IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009 4777 Ranging Energy Optimization for Robust Sensor Positioning Based on Semidefinite Programming Tao Wang, Geert Leus, Senior Member, IEEE, and Li Huang, Member, IEEE Abstract—Sensor positioning is an important task of location-aware wireless sensor networks. In most sensor positioning systems, sensors and beacons need to emit ranging signals to each other. Sensor ranging energy should be low to prolong system lifetime, but sufficiently high to fulfill prescribed accuracy requirements. This motivates us to investigate ranging energy optimization problems. We address ranging energy optimization for an unsynchronized positioning system, which features robust sensor positioning (RSP) in the sense that a specific accuracy requirement is fulfilled within a prescribed service area. We assume a line-of-sight (LOS) channel exists between the sensor and each beacon. The positioning is implemented by time-of-arrival (TOA) based two-way ranging between a sensor and beacons, followed by a location estimation at a central processing unit. To establish a dependency between positioning accuracy and ranging energy, we assume the adopted TOA and location estimators are unbiased and attain the associated Cramér–Rao bound. The accuracy requirement has the same form as the one defined by the Federal Communication Commission (FCC), and we present two constraints with linear-matrix-inequality form for the RSP. Ranging energy optimization problems, as well as a practical algorithm based on semidefinite programming are proposed. The effectiveness of the algorithm is illustrated by numerical experiments. Index Terms—Cramér–Rao bound, localization, semidefinite programming, wireless sensor networks. I. INTRODUCTION IRELESS sensor networks (WSNs) enable a rich variety of promising applications, and therefore have attracted intensive research interest lately . Typical WSNs consist of untethered sensors randomly deployed to collect application-specific measurements, as well as a few fusion centers for in-network data processing. For most WSNs, sensor positions have to be estimated first, because they are often indispensable to annotate sensed data. For instance, temperature data produced by an environment-monitoring WSN are W Manuscript received December 16, 2008; accepted June 18, 2009. First published July 21, 2009; current version published November 18, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Brian M. Sadler. Part of this paper was presented at International Conference on Acoustics, Speech and Signal Processing (ICASSP), Taipei, Taiwan, April 19–24, 2009. T. Wang was with the Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD Delft, The Netherlands. He is also with the Stichting IMEC Nederland, Holst Centre, Eindhoven, The Netherlands (e-mail: [email protected]). G. Leus is with the Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail: [email protected]). L. Huang is with the Stichting IMEC Nederland, Holst Centre, Eindhoven, The Netherlands (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2009.2028211 useless unless we know where they are sensed. Furthermore, some WSN applications, e.g., target tracking, require the sensor positions to be known a priori . Especially for large-scale WSNs, the sensor positions are required for self-organization, such as naming , routing , as well as ciphering , just to name a few. Therefore, sensor positioning has been a research topic of particular interest over the past few years. To support location-aware WSNs, sensor positions must be estimated reliably with prescribed accuracy requirements fulfilled. Perhaps the most well-known requirement is the one defined by the Federal Communication Commission (FCC) . More specifically, it requires the location estimation error to with probability higher than , have a length smaller than and have a prescribed value. Although this where both requirement has been introduced to regulate the localization of mobile users, it can be considered to prescribe an accuracy requirement for sensor positioning as well. In practice, it is costly to equip each sensor with a global positioning system (GPS). Instead, a few beacons, which are sensors or fusion centers with known positions, are encompassed in most WSNs for locating the rest of the sensors in two steps. In the first step, beacons and sensors are scheduled to emit ranging signals to each other, and some signal parameters related to sensor positions are measured. Possible parameters include time-of-arrival (TOA), time-difference-of-arrival (TDOA), received-signal-strength (RSS), as well as angle-of-arrival (AOA) , . In the second step, these measurements are transformed into distance or bearing information, from which the sensor positions are estimated by a specific algorithm with the use of the beacons’ positions. In practice, all the above position-dependent parameters carry information about the sensor position, and an optimal localization algorithm should make use of all information related to the sensor position. However, the complexity may be too high especially when the localization task is partly carried out by the sensor. As a result, practical two-step algorithms usually estimate one or several dominant position-dependent parameters in the first step, and then estimate the sensor position in the second step. This incurs a loss of position-related information, and therefore practical algorithms have suboptimal performance in general. However, they are preferred for WSNs where the reduction of implementation complexity is a big concern. To locate sensors with high accuracy, TOA measurements are preferred, since TDOA measurements require multiple synchronized beacons, RSS measurements need an accurate pathloss model, and antenna arrays are necessary for AOA measurements . Thanks to the superior penetration and resolution capability of ultrawideband (UWB), TOA-based ranging using 1053-587X/$26.00 © 2009 IEEE Authorized licensed use limited to: TU Delft Library. Downloaded on July 06,2010 at 12:01:11 UTC from IEEE Xplore. Restrictions apply. 