Linear Approaches

The basic idea of the linear localization methodology is to convert the nonlinear expressions of Equations （1） into a set of linear equations with zero - mean disturbances, assuming that the measurement errors are sufficiently small.

（1）

As the corresponding optimization cost functions are now unimodal, it is always guaranteed to obtain the global solution. Three linear positioning approaches, namely, LLS, WLLS, and subspace estimators, will be presented as follows.

Analogous to NLS and ML estimators, the WLLS method is a weighted version of the LLS scheme and it provides higher localization accuracy, although the mean and covariance of the errors in the linear equations are required for the weight computation. On the other hand, the subspace technique fi rst relates x with the squared pairwise distances among the source and receivers. Source localization is
then achieved using an eigenvalue decomposition ( EVD ) procedure.

LLS

The LLS approach attempts to reorganize Equations （1） into linear equations in x , and the position is then estimated by using the ordinary LS technique. For TOA, TDOA, and RSS measurements, we have to introduce an intermediate variable, which is a function of the source position in the linearization process. The LLS location estimators based on TOA, TDOA, RSS, and DOA information are developed one by one as follows.

LLS方法试图将方程（1）重组为x中的线性方程，然后使用普通LS技术估计位置。 对于 TOA 测量，我们必须引入一个中间变量，它是线性化过程中源位置的函数。 基于 TOA 信息的LLS位置估计器如下。

TOA - Based Positioning

To convert the TOA measurements into linear models in x , we first consider squaring both sides of Equation （1） to obtain

（2）

Let

（3）

be the noise component in Equation （2） and introduce a dummy variable R of the form

（4）

Substituting Equations （3） and （4） into （2） yields

（5）

Let

（6）

（7）

（8）

and

（9）

The matrix form for Equation （5） is then

（10）

where the observed of Equation

（11）

is now transformed to b , A is constructed from the known receiver positions, and θ contains the source location to be determined. When are sufficiently small such that

现在转换为 b，由已知的接收器位置构成，θ 包含要确定的源位置。当   足够小

（12）

can be considered a zero - mean vector; that is, E { q } ≈ 0 , we can approximate Equation （11）as

（13）

the LS cost function based on Equation （13） is

（14）

which is a quadratic function in , indicating that there is a unique minimum in .

The LLS estimate corresponds to

（15）

which can be easily computed by differentiating Equation （14） with respect to   and by setting the resultant expression to zero:

（16）

The LLS position estimate is simply extracted from the fi rst and second entries of ; that is,

（17）

In the literature, Equation （17） is also referred to as the LS calibration method  .

（5）

An alternative way for LLS TOA - based positioning is to eliminate R in Equation （5） by employing the differences between any two equations [18] . For simplicity but without loss of generality, subtracting the first equation of Equation （5） from the remaining ( L − 1) equations, R is removed and we have

（18）

or in matrix form

（19）

where A , q , and b are now modifi ed to

（20）

（21）

（22）

Assuming sufficiently small noise conditions and following Equations （12）——（16）, a variant of the LLS method using TOA measurements is

（23）