Geometry-Assisted Localization
Algorithms for Wireless Networks
Po-Hsuan Tseng, Member, IEEE, and Kai-Ten Feng, Member, IEEE
Abstract—Linear estimators have been extensively utilized for wireless location estimation for their simplicity and closed form property. In the paper, the class of linear estimator by introducing an additional variable, e.g., the well-adopted linear least squares (LLS) estimator, is discussed. There exists information loss from the linearization of location estimator to the nonlinear location estimation, which prevents the linear estimator from approaching the Crame´r-Rao lower bound (CRLB). The linearized location estimation problem-based CRLB (L-CRLB) is derived in this paper to provide a portrayal that can fully characterize the behavior for this type of linearized location estimator. The relationships between the proposed L-CRLB and the conventional CRLB are obtained and theoretically proven in this paper. As suggested by the L-CRLB, higher estimation accuracy can be achieved if the mobile station (MS) is located inside the convex hull of the base stations (BSs) compared to the case that the MS is situated outside of the geometric layout. This result motivates the proposal of geometry-assisted localization (GAL) algorithm in order to consider the geometric effect associated with the linearization loss. Based on the initial estimation, the GAL algorithm fictitiously moves the BSs based on the L-CRLB criteria. Two different implementations, including the GAL with two-step least squares estimator (GAL-TSLS) and the GAL with Kalman filter (GAL-KF), are proposed to consider the situations with and without the adoption of MS’s historical estimation. Simulation results show that the GAL-KF scheme can compensate the linearization loss and improve the performance of conventional location estimators.
Index Terms—Linear least squares (LLS) estimator, location estimation, Crame´r-Rao lower bound (CRLB), two-step least squares estimator, Kalman filter
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1
I
NTRODUCTIONW
IRELESS location technologies [1], [2], [3], which aredesignated to estimate the position of a mobile station (MS), have drawn a lot of attention over the past few decades. The location estimation schemes locate the posi-tion of an MS based on the measured radio signals from its neighborhood base stations (BSs). It is recognized that the distance measurements associated with the wireless loca-tion estimaloca-tion schemes are inherently nonlinear. The uncertainties induced by the measurement noises make it more difficult to acquire the MS’s estimated position with tolerable precision. A number of wireless positioning methods have been widely studied with various types of signal measurements, including time-of-arrival (TOA), time difference-of-arrival (TDOA), and the received signal strength (RSS). Since the RSS measurements can be highly inaccurate because of the shadowing effect in practice, the distance-based TOA measurements will be considered in this work.
There are several representative techniques which are widely used in practical localization systems to deal with the location estimation problem (LEP), such as the Taylor series expansion-based (TSE) [4] method and the linear least
squares (LLS) [5] method. The TSE method approximates the localization problem by taking the first two orders of Taylor expansion on the measurement inputs. Initial MS’s position estimate and the iterative processes are required to obtain a location estimate from the linearized system based on the TSE scheme. The major drawback of the TSE method is that it may suffer from the convergence problem due to an incorrect initial guess of the MS’s position. On the other hand, the original nonlinear estimation problem can be transformed into a linear relationship for the computation of MS’s position by introducing an additional variable. This type of linearized methods deal with the linearized location estimation problem (L-LEP) and will be the major discus-sion in the paper. In general, the LLS is one of the popular techniques to solve the L-LEP in practical localization systems, e.g., the Cricket system [6], and has been continuously investigated from research perspectives [7], [8], [9], [10], [11]. Moreover, the closed form characteristic of LLS estimator is suitable for real-time implementation due to its computational efficiency.
The first objective of this paper is to formulate the theoretic lower bound for the geometric analysis of the L-LEP. Note that the Crame´r-Rao lower bound (CRLB) serves as a benchmark of the non-Bayesian estimator. The CRLB for the conventional LEP is derived in [12], [13], and [14]. Since the LEP is inherently nonlinear, the original LEP is often transformed into an L-LEP by introducing an additional variable to transfer the nonlinear equation into a linear equation for the computation of MS’s position. This transformation leads to a different parametrization and the geometric analysis of the L-LEP has not been fully addressed in previous research work. The work presented
. P.-H. Tseng is with the Department of Electronic Engineering, National Taipei University of Technology, Taipei, Taiwan.
E-mail: phtseng@ntut.edu.tw.
. K.-T. Feng is with the Department of Electrical and Computer Engineering, National Chiao Tung University, Hsinchu, Taiwan. E-mail: ktfeng@mail.nctu.edu.tw.
Manuscript received 22 Apr. 2011; revised 15 Nov. 2011; accepted 14 Feb. 2012; published online 7 Mar. 2012.
For information on obtaining reprints of this article, please send e-mail to: tmc@computer.org, and reference IEEECS Log Number TMC-2011-04-0210. Digital Object Identifier no. 10.1109/TMC.2012.61.
in [15] provided a theoretic lower bound of L-LEP based on the maximum likelihood manner, which is equivalent to the CRLB under zero mean and independent Gaussian noises. In this paper, the performance analysis of L-LEP is conducted based on the theory of CRLB. The theoretical lower bound of the L-LEP is derived as the L-LEP-based CRLB (L-CRLB). Note that the main advantage of the derivation based on CRLB comparing to [15] is that the geometric properties can be captured from the formulation of L-CRLB. The closed form formulation of the Fisher information matrix (FIM) for the derived L-CRLB provides a comparison between the L-LEP and the conventional LEP. Since an additional variable other than the MS’s position is required to be estimated for the L-LEP, it can be proved that the value of L-CRLB is greater than or equal to the conventional CRLB. The geometric layout between the MS and the BSs for the L-CRLB to be equivalent to the CRLB is also derived. By comparing the difference between the CRLB and the L-CRLB, it can be inferred from the proposed L-CRLB that better performance can be obtained if the MS is located inside the geometry constrained by the BSs; while inferior performance is acquired if the MS is outside of the geometric layout. Based on the observation, the geometry-assisted localization (GAL) algorithm is proposed.
In this paper, the GAL scheme is proposed to enhance the estimation precision by incorporating the geometric infor-mation within the conventional two-step least squares (LS) algorithm [9]. Note that the LLS method is one of the methods to solve the L-LEP; while the two-step LS is an performance enhancing estimator based on the LLS method. Since the linearization loss exists in the first step of the two-step LS estimator, the properties derived for the L-LEP can be utilized to describe the geometric effect of the two-step LS estimator. Based on an initial estimate of the MS’s location, the GAL algorithm is proposed to fictitiously rotate (i.e., not to physically relocate) different BSs locations according to the L-CRLB criterion in order to achieve enhanced MSs location estimate. Reasonable location estimation can be acquired by adopting the GAL algorithm, especially feasible for the cases with poor geometric circumstances of the L-LEP, e.g., if the MS is located outside of the geometric layout confined by the BSs. Two different types of implementations are proposed for the GAL scheme, including the GAL with two-step LS estimator (GAL-TSLS) and the GAL with Kalman filter (GAL-KF) schemes. The GAL-TSLS can directly provide enhanced MS’s location estimate compared to the conventional two-step LS method, which will be validated in the simulation results. The GAL-KF approach further uses the Kalman filter to provide smoothing effect on the initial estimate with a two-stage location estimation architecture [16], [17]. Simulation results illustrate that the proposed GAL-KF scheme can achieve higher accuracy for the MS’s estimated location compared to the other existing methods in both line-of-sight (LOS) and none-line-of-sight (NLOS) environments.
The contributions of proposed GAL algorithms are summarized as follows: note that the conference version of this paper [18] introduces initial idea of fictitious BS rotation based on the geometric condition between the MS and BSs. In this paper, we extend this concept to several aspects as follows: comprehensive geometric properties between the BSs and MS are first analyzed in the presence
of linearization loss. The benefits of fictitious BS rotation within the GAL algorithms can be observed from the analysis of geometric properties between the BSs and MS. Based on the concept of fictitious BS rotation, the proposed GAL algorithms decouple the original LEP into 1) an L-LEP with closed form solution and 2) the remaining nonlinear part as the fictitiously moveable BS problem. The L-LEP can be efficiently solved by the two-step LS estimator to provide an initial estimation for the fictitiously moveable BS problem. Note that the cost function of fictitiously movable BS problem is basically a mixture of trigonometric functions depending on the geometric configuration between the MS and BSs. Therefore, the peak value of cost function can be obtained with low computational complexity. In other words, the GAL algorithms decouple the original problem into two subproblems with efficient computation; while it can still provides better performance compared to the other existing nonlinear methods.
