Reference-free time-based localization for an asynchronous target
- Yiyin Wang^{1}Email author and
- Geert Leus^{1}
https://doi.org/10.1186/1687-6180-2012-19
© Wang and Leus; licensee Springer. 2012
Received: 13 May 2011
Accepted: 26 January 2012
Published: 26 January 2012
Abstract
Low-complexity least-squares (LS) estimators based on time-of-arrival (TOA) or time-difference-of-arrival (TDOA) measurements have been developed to locate a target node with the help of anchors (nodes with known positions). They require to select a reference anchor in order to cancel nuisance parameters or relax stringent synchronization requirements. Thus, their localization performance relies heavily on the reference selection. In this article, we propose several reference-free localization estimators based on TOA measurements for a scenario, where anchor nodes are synchronized, and the clock of the target node runs freely. The reference-free LS estimators that are different from the reference-based ones do not suffer from a poor reference selection. Furthermore, we generalize existing reference-based localization estimators using TOA or TDOA measurements, which are scattered over different research areas, and we shed new light on their relations. We justify that the optimal weighting matrix can compensate the influence of the reference selection for reference-based weighted LS (WLS) estimators using TOA measurements, and make all those estimators identical. However, the optimal weighting matrix cannot decouple the reference dependency for reference-based WLS estimators using a nonredundant set of TDOA measurements, but can make the estimators using the same set identical as well. Moreover, the Cramér-Rao bounds are derived as benchmarks. Simulation results corroborate our analysis.
Keywords
1. Introduction
Localization is a challenging research topic under investigation for many decades. It finds applications in the global positioning system (GPS) [1], radar systems [2], underwater systems [3], acoustic systems [4, 5], cellular networks [6], wireless local area networks (WLANs) [7], wireless sensor networks (WSNs) [8, 9], etc. It is embraced everywhere at any scale. New applications of localization are continuously emerging, which motivates further exploration and attracts many researchers from different research areas, such as geophysics, signal processing, aerospace engineering, and computer science. In general, the localization problem can be solved by two steps [7–9]: firstly measure the metrics bearing location information, the so-called ranging or bearing, and secondly estimate the positions based on those metrics, the so-called location information fusion. There are mainly four metrics: time-of-arrival (TOA) or time-of-flight (TOF) [10], time-difference-of-arrival (TDOA) [4, 11], angle-of-arrival (AOA) [12], and received signal strength (RSS) [13]. The ranging methods using RSS can be implemented by energy detectors, but they can only achieve a coarse resolution. Antenna arrays are required for AOA-based methods, which encumbers their popularity. On the other hand, the high accuracy and potentially low cost implementation make TOA or TDOA based on ultra-wideband impulse radios (UWB-IRs) a promising ranging method [8].
Closed-form localization solutions based on TOAs or TDOAs are used to locate a target node with the help of anchors (nodes with known positions). They are appreciated for real-time localization applications, initiating iterative localization algorithms, and facilitating Kalman tracking [14]. They have much lower complexity compared to the optimal maximum likelihood estimator (MLE), and also do not require prior knowledge of noise statistics. However, a common feature of existing closed-form localization solutions is reference dependency. The reference here indicates the time associated with the reference anchor. For instance, in order to measure TDOAs, a reference anchor has to be chosen first [7]. The reference anchor is also needed to cancel nuisance parameters in closed-form solutions based on TOAs or TDOAs [15]. Thus, the localization performance depends heavily on the reference selection. There are some efforts to improve the reference selection [16–18], but they mainly rely on heuristics. Furthermore, when TOAs are measured using the one-way ranging protocol for calculating the distance between the target and the anchor, stringent synchronization is required between these two nodes in the conventional methods [7, 10]. However, it is difficult to maintain synchronization due to the clock inaccuracy and other error sources. Therefore, various closed-form localization methods resort to using TDOA measurements to relax this synchronization constraint between the target and the anchor. These methods only require synchronization among the anchors, e.g., the source localization methods based on TDOAs using a passive sensor array [4, 19–22].^{a}
In this article, we also relax the above synchronization requirement, and consider a scenario, where anchor nodes are synchronized, and the clock of the target node runs freely. However, instead of using TDOAs, we model the asynchronous effect as a common bias, and propose reference-free least-squares (LS), weighted LS (WLS), and constrained WLS (CWLS) localization estimators based on TOA measurements. Furthermore, we generalize existing reference-based localization solutions using TOA or TDOA measurements, which are scattered over different research areas, and provide new insights into their relations, which have been overlooked. We clarify that the reference dependency for reference-based WLS estimators using TOA measurements can be decoupled by the optimal weighting matrix, which also makes all those estimators identical. However, the influence of the reference selection for reference-based WLS estimators using a nonredundant set of TDOA measurements cannot be compensated by the optimal weighting matrix. But the optimal weighting matrix can make the estimators using the same set equivalent as well. Moreover, the Cramér-Rao bounds (CRBs) are derived as benchmarks for comparison.
