最小最大优化：恒定模数MIMO波形设计用于主动感应
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"这篇研究论文聚焦于用于主动感应的恒定模数多输入多输出（MIMO）波形的最小最大（Minimax）设计。在MIMO系统的设计中，波形优化是一个关键环节，旨在抑制自相关和互相关。本文提出了一种针对实际波形长度和系统规模的联合优化方法，利用最小最大准则来优化恒定模数波形和不匹配（或匹配）接收滤波器。由于问题规模大（涉及超过105个单模复变量和107个非线性约束），这给波形设计带来了挑战。此外，该问题非凸、非光滑，现有的波形设计算法和标准优化工具无法有效处理。论文中开发了一种效率高的基于原始-对偶类型的算法，具有较低的每迭代复杂度，解决了这个问题。数值比较表明，基于最小最大设计的波形在性能上优于现有方法产生的波形。"
详细知识点说明：
1. **主动感应**：主动感应是一种通过发射信号并接收回波来探测和识别目标的技术，广泛应用于雷达、无线通信和传感器网络等领域。它依赖于精心设计的波形以获取最佳的目标信息。
2. **恒定模数波形**：恒定模数波形是指幅度始终保持恒定的信号，这种波形在MIMO系统中可以提供良好的抗干扰能力和功率效率。它们通常用于满足系统对功率约束和线性特性的需求。
3. **MIMO系统**：多输入多输出系统是现代通信和雷达技术中的核心概念，通过多个天线同时发送和接收信号，能显著提高系统的容量和性能。
4. **波形优化**：波形优化是设计MIMO系统的重要步骤，目的是找到能够最大限度地减少自相关和互相关、提高信噪比或者满足其他特定性能指标的信号波形。
5. **最小最大设计准则**（Minimax设计）：这是一种优化策略，旨在最坏情况下最大化系统的性能。在本文的背景下，意味着要找到在所有可能的干扰或噪声条件下，性能最差但仍可接受的波形设计。
6. **非凸、非光滑优化问题**：这类问题的优化难度较高，因为没有全局最优解的简单路径，且函数可能包含尖锐的拐点，使得传统的优化算法难以收敛到全局最优。
7. **原始-对偶类型算法**：这种算法是解决优化问题的一种策略，通过同时更新原始问题和对偶问题的变量来逐步接近解。在处理大规模非凸、非光滑问题时，这种算法具有较好的性能。
8. **接收滤波器**：在MIMO系统中，接收滤波器用于从接收到的信号中提取有用信息，抑制噪声和干扰。与波形联合优化可以进一步提升系统性能。
9. **数值比较**：通过数值模拟或实验对比，作者证明了新提出的最小最大设计方法相对于现有方法的优势，表明了这种方法在实际应用中的潜力。
这篇论文介绍了一种新的方法来解决MIMO系统中复杂且规模庞大的波形优化问题，为主动感应系统的性能提升提供了新的途径。
IEEE SIGNAL PROCESSING LETTERS, VOL. 26, NO. 10, OCTOBER 2019 1531
Minimax Design of Constant Modulus MIMO
Waveforms for Active Sensing
Zhen Lin , Wenqiang Pu, and Zhi-Quan Luo , Fellow, IEEE
Abstract—Waveform optimization is a crucial step in the design
of a multiple-input multiple-output system. This letter considers
the joint optimization of constant modulus waveforms and mis-
matched (or matched) receive filters to suppress the auto- and cross
correlations using the minimax (
∞
-norm) design criterion. For
practical waveform length and system size, the waveform design
problem becomes quite challenging due to the large problem size
(more than 10
5
unimodular complex variables and 10
7
nonlinear
constraints). In addition to the large size, this problem is noncon-
vex, nonsmooth, and as such, cannot be handled effectively by the
existing waveform design algorithms or off-the-shelve optimization
tools. This letter develops an efficient primal–dual type algorithm
with low per-iteration complexity to solve this problem. Numerical
comparison shows that the waveforms based on the minimax de-
sign outperform those obtained from the existing
2
-norm design
by 4–5 dBs in terms of peak sidelobe levels.
Index Terms—MIMO system, mismatched (or matched) fil-
ters, unimodular waveforms, auto-/cross-correlation, primal-dual
method.
I. INTRODUCTION
M
ULTIPLE-INPUT multiple-output (MIMO) systems
have many important applications in radar active sensing
[1]–[3], such as target detection and localization [4], [5], tracking
[6], and radar imaging [7], [8]. A fundamental component of a
MIMO system is the set of signal waveforms used by its antenna
elements. The performance of a MIMO system, such as location
accuracy [4], detection performance [9] and parameter identifia-
bility [10], is known to be directly related to the good correlation
property of its signal waveforms. In particular, it is desirable to
have low auto-/cross-correlations among the signal waveforms.