4778 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009 UWB pulses has been regarded as the most promising candidate for WSNs. Especially for unsynchronized WSNs, TOAbased two-way ranging (TWR) is proposed and supported by the IEEE 802.15.4a standard . Recently, a variety of work on TOA-based ranging has been reported in –. Particularly, the associated CRBs have been derived to benchmark the performance of a particular algorithm , –. As for the second step, two types of positioning algorithms have been developed. There is a rich literature on noncollaborative positioning algorithms, which consider only the aforementioned ranging between the sensor and the beacons –. Recently, collaborative positioning algorithms, which employ additional inter-sensor ranging for performance improvement, have been developed –. The associated CRBs have been derived in , –, and – to benchmark the performance of a noncollaborative and a collaborative algorithm, respectively. For the above two-step algorithms, positioning accuracy improves if the ranging energy of sensors and beacons is enhanced. In real scenarios, a beacon might have a reliable power supply and its ranging energy can be easily increased, but the ranging energy of an untethered sensor must be reduced in order to prolong system lifetime. Therefore, the positioning accuracy is actually dominated by the sensor ranging energy, which should be small but sufficiently high to fulfill prescribed accuracy requirements. This motivates us to investigate the following ranging energy optimization problem: how to allocate the ranging energy to sensors and beacons, so that the sensor ranging energy is minimized and specific accuracy requirements are fulfilled as well? We will address this problem for an unsynchronized robust sensor positioning (RSP) system, which consists of power-supplied beacons connected to a central processing unit (CPU), as well as sensors randomly deployed within a prescribed service area. We assume a line-of-sight (LOS) channel exists between the sensor and each beacon. The positioning is implemented by TOA-based TWR between a sensor and the beacons, followed by a location estimation at the CPU with a noncollaborative algorithm. In particular, this system features RSP, in the sense that a specific accuracy requirement is fulfilled within a prescribed service area. To reduce the implementation complexity, the ranging energy of both the sensor and the beacons is fixed and determined during the system design phase. In order to establish a mathematical dependency between the positioning accuracy and the ranging energy, we assume the adopted TOA estimator achieves its CRB with an unbiased Gaussian distribution, and the positioning CRB is achieved by an unbiased location estimator. The motivation behind these assumptions is twofold. One is that although our assumptions seem too optimistic in practice, the associated positioning CRB has a very attractive mathematical structure which lends the proposed problems to be efficiently solved by semidefinite programming (SDP). The other is that the optimal ranging energy allocation for a real scenario is actually lower bounded by the result produced by SDP for the optimistically assumed scenario. Therefore, our assumptions lead to useful optimization results, which can not only be found efficiently by SDP, but also provide a sense of how much energy should be allocated at least. Fig. 1. An exemplary sensor positioning system, where the circular area and the small circles represent the service area and sensors, respectively. The rest of this paper is organized as follows. In the next section, we will derive the performance of the considered TOA and location estimators. Next, Section III will present the RSP constraints when the location estimate is Gaussian distributed, and when such statistical knowledge is unavailable. After that, we will propose ranging energy optimization problems and a practical algorithm based on SDP in Section IV. In Section V, we will illustrate the effectiveness of the proposed algorithm by numerical experiments. Finally, we will wrap up this paper by some conclusions in Section VI. II. SYSTEM SETUP AND PERFORMANCE ANALYSIS In this section, we will first describe a two-dimensional (2D) RSP system. Next, a TOA-based TWR procedure is introduced and its performance is derived. Finally, the performance of the adopted positioning algorithm, which is actually the positioning CRB, is derived. A. System Setup We study a 2D RSP system with beacons deployed and connected to a CPU through wired or radio links . is at a known coordinate , Beacon , and we will consider locating a sensor at an unknown within a prescribed service area . coordinate The distance between the sensor and beacon is denoted as , where represents the -norm operator. In addition, the