【笔记】定位算法性能分析

PERFORMANCE ANALYSIS FOR LOCALIZATION ALGORITHMS

CRLB给出了使用相同数据的任何无偏估计可获得的方差的下界，因此它可以作为与定位算法的均方误差（MSE）进行比较的重要基准。然而，有偏估计的MSE可能小于CRLB。

在第1节中提供了在存在高斯噪声的情况下使用TOA测量进行CRLB计算的过程。在第2节中，我们给出了定位估计量的理论均值和MSE表达式，其推导基于成本函数的最小化或最大化。

1 CRLB Computation

生成CRLB的关键是构造相应的Fisher信息矩阵（FIM）。 FIM逆的对角元素是可实现的最小方差值。考虑公式2.1的一般测量模型，使用以下步骤总结计算CRLB的标准程序：

或者，当测量误差为零 - 均值高斯分布时，I（x）也可以计算为[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

当位置估计器对应于最小化或最大化连续成本函数时， $\bold{\hat x}$ 的均值和MSE表达式可以使用泰勒级数展开产生如下[25]。设 $\bold{J(\tilde x)}$ 是 $\bold{\tilde x}$ 的一般连续函数，估计 $\bold{\hat x}$ 由其最小值或最大值给出。这意味着

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

在小的估计误差条件下，使得 $\bold{\hat x}$ 位于x 的理想解的合理接近处，使用泰勒级数将公式2.165扩展到 x 到一阶项，我们有

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 $\bold{ \triangledown (J(x))}$ 是在真实位置评估的相应Hessian矩阵和梯度向量。当Hessian矩阵内的二阶导数在 x 周围足够平滑时，我们有[29]：

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

采用公式2.167并取公式2.166的预期值得出 $\bold{\hat x}$ 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的后半部分是零向量。由于Hessian的逆不为零，那么梯度向量为零。

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.

对角线上的数是x,y的方差。

下面是具体的实例，以ML为例：

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