The remainder of this paper is organized as follows: Section 2 describes the properties that are derived from the CRLB and the L-CRLB metrics. The determination of fictitious BS’s locations based on the proposed GAL algorithm is explained in Section 3; while Section 4 demonstrates the GAL-TSLS and the GAL-KF schemes as the implementation of the GAL algorithm. Section 5 shows the performance evaluation of the proposed schemes. The conclusions are drawn in Section 6.
2
A
NALYSIS OFCRLB
ANDL-CRLB
2.1 Mathematical Modeling of Signal Sources The signal model for the TOA measurements is adopted for two-dimensional (2D) location estimation. The set rr contains all the available measured relative distance, i.e., rr¼ ½r1; . . . ; ri; . . . ; rN where N denotes the number of
available BSs. The measured relative distance between the MS and the ith BS can be represented as
ri¼ c ti¼ iþ ni for i¼ 1; 2; . . . ; N; ð1Þ
where c is the speed of light. The parameter tiindicates the
TOA measurement obtained from the ith BS, which is contaminated with the measurement noise ni. The noiseless
relative distance iin (1) between the MS’s true position and
the ith BS can be acquired as
i¼ kxx xxik for i¼ 1; 2; . . . ; N; ð2Þ
where xx¼ ½x; yTrepresents the MS’s true position and xxi¼
½xi; yiTis the location of the ith BS. The notation k:k denotes
the euclidean norm of a vector and ½:T represents the transpose operator.
Definition 1 (BS’s Orientation). Considering the MS as a vertex in geometry, the orientation of ith BS (i) is defined as
the angle between the MS to the ith BS and the positive x-axis. Without loss of generality, the index i of BSs are sorted such that the ith BS is located at the angle 1 2 i
N for i ¼ 1 to N.
Based on the definition of i, the following geometric
relationship can also be obtained as cos i¼ ðxi xÞ=iand
2.2 Properties of CRLB
Definition 2 (Location Estimation Problem). By collecting the measurements rr, the goal of the LEP is to generate a 2D estimate ^xx¼ ½^x; ^yT of the MS’s location.
The CRLB represents the theoretical lowest error variance of an unknown parameter for any unbiased estimator. Note that the CRLB in the rest of this paper refers to the CRLB for the conventional LEP. Therefore, based on the TOA-based LEP as in (1) and (2), the variance of the MS’s estimated position ^xxwill be greater or equal to the CRLB (C) as
Efk^xx xxk2g C ¼ ½I1
xx 11þ ½I1xx 22; ð3Þ
where the CRLB C ¼ ½I1xx 11þ ½I1xx 22 inherently represents the theoretical minimum mean square error (MMSE) of position. It is noted that ½I1
xx 11and ½I1xx 22correspond to the
first and second diagonal terms of the inverse of 2 2 FIM Ixx, which can be obtained as
Ixx¼ G I GT; ð4Þ where G¼@ @xx¼ x1 x 1 . . . xi x i . . . xN x N y1 y 1 . . . yi y i . . . y yN N 2 6 4 3 7 5 ð5Þ
¼ cos 1 . . . cos i . . . cos N sin 1 . . . sin i . . . sin N
; ð6Þ I¼ E @ @ln fðrrjÞ @ @ln fðrrjÞ T " # : ð7Þ The fðrrjÞ function in (7) denotes the probability density function for rr conditioning on , where ¼ ½1; . . . ; i; . . . ; N.
The matrices G and I are introduced as the change of
variables since Ixx is unobtainable owing to the unknown
MS’s true position xx.
Lemma 1.Considering the TOA-based LEP, the noise model for each measurement ri is an i.i.d. Gaussian distribution with
zero mean and a fixed set of variances 2
ri as ni N ð0;
2 riÞ.
The minimum CRLB Cm with respect to the angle i can be
achieved in [13] as Cm¼ 4 PN i¼112 ri ; ð8Þ
if the following two conditions hold: XN i¼1 1 2 ri sin 2i ¼ 0; XN i¼1 1 2 ri cos 2i¼ 0: 8 > > > > < > > > > : ð9Þ
Proof.Based on (1), fðrrjÞ can be obtained as fðrrjÞ /Y N i¼1 exp 1 22 ri ðri iÞ2 " # : ð10Þ
Therefore, the matrix I can be derived from (7) as
I¼ diagf½2r1; 2 r2; . . . ; 2 ri ; . . . ; 2 rNg. The 2 2 matrix Ixx
can be obtained from (4) and (7) as Ixx¼ ½Ixx11 ½Ixx12 Ixx ½ 21 ½ Ixx22 ¼ XN i¼1 1 2 ri cos2i XN i¼1 1 2 ri cos i sin i XN i¼1 1 2 ri cos i sin i XN i¼1 1 2 ri sin2i 2 6 6 6 6 4 3 7 7 7 7 5: ð11Þ
In order to obtain the minimum CRLB, (3) can further be derived as
Efð^xx xxÞ2g I1xx 11þI1xx 22¼ ½Ixx11þ ½Ixx22
½Ixx11 ½Ixx22 ½Ixx212
½Ixx11þ ½Ixx22
½Ixx11 ½Ixx22
:
ð12Þ
Noted that the second inequality in (12) is valid since the quadratic term ½Ixx212 0 for all i. Therefore, one of
the necessary conditions to achieve minimum CRLB will be ½Ixx12¼
PN i¼112
ricos i sin i¼ 0, which
vali-dates the first equation of (9). Moreover, since cos2 iþ
sin2
i¼ 1 for all i, the numerator in (12) becomes
½Ixx11þ ½Ixx22¼
PN
i¼11=2ri. Consequently, to acquire the
minimum value of CRLB corresponds to maximizing the denominator ½Ixx11 ½Ixx22 in (12). According to the
inequality of arithmetic and geometric means, the following relationship can be obtained:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½Ixx11 ½Ixx22 q ½Ixx11þ ½Ixx22 2 ¼ 1 2 XN i¼1 1 2 ri ; ð13Þ where the equality holds if and only if ½Ixx11¼ ½Ixx22, which
corresponds to the second equation in (9). By substituting (13) into (12), the minimum CRLB can be obtained as Cm¼ 4=ðPNi¼112
riÞ. This completes the proof. tu
Example 1 (Network Layout with Minimum CRLB). Following the requirement as in Lemma 1 with N ¼ 3 and all the variances are equivalent 2
ri ¼
2
rfor i ¼ 1 to 3,
the best geometric layout that can achieve the minimum CRLB Cm¼ 42r=3 is acquired at either the angle sets
f1; 2; 3g ¼ f; þ 120 ; þ 240 g or f1; 2; 3g ¼
f; þ 60 ; þ 120 g8 ¼ ½0 ; 360 Þ.
2.3 Properties of Proposed L-CRLB
Definition 3 (Linearized Location Estimation Problem).In order to estimate the MS’s position xx, the nonlinear terms x2
and y2 in (2) are replaced by a new parameter R ¼ x2þ y2.
The goal of the L-LEP is to generate an estimate ^ ¼ ½^xL; ^yL; ^RT based on the collecting measurements rr.
Note that the MS’s estimated position ^xxL¼ ½^xL; ^yLT of
the L-LEP is in general not optimal compared to the original LEP since an additional nonlinear parameter R is also estimated, which reduces the estimation precision for ^xxL
observation explains that the conventional CRLB cannot be achieved by the linearized location estimator for LEP. In order to appropriately describe the behavior of linearized location estimator, the L-CRLB is defined based on the relationships in (1) and (2) as follows:
Definition 4 (L-CRLB). The L-CRLB (CL) is defined for
linearized location estimation in terms of the estimated parameters ^xxL as
Efk^xxL xxk2g CL¼ ½I1 11þ ½I1 22; ð14Þ
where ½I1 11 and ½I1 22, respectively, denotes the first and
second diagonal terms of the inverse of 3 3 FIM matrix Ias
I¼ H I HT; ð15Þ
with
H¼@ @¼
cos 1 . . . cos i . . . cos N
sin 1 . . . sin i . . . sin N
1 21 . . . 1 2i . . . 1 2N 2 6 4 3 7 5; ð16Þ
and I obtained from (7).
Note that the derivation of the inequality (14) is neglected in this paper, which can be similarly referred from the derivation of CRLB in [19]. Based on the theory of CRLB, the closed forms of FIM in (15) can be formulated and the relevant matrix in (16) is derived. In other words, the proposed L-CRLB is adopted to denote the minimum variance for any estimator that estimates the parameter vector from the TOA measurements. In the following lemma, the fact that the L-CRLB is greater than or equal to the CRLB will be proved.
Lemma 2.Considering that there exists sufficient measurement inputs for location estimation with zero mean Gaussian noises, the L-CRLB is greater than or equal to the CRLB, i.e., CL C.