clocks of the sensor and the beacons are unsynchronized but run at the same pace. We assume that the two-sided power spectral density (PSD) of the additive white Gaussian noise (AWGN) at the sensor and the beacons is and , respectively. For illustration purposes, Fig. 1 shows an exemplary system with a circular service area and a few sensors. We assume a LOS channel exists between the sensor and and attenbeacon , which incurs a propagation delay . Here represents the signal propagation speed and uation , where and refer to the path gain at 1 m and the path-loss coefficient, respectively. We assume and are both known by system designers. Authorized licensed use limited to: TU Delft Library. Downloaded on July 06,2010 at 12:01:11 UTC from IEEE Xplore. Restrictions apply. WANG et al.: RSP BASED ON SEMIDEFINITE PROGRAMMING 4779 During the ranging phase, the sensor and beacon are scheduled to broadcast respectively the ranging signals and . In practice, a ranging signal can be the preamble part of a communication signal between the sensor and beacon . Note that these signals are not necessarily UWB in general. Finally, we assume the channel remains unchanged are during the ranging phase, and both the signal TOA and regarded as unknown deterministic parameters by the adopted TOA and location estimators. It has been shown in , , and  that the CRB of TOA estimation depends on the root-mean-square (RMS) angular frequency of the adopted ranging signals. We assume all beacon ranging signals have the same RMS angular frequency. To facilitate the following derivations, we define the RMS angular and respectively as frequency of – for scenarios similar to the one considered here. For self-consistency purposes, we put a brief derivation of the CRB is Gaussian disin Appendix A, where we show that tributed as , and is expressed by (4) Similarly, has , where the Gaussian is evaluated as distribution (5) and are independent since they We can see that is disare generated from independent signals. Therefore, tributed as , where is given by (6) (1) (7) (2) where and , respectively. represent the spectrum of and and . In fact, reprewith sents the TOA estimation accuracy of the sensor relative to that ( means that the TOA estiof beacon when mation of the sensor is more accurate). C. Performance of the Positioning Algorithm B. TOA-Based TWR and Performance TOA-based TWR proceeds as follows . First, the CPU schedules the beacons to broadcast ranging signals sequentially, so that they are separated when arriving at the sensor. Let’s say broadcasts of energy at time . At the beacon sensor, the LOS signal’s TOA is estimated and denoted . After the sensor has generated all the TOA estimates, as of energy at time to the beacons. At it broadcasts is estimated and debeacon , the LOS signal’s TOA noted as . We assume hardware calibration is perfectly and accomplished by the sensor and beacon , so that are precisely known by the sensor and beacon , respectively. and are recorded by the It is important to note that sensor with its internal clock, while and are recorded by beacon with its internal clock. Finally, all the processing delays produced by the sensor are first transmitted through data packets to the beacons, and then sent to generated by the CPU. Meanwhile, the total delay beacon is also transmitted to the CPU, which evaluates as (3) Ideally, if and are precisely estimated, is . However, the estimation performance is degraded equal to by the AWGN. It is well known that the variance of any unbiased estimator is lower-bounded by the CRB. We assume each TOA estimator for the TWR achieves the CRB with an unbiased Gaussian distribution. In fact, this can be asymptotically accomplished by the maximum-likelihood (ML) estimator , . The CRB has been derived in  when the distribution is known a priori, and in , of the propagation delay and After TWR, a set of independent measurements is available at the CPU for estimating . For any unbiased losets a lower-bound to its covarication estimator, the CRB ance. The CRBs for similar scenarios have been derived in  and –; however, they did not consider the presence of as nuisance parameters, and thus the results reported can be there can’t be used here. In Appendix B, we show evaluated according to (8) , and . Here, can be regarded as the effective energy that combines the joint effect and on . Note that , and of is an increasing function of both and . One may assume that the location estimate follows the . Such an assumption is attracGaussian distribution tive because the Gaussian distribution has salient mathematical properties and therefore can ease theoretical analysis. For instance, this assumption was adopted for localization performance analysis in  and . In addition, the Gaussian-distribution assumption can be justified since the CRB can be achieved with an unbiased Gaussian distribution by the ML estimator, provided that we have a large set of independent and identically distributed (i.i.d.) ranging measurements . Although such a large set is usually unavailable in practice, it has been shown in  that the estimator based on constrained weighted least-squares could approach the CRB if the ranging accuracy is sufficiently high, and the distribution of the location estimate resembles an unbiased Gaussian distribution. where Authorized licensed use limited to: TU Delft Library. Downloaded on July 06,2010 at 12:01:11 UTC from IEEE Xplore. Restrictions apply. 