## 8 Cramer-Rao Bound on Covariance Matrix of Localization Errors
### Discussion on RMSE of Direct Localization
#### 9.1.1 The First Set of Experiments  
In the first set of experiments, the location estimation is performed on a 2-D plane and a simple array error model is used, which corresponds to case 1 in Section 4 in [44]. Specifically, \(\widetilde{\boldsymbol{\varphi}}_{n}\) follows a circularly symmetric complex Gaussian distribution with second-order moments given by  
\[\mathrm{E}[\widetilde{\boldsymbol{\varphi}}_{n}\widetilde{\boldsymbol{ \varphi}}_{n}^{\mathrm{T}}]=\boldsymbol{\Phi}_{n}^{(1)}=\boldsymbol{O}_{M \times M}\,,\,\mathrm{E}[\widetilde{\boldsymbol{\varphi}}_{n}\widetilde{ \boldsymbol{\varphi}}_{n}^{\mathrm{H}}]=\boldsymbol{\Phi}_{n}^{(2)}=\sigma_{ \widetilde{\boldsymbol{\varphi}}}^{2}\boldsymbol{I}_{M}\quad(1\leq n\leq N)\,, \tag{103}\]  
where \(\sigma_{\widetilde{\boldsymbol{\varphi}}}\) is the standard deviation of the array model error.  
The location geometry of the first set of experiments is shown in Figure 1, where both base stations and transmitter lie on a plane. We consider four base stations with coordinates [0, 1000] m, [0,0] m, [0,1000] m, and [0, 3000] m, while the emitter position is fixed at [2000, 2000] m. The transmitted waveforms are realizations of a normal Gaussian random process, and are unknown to the receivers. Each base station is equipped with a uniform linear array. The channel attenuation magnitude is fixed at 1, and the channel phase is selected at random from a uniform distribution over \([-\pi,\pi)\). In addition, unless stated otherwise, we use the settings (1) \(K=64\) samples; (2) SNR of 5 dB; (3) \(M=5\) sensors; (4) \(\sigma_{\widetilde{\boldsymbol{\varphi}}}=0.1\); and (5) sensor elements are separated by a half wavelength. Figures 2-6 display the RMSEs of the DPD method, as functions of the SNR of the emitter signal, the standard deviation of array model error \(\sigma_{\widetilde{\boldsymbol{\varphi}}}\), the number of array elements \(M\), the ratio of the intersensor spacing to wavelength, and the number of snapshots \(K\).  
Figure 1: Location geometry for simulation.  
Figure 4: RMSE of DPD versus number of array elements.  
Figure 3: RMSE of DPD versus standard deviation of array model error.  
Figure 2: Root-mean-square-error (RMSE) of direct position determination (DPD) versus signal-to-noise ratio (SNR) of the emitter signal.  
Figures 2-6 reveal that the theoretical RMSE provided by (54) is in close agreement with the simulation result in the presence of array model errors. Consequently, the validity of the theoretical study in Section 6 is confirmed. Furthermore, when array model errors are absent, the empirical RMSE is very close to the CRB given by (92) and the theoretical RMSE in (55), which implies the asymptotical efficiency of the DPD method presented in [29], provided that the array is accurately calibrated. It is also seen that, as expected, the presence of array model errors leads to considerable deteriorations in location accuracy. Furthermore, Figures 2 and 6 show that the RMSE of the DPD method remains approximately constant no matter how much the SNR and sample number increase. The reason for this is that when the SNR or the sample number is large enough, the effects of sensor noise can be neglected and the localization errors are therefore primarily caused by array model errors, whose affects cannot be effectively eliminated in this DPD method yet. Additionally, we find that the RMSE performance in the presence of array uncertainties is significantly greater than the CRB provided by (102), especially when the standard deviation \(\sigma_{\widetilde{\varphi}}\) increases (see Figure 3). Consequently, a new DPD method that accounts for array model errors is needed to improve the location accuracy.  
Figure 5: RMSE of DPD versus ratio of intersensor spacing to wavelength.  
Figure 6: RMSE of DPD versus number of snapshots.