Proof.The 3 3 matrix I can be obtained from (7) and
(15) as I¼ Ixx B BT C ; ð17Þ where the matrices
B¼ X N i¼1 1 2 ri cos i 2i XN i¼1 1 2 ri sin i 2i " #T and C¼ X N i¼1 1 42 ri 2 i " # :
Note that the 2 2 matrix Ixxis the same as that obtained
from (4). Moreover, the inverse of the covariance matrix I can be represented as I1 ¼ ½I122 B 0 B0T C0 ; ð18Þ where the 22 submatrix ½I122of I1 can be obtained
as ½I122¼ ðIxx B C1 BTÞ1 based on the matrix
inversion lemma. Considering that there are sufficient measurement inputs for the linearized location estima-tion, i.e., N 3, both the covariance matrices I and Ixx
are nonsingular which corresponds to positive definite matrices. Consequently, the submatrix ½I22 and their
corresponding inverse matrices I1 , I1xx , and ½I122 are
positive definite. Furthermore, both C and C1 are positive definite since
C¼ X N i¼1 1 42 ri 2 i " # > 0:
Therefore, it can be shown that ½I22¼ ðIxx B C1
BTÞ I
xxsince C1 is positive definite and the equality
only occurs with zero matrix B. Given two positive definite matrices ½I22and Ixx, Ixx ½I22if and only if
½I122 I1xx > 0. Furthermore, since ½I122 I1xx , their
corresponding traces will follow as trace ð½I122Þ trace
ðI1
xx Þ which consequently results in ½I111 þ ½I122
½Ixx111 þ ½Ixx122. This completes the proof. tu
Corollary 1. The L-CRLB is equivalent to the CRLB if the following two conditions hold
XN i¼1 1 2 ri sin i i ¼ 0; XN i¼1 1 2 ri cos i i ¼ 0: 8 > > > > < > > > > : ð19Þ
Proof.As stated in Lemma 2, the necessary and sufficient condition for both L-CRLB and CRLB to be equivalent is that B is a zero matrix. Therefore, the two matrix elements in B, i.e., PNi¼1cos i=ð2riiÞ and
PN
i¼1sin i=
ð2
riiÞ, will be equal to zero. tu
It can be generalized from Corollary 1 that the two error terms "1¼PNi¼1cos i=ð2riiÞ and "2¼
PN
i¼1sin i=ð2riiÞ
will influence the value of L-CRLB, which consequently affect the precision of linearized location estimators. Under the geometric layout with smaller values of "1 and "2,
smaller difference between the CRLB and L-CRLB value can be obtained, which indicates that the linearization loss by adopting linearized location estimators is smaller. "1and
"2 can be mapping to the x- and y-direction vectors from
the MS to the BS. The noise variance terms can be regarded as the weighting of the direction vector. The minimum linearization loss for the linearized location estimator is achieved when the sum of the weighted direction vector from the MS to the BS is equal to zero. Besides, consider the case that the MS is situated outside of the polygon formed by the BSs, all the angles i will be in the range of ½0; 180
which results in larger value of the error terms "1 and "2.
As a result, the estimation errors acquired from the linearized location estimator will be comparably large in this type of geometric relationship. The following example is given to demonstrate the scenario where the L-CRLB is equal to CRLB.
Example 2 (Network Layout for Equivalent L-CRLB and CRLB). Assuming that the variances ri from all the
measurement noises are equivalent, the L-CRLB can achieve the CRLB if 1) the noiseless distances i from
the MS to all the corresponding BSs are equal, and 2) the orientation angles i from the MS to all the BSs are
uniformly distributed in ½0 ; 360 Þ as
i¼ 360 ði 1Þ=
Nþ , 8 ¼ ½0 ; 360 Þ and i ¼ 1 to N.
Proof.By substituting the conditions 1¼ 2¼ ¼ N and
r1¼ r2¼ ¼ rN into (19), the necessary condition for
the L-CRLB and the CRLB to be equivalent becomes PN
i¼1cos i ¼ 0 and PNi¼1sin i¼ 0. Based on the
as-sumptions as stated above, a vector can be utilized to represent the distance from the MS to the ith BS as i¼
½cos i; sin i for i ¼ 1 to N. In order to satisfy the
conditions for both PNi¼1cos i¼ 0 and PNi¼1sin i¼ 0,
the summation for projecting all unit vectors i for i ¼ 1
to N on the x-axis and y-axis respectively should be equal to zero. In order to achieve this condition, it can be verified that the angles i will be uniformly distributed
in ½0 ; 360 Þ with its value equal to
i¼ 360 ði 1Þ=
Nþ , 8 ¼ ½0 ; 360 Þ. This completes the proof. tu
Corollary 2. The minimum L-CRLB (CL;m) is achieved if the
conditions stated in (9) and (19) are satisfied.
Proof. It has been indicated that the minimum CRLB (Cm)
can be obtained if the conditions in (9) hold. Moreover, Corollary 1 proves that (19) should be satisfied for both L-CRLB and CRLB to be equivalent. Therefore, the minimum L-CRLB (CL;m) can be achieved if (9) and (19)
are satisfied. tu It can be observed from Corollary 2 that additional condition (19) should be satisfied for achieving minimum L-CRLB comparing with the minimum CRLB. The major difference is that the CRLB is affected by the angles i and
signal variances 2
ri; while the L-CRLB additionally depends
on the distance information i. Therefore, the performance
of the L-LEP is affected by the additional relative distance information between the MS and BSs. In order to provide intuitive explanation, the exemplified network layout for achieving minimum L-CRLB is shown as follows.
Example 3 (Network Layout with Minimum L-CRLB). Following the requirement as in Lemmas 1 and 2 with N ¼ 3 and all the variances are equivalent, i.e., 2
ri ¼
2 rfor
i¼ 1 to 3, and further assuming that the noiseless distances ifrom the MS to all the three BSs are equivalent,
the minimum L-CRLB can be achieved only at the angle sets f1; 2; 3g ¼ f; þ 120 ; þ 240 g8 ¼ ½0 ; 360 Þ.
Proof.Considering N ¼ 3 and r1¼ r2¼ r3¼ rin (9), the
following relationship is obtained:
sin 21þ sin 22þ sin 23¼ 0;
cos 21þ cos 22þ cos 23¼ 0:
ð20Þ It can be verified that both conditions in (20) are only satisfied at either one of the following angle sets: f1; 2; 3g ¼ f; þ 120 ; þ 240 g and f1; 2; 3g ¼
f; þ 60 ; þ 120 g8 ¼ ½0 ; 360 Þ. The corresponding
minimum CRLB can be calculated from (8) as Cm¼ 42r=3.
On the other hand, according to Corollary 2, the conditions (9) and (19) must be satisfied in order to
achieve minimum L-CRLB. Considering the N ¼ 3 case with 1¼ 2¼ 3, condition (19) is rewritten as
sin 1þ sin 2þ sin 3¼ 0;
cos 1þ cos 2þ cos 3¼ 0:
ð21Þ It can be verified that only the angle sets f1; 2; 3g ¼
f; þ 120 ; þ 240 g8 ¼ ½0 ; 360 Þ can satisfy all the
three conditions as defined in (20) and (21) for achieving the minimum value of L-CRLB. This com-pletes the proof. tu In other words, when the MS is positioned at the center of a regular polygon formed by the BSs, the proposed L-CRLB will be equivalent to the CRLB based on the conditions stated in (19). Example 3 describes the fact that minimum CRLB can be achieved under two different set of orientation angles; while the minimum L-CRLB is reached by one of its subset of angles. This indicates the situation that the L-CRLB provides a more stringent criterion compared to the CRLB for achieving its minimum value. Even though certain network layouts are suggested to achieve minimum CRLB, it does not guarantee that the corresponding L-CRLB can reach the same value. There-fore, the CRLB does not provide sufficient information to be used as the criterion for the linearized location estimator of the L-LEP; while the L-CRLB can be more feasible to reveal the geometric properties and requirements.
Example 4 (Contour Plots of CRLB and L-CRLB).In order to observe the difference between the CRLB and L-CRLB, their corresponding contour plots under the number of BSs N ¼ 3 are shown in Figs. 1a and 1b, respectively. Note that the three BSs are located at the vertexes of a regular triangular which are denoted with red circles in Figs. 1a and 1b. The positions of BSs are xx1¼ ½300; 200T
with 1¼ 0 , xx2¼ ½150; 286:6T with 2¼ 120 , and xx3¼
½150; 113:4T with 3¼ 240 . Based on the three BS’s
positions, each individual contour point represents the corresponding CRLB or L-CRLB value when the MS is situated at that location. The standard deviation of measurement noises ri is chosen as 1 m for simplicity.
