#### 2.2.1. Proposed localization algorithms

Note that (1) is a nonlinear equation with respect to (w.r.t.)

**x**. To solve it, a MLE can be derived, which is optimal in the sense that for a large number of data it is unbiased and approaches the CRB. However, the MLE has a high computational complexity, and also requires the unknown noise statistics. Therefore, low-complexity solutions are of great interest for localization. From

${\u2225{\mathbf{x}}_{i}-\mathbf{x}\u2225}^{2}={\u2225{\mathbf{x}}_{i}\u2225}^{2}-2{\mathbf{x}}_{i}^{T}\mathbf{x}+{\u2225\mathbf{x}\u2225}^{2}$, we derive that

$\mathbf{d}\odot \mathbf{d}={\psi}_{a}-2{\mathbf{X}}_{a}^{T}\mathbf{x}+{\u2225\mathbf{x}\u2225}^{2}{\mathbf{1}}_{M}$, where

${\psi}_{a}={\left[{\u2225{\mathbf{x}}_{1}\u2225}^{2},{\u2225{\mathbf{x}}_{2}\u2225}^{2},\dots ,{\u2225{\mathbf{x}}_{M}\u2225}^{2}\right]}^{T}$. Element-wise multiplication at both sides of (1) is carried out, which leads to

$\mathbf{u}\odot \mathbf{u}-2b\mathbf{u}+{b}^{2}{\mathbf{1}}_{M}={\mathit{\psi}}_{\mathbf{a}}-2{\mathbf{X}}_{a}^{T}\mathbf{x}+{\u2225\mathbf{x}\u2225}^{2}{\mathbf{1}}_{M}+2\mathbf{d}\odot \mathbf{n}+\mathbf{n}\odot \mathbf{n}.$

(2)

Moving knowns to one side and unknowns to the other side, we achieve

${\mathit{\psi}}_{\mathbf{a}}-\mathbf{u}\odot \mathbf{u}=2{\mathbf{X}}_{a}^{T}\mathbf{x}-2b\mathbf{u}+\left({b}^{2}-{\u2225\mathbf{x}\u2225}^{2}\right){\mathbf{1}}_{M}+\mathbf{m},$

(3)

where

**m** = -(2

**d** ⊙

**n** +

**n** ⊙

**n**). The stochastic properties of

**m** are as follows

$E\left[{\left[\mathbf{m}\right]}_{i}\right]=-{\sigma}_{i}^{2}\approx 0,$

(4)

$\begin{array}{ll}{[\mathbf{\Sigma}]}_{i,j}\hfill & =E[{[\mathbf{m}]}_{i}{[\mathbf{m}]}_{j}]-E[{[\mathbf{m}]}_{i}]E[{[\mathbf{m}]}_{j}]\hfill \\ =E[(2{d}_{i}{n}_{i}+{n}_{i}^{2})(2{d}_{j}{n}_{j}+{n}_{j}^{2})]-{\sigma}_{i}^{2}{\sigma}_{j}^{2}\hfill \\ =4{d}_{i}{d}_{j}E[{n}_{i},{n}_{j}]+E[{n}_{i}^{2}{n}_{j}^{2}]-{\sigma}_{i}^{2}{\sigma}_{j}^{2}\hfill \\ =\{\begin{array}{ll}4{d}_{i}^{2}{\sigma}_{i}^{2}+2{\sigma}_{i}^{4}\approx 4{d}_{i}^{2}{\sigma}_{i}^{2},\hfill & i=j\hfill \\ 0,\hfill & i\ne j\hfill \end{array},\hfill \end{array}$

(5)

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**.

Defining

ϕ=

ψ_{
a
}-

**u** ⊙

**u**,

**y** = [

**x**^{
T
},

*b, b*^{2} - ||

**x**||

^{2}]

^{
T
}, and

$\mathbf{A}=\left[2{\mathbf{X}}_{a}^{T},-2\mathbf{u},{\mathbf{1}}_{M}\right]$, we can finally rewrite (3) as

$\mathit{\varphi}=\mathbf{Ay}+\mathbf{m}.$

(6)

Ignoring the parameter relations in

**y**, an unconstrained LS and WLS estimate of

**y** can be computed respectively given by

$\widehat{\mathbf{y}}={\left({\mathbf{A}}^{T}\mathbf{A}\right)}^{-1}{\mathbf{A}}^{T}\mathit{\varphi},$