Orthogonality between waveforms promises many advantages
in different applications, e.g., higher location accuracy [4],
better detection performance [9], and parameter identifiability
[10]. For practical implementation, the MIMO waveforms are
required to be constant modulus (i.e., unimodular).
In view of the importance of waveform design, various math-
ematical formulations and algorithms have been proposed in
Manuscript received March 30, 2019; revised June 10, 2019; accepted June
23, 2019. Date of publication July 1, 2019; date of current version September 11,
2019. This work was supported in part by the National Natural Science Foun-
dation of China under Grants 61571384 and 61731018, in part by the Leading
Talents of Guangdong Province Program (00201501), and in part by the De-
velopment and Reform Commission of Shenzhen Municipality. This letter was
presented in part at the Asilomar Conference on Signals, Systems, and Comput-
ers, Pacific Grove, CA, USA, October 2018. The associate editor coordinating
the review of this manuscript and approving it for publication was Prof. Luca
Venturino. (Corresponding author: Zhi-Quan Luo.)
The authors are with the Shenzhen Research Institute of Big Data, The Chi-
nese University of Hong Kong, Shenzhen 518172, China (e-mail: 114020243
@link.cuhk.edu.cn; wenqiangpu@outlook.com; luozq@cuhk.edu.cn).
Digital Object Identifier 10.1109/LSP.2019.2926020
the past decade for the MIMO waveform design problem. In
[11], a customized alternating direction method of multipliers
(ADMM) algorithm was proposed to approximate desired beam-
patterns using constant modulus waveforms with low spacial
auto- and cross-correlation sidelobes. In [12], [13], an
2
-norm
minimization based waveform design (or the so called integrated
sidelobe levels (ISL)) was proposed to minimize the squared
sum of the auto- and cross-correlations of the waveforms. A
cyclic iterative algorithm called CAN [12] and a majorization-
minimization (MM) algorithm [13] were proposed to solve the
2
-norm based waveform design problem by exploiting the effi-
cient FFT representation of the minimization problem. An alter-
native
∞
-norm based design has also been considered in [14] to
minimize the so called peak auto-/cross-correlation (or sidelobe)
levels, PSL for short, so as to mitigate the effect of unknown in-
terferences in the worst-case. In particular, the reference [14]
derived theoretical bounds on ISL and PSL of sequence correla-
tions, and proposed an alternating projection (AP) algorithm for
the MIMO waveform design problem. However, each iteration
of the AP algorithm requires an eigenvalue decomposition of a
MN × MN matrix, where M is the number of waveforms and
N is the length of each waveform; such a step is prohibitively
expensive for large values of M,N (say, MN =10
5
). The MM
method proposed in [15] approximates
∞
-norm by an
p
-norm
of the correlations, for a large p, to address the nonsmoothness
of the
∞
-norm, but is only applicable to the design of a single
waveform (M =1). Apart from the aforementioned work on
ISL and PSL minimization for waveform design, various other
design criteria were proposed in different application contexts,
see [16]–[21]. Another factor affecting the achievable PSL of
MIMO waveforms is the use of mismatched receive filter [22],
[23]. The latter has been proposed to improve robustness to the
Doppler shifting [17], mitigate interferences [24] and clutters
[25], and approximate a desired beampattern [26], among others.
This letter considers the joint optimization of constant mod-
ulus waveforms and mismatched (or matched) receive filters
to suppress the auto- and cross-correlations using the minimax
design criterion. For practical system size (say, M = 128 an-
tenna elements) and waveform length (N>10
3
), the waveform
design problem becomes quite challenging due to the large
problem size (more than 10
5
complex unimodular variables,
and approximately 10
7
nonlinear constraints). In addition to the
large size, this optimization problem is nonconvex, nonsmooth,
and as such, can not be handled effectively by the existing
2
/
∞
-norm waveform design algorithms or off-the-shelve
optimization tools. The main contribution of this letter is an
efficient primal-dual type algorithm with low per-iteration
complexity to solve this minimax waveform design problem.
Numerical comparison shows that the newly designed wave-
forms outperform those obtained from the existing algorithms
by 4–5 dBs in terms of peak sidelobe level suppression,
1070-9908 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Authorized licensed use limited to: The Chinese University of Hong Kong CUHK(Shenzhen). Downloaded on March 25,2020 at 07:51:40 UTC from IEEE Xplore. Restrictions apply.
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