It can be observed from Fig. 1a that there are four minimum points for the CRLB value equal to Cm¼ 1:33
with MS’s positions as xx¼ ½200; 200T, ½100; 200T, ½260; 120T, and ½260; 280T. The conditions for minimum CRLB can be verified by substituting the corresponding parameters into the condition (9). The minimum CRLB value can also be validated to satisfy (8), which demonstrates the correctness of Lemma 1.
On the other hand, by comparing Figs. 1a and 1b, it is observed that the distribution of L-CRLB is different from that of CRLB. The only minimum L-CRLB value identical to that of the CRLB, i.e., CL;m¼ Cm¼ 1:33, is
located at the center of regular triangle formed by the three BSs, i.e., xx¼ ½200; 200T. Starting at the MS’s position with minimum L-CRLB, the L-CRLB value will increase in all directions. Except for the minimum L-CRLB at the center of the triangle, the relationship that CL>C can be observed from both Figs. 1a and 1b.
Moreover, the difference between the L-CRLB and CRLB inside the triangle is smaller than that outside of the triangle. The reason can be contributed to the estimation
of parameter R by adopting the L-CRLB criterion, which introduces the two terms "1 and "2. Owing to the
nonlinear behavior of location estimation, the additional consideration of R within the L-CRLB can better characterize the performance of linearized location estimator for the L-LEP. The correctness of minimum L-CRLB value obtained from Fig. 1b can also be verified by substituting corresponding parameters into the conditions stated in Lemma 2, i.e., the conditions (9) and (19) can all be satisfied. By comparing the results from Figs. 1a and 1b, Corollaries 1 to 2 and Examples 2 to 3 can all be validated by substituting the corresponding numerical values.
In order to provide better explanation on the properties of CRLB and L-CRLB, the definitions of several geometric relationships between the MS and the BSs are described as follows: note that Fig. 2 illustrates the BS’s orientation for Definition 1 and geometric relationship between BSs and MS for Definitions 5 to 8.
Definition 5 (BS’s Adjacent Included Angle).Based on the BS’s orientation i, the adjacent included angle between two
neighboring ith and ði þ 1Þth BSs is defined as i¼ iþ1 i
for i ¼ 1 to N 1, and N ¼ 360 þ 1 N.
Definition 6 (BS Polygon).Considering the locations of BSs as the vertices in geometry, the BS polygon is defined by connecting the adjacent BSs as the edges of the polygon from BS1to BSN.
Definition 7 (Inside-Polygon Layout (IPL)).Given the BS’s adjacent included angle set ¼ f1; . . . ; i; . . . ; Ng, an
inside-polygon layout is defined if the MS is located inside the BS polygon where 0 <
i< 180 8 i from 1 to N.
Definition 8 (Outside-Polygon Layout (OPL)). Given the BS’s adjacent included angle set ¼ f1, . . . ; i, . . . ; Ng, a
outside-polygon layout is defined if the MS is located outside the BS polygon where there exists an adjacent included angle 180 i< 360 8 i from 1 to N.
Lemma 3.Consider two types of layouts, IPL and OPL, between the MS and three BSs under the requirements with equivalent variances 2
ri and noiseless distances i for i ¼ 1 to 3. There
can exist both IPL and OPL that possess the same CRLB value; while the corresponding L-CRLB value of the IPL is smaller than that of the OPL.
Proof.Given an IPL, the set of BS’s adjacent included angle is defined as in¼ f1; 2; 3¼ 360 1 2g where
0 < i < 180 8 i ¼ 1 to 3. The set of BS’s orientation
between the MS and three BSs can be represented as
in¼ f1¼ 0; 2¼ 1; 3¼ 1þ 2g. Without lose of
generality, 1 is set with zero degree according to the
rotation property as proven in [13] for CRLB. In order to establish an OPL, the third BS is repositioned to the reflected side with respect to the MS, which results in its BS’s orientation as out¼ f1¼ 0; 2¼ 1; 03¼ 1þ
2 180 g. By substituting both IPL and OPL cases with
in¼ f0; 2; 3g and out¼ f0; 2; 3 180 g
respec-tively into (3), it can be observed that same value of CRLB is achieved by both OPL and OPL.
Moreover, in order to compare the L-CRLB for the IPL and OPL, i.e., CL;in and CL;out, the difference of L-CRLB
for both layouts is derived from (14) to (16) with the substitution of in and out as
¼ CL;in CL;out¼ ½I;in111 þ ½I;in122½I;out111 ½I;out122
¼ 1 DI;inDI;out 8 cos2 2 cos 23 2 2 ð1 cos 2Þ ½2 cos 2þ 2 cos 3cosð2 3Þ 4
;
ð22Þ
Fig. 2. Illustration for Definitions 5 to 8.
where DI;in and DI;out denote the determinants of the
FIM matrix I;inand I;outfor the L-CRLB of IPL and OPL
respectively. Since both I;in and I;out are positive
definite, their corresponding determinants DI;in and
DI;out will be positive values. Furthermore, the following
conditions hold since the BS’s orientation set in
corresponds to an IPL: 0 < 2< 180 , 0 < 3 2<
180 , and 180 <
3< 360 . Therefore, the following
conditions hold for the numerator terms in (22): cosð2=2Þ > 0 since 0 < 2=2 < 90 , cos½ð23 2Þ=2 <
0 since 90 <ð2
3 2Þ=2 < 270 , ð1 cos 2Þ > 0 since
1 < cos 2< 1, and 2 cos 2þ 2 cos 3cosð2 3Þ < 4
since 1 < cos 2< 1 and 1 < cos 3cosð2 3Þ < 1.
As a consequence, the difference ¼ CL;in CL;out< 0
which corresponds to the result that the L-CRLB of the IPL is smaller than that of the OPL. This completes the proof. tu A key contribution of this paper is obtained from Lemma 3 that the proposed L-CRLB can describe the geometric relationship between the MS and its correspond-ing BSs, i.e., either the IPL or OPL; while the conventional CRLB criterion observes the same value for both cases. It is found in Lemma 3 that the L-CRLB for MS to locate inside the BS polygon will be smaller than that for MS situated outside the BS polygon. This result implicitly indicates that the estimation accuracy from a linearized location estima-tor will be higher for the IPL compared to the OPL case. Based on this fact, the main objective of the remaining sections is to propose a localization algorithm to enhance the estimation accuracy of the linearized location estimator for the OPL.
3
P
ROPOSEDG
EOMETRY-A
SSISTEDL
OCALIZATIONA
LGORITHMThe main objective of the proposed GAL scheme is to enhance the LLS-based algorithms by considering the geometric effect to the location estimation accuracy. The core component of the GAL scheme is to acquire the positions of the fictitious BSs to achieve the minimum L-CRLB value with respect to the MS’s initial location estimate ^xxo. Note that the position of the ith fictitious BS
with respect to the MS’s initial location estimate can be represented based on the measurement distance riand the
BS’s orientation i. Since ri is available as the measured
information, the determination of fictitious BS’s position corresponds to the adjustment of the BS’s orientation i. The
position information of these fictitious BSs will be utilized to replace that of the original BSs in order to achieve better geometric layout for location estimation, which will be discussed as the implementation of the GAL scheme in Section 4. In this section, the core mechanism of the GAL algorithm to identify which BSs should be fictitiously rotated will be demonstrated. The subschemes of the GAL algorithm with different numbers of fictitious BSs will be stated, i.e., the GAL with one fictitiously movable BS scheme (GAL(1BS)) and the GAL with two fictitiously movable BSs scheme (GAL(2BS)) in Sections 3.1 and 3.2, respectively. Section 3.3 describes the combined schemes of the GAL algorithm by selecting among different numbers of fictitious BSs based on the minimum L-CRLB requirement.
3.1 GAL with One Fictitiously Movable BS Scheme The GAL(1BS) scheme is designed to fictitiously relocate the position of one BS according to the criterion for achieving the optimal geometric layout, i.e., the minimum L-CRLB. Note that only one BS is allowed to be fictitiously movable and the others remain fixed in this case. The objective of the GAL(1BS) scheme is to decide which BS should be fictitiously moved in order to obtain the minimum L-CRLB. The GAL(1BS) problem is defined as
jm¼ arg min j¼1;...;N CL ~ m j subject to CL ~ mj <CL; j¼ 1; . . . ; N; ð23Þ where ~m
j represents the orientation of the j
mth fictitiously
moveable BS that achieves the minimum L-CRLB. Provid-ing that all the BSs within GAL(1BS) scheme cannot result in lowered L-CRLB values compared to the original CL, the
constraint defined in (23) will not be satisfied. In other words, the original L-CRLB has already been the lowest under the given measurement conditions, where none of the BSs is required to be fictitiously moved and the initial MS’s location estimate will become the final estimate. By observing from the problem defined in (23), the optimal rotated angle ~m
j of a single BS should be determined first.