(7)

and

$\widehat{\mathbf{y}}={\left({\mathbf{A}}^{T}\mathbf{WA}\right)}^{-1}{\mathbf{A}}^{T}\mathbf{W}\mathit{\varphi},$

(8)

where

**W** is a weighting matrix of size

*M* ×

*M*. Note that

*M* ≥

*l* + 2 is required in (7) and (8), which indicates that we need at least four anchors to estimate the target position on a plane. The optimal

**W** is

**W*** =

**Σ**^{-1}, which depends on the unknown

**d** as we mentioned before. Thus, we can update it iteratively, and the resulting iterative WLS can be summarized as follows:

- (1)
Initialize **W** using the estimate of **d** based on the LS estimate of **x**;

- (2)

- (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.

The typical stopping criteria are discussed in [

24]. We stop the iterations when

$\u2225{\widehat{\mathbf{y}}}^{\left(k+1\right)}-{\widehat{\mathbf{y}}}^{\left(k\right)}\u2225\le \epsilon $, where

**ŷ**^{(k)}is the estimate of the

*k* th iteration and

*ϵ* is a given threshold [

25]. An estimate of

**x** is finally given by

$\widehat{\mathbf{x}}=\left[{\mathbf{I}}_{l}\phantom{\rule{1em}{0ex}}{\mathbf{0}}_{l\times 2}\right]\widehat{\mathbf{y}}.$

(9)

To accurately estimate

**y**, we can further explore the relations among the parameters in

**y**. A CWLS estimator is obtained as

$\widehat{\mathbf{y}}=\text{arg}\underset{\widehat{\mathbf{y}}}{\text{min}}{\left(\varphi -\mathbf{Ay}\right)}^{T}\mathbf{W}\left(\mathit{\varphi}-\mathbf{Ay}\right)$

(10)

subject to

${\mathbf{y}}^{T}\mathbf{Jy}+{\mathit{\rho}}^{T}\mathbf{y}=0,$

(11)

where

$\rho ={\left[{\mathbf{0}}_{l+1}^{T},1\right]}^{T}$ and

$\mathbf{J}=\left[\begin{array}{ccc}{\mathbf{I}}_{l}\hfill & {\mathbf{0}}_{l}\hfill & {\mathbf{0}}_{l}\hfill \\ {\mathbf{0}}_{l}^{T}\hfill & -1\hfill & 0\hfill \\ {\mathbf{0}}_{l}^{T}\hfill & 0\hfill & 0\hfill \end{array}\right].$

(12)

Solving the CWLS problem is equivalent to minimizing the Lagrangian [

4,

10]

$\mathcal{\mathcal{L}}\left(\mathbf{y},\lambda \right)={\left(\mathit{\varphi}-\mathbf{Ay}\right)}^{T}\mathbf{W}\left(\mathit{\varphi}-\mathbf{Ay}\right)+\lambda \left({\mathbf{y}}^{T}\mathbf{Jy}+{\mathit{\rho}}^{T}\mathbf{y}\right),$

(13)

where

*λ* is a Lagrangian multiplier. A minimum point for (13) is given by

$\widehat{\mathbf{y}}={\left({\mathbf{A}}^{T}\mathbf{WA}+\lambda \mathbf{J}\right)}^{-1}\left({\mathbf{A}}^{T}\mathbf{W}\mathit{\varphi}-\frac{\lambda}{2}\mathit{\rho}\right),$

(14)

where

*λ* is determined by plugging (14) into the following equation

${\widehat{\mathbf{y}}}^{T}\mathbf{J}\widehat{\mathbf{y}}+{\mathit{\rho}}^{T}\widehat{\mathbf{y}}=0.$

(15)

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.

Moreover, we propose the subspace minimization (SM) method [

22] to achieve a REFF-USR-LS estimate of

**x** alone, which is identical to

$\widehat{\mathbf{x}}$ in (7), but shows more insight into the links among different estimators. Treating

*b* and

*b*^{2} - ||

**x**||

^{2} as nuisance parameters, we try to get rid of them by orthogonal projections instead of random reference selection. We first use an orthogonal projection

$\mathbf{P}={\mathbf{I}}_{M}-\frac{1}{M}{\mathbf{1}}_{M}{\mathbf{1}}_{M}^{T}$ of size

*M* ×

*M* onto the orthogonal complement of

**1**_{
M
}to eliminate (

*b*^{2} - ||

**x**||

^{2})

