*4.1. MSE and Sample MSE*

We estimate the *<sup>x</sup>* coordinate using noisy measurements at a discrete set {*xk*}*Nx <sup>k</sup>*=<sup>1</sup> of values. Let *x*ˆ*k*,*<sup>m</sup>* denote the estimate of *xk* in the *m*th Monte Carlo run. Subsequently, the error *x*˜*k*,*<sup>m</sup>* in *x*ˆ*k*,*<sup>m</sup>* is defined by

$$\pounds\_{k,m} := \mathbf{x}\_k - \pounds\_{k,m}, k = 1, 2, \dots, N\_{\mathbf{x}}, m = 1, 2, \dots, M,\tag{77}$$

where *M* is the number of Monte Carlo runs. The MSE at *xk* is given by

$$\text{MSE}\_k = E[(\mathfrak{x}\_{k,m})^2], \quad k = 1, 2, \dots, N\_{\mathbf{x}}.\tag{78}$$

The sample MSE (SMSE) at *xk* is defined by

$$\text{MSESE}\_k := \frac{1}{M} \sum\_{m=1}^{M} \left( \tilde{\mathbf{x}}\_{k,m} \right)^2, \quad k = 1, 2, \dots, N\_{\mathbf{x}}.\tag{79}$$

Let *L*CRLB(*x*) denote the log10 of the CRLB,

$$L\_{\rm CRLB}(\mathbf{x}) := \log\_{10} \mathbf{CRLB}\_{\mathbf{x}}.\tag{80}$$

Taking the log of CRLB*x* in (32) we get

$$L\_{\rm CRLB}(\mathbf{x}) = \log\_{10} \left( \frac{\sigma^2}{N n^2 a^2} \right) - 2(n - 1) \log\_{10} \mathbf{x}. \tag{81}$$