The one fictitiously movable BS problem is defined to obtain the optimal rotated angle of the jth BS as
~
mj ¼ arg min
8 ~j
CLð ~jÞ; 8 ~j¼ ½0 ; 360 Þ: ð24Þ
Note that the original H matrix in (16) for the computation of CL cannot be obtained owing to the required true MS’s
position and noiseless relative distances. The estimated matrix ^Hcan be calculated based on the initial estimate ^xxo
and the measurement distance rr as
^ H¼ x1 ^xo r1 . . . xi ^x o ri . . . x~j ^x o rj . . . xN ^x o rN y1 ^yo r1 . . . yi ^y o ri . . . y~j ^y o rj . . . yN ^y o rN 1 2r1 . . . 1 2ri . . . 1 2rj . . . 1 2rN 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ¼
cos ^1 . . . cos ^i . . . cos ~j . . . cos ^N
sin ^1 . . . sin ^i . . . sin ~j . . . sin ^N
1 2r1 . . . 1 2ri . . . 1 2rj . . . 1 2rN 2 6 6 6 4 3 7 7 7 5; ð25Þ where ^i in (25) represents the ith BS’s estimated
orienta-tion based on the initial estimate ^xxo. The parameters ~xx j¼
ð~xj; ~yjÞ and ~jin (25) denote the position and orientation of
the jth fictitiously moveable BS, respectively. It is noticed that the design of proposed GAL scheme also considers the effect coming from the approximation of matrix ^H. As shown in Fig. 3b, the orientation of the fictitiously moved BS is considered to center at the MS’s estimated position instead of the true MS’s position as in Fig. 3a. It can be observed that the selection of fictitiously moveable BS may induce additional error since it is designed based on the imperfect initial estimate even though it can provide better geometry for location estimation. This demonstrates the
situation that the fictitiously moveable BS may not always result in lowered L-CRLB value than the original network layout as stated in the constraint of problem (23). Note that the initial estimation error can be approximated as a Gaussian noise distribution with the standard deviation of the initial estimation error re, i.e., N ð0;
2
reÞ where re
depends on the precision of the initial estimate ^xxo.
Providing that the LLS-based estimation is used for the initial estimation, the matrix I for the computation of CL
can be derived based on the precision of MS’s initial estimate re as I¼ diagf½ 2 r1; . . . ; 2 ri ; . . . ;ð 2 rjþ 2 reÞ 1 ; . . . ; 2r
Ng. The L-CRLB CLð ~jÞ in (24) can thus be derived as
CLð ~jÞ ¼ 1 D " XN i¼1 ði6¼jÞ sin ^i 22 ri ri þ sin ~j 2ð2 rjþ 2 reÞ rj !2 þ X N i¼1 ði6¼jÞ cos ^i 22 ri ri þ cos ~j 2ð2 rjþ 2 reÞ rj !2 þX N i¼1 ði6¼jÞ N 42 ri r 2 i þ N 4ð2 rjþ 2 reÞ r 2 j # ; ð26Þ
where Ddenotes for the determinant of FIM matrix I. By
neglecting the terms in (26) that are not related to the parameter ~j, the equivalent one fictitiously movable BS
problem as presented in problem (24) can be obtained as ~ mj ¼ arg min 8 ~j f1ð ~jÞ ¼ arg min 8 ~j 1 D " XN i¼1 ði6¼jÞ sin ^i 22 ri ri þ sin ~j 2ð2 rjþ 2 reÞ rj !2 þ X N i¼1 ði6¼jÞ cos ^i 22 ri ri þ cos ~j 2ð2 rjþ 2 reÞ rj !2# ; ð27Þ
where f1ð ~jÞ can be regarded as the cost function of the
considered problem. It can be observed that the solution of
problem (27) can be acquired if the following conditions on the first and second derivatives of f1ð ~jÞ are satisfied, i.e.,
@f1ð ~jÞ @ ~j ~ j¼ ~mj ¼ 0; ð28Þ @2f1ð ~jÞ @2~ j ~ j¼ ~mj > 0: ð29Þ Due to the complex formulation of (27)-(29), there does not exist closed form for obtaining the optimal value of ~m
j. In
order to solve the optimum rotated angle ~m
j, root-finding
algorithms can be utilized to find suitable solution candidates between ½0; 360 Þ in (28), and these solutions
will further be examined to satisfy the requirement of (29). If there are still multiple candidates that fits all the requirements, i.e., there are multiple local minimums for problem (24), the angle ~m
j that possesses with the global
minimum L-CRLB value will be chosen from (26) among those solution candidates.
Furthermore, in order to reduce the computation com-plexity, a cost function g1ð ~jÞ is defined to simplify the
original problem (24) without the consideration of D in
(27). An approximate one fictitiously movable BS problem can, therefore, be obtained as ~ mj ¼ arg min 8 ~j g1ð ~jÞ ¼ arg min 8 ~j " XN i¼1 ði6¼jÞ sin ^i 22 ri ri þ sin ~j 2ð2 rjþ 2 reÞ rj !2 þ X N i¼1 ði6¼jÞ cos ^i 22 ri ri þ cos ~j 2ð2 rjþ 2 reÞ rj !2# : ð30Þ
It is interesting to notice that the cost function g1ð ~jÞ can
be closely related to the conditions stated in (19) for Corollary 1. Providing that the minimum value of g1ð ~jÞ
approaches zero, the two conditions in (19) can be satisfied by solving problem (30). In other words, by fictitiously moving the jth BS via angle ~m
j with the consideration of
re for initial estimate, the layout with the smallest
linearization loss can possibly be achieved where the L-CRLB is equivalent to the CRLB. Therefore, based on the design of fictitious movable BS, the two square terms within g1ð ~jÞ can be treated as the extension of the two
error terms "1 and "2 as described after Corollary 1 that
affect the precision of linearized location estimators. By considering the first derivative of g1ð ~jÞ equal to zero, the
rotated angle ~m
j for this problem can be derived as
~ mj ¼ tan1 XN i¼1 ði6¼jÞ cos ^i 22 ri ri 0 B @ 1 C A , XN i¼1 ði6¼jÞ sin ^i 22 ri ri 0 B @ 1 C A: ð31Þ
Note that the angle ~m
j does not depend on any information
from the jth BS, i.e., the measurement of the jth BS. The angle ~m
j lies in the domain of arc tangent function between
ð90 ; 90 Þ which is half of the domain of ½0; 360 Þ. Since both
angles ~mj and ~mj þ 180 can be the local minimum of the
subproblem (30), these two angles will be further substituted into (26) to choose the angle with smaller L-CRLB value.
Following the procedures as stated above, all the BSs can be fictitiously moved and the associated rotated angle ~m j
can be obtained. The jmth BS with the minimum L-CRLB
value will be selected to be the fictitiously moveable BS for the GAL(1BS) scheme in problem (23). For example as shown in Fig. 3, only the third BS, i.e., jm¼ 3, is fictitiously
adjusted and the other two BSs remain at the same position. The jmth fictitiously moved BS will be relocated to the
coordinate as ~ xjm ¼ rjmcosð ~m jÞ; ~ yjm¼ rjmsinð ~m j Þ: ð32Þ Note that the measurement of the jmth BS remains the same
as rjm. The noise variance of this measurement is
recalcu-lated as 2 rm
j þ
2
re. Based on the new set of BSs adjusted by
the proposed GAL(1BS) scheme, the LLS-based estimation can be adopted to obtain the final estimation of MS’s position.