**1**_{
M
}. Sequentially, we employ a second orthogonal projection

**P**_{
u
}of size

*M* ×

*M* onto the orthogonal complement of

**Pu** to cancel -2

*b* **Pu**, which is given by

${\mathbf{P}}_{u}={\mathbf{I}}_{M}-\frac{\mathbf{Pu}{\mathbf{u}}^{T}\mathbf{P}}{{\mathbf{u}}^{T}\mathbf{Pu}}.$

(16)

Thus, premultiplying (3) with

**P**_{
u
}**P**, we obtain

${\mathbf{P}}_{u}\mathbf{P}\mathit{\varphi}=2{\mathbf{P}}_{u}{\mathbf{PX}}_{a}^{T}\mathbf{x}+{\mathbf{P}}_{u}\mathbf{Pm},$

(17)

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 -2*b* **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.

Based on (17), the LS and WLS estimate of

**x** is respectively given by,

$\widehat{\mathbf{x}}=\frac{1}{2}{\left({\mathbf{X}}_{a}\mathbf{P}{\mathbf{P}}_{u}{\mathbf{PX}}_{a}^{T}\right)}^{-1}{\mathbf{X}}_{a}\mathbf{P}{\mathbf{P}}_{u}\mathbf{P}\mathit{\varphi},$

(18)

and

$\widehat{\mathbf{x}}=\frac{1}{2}{\left({\mathbf{X}}_{a}{\mathbf{QX}}_{a}^{T}\right)}^{-1}{\mathbf{X}}_{a}\mathbf{Q}\mathit{\varphi},$

(19)

where

**Q** is an aggregate weighting matrix of size

*M* ×

*M*. The optimal

**Q** is given by

${\mathbf{Q}}^{*}=\mathbf{P}{\mathbf{P}}_{u}{\left({\mathbf{P}}_{u}\mathbf{P}\mathbf{\Sigma}\mathbf{P}{\mathbf{P}}_{u}\right)}^{\u2020}{\mathbf{P}}_{u}\mathbf{P}$

(20)

$={\left({\mathbf{P}}_{u}\mathbf{P}\mathbf{\Sigma}\mathbf{P}{\mathbf{P}}_{u}\right)}^{\u2020},$

(21)

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

As we mentioned before, all the REFB-USR-LS methods suffer from a poor reference selection. There are some efforts to improve the reference selection [

16–

18]. In [

16], the operation employed to cancel ||

**x**||

^{2}**1**_{
M
}is equivalent to the orthogonal projection

**P**. All anchors are chosen as a reference once in [

17] in order to obtain

*M*(

*M* - 1)/2 equations in total. A reference anchor is chosen based on the criterion of the shortest anchor-target distance measurement in [

18]. However, reference-free methods are better than these heuristic reference-based methods in the sense that they cancel nuisance parameters in a systematic way. To clarify the relations between the REFB-USR and the REFF-USR approaches, we generalize the reference selection of all the reference-based methods as a linear transformation, which is used to cancel nuisance parameters, similarly as an orthogonal projection. To eliminate (

*b*^{2} - ||

**x**||

^{2})

**1**_{
M
}, the

*i* th anchor is chosen as a reference to make differences. As a result, the corresponding linear transformation

**T**_{
i
}of size (

*M* - 1) ×

*M* can be obtained by inserting the column vector -

**1**_{M-1}after the (

*i*-1)th column of

**I**_{M-1}, which fulfills

**T**_{
i
}**1**_{
M
}=

**0**_{M-1},

*i* ∈ {1,...,

*M*}. For example, if the first anchor is chosen as a reference, then

**T**_{1} = [-

**1**_{M-1},

**I**_{M-1}]. Furthermore, we can write

**T**_{
i
}**d** =

**T**_{i 1}**d** *- d*_{
i
}**1**_{m- 1}, where

**T**_{i 1}is achieved by replacing the

*i* th column of

**T**_{
i
}with the column vector

**0**_{M- 1}. Applying

**T**_{
i
}to both sides of (3), we arrive at

${\mathbf{T}}_{i}\mathit{\varphi}=2{\mathbf{T}}_{i}{\mathbf{X}}_{a}^{T}\mathbf{x}-2b{\mathbf{T}}_{i}\mathbf{u}+{\mathbf{T}}_{i}\mathbf{m}.$

(22)