4780 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009 On the other hand, although the location estimators based on approximate ML , , and multidimensional scaling (MDS)  can also achieve the CRB, they do not necessarily follow a Gaussian distribution. Based on the above considerations, we assume is unbi, while leaving the assumpased and its covariance attains tion about the distribution of open. In the next section, we will derive constraints for RSP when is Gaussian distributed, as well as when the knowledge about the distribution of is unavailable. location-dependent functions. We can see that a higher accuracy and increasing . level is expected by reducing In the following subsections, we will propose constraints for RSP when is Gaussian distributed, as well as when the knowledge about the distribution of is unavailable. A. Gaussian Distribution of the Location Estimate In this case, a sufficient condition to satisfy the accuracy re, quirement at has been introduced in  as where is the minimal eigenvalue of , and is the threshold translated from and according to D. Comments It is important to examine more closely the assumptions we made. First, we assumed the TOA estimation CRB can be achieved. However, it was shown in ,  that this CRB is in nature a local bound and not tight for scenarios with low signal-to-noise ratio (SNR). Alternatively, one may consider the improved Ziv–Zakai bound (ZZB) as a better metric. Second, we assumed a LOS channel exists between the sensor and each beacon, which may be too optimistic for real scenarios. Especially in indoor or dense urban environments, the LOS path may not exist and even if it exists, there may exist strong multipath interference on the LOS signal. Third, we assumed each ranging estimate is only degraded by the TOA estimation error and thus Gaussian distributed. In practice, it may be further degraded by the random behavior of queuing the ranging signals at the sensor and beacons. Our assumptions are motivated by the fact that if we take other bounds, such as the ZZB, or more realistic assumptions, finding the optimal ranging energy allocation for the proposed problems is generally very difficult. Although our assumptions seem too optimistic in practice, the associated CRB computed from (8) has a very attractive mathematical structure which lends the proposed problems to be efficiently solved by SDP. Furthermore, the optimal ranging energy allocation for a real scenario is actually lower bounded by the result produced by SDP for the optimistically assumed scenario. Therefore, our assumptions lead to useful optimization results, which can not only be found efficiently by SDP, but also provide a sense of how much energy should be allocated at least. Note that it is very attractive to investigate the gap between the optimal energy allocation under our assumptions and that for real scenarios, when more knowledge is available about the statistics of the channels, the signal queuing, and the implemented TOA and position estimation algorithms. Intuitively, this gap should be translated from the localization performance loss due to the impairments present in a real scenario but ignored by our assumptions. This topic is part of our future work. III. CONSTRAINTS FOR RSP To support location-awareness reliably, we require the sensor to be located with sufficiently high accuracy. To this end, we impose an RSP requirement on the considered system using the FCC definition. More specifically, we require that for every within , the location estimation error falls into the with probability higher than origin-centered circle of radius , i.e., , . Note that and can be prescribed according to the accuracy level expected by (9) This sufficient condition can be justified as follows. Since is distributed as , it was shown in  that falls within an ellipse expressed by with probability . This ellipse has a major principal axis . When of length is satisfied, is contained within the circle , which means that . Therefore, is sufficient to fulfill the accuracy requirement at . Note that is equivalent to the linear matrix inequality (LMI) , where is a 2 2 identity matrix, and means that is a positive semidefinite matrix. According to the convex optimization theory, the set of ’s satisfying this LMI is convex . With this constraint, convex optimization methods can be used to effectively solve the optimization problems proposed in Section IV. To satisfy the RSP requirement, we require the considered system to fulfill this sufficient condition for every position within , namely , . B. Unknown Distribution of the Location Estimate In this case, a sufficient condition to satisfy the accuracy re. In the folquirement at will be derived as lowing, we will show that can be translated from and according to (10) This sufficient condition can be derived according to the Chebyshev’s inequality, which is of the form for the considered system denotes the trace of and is equal . Here, to the mean square error of . Note that this inequality holds for any particular distribution of , and thus can be used to develop a sufficient condition when the knowledge about the distribution of is unavailable. As a result, , and it is sufficient to satisfy the accuracy requirement if , or equivalently . Referring to (8), each eigenvalue of is greater than , which means that each eigenvalue of is upper bounded by . As a result, . Therefore, a stronger sufficient condition is , or equivalently . Authorized licensed