JOINT WA VEFORM AND PASSIVE BEAMFORMER DESIGN IN MULTI-IRS-AIDED RADAR Zahra Esmaeilbeig1 Arian Eamaz1 Kumar Vijay Mishray and Mojtaba Soltanalian ECE Department University of Illinois Chicago Chicago IL 60607 USA

2025-05-05 0 0 472.25KB 6 页 10玖币

侵权投诉

JOINT WAVEFORM AND PASSIVE BEAMFORMER DESIGN IN MULTI-IRS-AIDED RADAR

Zahra Esmaeilbeig1?, Arian Eamaz1?, Kumar Vijay Mishra†, and Mojtaba Soltanalian?

?ECE Department, University of Illinois Chicago, Chicago, IL 60607, USA

†United States DEVCOM Army Research Laboratory, Adelphi, MD 20783, USA

ABSTRACT

Intelligent reﬂecting surface (IRS) technology has recently attracted

a signiﬁcant interest in non-light-of-sight radar remote sensing. Prior

works have largely focused on designing single IRS beamformers for

this problem. For the ﬁrst time in the literature, this paper considers

multi-IRS-aided multiple-input multiple-output (MIMO) radar and

jointly designs the transmit unimodular waveforms and optimal IRS

beamformers. To this end, we derive the Cram´

er-Rao lower bound

(CRLB) of target direction-of-arrival (DoA) as a performance met-

ric. Unimodular transmit sequences are the preferred waveforms

from a hardware perspective. We show that, through suitable trans-

formations, the joint design problem can be reformulated as two uni-

modular quadratic programs (UQP). To deal with the NP-hard nature

of both UQPs, we propose unimodular waveform and beamforming

design for multi-IRS radar (UBeR) algorithm that takes advantage of

the low-cost power method-like iterations. Numerical experiments

illustrate that the MIMO waveforms and phase shifts obtained from

our UBeR algorithm are effective in improving the CRLB of DoA

estimation.

Index Terms—Beamforming, IRS-aided radar, non-line-of-

sight sensing, unimodular sequences, waveform design.

1. INTRODUCTION

An intelligent reﬂecting surface (IRS) is composed of a large array of

scattering meta-material elements, which reﬂect the incoming signal

after introducing a pre-determined phase shift [1, 2]. Recently, the

beneﬁts of IRS have been investigated for future wireless communi-

cations [3–5] applications, including multi-beam design [6], secure

parameter estimation [7] and joint sensing-communications [8–10].

In this paper, we focus on the IRS-aided radar, where combined pro-

cessing of line-of-sight (LoS) and non-LoS (NLoS) paths has shown

improvement in target estimation and detection [11–14] through an

optimal design of IRS phase shifts.

Target detection via multiple-input multiple-output (MIMO)

IRS-aided radar was studied extensively in [11]. In our earlier works

on target estimation [12, 15], we derived the optimal IRS phase

shifts based on the mean-squared-error of the best linear unbiased

estimator (BLUE) for complex target reﬂection factor [12] and the

Cram´

er-Rao lower bound (CRLB) of Doppler estimation for mov-

ing targets [15]. Recent studies [13, 16] focused on optimization

of IRS beamforming based on CRLB of direction-of-arrival (DoA)

1Equal contribution. This work was sponsored in part by the National

Science Foundation Grant ECCS-1809225, and in part by the Army Re-

search Ofﬁce, accomplished under Grant Number W911NF-22-1-0263. The

views and conclusions contained in this document are those of the authors

and should not be interpreted as representing the ofﬁcial policies, either ex-

pressed or implied, of the Army Research Ofﬁce or the U.S. Government.

The U.S. Government is authorized to reproduce and distribute reprints for

Government purposes notwithstanding any copyright notation herein.

estimation for a single IRS-aided radar. More recent works [15, 17]

demonstrate the beneﬁts of deploying multiple IRS platforms instead

of a single IRS.

Similar to a conventional radar [18], a judicious design of

transmit waveforms improves the performance of IRS-aided radar.

Whereas designing radar probing signals is a well-studied prob-

lem [18–22], it is relatively unexamined for IRS-aided radar. In

this context, transmit sequences that mitigate the non-linearities of

ampliﬁers and yield a uniform power transmission over time are of

particular interest. Unimodular sequences with the minimum peak-

to-average power ratio exhibit these properties and have been studied

in previous non-IRS works for radar applications [21]. In this paper,

we jointly design unimodular sequences and IRS beamformers.

Multipath propagation through multiple IRS platforms increases

the spatial diversity of the radar system [23]. To this end, we inves-

tigate the beneﬁts of multipath processing for multi-IRS-aided tar-

get estimation. We ﬁrst derive the CRLB of DoA estimation for a

multi-IRS-aided radar. Then, we formulate the unimodular wave-

form design problem based on the CRLB minimization for IRS-

aided radar as a unimodular quadratic program (UQP). The uni-

modularity constraint makes the UQP an NP-hard problem. In gen-

eral, UQP may be relaxed via a semi-deﬁnite program (SDP) for-

mulation but the latter has a high computational complexity as well

[24,25]. Inspired by the power method that has the advantage of sim-

ple matrix-vector multiplications, [22, 26] proposed power method

like iterations (PMLI) algorithm to approximate UQP solutions lead-

ing to a low-cost algorithm. We formulate the IRS beamforming

design as a unimodular quartic programming (UQ2P). Prior works

[19,27] on unimodular waveform design with good correlation prop-

erties also lead to UQ2Ps, for which they employ a more costly

majorization-minimization technique. On the contrary, we use a

quartic to bi-quadratic transformation to solve UQ2P by splitting

it into two quadratic subproblems. Our unimodular waveform and

beamforming design for multi-IRS radar (UBeR) algorithm is based

on the cyclic application of PMLI and provides the optimized CRLB.

