the square region: {(x, y)| − 100 ≤ x ≤ 100, −100 ≤ y ≤ 100}. For a specific N , we sample 2000 random MS locations uniformly, calculate the location error for each MS location and then obtain the RMSE. Fig. 3 shows the RMSEs versus different number of hearable BSs when σ = 3dB. It can be observed that when N ≤ 4,


proposed SDP estimator performs almost the same as (and sometimes even better than) the asymptotically optimal ML estimator.

200 LLS SDP ML SDPRSS

180

RMSE (m)

160 140

R EFERENCES

120

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100 80 60 40 20 0

1

2

Fig. 2.


5

6

RMSE versus σ when N = 4.

85 LLS SDP ML SDPRSS

80 75

RMSE (m)

70 65 60 55 50 45 40 35

3

4

Fig. 3.

5 6 Number of hearable BSs

7

8

RMSE versus N when σ = 3dB.

SDPRSS performs best, while when N > 4, ML is the best. As N increases, the performance of all the estimators turns better, and ML exhibits the greatest extent of improvement. No matter what the value of N is, SDPRSS is always better than SDP and LLS. These results indicate that SDPRSS is more suitable for small N , i.e., the case of limited BSs rather than rich sources. This is definitely a very good property since in most scenarios, we can not expect there are abundant BSs available. On the other hand, this is also a drawback of SDPRSS since it cannot benefit from redundant BSs as efficiently as ML, probably due to relaxation. Alternatively, the solution produced by SDPRSS can serve as a good initial point for ML for larger N . VII. C ONCLUSIONS To circumvent the nonconvexity of conventional ML estimator (requiring reasonable initialization), we apply the SDP relaxation technique and design a convex SDP estimator for the RSS-based wireless localization system. The proposed SDP estimator exhibits excellent performance in the RSS-based wireless localization system and it is much better than the general SDP estimator and the LLS estimator. Moreover, in the case of very limited BSs hearable, the