use limited to: TU Delft Library. Downloaded on July 06,2010 at 12:01:11 UTC from IEEE Xplore. Restrictions apply. WANG et al.: RSP BASED ON SEMIDEFINITE PROGRAMMING 4781 Note that is a sufficient condition as well. However, the associated set of feasible ’s forms a nonconvex set in general, and this will make it difficult to solve the optimization problems proposed in Section IV. Therefore, we choose the stronger condition . To satisfy the RSP requirement, we require the considered system to fulfill the proposed sufficient condition at every posi, . tion within , namely . We present the following theorem to compute , where Theorem 1: denotes the minimal eigenvalue of . Proof: Let’s consider the following optimization problem: C. Comments (12) The second RSP constraint is stronger than the first one, since for any . This is because the first one is derived with extra knowledge about the error ellipse of a Gaussian distribution. With the second constraint, the RSP can be achieved no matter how is distributed. This means that using the second constraint guarantees the RSP even though the knowledge about the distribution of is unavailable. To facilitate the discussion in the following sections, we for, , mulate a general RSP constraint as: if we know is Gaussian distributed, or where if we have no knowledge about the distribution of . and are the only optimization variables Note that in (12). Alternatively, (12) can be regarded as a simapproaches infinity. plification of (11) when every , we can With the equality constraint easily find that the inequality constraint in (12) reduces to . This means that satisfies each feasible . As a result, the optimal objective . value is We can express and alternatively as (13) IV. RANGING ENERGY OPTIMIZATION In this section, we will first formulate a few ranging energy optimization problems of interest under the RSP constraint proposed in Section III. Then, we will present a practical algorithm to solve them based on semidefinite programming (SDP). Note , are all treated as optimization variables that , on which the equality constraints , are imposed. A. Sensor Ranging Energy Optimization Problem We first prove that is a nondecreasing function , . Suppose is increased by a positive of . As a consequence, , and bevalue come , and , is a positive semidefinite matrix, respectively. Since according to Corollary 4.3.3 in , is nondecreasing which justifies our claim. Since with , the RSP constraint can be fulfilled by increasing entries of , which in turn is accomplished by enhancing and . , where It is important to note that denotes the -norm of , and the equality increases to be infinity. To prolong holds when at least one should be reduced as much as possible. system life time, is reduced too much, every entry of will However, in case be bounded by a too small to fulfill the RSP constraint. This for , above which RSP motivates us to find a threshold can be computed as the becomes possible. Mathematically, optimal objective value of the following optimization problem: (11) where and are the feasible sets of for the problems (11) and (12), respectively. since . Suppose now that Apparently, . This means that there exists at least one particular that fulfills the constraints of (11) and . We can build a vector . Obviously, each entry of is no smaller than that of . Since is nonde, , and therecreasing with is feasible for (12). This means is a possible fore objective value of (12), but this value is smaller than , and is the minimal objective value of (12) is thus the fact that . Therefore, violated. This contradicts the assumption . We can see that is actually the minimal -norm of any fulfilling the RSP constraint. This implies that at least one has to be infinitely high, in case is reduced to its lowest level . It is interesting to observe that, if is increased from to , has to be increased by dBJ, where . In effect, this increase can be regarded as the extra sensor energy required to compensate for the unavailable knowledge about the distribution of when ideally approaches infinity. It is also interesting to study the impact of the path loss on . In order to facilitate the following coefficient and, discussion, let’s denote thus, . In fact, where . Suppose after increases, and change to and , respectively. Since reis positive semidefinite, duces and each holds according to Corollary 4.3.3 in . Suppose , we have . This means that is nonincreasing with . Therefore, the threshold sensor energy Authorized licensed use limited to: TU Delft Library. Downloaded on July 06,2010 at 12:01:11 UTC from IEEE Xplore. Restrictions apply. 4782 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009 will not reduce, if the signal energy decays faster with respect to the distance. B. Sensor and Beacon Ranging Energy Optimization Problems In real scenarios, we can not simply adopt the energy alloca, for at least one beacon, because tion is usually constrained by a prescribed value due to implementation difficulties, e.g., short ranging duration or limited power due to power-amplifier nonlinearity. Note that the semust be no smaller than in order to fulfill the lected since RSP constraint. This is because and hold. Besides, must be satisfied in order to fulfill the RSP constraint. Theremust be no smaller than in order to fulfill the fore, RSP constraint. and The optimization problem now is to find the minimal the associated which meet the RSP constraint. Mathematically, this