Comparison of Nonlinear and Linear Approaches with CRLB for TOA - Based Positioning for Different SNRs

上篇博文：【 筆記 】定位演算法效能分析 給出了各種定位演算法效能分析的理論知識。

這篇博文對它進行模擬分析，主要是求CRLB，我們這篇博文延續以前博文的慣例，求解各種定位演算法的RMSE，這樣的話，我們求得的CRLB在畫圖的時候也加上一個平方根。

條件：

Consider a 2 - D geometry of L = 4 receivers with known coordinates at (0, 0), (0, 10), (10, 0), and (10, 10), while the unknown source position is ( x , y ) = (2, − 3). Note that the source is located outside the square bounded by the four receivers. The range error variance,

We compute the MSPE based on 1000 independent runs. The NLS and ML estimators are realized by the Newton –Raphson scheme, and their initial guesses are provided by the LLS and WLLS algorithms, respectively. The MSPEs of the nonlinear and linear approaches are shown in Figures 2.7 and 2.8 , respectively. In Figure 2.7 , we observe that the ML estimator is superior to the NLS method and its RMSE  can attain the sqrt of CRLB for sufficiently high SNR conditions, namely, SNR ≥ 35 dB, which agrees with Equation 2.173 . In Figure 2.8 , it is seen that the two - step WLS estimator achieves optimal estimation performance at SNR ≥ 25 dB, while the LLS, WLLS, and subspace methods can only give suboptimal accuracy. Note that the MSPEs of the ML and two - step WLS estimators can be less than CRLB when SNR ≤ 0 dB because their estimates become biased for sufficiently large noise conditions.

下圖(fig 2.5)是非線性演算法的RMSE以及CRLB的對比：

下面(fig 2.6)是線性演算法的RMSE及CRLB：

It is worthy to point out that the results of Figures 2.5 and 2.6 can also be produced from Example by modifying the source location. In doing so, we will again observe the optimality of the ML and two - step WLS estimators and the suboptimality of the NLS, LLS, WLLS, and subspace schemes. Note that the CRLB for ( x , y ) = (2, 3) is smaller than that of ( x , y ) = (2, − 3). This aligns with the conventional wisdom [30, 31] that better estimation performance can be achieved when the source location falls within the convex hull of the receivers.

當源位置變為： ( x , y ) = (2, 3)

上面的模擬對比圖如下：

下面比較，源位置分別位於（2,3）與（2，-3）時候的CRLB的平方根曲線：

Note that the CRLB for ( x , y ) = (2, 3) is smaller than that of ( x , y ) = (2, − 3).

這說明，源位置對定位的精度也是有影響的，這符合我們的常識，四個測量站圍著一個目標源位置，那麼定位誤差肯定要小一點。上圖也說明了這個問題。

用官方的話說：

Note that the CRLB for ( x , y ) = (2, 3) is smaller than that of ( x , y ) = (2, − 3). This aligns with the conventional wisdom [30, 31] that better estimation performance can be achieved when the source location falls within the convex hull of the receivers.

這符合傳統觀點[30,31]，當源位置落入接收器的凸包內時，可以實現更好的估計效能。