The rest of this article is organized as follows. In Section 2, different kinds of reference-free TOA-based estimators are proposed, as well as existing reference-based estimators using TOA measurements. Their relations are thoroughly investigated. In Section 3, we generalize existing reference-based localization algorithms using TDOA measurements, and shed light on their relations as well. Simulation results and performance bounds are shown in Section 4. Conclusions are drawn at the end of the article.
Notation: We use upper (lower) bold face letters to denote matrices (column vectors). [X]_{m,n}, [X]_{m,:} and [X]_{:},_{ n }denote the element on the m th row and n th column, the m th row, and the n th column of the matrix X, respectively. [x]_{ n }indicates the n th element of x. 0_{ m }(1_{ m }) is an all-zero (all-one) column vector of length m. I_{ m }indicates an identity matrix of size m × m. Moreover, (·)^{ T }, || · ||, and ⊙ designate transposition, ℓ_{2} norm, and element-wise product, respectively. All other notation should be self-explanatory.
2. Localization based on TOA measurements
Considering M anchor nodes and one target node, we would like to estimate the position of the target node. All the nodes are distributed in an l-dimensional space, e.g., l = 2 (a plane (2-D)) or l = 3 (a space (3-D)). The coordinates of the anchor nodes are known and defined as X_{ a }= [x_{1}, x_{2}, ..., x_{ M }], where the vector x_{ i }= [x_{1,i}, x_{2},_{ i }, . . ., x_{l,i}]^{ T }of length l indicates the known coordinates of the i th anchor node. We employ a vector x of length l to denote the unknown coordinates of the target node. Our method can also be extended for multiple target nodes. We remark that in a large scale WSN, it is common to localize target nodes in a sequential way [23]. The target nodes that have enough anchors are localized first. Then, the located target nodes can be viewed as new anchors that can facilitate the localization of other target nodes. Therefore, the multiple-anchors-one-target scenario here is of practical interest. We can even consider a case with a moving anchor, in which a ranging signal is periodically transmitted by the target node, and all the positions where the moving anchor receives the ranging signal are viewed as the fixed positions of some virtual anchors. We assume that all the anchors are synchronized, and their clock skews are equal to 1, whereas the clock of the target node runs freely. Furthermore, we assume that the target node transmits a ranging signal, and all the anchors act as receivers. We remark that other systems may share the same data model such as a passive sensor array for source localization or a GPS system, where a GPS receiver locates itself by exploring the received ranging signals from several satellites [1]. All the satellites are synchronized to an atomic clock, but the GPS receiver has a clock offset relative to the satellite clock. Note that this is a stricter synchronization requirement than ours, as we allow the clock of the target node to run freely. Every satellite sends a ranging signal and a corresponding transmission time. The GPS receiver measures the TOAs, and calculates the time-of-flight (TOF) plus an unknown offset. In this section, TOA measurements are used, and TDOA measurements are employed in the next section.