3.2 GAL with Two Fictitiously Movable BSs Scheme The GAL(2BSs) scheme is designed to fictitiously relocate the position of two BSs according to the minimal L-CRLB layout criterion. Under this condition, two BSs are defined to be fictitiously movable and the others are fixed. The objective of the GAL(2BSs) scheme is to select the specific two BSs that should be fictitiously moved in order to achieve the layout with the minimum L-CRLB. The GAL(2BSs) problem is defined as
fjm; kmg ¼ arg min j¼1;...;N;k¼1;...;N;j6¼k CLð ~ m j; ~mkÞ subject to CLð ~mj; ~ m kÞ < CL; ð33Þ where ~m
j and ~mk represent the orientation of two
fictitiously moveable BSs. The constraint in (33) is to verify if the original L-CRLB has already been the lowest under the given measurement conditions. Before solving problem
(33), the optimum rotated angles ~mj and ~mk of the two
fictitiously movable BSs should be decided first. The two fictitiously movable BSs problem is defined to find the optimum rotated angles as
f ~mj ; ~mkg ¼ arg min
8 ~j; ~k;j6¼k
CLð ~j; ~kÞ; 8 ~j; ~k¼ ½0 ; 360 Þ: ð34Þ
Note that the matrix I in CLð ~j; ~kÞ can be obtained as
I¼ diagf½2r1, . . . , ð 2 rjþ 2 reÞ 1 , . . . , ð2 rkþ 2 reÞ 1 , . . . , 2 rNg,
where the standard deviation re of MS’s initial estimate is
considered in both the jth and kth fictitiously movable BSs. Therefore, the L-CRLB can be derived as
CLð ~j; ~kÞ ¼ 1 D " XN i¼1 ði6¼j;kÞ sin ^i 22 ri ri þ sin ~j 2ð2 rjþ 2 reÞ rj þ sin ~k 2ð2 rkþ 2 reÞ rk !2 þ X N i¼1 ði6¼j;kÞ cos ^i 22 ri ri þ cos ~j 2ð2 rjþ 2 reÞ rj þ cos ~k 2ð2 rkþ 2 reÞ rk !2 þ X N i¼1 ði6¼j;kÞ N 42 ri r 2 i þ N 4ð2 rjþ 2 reÞ r 2 j þ N 4ð2 rkþ 2 reÞ r 2 k # : ð35Þ
Similar to the GAL(1BS) scheme, the equivalent two fictitiously movable BSs problem for problem (34) can also be acquired as
f ~mj; ~mkg ¼ arg min
8 ~j; ~k;j6¼k
f2ð ~j; ~kÞ; ð36Þ
where f2ð ~j; ~kÞ is regarded as the cost function of problem
(36) as f2ð ~j; ~kÞ ¼ 1 D " XN i¼1 ði6¼j;kÞ sin ^i 22 ri ri þ sin ~j 2ð2 rjþ 2 reÞ rj þ sin ~k 2ð2 rkþ 2 reÞ rk !2 þ X N i¼1 ði6¼j;kÞ cos ^i 22 ri ri þ cos ~j 2ð2 rjþ 2 reÞ rj þ cos ~k 2ð2 rkþ 2 reÞ rk !2# : ð37Þ
It can be observed that problem (36) can be solved if the following conditions on the first derivatives of f2ð ~j; ~kÞ are
satisfied, i.e., @f2ð ~j; ~kÞ @ ~j ~ j¼ ~mj ~ k¼ ~mk ¼ 0; @f2ð ~j; ~kÞ @ ~k ~ j¼ ~mj ~ k¼ ~mk ¼ 0; ð38Þ and the conditions on the second derivatives of f2ð ~j; ~kÞ
are also fulfilled, i.e., a > 0 and ac b2> 0, where
a¼ @ 2f 2ð ~j; ~kÞ @2~ j ~ j¼ ~mj ~ k¼ ~mk ; b¼ @ 2f 2ð ~j; ~kÞ @ ~j@ ~k ~ j¼ ~mj ~ k¼ ~mk ; c¼ @ 2f 2ð ~j; ~kÞ @2~ k ~ j¼ ~mj ~ k¼ ~mk ; ð39Þ
Since the closed form solution of ~m
j and ~mk can not be
acquire the optimal angles ~mj and ~mk for achieving
minimum L-CRLB for problem (36). To reduce the computation complexity, a cost function g2ð ~j; ~kÞ is
defined to simplify the original problem (34) into an approximate two fictitiously movable BSs problem
f ~mj ; ~mkg ¼ arg min 8 ~j; ~k;j6¼k g2ð ~j; ~kÞ; ð40Þ where g2ð ~j; ~kÞ ¼ XN i¼1 ði6¼j;kÞ sin ^i 22 ri ri þ sin ~j 2ð2 rjþ 2 reÞ rj þ sin ~k 2ð2 rkþ 2 reÞ rk !2 þ X N i¼1 ði6¼j;kÞ cos ^i 22 ri ri þ cos ~j 2ð2 rjþ 2 reÞ rj þ cos ~k 2ð2 rkþ 2 reÞ rk !2 ; ð41Þ The closed form solution of (40) for the N ¼ 3 case is illustrated as follows: For ease of computation, the three BSs’ orientation can be represented by their adjacent included angles as ^¼ f ^1¼ ^2 ^1; ^2; ^3¼ ^2þ ^2g as
shown in Fig. 3. The GAL(2BSs) scheme fictitiously relocates the positions of two BSs among the three which can be denoted as ~¼ f ^2 ~1; ^2; ^2 ~2g. Furthermore,
according to the rotation property, the orientation of the fictitiously moved BSs can be transformed as ~¼ f ~1;
0 ; ~2g. By considering the first derivative equation in (38),
the BSs’ adjacent included angles for achieving the minimum CRLB are calculated as
~ m1 ¼ cos1 2 r1þ 2 re 2 r214r2r22 4 r2r 2 2 2r3þ 2 re 2 r23 2r1þ 2 re 2 r21 2r3þ 2 re 2 r23= 2 2r 1þ 2 re r1 2r2r2 2r 3þ 2 re 2 r23; ð42Þ ~ m2 ¼ cos1 4r2r 2 2 2r3þ 2 re 2 r23 2r1þ 2 re 2 r21 2r3þ 2 re 2 r23 2r1þ 2 re 2 r214r2r22 2 2r1þ 2 re 2 r21 2 r2r2 2r3þ 2 re r3 ; ð43Þ
where the angle ~m
j for j ¼ 1 and 2 lies in the domain of
arc cosine function from ½0; 180 which is half of the domain
½0; 360 Þ. Both angles ~m
j and ~mj þ 180 are considered the
local minimums of subproblem (40). Therefore, these angles will be substituted into (35) to determine the angle with smaller L-CRLB value. After the adjacent included angles are calculated, the BSs’ orientation of the GAL(2BSs) scheme can be acquired as ~m¼ f ~m
1 ¼ ^2 ~1m; ^2; ~m3 ¼ ^2þ ~2mg.
Accordingly, all the BSs can be fictitiously moved and the associated rotated angle ~m
j and ~mk will be obtained.
The jmth and kmth BSs with the minimum L-CRLB value are
decided to be the fictitiously moveable BS for the GAL(2BSs) scheme in problem (33). The positions of jmth and kmth BSs
can be fictitiously relocated and computed based on (32). Note that the measurement remains the same while the noise variance of the measurement is recalculated by considering the initial estimation error. With the new set of BSs obtained from the GAL(2BSs) scheme, the LLS-based estimation algorithms can be adopted to obtain the final
MS’s location estimation. Based on the derivation of proposed GAL(2BSs) scheme, the number of fictitiously movable BSs can also be increased by extending the GAL scheme with multivariable optimization, i.e., GAL(3BSs), GAL(4BSs).
3.3 GAL Scheme
Based on Sections 3.1 and 3.2, it can be observed that the GAL(2BSs) scheme provides one more degree of freedom compared to the GAL(1BS) scheme, which should increase the precision of location estimation owing to the enhance-ment from the geometric effect. However, the two ficti-tiously moveable BSs associate with the initial MS’s estimation error may degrade the performance for location estimation. Therefore, the GAL scheme for the N ¼ 3 case is designed to select between the two sub-schemes, i.e., GAL(1BS) and GAL(2BSs), in order to achieve minimum L-CRLB value among all different cases. The GAL problem can be defined as ~ m¼ arg min ~ m 1BS;~ m 2BS ½CLð~m1BSÞ; CLð~m2BSÞ subject to CLð~m1BSÞ; CLð~m2BSÞ < CL; ð44Þ where ~m
1BS and ~m2BS represent the sets of BS’s orientation
with the lowest L-CRLB by adopting the GAL(1BS) and GAL(2BSs) schemes, respectively. Each of the fictitiously moved BS set is selected according to the minimum L-CRLB criteria. Note that if both the GAL(1BS) and GAL(2BSs) schemes cannot provide a lower L-CRLB scenario com-pared to the original unmoved version, the GAL scheme will choose the original BS’s positions for MS’s location estimation according to the constraint in (44).