Sequentially, we investigate the second linear transformation

**M**_{
j
}of size (

*M* - 2) × (

*M* - 1), which fulfills

**M**_{
j
}**T**_{
i
}**u** =

**0**_{M-2},

*j* ∈ {1,...,

*M*} and

*j* ≠

*i*. As a result, the nullspace

$\mathcal{N}\left({\mathbf{M}}_{j}{\mathbf{T}}_{i}\right)=\text{span}\left({\mathbf{1}}_{M},\mathbf{u}\right)=\mathcal{N}\left({\mathbf{U}}^{T}\right)$, and

$\mathcal{R}\left({\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}\right)=\mathcal{R}\left(\mathbf{U}\right)$. Note that

*b* = 0 in [

7,

16–

18,

22,

26], which means that there is no need to apply

**M**_{
j
}in these works. But the double differencing method in [

15] is equivalent to employing

**M**_{
j
}, and thus the results of [

15] can be used to design

**M**_{
j
}. Let us first define a matrix

${\stackrel{\u0304}{\mathbf{T}}}_{j1}$ of size (

*M -* 2) × (

*M -* 1) similarly as

**T**_{i 1}using the column vector

**0**_{M- 2}instead of

**0**_{M- 1}. When the

*j* th anchor is chosen as a reference and

*j* <

*i*,

**M**_{
j
}can be obtained by inserting the column vector

*-*(1/(

*u*_{
j
}-

*u*_{
i
}))

**1**_{M-2}after the (

*j -* 1)th column of the matrix

$\text{diag}\left({\stackrel{\u0304}{\mathbf{T}}}_{j1}\left({\mathbf{1}}_{M-1}\oslash \left({\mathbf{T}}_{i}\mathbf{u}\right)\right)\right)$, where ∅ is element-wise division. If

*j* >

*i*, then

**M**_{
j
}can be obtained by inserting the column vector

*-*(1/(

*u*_{
j
}-

*u*_{
i
}))

**1**_{M-2}after the (

*j* - 2)th column of the matrix

$\text{diag}\left({\stackrel{\u0304}{\mathbf{T}}}_{\left(j-1\right)1}\left({\mathbf{1}}_{M-1}\oslash \left({\mathbf{T}}_{i}\mathbf{u}\right)\right)\right)$. For example, if the first anchor is chosen to cancel out (

*b*^{2} - ||

**x**||

^{2})

**1**_{
M
}(

**T**_{1} is used), and the second anchor is chosen to eliminate

**T**_{1}**u**, then

**M**_{2} is given by

${\mathbf{M}}_{2}=\left[\begin{array}{cc}-1/\left({u}_{2}-{u}_{1}\right)\hfill & 1/\left({u}_{3}-{u}_{1}\right)\hfill \\ -1/\left({u}_{2}-{u}_{1}\right)\hfill & 1/\left({u}_{4}-{u}_{1}\right)\hfill \\ \vdots \hfill & \ddots \hfill \\ -1/\left({u}_{2}-{u}_{1}\right)\hfill & 1/\left({u}_{M}-{u}_{1}\right)\hfill \end{array}\right].$

(23)

Premultiplying

**M**_{
j
}**T**_{
i
}to both sides of (3), we achieve

${\mathbf{M}}_{j}{\mathbf{T}}_{i}\varphi =2{\mathbf{M}}_{j}{\mathbf{T}}_{i}{\mathbf{X}}_{a}^{T}\mathbf{x}+{\mathbf{M}}_{j}{\mathbf{T}}_{i}\mathbf{m}.$

(24)

Consequently, the general form of the REFB-USR-LS and the REFB-USR-WLS estimates are derived in the same way as (18) and (19) by replacing

**PP**_{
u
}**P** and

**Q** with

${\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}{\mathbf{M}}_{j}{\mathbf{T}}_{i}$ and

**Q**_{
i,j
}, respectively. We do not repeat these equations for the sake of brevity. Note that

Q_{i,j}is an aggregate weighting matrix of size

*M* ×

*M*. The optimal

**Q**_{i,j}is given by

${\mathbf{Q}}_{i,j}^{*}={\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}{\left({\mathbf{M}}_{j}{\mathbf{T}}_{i}{\mathbf{\Sigma}\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}\right)}^{-1}{\mathbf{M}}_{j}{\mathbf{T}}_{i}$

(25)

$={\left[{\left({\mathbf{M}}_{j}{\mathbf{T}}_{i}\right)}^{\u2020}{\mathbf{M}}_{j}{\mathbf{T}}_{i}{\mathbf{\Sigma}\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}{\left({\mathbf{T}}_{i}^{T}{\mathbf{M}}_{j}^{T}\right)}^{\u2020}\right]}^{\u2020},$

(26)

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.