2.1. System model
where d = [d_{1}, d_{2}, ..., d_{M}]^{T}, with d_{ i }= ||x_{ i }- x|| the true distance between the i th anchor node and the target node, and n = [n_{1}, n_{2}, ..., n_{ M }]^{ T }with n_{ i }the distance error term corresponding to the TOA measurement error at the i th anchor, which can be modeled as a random variable with zero mean and variance ${\sigma}_{i}^{2}$, and which is independent of the other terms (E[n_{ i }n_{ j }] = 0, i ≠ j). We remark that instead of using TDOAs to directly get rid of the distance bias, we use TOAs and take the bias into account in the system model.
2.2. Localization based on squared TOA measurements
2.2.1. Proposed localization algorithms
where we ignore the higher order noise terms to obtain (5) and assume that the noise mean E[[m]_{ i }] ≈ 0 under the condition of sufficiently small measurement errors. Note that the noise covariance matrix Σ depends on the unknown d.
- (1)
Initialize W using the estimate of d based on the LS estimate of x;
- (2)
Estimate ŷ using (8);
- (3)
Update $\mathbf{W}={\widehat{\mathbf{\Sigma}}}^{-1}$ where $\widehat{\mathbf{\Sigma}}$ is computed using ŷ ;
- (4)
Repeat Steps (2) and (3) until a stopping criterion is satisfied.
We could find all the seven roots of (15) as in [4, 10], or employ a bisection algorithm as in [26] to look for λ instead of finding all the roots. If we obtain seven roots as in [4, 10], we discard the complex roots, and plug the real roots into (14). Finally, we choose the estimate ŷ , which fulfills (10). The details of solving (15) are mentioned in Appendix 1. Note that the proposed CWLS estimator (14) is different from the estimators in [4, 10]. The CLS estimator in [4] is based on TDOA measurements, and the CWLS estimator in [10] is based on TOA measurements for a synchronous target (b = 0). Furthermore, we remark that the WLS estimator proposed in [27] based on the same data model as (1), is labeled as an extension of Bancroft's algorithm [28], which is actually similar to the spherical-intersection (SX) method proposed in [29] for TDOA measurements. It first solves a quadratic equation in b^{ 2 }- ||x||^{ 2 }, and then estimates x and b via a WLS estimator. However, it fails to provide a solution for the quadratic equation under certain circumstances, and performs unsatisfactorily when the target node is far away from the anchors [29].
Many research works have focused on LS solutions ignoring the constraint (11) in order to obtain low-complexity closed-form estimates [7]. As squared range (SR) measurements are employed, we call them unconstrained SR-based LS (USR-LS) approaches, to be consistent with [26]. Because only x is of interest, b and b^{2} - ||x||^{2} are nuisance parameters. Different methods have been proposed to get rid of them instead of estimating them. A common characteristic of all these methods is that they have to choose a reference anchor first, and thus we label them reference-based USR-LS (REFB-USR-LS) approaches. As a result, the performance of these REFB-USR-LS methods depends on the reference selection [7]. However, note that the unconstrained LS estimate of y in (7) does not depend on the reference selection. Thus, we call (7) the reference-free USR-LS (REFF-USR-LS) estimate, (8) the REFF-USR-WLS, and (14) the REFF-SR-CWLS estimate.
which is linear w.r.t. x. The price paid for applying these two projections is the loss of information. The rank of P_{ u }P is M - 2, which means that M ≥ l + 2 still has to be fulfilled as before to obtain an unconstrained LS or WLS estimate of x based on (17). In a different way, P_{ u }P can be achieved directly by calculating an orthogonal projection onto the orthogonal complement of [1_{ M },u]. Let us define the nullspace $\mathcal{N}\left({\mathbf{U}}^{T}\right)=\text{span}\left({\mathbf{1}}_{M},\mathbf{u}\right)$, and $\mathcal{R}\left(\mathbf{U}\right)\oplus \mathcal{N}\left({\mathbf{U}}^{T}\right)={\mathbb{R}}^{M}$, where $\mathcal{R}\left(\mathbf{U}\right)$ is the column space of U, ⊕ denotes the direct sum of two linearly independent subspaces and ℝ^{ M }is the M-dimensional vector space. Therefore, P_{ u }P is the projection onto $\mathcal{R}\left(\mathbf{U}\right)$. Note that the rank of ${\mathbf{P}}_{u}{\mathbf{PX}}_{a}^{T}$ has to be equal to l, which indicates that the anchors should not be co-linear for both 2-D and 3-D or co-planar for 3-D. A special case occurs when u = k 1_{ M }, where k is any positive real number. In this case, P can cancel out both (b^{2} - ||x||^{2})1_{ M }and -2b u, and one projection is enough, leading to the condition M ≥ l + 1. The drawback though is that we can then only estimate x and b^{2} - ||x||^{2}- 2bk due to the dependence between u and 1_{ M }according to (3). The SM method indicates all the insights mentioned above, which cannot be easily observed by the unconstrained estimators.