4
I
MPLEMENTATIONS OFP
ROPOSEDG
EOMETRY-A
SSISTEDL
OCALIZATIONA
LGORITHMAs described in previous section, the main objective of the proposed GAL scheme is to acquire the positions of fictitiously movable BSs in order to provide better geo-metric layout for MS’s location estimation. Since the GAL scheme is designed based on the initial estimate of the MS, there can be different implementations to adopt the GAL scheme for location estimation. Fig. 4 illustrates the schematic diagrams for the implementations of the pro-posed GAL algorithm. The GAL-TSLS scheme as shown in Fig. 4a is proposed to calculate both the initial and final estimates of the MS’s position based on the two-step LS estimator [9]. On the other hand, a two-stage architecture named GAL-KF scheme as shown in Fig. 4b, i.e., a two-step LS estimator with a Kalman filter, is proposed to enhance the initial estimate with the historical information from Kalman filter. These two types of implementations of GAL scheme are explained in the following two sections. 4.1 GAL with Two-Step LS Estimator
As shown in Fig. 4a, the MS’s initial estimate ^xxo can be obtained by performing the two-step LS method. The concept of the two-step LS method is to acquire an intermediate location estimate in the first step by assuming that xxand R are not correlated. Note that this first step is
exactly the LLS method [5] which solves the L-LEP instead of the conventional LEP. By combining (1) and (2), the elements within the matrix formulation (i.e., M1¼ J1) for
the first step of the estimator can be obtained as
M1¼ 2x1 2y1 1 2x2 2y2 1 : : : 2xN 2yN 1 2 6 6 4 3 7 7 5 J1¼ r2 1 1 r2 2 2 : r2 N N 2 6 6 4 3 7 7 5; where i¼ x2i þ y2i. The parameter set ^ ¼ ½xð1Þyð1ÞRð1Þ
T
that is to be estimated at the first step can be acquired as ^
¼ MT111 M1 1
MT111 J1: ð45Þ
The error of the first step estimation is obtained as 1¼
M1^ J1 ¼ 2B1n1 þ n21 where n1¼ ½n1; . . . ; ni; . . . ; nNT
represents the measurement noise vector in (1). Therefore, the weighting matrix 1 in (45) can be acquired by
neglecting the square term n2 1 as
1¼ E½ 1 T1 ¼ 4B1En1nT1
B1¼ 4B1IB1; ð46Þ
where B1¼ diagf½r1; . . . ; ri; . . . ; rNg. The second step of the
two-step LS method releases this assumption that xxand R are uncorrelated by adjusting the intermediate result to obtain an improved location estimate. Therefore, the correlation relationship can be applied and the elements within the second step of the two-step LS estimator formulation, i.e., M2x^xð2Þ¼ J2, are obtained as
M2¼ 1 0 0 0 1 0 1 1 0 2 4 3 5 J2¼ ðxð1ÞÞ2 ðyð1ÞÞ2 Rð1Þ 2 4 3 5; ð47Þ with ^xxð2Þ¼ ½ðxð2ÞÞ2; ðyð2ÞÞ2
T. It can also be solved by the weighted LS formulation with the weighting matrix 2 of
the second step as
2¼ 4B2covð^ÞB2¼ 4B2
M1T11M1
1
B2; ð48Þ
where B2¼ diagf½xð1Þ; yð1Þ; 1=2g. With the relationship of
R¼ x2þ y2, it can be observed that the variable R is
removed from ^ such as to form the reduced dimension vector of xxð2Þ. The MS’s position estimate by using the
two-step LS estimator can be obtained by taking element-wise square root as ^xx¼ ð^xxð2ÞÞ1=2¼ ½xð2Þ; yð2ÞT
.
As shown in Fig. 4a, the two-step LS estimator is performed to obtain an initial estimate ^xxo. With the initial
estimate, the fictitious BS set can be calculated based on the proposed GAL scheme targeting on achieving the minimum L-CRLB requirement. The two-step LS is performed for the second time with the adjusted BSs and the received measurements to obtain the MS’s location estimate for the GAL-TSLS scheme.
4.2 GAL with Kalman Filter
In order to provide enhanced location estimate, the proposed GAL-KF scheme as shown in Fig. 4b is suggested to estimate the MS’s position using a two-stage estimator, i.e., a two-step LS estimator with a Kalman filter. The Kalman filtering technique is employed to estimate the MS’s position based on its previously estimated data. The measurement and state equations at the kth time step for the Kalman filter can be represented as
zk¼ E^xxfkþ mk; ð49Þ
^ x
xfk¼ F^xxfk1þ pk; ð50Þ
where ^xxfk represents the output and zk denotes the
measurement input of the Kalman filter. Note that for the two-stage location estimation, the input of the Kalman filter is obtained from the result of the GAL scheme as zk¼ ^xxGk
in Fig. 4b. The variables mk and pk denote the
measure-ment and the process noises associated with the covariance matrices R and Q within the Kalman filtering formulation. Note that the matrix R can be determined by the FIM of L-CRLB and Q is set to be an identity matrix. Furthermore, the matrix E and the state transition matrix F in (49) and (50) respectively can be obtained as E ¼ F ¼ I22.
As shown in Fig. 4b, the execution process of the proposed GAL-KF scheme consists of two phases, includ-ing the transition period (Tp) and the stable period. During
the transient period t < Tp, the GAL-KF scheme adopts the
two-step LS method to provide the initial estimate of the MS. After the tracking time is longer than Tp, the GAL-KF
scheme starts to adopt the prediction from the output of the Kalman filter, i.e., ^xxfkjk1, which serves as the updated initial MS’s estimate for the GAL scheme. By adopting the Kalman filter, it can be observed that the GAL-KF scheme only requires to perform a single round of location estimation compared to the GAL-TSLS scheme after the system is executed in the stable state. The Kalman filter can refine the MS’s position estimation with the historical measurements based on the initial estimate, which should provide better estimation accuracy by adopting the GAL-KF scheme compared to the GAL-TSLS method.
5
P
ERFORMANCEE
VALUATIONSimulations are performed to show the effectiveness of the GAL algorithms (i.e., GAL-TSLS and GAL-KF) under different network topologies and the MS’s positions. The number of BSs is considered as three in the examples since
three BSs is the minimum sufficient number for the localization problem. The model for the LOS measurement noise of the TOA signals is selected as the Gaussian distribution with zero mean and standard deviation as n
meters in different cases, i.e., ni;k N ð0; 2nÞ. In the
following examples 5 and 6, the GAL-TSLS algorithm is simulated to validate the effectiveness of the fictitiously movable BS schemes on the two-step LS based estimation. Example 5 (Validation on Approximate One Fictitiously
Movable BS Problem).The purpose of this example is to compare and validate the difference between the original and approximate one fictitiously movable BS problems. Consider an array of three sensors whose coordinates are xx1¼ ½50 cos 0 ; 50 sin 0 T, xx2¼ ½30 cos 2; 30 sin 2T, and
xx3¼ ½20 cos 140 ; 20 sin 140 T where xx2 is function of
2with its range as indicated in the x-axis of Fig. 5. The
MS’s true position is assumed to be placed at the origin, i.e., xx¼ ½0; 0T. Note that all the layouts formed by the three sensors with the change of 2 are designed to be
OPLs for validation purpose. More realistic network scenarios will be considered in the following examples. The standard deviation of the Gaussian noises is chosen as n¼ 1 in this example. The root mean square error
(RMSE) is defined as RMSE¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XM i¼1 k^xx xxk2=M v u u t ;
where M denotes the number of trials as 1,000. Fig. 5a shows the original one fictitiously movable BS problem obtained by exhaustively solving (24); while Fig. 5b illustrates the approximate one fictitiously movable BS problem acquired from (30) and (31), which are respectively denoted as “GAL(1BS)-TSLS BSi moved”
in both plots, for i ¼ 1, 2, and 3. In each of the three cases, i.e., i ¼ 1, 2, and 3, BSiis fictitiously moved for obtaining
the optimal angle ~m
i that can achieve the minimum
value of L-CRLB. Moreover, the “GAL(1BS)-TSLS” curves in both plots respectively denote the GAL(1BS) problem as defined in (23) by selecting among different
fictitiously movable angles from the original problem (24) in Fig. 5a and approximate problem (30) in Fig. 5b. Since the “GAL(1BS)-TSLS” method acquires the posi-tions of the fictitious BSs based on the MS’s initial estimate instead of the true position, the RMSE perfor-mance is not necessarily the lowest compared to the “GAL(1BS)-TSLS BSimoved” scheme for i ¼ 1, 2, and 3.
Both the CRLB and the conventional two-step LS scheme are also illustrated in both plots for comparison purpose. Notice that the derived L-CRLB is to characterize the geometric property for linearized location estimation, i.e., the initial estimation of GAL algorithms. By fictitiously moving the BS locations, the GAL algorithm improves the estimation accuracy of linearized algorithm which is originally bounded by L-CRLB. In other words, the MS’s location estimation error can further be reduced such that the CRLB bound will possibly be achieved. Therefore, the benchmark of localization problem should still be CRLB instead of L-CRLB. It can be observed that even though the approximate problem will be differ from the original problem by individually moving one of the three BSs fictitiously, the resulting problem (23), i.e., the GAL(1BS)-TSLS scheme, obtained from (30) will be closely match with (24) as shown in both plots. Furthermore, it can be seen that the GAL(1BS)-TSLS scheme can provide better RMSE performance compared to the conventional two-step LS scheme.
