基于自然对数复合函数近似l0范数的DOA估计

doi:10.12382/bgxb.2022.0025

摘要/Abstract

摘要：

针对现有基于压缩感知的DOA估计算法收敛速度慢、精度不高等问题,提出一种基于自然对数复合函数近似l₀范数的DOA估计算法。新算法采用一种自然对数复合函数来近似l₀范数,将求解l₀范数问题转化为近似l₀范数的最优化问题。采用牛顿迭代法获得自然对数复合函数(即近似l₀范数)的迭代表达式,通过内外双层循环的方法获得牛顿迭代的最优解,即通过外层循环控制函数逼近因子σ的大小,内层循环采用最陡梯度法对牛顿迭代表达式进行求解,经有限次迭代即可获得近似l₀范数的最优解,进而得到DOA的估计值。通过仿真实验验证新算法的有效性,结果表明新算法在单快拍条件下即可实现DOA有效估计,且与平滑l₀范数算法及其改进算法相比具有更快的计算速度和更高的估计精度。

关键词: 波达方向估计, 压缩感知, 近似l₀范数, 自然对数复合函数

Abstract:

A DOA estimation algorithm based on a natural logarithm compound function approximating l₀ norm is proposed to address the issues of slow convergence and low accuracy of existing compressed-sensing-based DOA estimation algorithms. We transform the problem of solving the l₀ norm into an optimization problem of approximating the l₀ norm by using a natural logarithm compound function. The Newton iteration method is employed to obtain the iteration expression of the natural logarithm compound function (for approximating l₀ norm).The optimal solution of Newton iteration is obtained through inside and outside double-loop iteration, where the outside loop controls the magnitude of function approximation factor σ, and the inside loop solves the Newton iteration expression using the steepest descent method. The optimal solution of the approximate l₀ norm is obtained after finite iterations, and the estimated value of DOA can be obtained. Simulation results show that the proposed algorithm achieves effective DOA estimation under the condition of a single snapshot, and outperforms the existing smoothened l₀ norm algorithm and its improved algorithms in terms of speed and accuracy.

Key words: DOA estimation, compressed sensing, approximate l₀ norm, natural logarithm composite function

单泽彪, 常立民, 刘小松, 王宇祥. 基于自然对数复合函数近似l₀范数的DOA估计[J]. 兵工学报, 2023, 44(5): 1521-1528.

SHAN Zebiao, CHANG Limin, LIU Xiaosong, WANG Yuxiang. DOA Estimation Based on Approximate l₀ Norm of Natural Logarithm Composite Function[J]. Acta Armamentarii, 2023, 44(5): 1521-1528.

图/表 11

图1 基于NLSL0算法的三信源DOA估计

Fig.1 DOA estimation of three sources based on NLSL0 algorithm

图2 基于NLSL0算法的四信源DOA估计

Fig.2 DOA estimation of four sources based on NLSL0 algorithm

图3 基于NLSL0算法的五信源DOA估计

Fig.3 DOA estimation of five sources based on NLSL0 algorithm

表1 幅相不一致时DOA估计均方根误差及成功概率

Table 1 RMSEs and success probabilities of DOA estimation with amplitude-phase inconsistency

幅度误差	相位误差	均方根误差/(°)	成功率/%
N(0,0.1)	N(0,10)	1.5845	91.17
N(0,0.2)	N(0,10)	2.1935	90.57
N(0,0.1)	N(0,20)	1.5743	91.10

表2 离格信号DOA估计均方根误差

Table 2 RMSEs of DOA estimation for off-grid signals

DOA	30.0°	30.2°	30.5°	30.7°	30.9°
0°	0.8488°	0.8763°	0.9042°	0.8609°	0.8395°
0.2°	0.8521°	0.8795°	0.9523°	0.8830°	0.8621°
0.5°	1.0237°	1.0392°	1.0416°	1.0211°	1.0199°

图4 不同信噪比时各算法的DOA估计均方根误差

Fig.4 RMSEs of DOA estimation using different algorithms versus SNR

图5 不同信噪比时各算法的DOA估计成功概率

Fig.5 Success probability of DOA estimation using different algorithms versus SNR

图6 不同阵元数时各算法的DOA估计均方根误差

Fig.6 RMSEs of DOA estimation of different algorithms versus array elements

图7 不同阵元数时各算法的DOA估计成功概率

Fig.7 Success probability of DOA estimation of different algorithms versus array elements

表3 不同算法所需迭代次数及均方根误差

Table 3 Number of iterations and RMSEs required by different algorithms

算法	迭代次数/次	均方根误差/(°)
NLSL0	6	0.3134
SL0	8	0.3873
ISL0	16	0.3742
WSL0	19	0.3317

表4 不同算法运算时间及均方根误差

Table 4 Calculation time and RMSEs of different algorithms

参数	算法
参数	NLSL0	SL0	ISL0	WSL0	OMP
时间/s	0.0157	0.0189	0.0185	0.0233	0.0132
误差/(°)	0.7483	1.9039	1.0392	0.8573	4.5706

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基于自然对数复合函数近似l₀范数的DOA估计

DOA Estimation Based on Approximate l₀ Norm of Natural Logarithm Composite Function

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