where the pseudo-inverse (†) is employed, because the argument is rank deficient. Note that P_{ u }P is the projection onto $\mathcal{R}\left(\mathbf{U}\right)$, and is applied to both sides of Σ. Thus, (P_{ u }PΣPP_{ u })^{†} is still in $\mathcal{R}\left(\mathbf{U}\right)$, and would not change with applying the projection again. As a result, we can simplify (20) as (21). Consequently, Q* is the pseudo-inverse of the matrix obtained by projecting the columns and rows of Σ onto $\mathcal{R}\left(\mathbf{U}\right)$, which is of rank M - 2. We remark that $\widehat{\mathbf{x}}$ in (18) (or (19)) is identical to the one in (7) (or (8)) according to [22]. The SM method and the unconstrained LS (or WLS) method lead to the same result. Therefore, $\widehat{\mathbf{x}}$ in (18) and (7) (or in (19) and (8)) are all REFF-USR-LS (or REFF-USR-WLS) estimates.
2.2.2. Revisiting existing localization algorithms
where ${\left({\mathbf{M}}_{j}{\mathbf{T}}_{i}\right)}^{\u2020}{\mathbf{M}}_{j}{\mathbf{T}}_{i}={\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}{\left({\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}\right)}^{\u2020}={\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}{\left({\mathbf{M}}_{j}{\mathbf{T}}_{i}{\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}\right)}^{-1}{\mathbf{M}}_{j}{\mathbf{T}}_{i}$, which is also the projection onto $\mathcal{R}\left(\mathbf{U}\right)$, and thus is equivalent to P_{ u }P. The equality between (25) and (26) can be verified using a property of the pseudo-inverse.^{b} Hence, ${\mathbf{Q}}_{i,j}^{*}$ is of rank M - 2, and ${\mathbf{Q}}_{i,j}^{*}={\mathbf{Q}}^{*},i,j\in \left\{1,\dots ,M\right\}$ with i ≠ j. As a result, the REFB-USR-WLS estimate and the REFF-USR-WLS estimate are identical if the optimal weighting matrix is used. Hence, the optimal weighting matrix can compensate the impact of random reference selection. However, since Σ depends on the unknown d, the optimal weighting matrix can only be approximated iteratively. Also note that the REFB-USR-LS estimate suffers from the ad-hoc reference selection, while the REFF-USR-LS estimate is independent of the reference selection.
2.3. Localization based on squared differences of TOA measurements
2.3.1. Proposed localization algorithms
the right hand side of which is exactly the same as the one in (17), and thus we can state P_{ u }P(ψ_{ a }- (Pu) ⊙ (Pu)) = P_{ u }P ϕ. Note that although (30) is different from (3), we find that (31) and (17) become equivalent after premultiplying P_{ u }P. Furthermore, (Pu) ⊙ (Pu) can be labeled as a SR difference (SRD) term. As a result, the unconstrained LS and WLS estimate of x based on (31), which are named the reference-free USRD-LS (REFF-USRD-LS) estimate and the REFF-USRD-WLS estimate, are exactly the same as the REFF-USR-LS estimate (18) and the REFF-USR-WLS estimate (19), respectively. We do not repeat them here in the interest of brevity. Moreover, the constrained LS and WLS based on (30), namely the REFF-SRD-CLS estimate and the REFF-SRD-CWLS estimate, are identical to the REFF-SR-CLS and the REFF-SR-CWLS estimate (14) as well.
