基于半正定松弛的到达频率差目标定位方法

doi:10.11959/j.issn.1000-0801.2024076

摘要/Abstract

摘要：

采用频率测量实现目标定位具有成本低、可靠性高的特点，仅利用到达频率差（frequency difference of arrival，FDOA）测量，提出了一种静态目标位置的精确定位方法。针对所建立的频率测量方程的高度非线性这一问题，通过引入辅助变量，将其转化为矩阵形式的伪线性方程；然后利用半正定松弛（semi-definite relaxation，SDR）方法将非凸的加权最小二乘（weighted least square，WLS）问题松弛为半正定规划（semidefinite programming，SDP）问题，从而进一步精确估计未知变量；最后对所提出方法的均方根误差（rootmean-square error，RMSE）进行了分析，以验证其性能。仿真结果表明，在较低的高斯噪声水平下，所采用的半正定松弛方法的性能能够达到克拉美罗下界（Cramer-Rao lower bound，CRLB），且该算法对几何形状具有较高的鲁棒性；此外，在使用较少数量的传感器时，其RMSE性能要优于两阶段加权最小二乘（two-stage weighted least square，TSWLS）法。

关键词: 频率测量, 到达频率差, 半正定松弛, 半正定规划, 静态目标定位

Abstract:

The use of frequency measurements to achieve target positioning is characterized by low cost and high reliability.Using only frequency difference of arrival (FDOA) measurements, a precise localization method for static target position was proposed.To address the highly nonlinear of the established frequency measurement equation, it was transformed into a pseudo-linear equation in matrix form by introducing auxiliary variables.Then the non-convex weighted least squares (WLS) problem was relaxed into a semi-definite programming (SDP) problem by using the semi-definite relaxation (SDR) method, so as to further accurately estimate the position of unknown variables.Finally, the root mean square error (RMSE) of the proposed method was analyzed to verify its performance.The simulation results show that the performance of the adopted semi-definite relaxation method is able to reach the Cramer-Rao lower bound (CRLB) at lower Gaussian noise levels and the algorithm is highly robust to geometry.In addition, its RMSE performance is better than that of the two-stage weighted least square (TSWLS) method for a smaller number of sensors.

Key words: frequency measurement, frequency difference of arrival, semi-definite relaxation, semi-definite programming, static target positioning

中图分类号:

TN929.5

毛晓婷, 吴晓平. 基于半正定松弛的到达频率差目标定位方法[J]. 电信科学, 2024, 40(3): 104-115.

Xiaoting MAO, Xiaoping WU. Target localization method based on semi-definite relaxation with frequency difference of arrival[J]. Telecommunications Science, 2024, 40(3): 104-115.

图/表 10

图1

表1

半正定松弛方法与复杂度有关的参数"

方案	m	N_sd	$n_{i}^{sd}$
半正定松弛（SDR）	10	1	L₂

表1

表2

表3

表4

传感器的速度（/m·s-1）"

编号	${\dot{x}}_{i}$	${\dot{y}}_{i}$	编号	${\dot{x}}_{i}$	${\dot{y}}_{i}$
1	2	-13	9	2	11
2	-4	15	10	22	3
3	-3	-10	11	8	18
4	22	-13	12	1	20
5	24	-8	13	23	0
6	-1	10	14	-15	-22
7	7	10	15	8	7
8	-8	8	16	12	-24

表4

图2

表5

图3

图4

图5

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方案	m	n
第一阶段加权最小二乘（OSWLS）	N-1	L₁
第二阶段加权最小二乘（TSWLS）	L₁	k

编号	x_i	y_i	编号	x_i	y_i
1	-1 135	-1 872	9	-1 828	191
2	1 214	-1 843	10	-1 995	899
3	964	-643	11	475	-511
4	-336	-1 225	12	-824	-1 780
5	1 789	-1 458	13	803	1 572
6	-123	1 406	14	904	342
7	-1 013	-274	15	1 809	-557
8	1 977	1 711	16	-1 635	-52

编号	x^o	y^o	编号	x^o	y^o
1	400	300	5	-300	560
2	500	100	6	680	70
3	800	-100	7	-260	530