Example 6 (Validation on Approximate Two Fictitiously Movable BSs Problem).This example is to compare and validate the difference between the original and approx-imate two fictitiously movable BSs problems in Figs. 6a and 6b, respectively. Same network layout and noise variance as in example 5 are used in this example. The curves named “GAL(2BSs)-TSLS BSi fixed” refer to
the problems that the ith BS is fixed while the other two BSs are movable, i.e., the angle set f ~m
2; ~m3g of the
curve “GAL(2BSs)-TSLS BS1fixed” are obtained via (34)
and (40) for the original (Fig. 6a) and approximate (Fig. 6b) problems, respectively. Moreover, the curve “GAL(2BSs)-TSLS” denotes for the problem in (33) to select among different movable angles from the original
problem (34) in Fig. 6a and the approximate problem (40) in Fig. 6b. Similar to the previous example, the final GAL(2BSs)-TSLS scheme of both problems are observed to be consistent with each other from the simulation results. Meanwhile, the effectiveness of problem (44) for the GAL scheme is also validated by selecting among the GAL(1BS)-TSLS and the GAL(2BSs)-TSLS schemes. By observing both Figs. 5 and 6, the GAL-TSLS scheme with the problem (44) can achieve the lowest RMSE compared to the other methods.
It is intuitive that the closed form property of the approximate problem can provide efficiency in computa-tional complexity compared to the original problem. There-fore, the approximate problem with the GAL scheme will be adopted in the rest of the examples for performance comparison. In order to provide better estimation precision for MS’s location estimate, the GAL-KF algorithm is simulated to compare with the existing two-step LS [9], Beck [20], and SDR [21] algorithms, which are named as TSLS-KF, Beck-KF, and SDR-KF, respectively. Note that these three algorithms are also cascaded with the Kalman filters to perform two-stage estimation in order to provide fair comparison.
Example 7 (A Special Case of GAL-KF Scheme). In this example, a special network scenario is simulated to provide performance comparison for the GAL-KF scheme. Consider an array of three sensors in the OPL whose coordinates are xx1¼ ½20 cos 0 ; 20 sin 0 T, xx2¼
½30 cos 80 ; 30 sin 80 T
, and xx3¼ ½50 cos 140 ; 50 sin 140 T,
while the MS’s true position is fixed at xx¼ ½0; 0T. Fig. 7a demonstrates the performance comparison of mean position error (MPE) for the simulation time interval k¼ 1;000, where each time instant is run with 1,000 simulation samples. Note that the MPE at the kth time instant is defined as MPEk¼PMi¼1k^xxk xxkk2=M. The
transient period Tp is chosen as 20 which means that
the GAL-KF scheme starts to adopt the prediction from the Kalman filter at the time instant k ¼ 21. The standard deviation of Gaussian noises nis chosen as 2 in Fig. 7a.
Since the Kalman filter is effective in smoothing the estimation result, it can be observed that the MPE is decreased and converges with the increment of time instant k for all the schemes in Fig. 7a. With the consideration of the L-CRLB in the algorithm design, the proposed GAL-KF implementation can achieve lowered MPE compared to the other schemes; while the SDR-KF has the worst performance among all
Fig. 7. Performance comparison of example 7.
the algorithms. Since the two-stage architecture does not provide smoothing gain to the KF method, the SDR-KF scheme will not be further considered in the rest of the simulation examples. Fig. 7b illustrates the perfor-mance comparison of RMSE at the simulation time instant k ¼ 1;000 under different standard deviations of the Gaussian noises. It can be observed that the GAL-KF scheme can provide a significant gain over the other methods under different noise values. Note that the Beck-KF scheme is observed to be sensitive to the noise which results in higher RMSE compared to the TSLS-KF scheme under larger noise condition. For example, compared to the TSLS-KF and Beck-KF methods, the proposed GAL-KF scheme will result in 2.8 and 6 meters less of RMSE respectively under n¼ 10 meter as shown
in Fig. 7.
Example 8 (A General Case of GAL-KF Scheme under LOS Environment). This example illustrates a general scenario of wireless sensor network (as shown in Fig. 8) for performance comparison under LOS environment. The BSs’ coordinates are selected as xx1¼ ½50; 35:36T,
xx2¼ ½50; 35:36T, and xx3¼ ½50; 35:36T, and there are
100 MSs randomly deployed in a 100 100 meter square space. Note that the number of MSs located in the OPL is
larger than that in the IPL in this example in order to demonstrate that the OPL may frequently occur in a sensor network environment. The performance of the IPL and the OPL under the LOS condition are separately examined under different noise standard deviation in Figs. 9a and 9b, respectively. Since the difference between the L-CRLB and the CRLB is considered small in the IPL case, similar RMSE performance is observed among the three compared schemes as illustrated in Fig. 8a. On the other hand, with the OPL scenario as shown in Fig. 8b, the GAL-KF scheme can outperform the other two methods under different noise environ-ments, e.g., the GAL-KF approach will result in 2.9 and 3.4 meters less of RMSE compared to the Beck-KF and TSLS-KF schemes, respectively, under n¼ 20 meter in
Fig. 9b, which is considered the major contribution of the proposed GAL-KF scheme.
Example 9 (A General Case of GAL-KF Scheme under Realistic Environment). In this example, the perfor-mance comparison is conducted for the GAL-KF scheme under the realistic environment. The same network layout setting as example 8 is adopted; while the noise setting is different by considering the NLOS in this example. In order to include the influence from the NLOS noise, the TOA model in (1) is rewritten as ri¼ iþ niþ ei, where ei is denoted as the NLOS noise.
Note that a range of NLOS mitigation algorithms [22], [23], [24] have been proposed in the literature to improve positioning accuracy. The standard deviation of the Gaussian noise is chosen as n¼ 10. An exponential
distribution peið Þ is assumed for the NLOS noise model
of the TOA measurements as peið Þ ¼ 1 i exp i ; > 0; 0; otherwise; 8 < : ð51Þ
where i¼ c i¼ c mðiÞ". The parameter i is the
RMS delay spread between the ith BS and the MS. m
represents the median value of i, which is utilized as the
x-axis in Figs. 10a and 10b. " is the path loss exponent which is assumed to be 0.5. The shadow fading factor is
Fig. 8. Network layout of example 8.
a log-normal random variable with zero mean and standard deviation chosen as 4 dB in the simulations.
Note that the median value m¼ 0:1 s primarily fulfills
the environment where the MS is located in the rural area [25].
Fig. 10a illustrates the performance comparison for the three schemes in the IPL under NLOS environment. It can be observed that the proposed GAL-KF approach can provide the smallest RMSE compared to the other two schemes, e.g., the GAL-KF scheme will result in 3.1 and 5.2 meters less of RMSE respectively under m¼ 0:2 s compared to the Beck-KF and TSLS-KF
methods. The reason is that the GAL-KF scheme is designed based on the minimization of L-CRLB by fictitiously adjusting the locations of BS. As mentioned in Section 2.3, the value of L-CRLB is affected by the distance between the BS and MS, which will be greatly influenced by the NLOS noises. Therefore, the effect from the NLOS noises has been implicitly considered in the design of GAL-KF scheme, which improves the location estimation performance. Moreover, Fig. 10b illustrates the performance comparison for the OPL scenario under NLOS environment. The proposed GAL-KF scheme can still outperform the other two methods, e.g., around 1.9 and 6 meters less of RMSE in comparison with the Beck-KF and TSLS-KF schemes under m¼ 0:2 s in Fig. 9b. The merits of proposed
GAL-KF scheme can therefore be observed.
6
C
ONCLUSIONThe properties of linearized location estimation algorithms by introducing an additional variable are analyzed from the geometric point of view. By proposing the linearized location estimation problem-based CRLB (L-CRLB), the linearization loss from the linearized location estimation algorithms can be observed. In order to minimize the linearization loss, the geometry-assisted localization (GAL) algorithm is proposed in the paper by fictitiously moving the base stations in order to achieve the new geometric layout with minimum L-CRLB value. The GAL with two-step least
squares implementation can enhance the estimation perfor-mance of the conventional two-step least squares estimator. By improving the initial estimation with the adoption of historical information, the GAL with Kalman filter scheme further outperforms the other location estimators with similar two-stage estimation structure.
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CKNOWLEDGMENTSThis work was in part funded by the Aiming for the Top University and Elite Research Center Development Plan, NSC 99-2628-E-009-005, the MediaTek research center at National Chiao Tung University, and the Telecommunica-tion Laboratories at Chunghwa Telecom Co. Ltd, Taiwan.
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