2.3.2. Revisiting existing localization algorithms
which is different from (30), and has only one nuisance parameter d_{ i }at the right hand side. Ignoring the relation between x and d_{ i }, we still have two ways to deal with d_{ i }. The first one is to estimate x and d_{ i }together [22], which means we only use a reference once for calculating the TDOAs. The second one is again to apply M_{ j }, which fulfills M_{ j }T_{ i }u = 0_{M- 2}. It employs two different references, one for calculating the TDOAs, and the other for eliminating the nuisance parameter. In order to distinguish these two, we call them the REFB-USRD-LS(1) and the REFB-USRD-LS(2) estimate, respectively, where the number between brackets indicates the number of references used in the approach. In the same way as we clarified the equivalence between the REFF-USRD-LS and the REFF-USR-LS estimate in the previous subsection, we can easily confirm the equivalence between the REFB-USRD-LS(2) (or the REFB-USRD-WLS(2)) and the REFB-USR-LS (or the REFB-USR-WLS) estimate of Section 2.2.2. We omit the details for the sake of brevity. Furthermore, we recall that similarly as above we could have dealt with -2b T_{ i }u in (22) in two different ways. But since b = 0 in [7, 16–18, 22, 26], there are no discussions about these two different ways in literature, and we do not distinguish between them in the REFB-USR-LS method.
where V_{ i }is of size M × (M - 2), and collects the right singular vectors corresponding to the M - 2 nonzero singular values of P_{ i }T_{ i }. We derive (35) in Appendix 3, and prove that ${\mathbf{V}}_{i}{\mathbf{V}}_{i}^{T}$ is the projection onto $\mathcal{R}\left(\mathbf{U}\right)$. As a result, ${\mathbf{Q}}_{i}^{*}={\mathbf{Q}}_{i,j}^{*}={\mathbf{Q}}^{*},i,j\phantom{\rule{2.77695pt}{0ex}}\in \left\{1,\dots ,M\right\}$ and i ≠ j.
Based on the above discussions, we achieve the important conclusion that the REFF-USRD-WLS, the REFB-USRD-WLS(1), the REFB-USRD-WLS(2), the REFF-USR-WLS, and the REFB-USR-WLS estimate are all identical if the optimal weighting matrix is adopted. The optimal weighting matrix releases the reference-based methods from the influence of a random reference selection. Moreover, the REFF-USR-LS and the REFF-USRD-LS estimate are identical, and free from a reference selection, whereas the REFB-USR-LS and the REFB-USRD-LS(2) estimate are equivalent, but still suffer from a poor reference selection.
The method to solve this CWLS problem is proposed in [26]. We do not review it for the sake of brevity. Note that there are two constraints for (36) compared to one for (10), thus the method to solve (36) is different from the one to solve (10).
