【筆記】定位演算法效能分析

阿新 • • 發佈：2018-11-22

1 CRLB Computation

2 Mean and Variance Analysis

PERFORMANCE ANALYSIS FOR LOCALIZATION ALGORITHMS

CRLB給出了使用相同資料的任何無偏估計可獲得的方差的下界，因此它可以作為與定位演算法的均方誤差（MSE）進行比較的重要基準。然而，有偏估計的MSE可能小於CRLB。

在第1節中提供了在存在高斯噪聲的情況下使用TOA測量進行CRLB計算的過程。在第2節中，我們給出了定位估計量的理論均值和MSE表示式，其推導基於成本函式的最小化或最大化。

1 CRLB Computation

生成CRLB的關鍵是構造相應的Fisher資訊矩陣（FIM）。 FIM逆的對角元素是可實現的最小方差值。考慮公式2.1的一般測量模型，使用以下步驟總結計算CRLB的標準程式：

或者，當測量誤差為零 - 均值高斯分佈時，I（x）也可以計算為[20]

where C denotes the covariance matrix for n . We now utilize Equation 2.154 to determine the FIMs for positioning with TOA measure -ments based on Equations 2.8 and their associated noise covariance matrices, respectively. The FIM based on TOA measurements of Equation 2.5 , denoted by $\bold{I_{TOA}(x)}$

, is

注：

It is straightforward to show that

Employing Equations 2.156 and 2.11 , Equation 2.155 becomes

注：

2 Mean and Variance Analysis

When the position estimator corresponds to minimizing or maximizing a continuous cost function, the mean and MSE expressions of $\bold{\hat x}$

can be produced with the use of Taylor ’ s series expansion as follows [25] . Let $\bold{J(\tilde x)}$ be a general continuous function of $\bold{\tilde x}$ and the estimate $\bold{\hat x}$ is given by its minimum or maximum. This implies that

當位置估計器對應於最小化或最大化連續成本函式時， $\bold{\hat x}$ 的均值和MSE表示式可以使用泰勒級數展開產生如下[25]。設 $\bold{J(\tilde x)}$ 是 $\bold{\tilde x}$ 的一般連續函式，估計 $\bold{\hat x}$ 由其最小值或最大值給出。這意味著

At small estimation error conditions, such that $\bold{\hat x}$ is located at a reasonable proximity of the ideal solution of x , using Taylor ’ s series to expand Equation 2.165 around x up to the first - order terms, we have

在小的估計誤差條件下，使得 $\bold{\hat x}$ 位於x 的理想解的合理接近處，使用泰勒級數將公式2.165擴充套件到 x 到一階項，我們有

where $\bold{H(J(x))}$ and $\bold{ \triangledown (J(x))}$ are the corresponding Hessian matrix and gradient vector evaluated at the true location. When the second- order derivatives inside the Hessian matrix are smooth enough around x , we have [29] :

其中 $\bold{H(J(x))}$ 和 $\bold{ \triangledown (J(x))}$ 是在真實位置評估的相應Hessian矩陣和梯度向量。當Hessian矩陣內的二階導數在 x 周圍足夠平滑時，我們有[29]：

Employing Equation 2.167 and taking the expected value of Equation 2.166 yield the mean of $\bold{\hat x}$ :

採用公式2.167並取公式2.166的預期值得出 $\bold{\hat x}$ 平均值：

When $\bold{\hat x}$ is an unbiased estimate of x , $E\{\bold{\hat x}\} = \bold x$ indicating that the last term in Equation 2.168 is a zero vector.

式子2.168的後半部分是零向量。由於Hessian的逆不為零，那麼梯度向量為零。

Utilizing Equations 2.166 and 2.167 again and the symmetric property of the Hessian matrix, we obtain the covariance for $\bold{\hat x}$ denoted by $\bold{C_x}$

that is, the variances of the estimates of x and y are given by $[\bold{C_x}]_{1,1}$ and $[\bold{C_x}]_{2,2}$ , respectively.

對角線上的數是x,y的方差。

下面是具體的例項，以ML為例：

We first take the ML cost function for TOA-based positioning in Equation 2.59 , namely, $\bold{J}_{ML,TOA}(\bold{\tilde x})$ , as an illustration. As $E{r_{TOA,l}}= d_l$ , the expected value of $\bold{H(J_{ML,TOA}(x))}$ can be easily determined with the use of Equations 2.61 – 2.64 as