problem can be cast into the following form: contains more than one element, since using any other , another solution with less beacon energy usage can than be constructed. For all those solutions, the optimal value of remains fixed at . Apparently, this , and the gap between and value is greater than can be computed in dBJ as follows: (18) We can see that reduces as increases, which means can approach closer if increases. In practice, that and , (18) can be used to establish a tradeoff between which is helpful to guide ranging hardware design of beacons , increases and sensors. We can also see that for a fixed increases. This means that if changes from to , as and becomes greater. the gap between Instead of using the solution in (17), another solution with a more efficient use of beacon energy can be constructed by using obtained by solving the following problem: in (16) the (19) (14) , Let’s denote the optimal solution to (14) by and . We present a general solution to the above problem with the following theorem. Theorem 2: The general solution to (14) is (15) (16) , which belongs to the set . Proof: First of all, after some simple mathematical manipand given above satisfy the ulations we can see that the must be no less than equality constraint in (14). Note that in order to fulfill the RSP constraint, and thus is nonnegative. Besides, since , and hence is also nonnegative. Therefore, and belong to the feasible set of (14). Second, is the optimal value of , bebecomes smaller than that value, the RSP concause once . The reason for straint can not be satisfied since , this is that for any when and , since is and . increasing with Third, when , . To fulfill the RSP must be equal to and thus must belong constraint, can be found by (16). to . Therefore, At least one solution to (14) exists and is given by where is the th entry of (17) In fact, this solution is constructed with in . This solution is the worst one if where is a weighting vector. We assume it is designed under the constraints that and . Particularly, assigning a greater repre. One special case sents a stronger expectation to reduce of interest is to set all entries of to zero except for the th and if . Using the associated entry, i.e., ’s by (16) reduce the optimal for (19), the constructed ranging energy of beacon to its minimal possible value with the RSP constraint satisfied. In general, no closed-form solution exists for (19). is equivalent to the LMI Nevertheless, , and, thus, (19) actually belongs to the class of SDP problems, which can be solved numerically with convex optimization techniques . C. A Practical Algorithm To evaluate and solve (19), the main difficulty lies in the fact that is in general a continuous area. A practical algorithm is realized by replacing with a discrete grid set , is the th grid point within . Then, we can evaluate where and solve (19) using numerical convex optimization software, such as Sedumi . Suppose that is generated by sampling uniformly in both . Ideally, vertical and horizontal directions with a spacing should approach zero as close as possible. But if is too small, the optimization problems will go beyond the processing ability of the optimization software, since too many LMI constraints are produced. This motivates us to choose a reasonably small . To this end, let’s consider and , which devalue for note the sets of fulfilling the RSP constraint over and , , and should be suffirespectively. Obviously, . ciently small such that , consider a square cell To find a principle for choosing of lateral length centered at a grid point . Such a Authorized licensed use limited to: TU Delft Library. Downloaded on July 06,2010 at 12:01:11 UTC from IEEE Xplore. Restrictions apply. WANG et al.: RSP BASED ON SEMIDEFINITE PROGRAMMING Fig. 2. An exemplary square cell C 4783 and its position relative to three beacons. cell is shown in Fig. 2 for illustration purposes. Within this cell, has the following property. is approximately a concave function Theorem 3: , provided that , where of refers to the distance of to the closest holds at four corner beacon. In addition, if points , for every . , , Proof: When and and thus hold for every . Therefore, can be approximated as follows: (20) and . where operator is an Obviously, the function inside the last affine function of parameterized by and , so is approximately a pointwise minimum of this function over . is approximately a concave function Therefore, . of In fact, is the convex hull formed by its corner points . This means that , there exists a set where and , such that . If holds, is satisfied within because of the concavity of over (21) Based on Theorem 3, if the RSP constraint is fulfilled at all corner points, the RSP constraint will be satisfied over the whole . As a result, should be chosen according to Fig. 3. System setup and subregions. . Under this condition, using any fulfilling the RSP constraint over , the RSP constraint is satisfied over as well, because the RSP constraint is satisfied within each cell formed by four adjacent points in , and those cells cover . To use the above sampling condition, we assume that all beacons lie outside . It is important to note that, if one beacon is will be very small. In order to very close to , reduce the size of , we can sample nonuniformly with the following method. First, divide into subregions . Then we sample each region uniformly with a spacing to produce a discrete set . Finally, is pro’s as: . Based