LS estimators based on TOAs for locating an asynchronous target
REFF-USR-LS | REFB-USR-LS | REFF-USRD-LS | REFB-USRD-LS(1) | REFB-USRD-LS(2) | |
---|---|---|---|---|---|
Relations | The REFF-USR-LS and the REFF-USRD-LS estimate are identical The REFB-USR-LS and the REFB-USRD-LS(2) estimate are identical | ||||
No. of references | 0 | 2 | 0 | 1 | 2 |
Reference dependency | No | Yes | No | Yes | Yes |
Literature | Proposed | Proposed | [15] | ||
Min. no. of anchors, x of length l | l + 2 |
WLS estimators based on TOAs for locating an asynchronous target
REFF-USR-WLS | REFB-USR-WLS | REFF-USRD-WLS | REFB-USRD-WLS(1) | REFB-USRD-WLS(2) | |
---|---|---|---|---|---|
Relations | The REFB-USR-WLS and the REFB-USRD-WLS(2) estimate are identical They are all identical with optimal weighting matrices ${\mathbf{Q}}^{*}={\mathbf{Q}}_{i,j}^{*}={\mathbf{Q}}_{i}^{*}$ | ||||
No. of references | 0 | 2 | 0 | 1 | 2 |
Reference dependency | No | Yes, with Q_{i,j} | No | Yes, with Q_{ i } | Yes, with Q_{i,j} |
No, with ${\mathbf{Q}}_{i,j}^{*}$ | No, with ${\mathbf{Q}}_{i}^{*}$ | No, with ${\mathbf{Q}}_{i,j}^{*}$ | |||
Literature | Proposed | Proposed | |||
Min. no. of anchors, x of length l | l+2 |
CLS estimators based on TOAs for locating an asynchronous target
REFF-SR-CWLS | REFF-SRD-CWLS | REFB-SRD-CWLS | |
---|---|---|---|
Equations | (14) | (36) | |
No. of references | 0 | 0 | 1 |
Reference dependency | No | No | Yes, with W_{ i } |
No, with ${\mathbf{W}}_{i}^{*}$ | |||
Literature | Proposed | Proposed | [26] |
Min. no. of anchors, x of length l | l+2 |
3. Localization based on TDOA measurements
3.1. System model
where we ignore the higher order noise terms to obtain (43) and assume that the noise mean E[[m]_{ i }] ≈ 0 under the condition of sufficiently small measurement errors. Note that the noise covariance matrix Σ_{ i }of size (M - 1) × (M - 1) depends on the unknown d as well.
3.2. Localization based on squared TDOA measurements
where ${\tilde{\mathbf{Q}}}_{i}^{*},i\in \left\{1,\dots ,M\right\}$ is the pseudo-inverse of the matrix achieved by projecting the columns and rows of Σ_{ i }onto $\mathcal{R}\left({\tilde{\mathbf{U}}}_{i}\right)$, which is of rank M - 2. We remark that the REFB-USRD-LS(1) estimate (44) is equivalent to the ones in [22, 32].
where ${\left({\tilde{\mathbf{M}}}_{j}\right)}^{\u2020}{\tilde{\mathbf{M}}}_{j}={\tilde{\mathbf{M}}}_{j}^{T}{\left({\tilde{\mathbf{M}}}_{j}^{T}\right)}^{\u2020}={\tilde{\mathbf{M}}}_{j}^{T}{\left({\tilde{\mathbf{M}}}_{j}{\tilde{\mathbf{M}}}_{j}^{T}\right)}^{-1}{\tilde{\mathbf{M}}}_{j}$ is also the projection onto $\mathcal{R}\left({\tilde{\mathbf{U}}}_{i}\right)$, which means that ${\tilde{\mathbf{Q}}}_{i,j}^{*}={\tilde{\mathbf{Q}}}_{i}^{*},i,j\in \left\{1,\dots ,M\right\}$ and i ≠ j. The REFB-USRD-LS(2) estimate and the REFB-USRD-WLS(2) estimate based on TDOA measurements are generalizations of the estimators proposed in [20]. However, the noise covariance matrix in [20] is a diagonal matrix, and the noise covariance matrix Σ_{ i }here is a full matrix.
We remark here that with the optimal weighting matrix, the REFB-USRD-WLS(1) estimate (45) and the REFB-USRD-WLS(2) estimate based on the same set of TDOA measurements are identical. However, the optimal weighting matrix cannot decouple the reference dependency. The performance of all the estimates still depends on the reference selection, since the reference dependency is an inherent property of the available measurement data. To further improve the localization performance, the REFB-SRD-CWLS estimate based on (41) can be derived in the same way as the estimate (36) by replacing ϱ_{ i }and B_{ i }with φ_{ i }and $2\left[{\mathbf{T}}_{i}{\mathbf{X}}_{a}^{T},{\mathbf{r}}_{i}\right]$, respectively. A solution to this CLS problem is presented in [26].
As a result, a LS estimator of x and d can be derived based on (52). We do not detail it in the interest of brevity.
