MIMO雷达信号处理：新型CE波形设计与算法

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本文主要探讨了多输入多输出(MIMO)雷达信号处理中的一个重要主题——恒包络波形设计，特别关注的是在近距离部署的MIMO雷达系统中实现给定协方差矩阵的方法。作者Ahmed、Thompson和Mulgrew，以及Petillot，分别来自英国爱丁堡大学和赫里奥特-瓦特大学的工程学院和物理科学学院，他们提出了一个新颖的策略。传统的MIMO雷达信号处理算法通常依赖于复杂的数学模型和算法来生成满足特定性能要求的波形，如低截获概率或抗干扰性。然而，该研究提出了一种创新的途径，即通过非线性函数将高斯随机变量(Gaussian RVs)映射到具有恒定包络的非高斯随机变量(CE RVs)，如二进制相移键控(BPSK)和正交相移键控(QPSK)等有限字母集。这种方法的优势在于它提供了封闭形式的解决方案，使得波形选择更加直观且易于实现。文中详细阐述了非线性函数如何作用于输入的高斯分布，进而影响输出的非高斯分布特性，以及这些特性与目标协方差矩阵之间的关系。作者还展示了通过模拟实验验证这一方法的有效性，结果显示，采用恒包络波形的MIMO雷达能够在保持信号质量的同时，提高系统的灵活性和抗干扰能力。关键词包括：共站天线、恒包络波形、MIMO雷达、 Hermite多项式。这篇文章不仅对现有技术有所突破，也为MIMO雷达的设计者提供了一个新的工具箱，以应对日益复杂的应用需求。理解并应用这种恒包络波形设计方法对于提升MIMO雷达的性能和效率具有重要意义。

CONSTANT ENVELOPE WAVEFORM DESIGN FOR MIMO RADAR

S. Ahmed, J. S. Thompson and B. Mulgrew

School of Engineering

University of Edinburgh

Edinburgh EH8 3JL, UK

Y. Petillot

School of Engineering and Physical Sciences

Heriot-Watt University

EdinburghEH144AS,UK

ABSTRACT

A method for generating constant envelope (CE) waveforms

to realise a given covariance matrix for a closely spaced

MIMO radar system is proposed. In contrast to available

algorithms, the technique provides closed form solutions for

ﬁnding the required waveforms and suggests that waveforms

can be chosen from ﬁnite alphabets such as binary-phase shift

keying (BPSK) and quadrature-phase shift keying (QPSK).

Gaussian random-variables (RV’s) are mapped onto CE non-

Gaussian RV’s using memoryless non-linear functions. The

relationship between the correlation of Gaussian RV’s at the

input to the nonlinear functions and non-Gaussian RV’s at

their output is established. Simulation results are presented

to demonstrate the effectiveness of the methodology.

Index Terms— co-located antennas, constant-envelope

waveforms, MIMO radar, Hermite polynomials

1. INTRODUCTION

In contrast to phased-array radars, multiple-input multiple-

output (MIMO) radars allow each transmitting antenna

to transmit independent waveforms, thus providing extra

degrees-of-freedom (DOF) [1]. MIMO radars can be clas-

siﬁed as either widely spaced [2] or co-located [3, 4]. In

the fo rmer, the transmitting an tennas are widely separated

and each antenna may view a different aspect of the target.

This topology can increase the spatial diversity of the system.

In co-located MIMO radars, the transmitting antennas are

closely spaced (on the order of half wavelength of the carrier

frequency) and all the transmit antennas view the same aspec t

of the target. Co-located antennas cannot provide improved

spatial diversity but can increase the spatial resolution of the

system. Moreover, compared to phased-array radars extra

DOFs of a co-located MIMO radar can provide better control

of the transmit beampattern [3].

In [3–5] the covariance matrix of the waveforms for co-

located MIMO radars is synthesised to achieve the desired

beampattern. The work of [3] is extended in [4] to design the

actual waveforms. This work is b ased on the assumption that

This work is funded through EPSRC/DSTL joint grant scheme

signals that have CE and take on values in a ﬁnite alphabets

may fail to realise a given covariance matrix. The compro-

mise on the requirement for CE waveforms is to allow small

variations in the amplitudes of the waveforms and satisfy a

less restrictive condition of low peak-to -average p ower ratio

(PAPR). To design the waveforms a cyclic algorithm [4] is

used that requires a number of iterations.

In this paper an alternative solution is proposed that guar-

antees the CE property. It is based on the principle that wave-

forms with sp eciﬁed correlation properties can be easily gen-

erated using independent Gaussian random variables and a

square root of the associated covariance matrix. Memoryless

nonlinearities are then used to map the Gaussian random pro-

cesses to random processes with the d esired density function

- in this case random variables with a CE. Although the ac-

tion of memoryless nonlinearities also alters the correlation

between the waveforms and changes the covariance matrix,

this change is well deﬁned in terms of a series of Hermite

polynomials [7] and can be inverted. Thus we can deﬁne a

covariance matrix at the input to the memoryless nonlineari-

ties that would produce the desired covariance matrix at their

outputs. A similar approach was used to generate correlated

Gamma processes in [6] to simulate radar sea clutter. Here

we extend this work to generate a variety of CE waveforms

with given cross-correlation properties.

The organisation of the paper is as follows. In Section 2

the problem is formulated, Section 3 discusses the generation

of CE waveforms. Simulation results are presented in Section

4, followed by conclusions in Section 5.

Notation: Bold upper case letters, X, and calligraphic

upper case letters, X, denote matrices while bold lower case

letters, x, denote vectors. Conjugate and conjugate transposi-

tion are respectively denoted by (.)

∗

and (.)

.Themth row

and nth column element of a matrix X is denoted by X(m, n)

and E{.} denote statistical expectation.

2. PROBLEM FORMULATION

Consider a closely spaced MIMO radar with M transmit an-

tennas. The sequence of N transmitted symbols from antenna

m is denoted by the column vector x

.The(N ×M ) matrix

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