on a duced by combining all similar analysis as that for uniform sampling, we can then jus. Apparently, , so has a size no greater tify than that produced by uniform sampling. V. NUMERICAL EXPERIMENTS For illustration purposes, numerical experiments have been conducted using the system setup shown in Fig. 3. Suppose is a square area centered at (0,0) with lateral length 2 m. There , , are three beacons located at , respectively. Note that the unit for all those and , values is meter. The system parameters are set as: , , and . We will set in the following experiments unless otherwise stated. All computations are performed with Matlab v7.1 and Sedumi v1.1 on a laptop equipped with an AMD Turion CPU of speed 2.2 GHz and a memory of 2 G Bytes. to be fixed at 6 cm. For the RSP constraint, we prescribe from 0.7 As we expect a better accuracy level by increasing to 0.95, increases from 28.3 to 32.2 dB as shown in Fig. 4. to , the increase of in dBJ is When changes from , which is shown in Fig. 4 as well. In equal to effect, this increase is the extra sensor energy required to compensate for the unavailable knowledge about the distribution of Authorized licensed use limited to: TU Delft Library. Downloaded on July 06,2010 at 12:01:11 UTC from IEEE Xplore. Restrictions apply. 4784 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009 Fig. 4. The computed results for and = when R = 6 cm. Fig. 5. Number of constraints and computation time. , when attains its lowest level and at least one approaches infinity. It is shown that this extra energy is between 4.4 and 8.2 dBJ, and a higher demands a higher extra energy. which corIn the following experiments, we choose responds to an extra energy of about 5.6 dBJ. , should be replaced with a discrete set . To evaluate Both uniform and nonuniform sampling are performed to produce . For the uniform sampling, we use a spacing where . For the nonuniform sampling, we divide into four subregions by and as is sampled uniformly with a spacing shown in Fig. 3, and where . Here, is a parameter for tuning the sampling spacing. During the evaluation, we the computed remains essentially find that when and , unchanged as 4.32 and 1.14 dBJ for respectively. This implies that is quite close to when is below 1%. When the nonuniform sampling is used instead of Fig. 6. The computed results for E Fig. 7. The computed results for , respectively. = E = with respect to . with respect to E when = and the uniform sampling, the number of constraints and computation time is reduced by about 17% as shown in Fig. 5. Hence, we replace by produced with the nonuniform sampling using for the following experiments. , we have evaluated In order to show the effect of on when varies from 2 to 4, and the results are shown in increases with . This means for Fig. 6. It is shown that will increase when signal energy decays faster a fixed , with respect to distance. increases from 5 to 12 dBJ, we have computed When when and , respectively. The results are increases, reduces. shown in Fig. 7. It is shown that as , is significantly higher than when . When , which In the following experiments, we choose corresponds to and 0.60 dBJ, or equivalently and 1.75 dBJ, when and , respectively. Authorized licensed use limited to: TU Delft Library. Downloaded on July 06,2010 at 12:01:11 UTC from IEEE Xplore. Restrictions apply. WANG et al.: RSP BASED ON SEMIDEFINITE PROGRAMMING 4785 should be increased from 7.11 to 9.79 dBJ. These energy increases correspond to the extra sensor and beacon energy required to compensate for the unavailable knowledge about the distribution of , when the energy of beacon 1 is used most efficiently. VI. CONCLUSION Fig. 8. The computed energy allocation to E and E when w = [1; 0; 0] . We have addressed ranging energy optimization for an unsynchronized positioning system, which features RSP in the sense that a specific accuracy requirement is fulfilled within a prescribed service area. We assume a LOS channel exists between the sensor and each beacon. The positioning is implemented by TOA-based TWR between a sensor and beacons, followed by a location estimation at a CPU. To establish a dependency between positioning accuracy and ranging energy, we assume the adopted TOA and location estimators are unbiased and attain the associated CRB. The accuracy requirement has the same form as that defined by the FCC, and we have presented two constraints with LMI form for RSP. Under these constraints, ranging energy optimization problems, as well as a practical algorithm based on SDP have been proposed. We have illustrated the effectiveness of the algorithm by numerical experiments. Although only a 2D system is considered here, the proposed methods can be easily extended to a 3D RSP system. Besides, these methods are not just for UWB based positioning systems, but can be generally applied to other systems as well, as long as the system models and assumptions considered here hold for those systems. APPENDIX A DERIVATION OF THE CRB FOR ESTIMATING Fig. 9. Error ellipses 8 and circles 8 Let’s consider the estimation of at the sensor. The received signal can be expressed as , where is the AWGN with PSD . Let’s stack all unknown parameters into a vector . The log-likelihood function of can be expressed as  for randomly chosen points. (22) To find a better beacon energy allocation than , we construct a solution to (14) by (16) using the that solves subject to the RSP con(19). Let’s say we want to