Furthermore, as indicated in [31], an optimal nonredundant set can be achieved by the optimum conversion of the full TDOA set in order to approach the same localization performance, and the use of this optimal nonredundant set is recommended to reduce the complexity. Because [31] relies on the assumption that the received signals at the anchors are corrupted by noise with equal variances, the optimal nonredundant set can be estimated by a LS estimator. This is not the case here however, where it should be estimated by a WLS estimator, which requires the knowledge of the stochastic properties of the noise.
LS, WLS, and CWLS estimators based on TDOAs for locating an asynchronous target
REFB-USRD-LS(1) | REFB-USRD-WLS(1) | REFB-USRD-LS(2) | REFB-USRD-WLS(2) | REFB-SRD-CWLS | |
---|---|---|---|---|---|
Relations | The REFB-USRD-WLS(1) and the REFB-USRD-WLS(2) estimate are identical with the optimal weighting matrices ${\tilde{\mathbf{Q}}}_{i}^{*}={\tilde{\mathbf{Q}}}_{i,j}^{*}$ | ||||
No. of references | 1 | 1 | 2 | 2 | 1 |
Reference dependency | Yes | Yes | Yes | Yes | Yes |
Literature | [21] | [20] | [20] | ||
Min. no. of anchors, x of length l | l + 2 |
4. Numerical results
4.1. Noise statistics
where N is the number of samples, κ is a constant parameter, s(n) is the source signal, and e_{ i }(n) and τ_{ i }are respectively the additive noise and the delay at the i th node. We assume that s(n) is a zero-mean white sequence with variance ${\sigma}_{s}^{2}$, and e_{ i }(n) is also a zero-mean white sequence with variance ${\sigma}_{e}^{2}$, independent from the other noise sequences and s(n).
In the simulations, we generate n_{ i }and n_{i,j}as zero-mean Gaussian random variables with covariance matrices specified as above.
4.2. Performance evaluation
As a well-adopted lower bound, the CRB is derived for localization estimators based on TOA measurements and TDOA measurements, respectively. Note that the estimators derived in this paper are biased. We remark that although the CRB is a bound for unbiased estimators, it still is interesting to compare it with the proposed biased estimators. Here, we exemplify the CRBs for location estimation on a plane, e.g., we take l = 2. We assume that n_{ i }and n_{i,j}are Gaussian distributed. The Fisher information matrix (FIM) I_{1}(θ) based on model (1) in Section 2 for TOA measurements is derived in Appendix 4, where θ= [x^{ T }, b]^{ T }, and x = [x_{1}, x_{2}]^{ T }. Consequently, we obtain $\text{CRB}\left({x}_{1}\right)={\left[{\mathbf{I}}_{1}^{-1}\left(\theta \right)\right]}_{1,1}$. We observe that b is not part of ${\mathbf{I}}_{1}^{-1}\left(\theta \right)$. Therefore, no matter how large b is, it has the same influence on the CRB for TOA measurements. The FIM I_{2}(x) and I_{3}(x) based on model (39) in Section 3 are derived in Appendix 5 for the nonredundant set and the full set of TDOA measurements, respectively.
We consider three simulation setups. In Setups 1 and 2, eight anchors are evenly located on the edges of a 100 m × 100 m rectangular. Meanwhile the target node is located at [200 m, 30 m] and [10 m, 20 m] for Setups 1 and 2, respectively. Thus, the target node is far away from the anchors in Setup 1, but close to them in Setup 2. In Setup 3, all anchors and the target node are randomly distributed on a grid with cells of size 1 m × 1 m inside the rectangular. The performance criterion is the root mean squared error (RMSE) of $\widehat{\mathbf{x}}$ versus a reference range $\left(\text{SN}{\text{R}}_{r}=\frac{N{\pi}^{2}{\kappa}^{2}}{3{c}^{2}}\text{SNR}\right)$, which can be expressed as $\sqrt{1/{N}_{exp}{\sum}_{j=1}^{{N}_{exp}}\left|\right|{\widehat{\mathbf{x}}}^{\left(j\right)}-\mathbf{x}|{|}^{2}}$, where ${\widehat{\mathbf{x}}}^{\left(j\right)}$ is the estimate obtained in the j th trial. Each simulation result is averaged over N_{exp} = 1,000 Monte Carlo trials. The bias b corresponding to the clock offset is randomly generated in the range of [0 m, 100 m] in each Monte Carlo run. We would like to compare all the REFF and REFB estimators, as well as the estimator proposed in [27] (first iteration) using TOA measurements, labeled the LS1 estimator, and the estimator proposed in [33] using the full TDOA set, namely the REFF-LS2 estimator.
