基于修正牛顿法的大规模MIMO低复杂度混合预编码算法

doi:10.11959/j.issn.1000-0801.2023186

摘要/Abstract

摘要：

针对大规模多输入多输出（multiple-input multiple-output，MIMO）系统，提出了一种基于修正牛顿（modified Newton，MN）法的相位跟踪算法，有效地解决了传统高性能混合预编码方案中的高计算复杂度问题。该算法从子维度向量恢复的角度优化模拟预编码矩阵。在每个子维度优化中，采用相位跟踪方法将模拟预编码向量的恢复转化为无约束的非线性优化问题，并利用 MN 法进行求解。同时，应用 Gerschgorin’s Disk 定理和Hermitian矩阵分块求逆引理，分别降低了MN法中计算修正因子以及Hessian矩阵求逆的计算复杂度。实验结果表明，与仿真中几种传统的高性能混合预编码方案相比，所提算法具有更高的频谱效率和更低的计算复杂度。

关键词: 大规模MIMO, 混合预编码, 相位跟踪, 修正牛顿法

Abstract:

A phase tracking algorithm was proposed for massivemultiple-input multiple-output(MIMO) systems based on the modified Newton (MN) method, which effectively reduced the high computational complexity in traditional high-performance hybrid precoding schemes.The algorithm optimized the analog precoding matrix from the perspective ofsub-dimensional vector recovery.In each sub-dimension optimization, the phase tracking method was used to transform the recovery of the analog precoding vectors into an unconstrained nonlinear optimization problem, which was then solved using the MN method.Concurrently, this strategy led to a marked reduction in the computational intricacy pertaining to both the computation of correction factors and the inversion of the Hessian matrix within the framework of the MN method.This was achieved through the insightful incorporation of Gerschgorin’s Disk theorem and the Hermitian matrix block-inverse lemma.Simulation results show that the proposed algorithm has higher spectral efficiency and lower computational complexity than several conventional high-performance hybrid precoding schemes.

Key words: massive MIMO, hybrid precoding, phase tracking, modified Newton method

中图分类号:

TN928

胡博, 王安定, 魏贵义. 基于修正牛顿法的大规模MIMO低复杂度混合预编码算法[J]. 电信科学, 2023, 39(11): 80-95.

Bo HU, Anding WANG, Guiyi WEI. Low complexity hybrid precoding algorithm for massive MIMO based on modified Newton method[J]. Telecommunications Science, 2023, 39(11): 80-95.

图/表 12

图1

表1

MN-AltMin算法每步操作所需的复数乘法和除法个数"

操作		复数乘法和除法个数
算法1（外部迭代）	$F_{RF}^{(k)^{†}} F_{opt}$	$N_{t} (N_{t}^{RF})^{2} + N_{t} N_{t}^{RF} N_{s} + f_{inv} (N_{t}^{RF})$
算法2（内部子迭代）	$g_{i}^{(k)}$	$(N_{t}^{RF})^{2} (N_{s} + 1) + N_{t}^{RF} N_{s} + f_{\exp} (N_{t}^{RF})$
	$H_{i}^{(k)}$	$\frac{1}{2} (N_{t}^{RF})^{2}$
	$d_{i}^{(k)}$	$α (\frac{1}{6} (4 N_{t}^{RF} - 1) (N_{t}^{RF} + 1) \cdot N_{t}^{RF} + (N_{t}^{RF})^{2} \begin{array}{l} \end{array})$

表1

表2

MN-AltMin算法与多种经典的交替优化算法的计算开销"

算法	计算开销
MN-AltMin	$N_{iter}^{o} (N_{t} (N_{t}^{RF})^{2} + N_{t} N_{t}^{RF} N_{s} + f_{inv} (N_{t}^{RF})) +$
	$N_{iter}^{i} N_{t} (\frac{1}{6} (N_{t}^{RF})^{3} + (N_{s} + \frac{15}{8}) (N_{t}^{RF})^{2} +$
	$(N_{s} - \frac{1}{24}) N_{t}^{RF} + f_{\exp} (N_{t}^{RF}))$
CG-AltMin	$N_{iter}^{o} (N_{t} (N_{t}^{RF})^{2} + N_{t} N_{t}^{RF} N_{s} + f_{inv} (N_{t}^{RF})) +$
	$N_{iter}^{i} N_{t} (4 N_{s} N_{t} N_{t}^{RF} + 11 N_{t}^{RF})$
BB-AltMin	$N_{iter}^{o} (N_{t} (N_{t}^{RF})^{2} + N_{t} N_{t}^{RF} N_{s} + f_{inv} (N_{t}^{RF})) +$
	$N_{iter}^{i} N_{t} (2 N_{s} N_{t} N_{t}^{RF} + 8 N_{t}^{RF})$
GP-AltMin	$N_{iter}^{o} (N_{t} (N_{t}^{RF})^{2} + N_{t} N_{t}^{RF} N_{s} + f_{inv} (N_{t}^{RF})) +$
	$N_{iter}^{i} N_{t} (2 N_{s} N_{t} N_{t}^{RF} + 2 N_{t}^{RF})$

表2

图2

图3

图4

表3

不同算法收敛时所需的平均迭代次数"

N_RF	MN-AltMin		CG-AltMin		GP-AltMin		BB-AltMin
N_RF	$T_{\max}^{i}$	$N_{iter}^{o}$	$N_{iter}^{i}$	$N_{iter}^{o}$	$N_{iter}^{i}$	$N_{iter}^{o}$	$N_{iter}^{i}$	$N_{iter}^{o}$
2	30.69	9.31	226.97	9.28	420.50	9.41	265.17	9.28
3	57.01	6.77	286.31	6.73	310.91	7.22	451.69	6.87
4	368.86	5.70	656.80	5.73	269.60	6.39	1 726.77	5.72
5	390.89	4.96	511.46	4.92	233.52	5.67	1 276.47	4.91
6	398.55	4.45	482.52	4.50	241.25	5.82	1 229.23	4.49

表3

图5

图6

表4

NRF=3时MN-AltMin算法在不同迭代停止条件下的平均迭代次数"

ε		$T_{\max}^{i}$
ε		2	4	8	16	32	64	128	512
10^-2	$N_{iter}^{i}$	8.36	11.98	15.56	18.23	20.62	21.40	22.82	23.55
	$N_{iter}^{o}$	5.18	4.09	3.74	3.69	3.67	3.63	3.68	3.67
10^-3	$N_{iter}^{i}$	17.60	24.64	31.84	37.66	40.96	45.52	47.33	46.16
	$N_{iter}^{o}$	9.81	7.77	7.28	7.006	6.74	6.85	6.80	6.84
10^-4	$N_{iter}^{i}$	47.14	63.89	81.18	93.50	104.88	108.83	130.32	125.36
	$N_{iter}^{o}$	24.57	20.17	18.86	18.39	18.03	17.70	18.78	17.36
10^-5	$N_{iter}^{i}$	131.88	170.87	227.27	243.51	289.60	326.18	326.11	319.09
	$N_{iter}^{o}$	66.85	56.60	56.34	50.69	52.35	53.64	51.96	50.23

表4

图7

图8

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