minimize . straint, so we prescribe the weighting vector as The associated problem (19) was solved with Sedumi v1.1, and ’s constructed with the produced are shown in Fig. 8 the and , respectively. The consumed CPU when computation time is 11.06 and 11.67 s, respectively. It is shown is 7.11 and 9.79 dBJ, respectively, while that the computed and are always 10 dBJ. To show the effectiveness of the energy allocation 4.15 dBJ, 7.11 dBJ, 10 dBJ, and 10 dBJ when , we randomly select a set of points within . We plot the error ellipse and the circle for each point, using the given energy allocation. It is shown in Fig. 9 that each ellipse is enclosed by the associated circle, which indicates that the RSP constraint is indeed satisfied for those randomly chosen to , the energy allocation to points. When changes from should be increased from 4.15 to 1.75 dBJ, and that to where keeps unchanged when changes. The Fisher information matrix (FIM) is , where represents the ensemble average is the Hessian matrix of with operator, and can respect to . Under the assumptions made in Section II, be reduced to (23) where is the energy of the differential of . The CRB can then be evaluated as , and thus (24) Since we assume the above CRB is achieved with an is distributed as unbiased Gaussian distribution, . Authorized licensed use limited to: TU Delft Library. Downloaded on July 06,2010 at 12:01:11 UTC from IEEE Xplore. Restrictions apply. 4786 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 12, DECEMBER 2009 APPENDIX B DERIVATION OF THE CRB FOR THE LOCATION ESTIMATOR . Note Let’s consider the estimation of using that play the role of nuisance parameters. The log-likelihood function of the unknown parameter vector can be expressed as The Fisher information matrix (FIM) can be evaluated as , where is the Hessian matrix of with respect to . By some arrangements, we can show : that the following equalities hold (25) Using the above equalities, can be reduced to (26) where is the Hessian matrix of with respect to , is an all-zero matrix, and is the submatrix at the right-bottom corner. It is shown in  that (27) where is the gradient vector of with respect to (28) Inserting (28) into (27), we can show that (29) The CRB for the location estimator is actually the 2 2 sub. By some rearrangement, matrix at the upper-left corner of can be evaluated according to (8). ACKNOWLEDGMENT The authors would like to thank D. Neirynck, F. Shu, and G. Dolmans at Holst Centre for their suggestions. The authors would also thank the anonymous reviewers for their valuable comments. REFERENCES  I. F. Akyildiz and W. 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Sturm, “Using SeDuMi 1.02 a Matlab toolbox for optimization over symmetric cones,” Optimiz., Methods Softw., no. 10–12, pp. 625–653, 1999. Tao Wang received the B.E. and Do.E. degrees in electronic engineering from Zhejiang University, China, in June 2001 and June 2006, respectively. Since August 2000 to February 2001, he had an internship on the design and implementation of a microcontrol unit with Motorola Electronics Ltd. Suzhou Branch, China. In October 2004 to March 2005, he was a visiting student with the Institute for Infocomm Research, Singapore, working on ultrawideband antenna design. His current research interests are in the design of statistical signal processing algorithms, as well as optimization of wireless localization and communication systems. Geert Leus (M’01–SM’05) was born in Leuven, Belgium, in 1973. He received the Electrical Engineering and Ph.D. degrees in applied sciences from the Katholieke Universiteit Leuven, in June 1996 and May 2000, respectively. He has been a Research Assistant and a Postdoctoral Fellow of the Fund for Scientific Research—Flanders, Belgium, from October 1996 until September 2003. During that period, he was with the Electrical Engineering Department, Katholieke Universiteit Leuven. Currently, he is an Associate Professor with the Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, The Netherlands. During summer 1998, he visited Stanford University, Stanford, CA, and from March 2001 until May 2002, he was a Visiting Researcher and Lecturer with the University of Minnesota, Minneapolis. His research interests are in the area of signal processing for communications. Dr. Leus received a 2002 IEEE Signal Processing Society Young Author Best Paper Award and a 2005 IEEE Signal Processing Society Best Paper Award. He is the Chair of the IEEE Signal Processing for Communications Technical Committee, and an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and the EURASIP Journal on Applied Signal Processing. In the past, he has served on the Editorial Board of the IEEE SIGNAL PROCESSING LETTERS and the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. Li Huang (S’04–M’08) received the B.Eng. degree in electronics and information engineering in 2002 from Huazhong University of Science Technology (HUST), Wuhan, China, and the Ph.D. degree from both the National University of Singapore (NUS) and the Technical University Eindhoven (TU/e), The Netherlands, in 2008. Since 2006, he has been with IMEC-NL, Holst Centre, Eindhoven, working on system-level design for ultralow power wireless systems. His main research interests include statistical signal processing, channel modeling, medium access control (MAC) design, wireless communications, and magnetic and optical recording. 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