4.2.1. Estimators using TOA measurements
4.2.2. Estimators using TDOA measurements
5. Conclusions
- (1)
Applying a projection is always preferred over making differences with a reference to get rid of nuisance parameters.
- (2)
The optimal weighting matrix can compensate for the impact of the reference selection for reference-based WLS estimators using TOA measurements, and make all those estimators equivalent. However, the optimal weighting matrix cannot release the reference influence for reference-based WLS estimators using a nonredundant set of TDOA measurements, but can make the estimators using the same set identical as well.
- (3)
There are corresponding equivalences between the SR-based and the SR-difference-based methods, which are all using TOA measurements.
- (4)
Beyond some SNR threshold, there are no obvious differences among the CRBs using TOA measurements, the nonredundant set and the full set of TDOA measurements, respectively.
- (5)
The performance of the reference-free LS estimators is neither too bad nor too good, but they do not suffer from a poor reference selection.
- (6)
The concrete value of the distance bias caused by the inaccurate clock does not affect the localization performance of the LS or WLS estimators.
Appendix 1 Derivation of λ for CLS
After obtaining the seven roots of (69), we discard the complex roots, and plug the real roots into (14). Finally, we choose the estimate ŷ , which fulfills (10). Note that (14) is a CLS estimate of y with W = I. Since the optimal W* depends on the unknown d, the CWLS problem can be solved in a similar way by iteratively updating the weights and the estimates, thus we do not repeat it here.
Appendix 2 Proof of P_{ i }((T_{ i }u) ⊙ (T_{ i }u)) = P_{ i }T_{ i }(u ⊙ u)
Appendix 3 Derivation of (35)
where ${\mathbf{V}}_{i}{\mathbf{V}}_{i}^{T}$ is the projection onto $\mathcal{R}\left(\mathbf{U}\right)$.
Appendix 4 CRB derivation for localization based on TOA measurements
Appendix 5 CRB derivation for localization based on TDOA measurements
Furthermore, let us define $\mathit{\mu}={\left[{\mathit{\mu}}_{1}^{T},{\left[{\mathit{\mu}}_{2}^{T}\right]}_{2:M},\dots ,{\left[{\mathit{\mu}}_{M-1}^{T}\right]}_{M-1}\right]}^{T}$, where μ_{ i }= [ μ_{i,1},..., μ_{i,i-1}, μ_{i,i+1},..., μ_{i,M}]^{ T }, and C as the covariance matrix of this full set of TDOA measurements. Then the FIM I_{3}(x) for the full set can also be derived based on (78) by replacing μ_{ i }and C_{ i }with μ and C, respectively. We can obtain [μ]_{ k }= μ_{i,j}, where k = (i -1)M-i^{2}/2-i/2 + j, k ∈ {1,2,...,M(M-1)/2}, i ∈ {1, 2,..., M-1}, j ∈ {2, 3,..., M} and j > i.
In the same way, [C]_{k,l}= cov(n_{i,j}, n_{p,q}), where l = (p - 1)M -p^{2}/2 - p/2 + q, l ∈ {1,2,..., M(M -1)/2}, p ∈ {1,2,..., M - 1}, q ∈ {2, 3,..., M} and q > p.
Endnotes
^{a}The sensor elements of a passive sensor array are equivalent to the anchor nodes here. ^{b}Given the matrix C of size n × r and the matrix D of size r × m both of rank r, then if A = CD, it holds that A^{†} = D^{†}C^{†}[35].
Declarations
Acknowledgements
This research was supported in part by STW under the Green and Smart Process Technologies Program (Project 7976).
Authors’ Affiliations
References
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