In summary, the contributions of our work are introducing the signal

model for a multi-IRS-aided radar system, derivation of the Fisher

information for the DoA estimation and developing our algorithm

called UBeR for joint Unimodular waveform and beamforming de-

Throughout this paper, we use bold lowercase and bold upper-

case letters for vectors and matrices, respectively. We represent a

vector x∈CNin terms of its elements {xi}as x= [xi]N

i=1. The

mn-th element of the matrix Bis [B]mn. The sets of complex

and real numbers are Cand R, respectively; (·)>,(·)∗and (·)H

are the vector/matrix transpose, conjugate and the Hermitian trans-

pose, respectively; trace of a matrix is Tr(.); the function diag(.)

returns the diagonal elements of the input matrix; and Diag(.)

produces a diagonal/block-diagonal matrix with the same diago-

nal entries/blocks as its vector/matrices argument. The Hadamard

arXiv:2210.14458v3 [cs.IT] 12 Mar 2023

(element-wise) and Kronecker products are and ⊗, respectively.

The vectorized form of a matrix Bis written as vec (B). The

s-dimensional all-ones vector, all-zeros vector, and the identity ma-

trix of size s×sare 1s,0N, and Is, respectively. The minimum

eigenvalue of Bis denoted by λmin(B). The real, imaginary, and

angle/phase components of a complex number are Re (·),Im (·),

and arg (·), respectively. vec−1

K,L (c)reshapes the input vector

c∈CKL into a matrix C∈CK×Lsuch that vec (C) = c.

2. MULTI-IRS-AIDED RADAR SYSTEM MODEL

Consider a colocated MIMO radar with Nttransmit and Nrreceive

antennas, each arranged as uniform arrays (ULA) with inter-element

spacing d. The MIRS platforms indexed as IRS1, IRS2,...,IRSM,

are implemented at stationary and known locations, each equipped

with Nmreﬂecting elements arranged as ULA, with element spac-

ing of dmbetween the antennas/reﬂecting elements of IRSm. The

continuous-time signal transmitted from the n-th antenna at time

instant tis xn(t). Denote the Nt×1vector of all transmit sig-

nals as x(t) = [xi(t)]Nt

i=1 ∈ΩNt, where the set of unimodular

sequences is Ωn=s∈Cn|s= [ejωi]n

i=1, ωi∈[0,2π]. The

steering vectors of radar transmitter, receiver and the m-th IRS

are, respectively, at(θ) = [1, ej2π

λdsinθ,...,ej2π

λd(Nt−1)sinθ]>,

ar(θ) = [1, ej2π

λdsinθ,...,ej2π

λd(Nr−1)sinθ]>, and bm(θ) =

[1, ej2π

λdmsinθ,...,ej2π

λdm(Nm−1)sinθ]>, where λ, is the carrier

wavelength and dand dmare usually assumed to be half the carrier

wavelength. Each reﬂecting element of IRSmreﬂects the incident

signal with a phase shift and amplitude change that is conﬁgured via

a smart controller [28]. We denote the phase shift vector of IRSm

by vm= [ejφm,1,...,ejφm,Nm]>∈CNm, where φm,k ∈[0,2π]

is the phase shift associated with the k-th passive element of IRSm.

Denote the angle between the radar-target, radar–IRSm, and

target-IRSmby θtr,θri,m , and θti,m, respectively. Denote target-

radar channel by Htr =ar(θtr)∈CNr×1; and radar-target by

Hrt =at(θtr )>∈C1×Nt. The LoS or radar-target-radar channel

matrix is Hrtr =ar(θtr )at(θtr )>∈CNr×NT. Analogously, for

the multi-IRS aided radar the NLoS channel matrices associated with

IRSmare deﬁned as Hri,m =bm(θri,m)a>

t(θri,m)∈CNm×NT

for radar-IRSm;Hit,m =b>

m(θti,m)∈C1×Nmfor IRSm-

target; Hti,m =bm(θti,m)∈CNm×1for target-IRSm; and

Hir,m =ar(θri,m)b>

m(θri,m)∈CNr×Nmfor IRSm-radar paths.

The received signal back-scattered from a single target is the

superimposition of echoes from both LoS and NLoS paths as

y(t) = αrtr Hrtrx(t−τrtr)

m=1

αritr,m HtrHit,m ΦmHri,mx(t−τritr,m)

m=1

αrtir,m Hir,mΦmHti,mHrtx(t−τrtir,m)

m=1

αritir,m Hir,mΦmHti,mHit,mΦmHri,m

x(t−τritir,m) + w(t),∈CNr,(1)

where Φm= Diag (vm),α(·),m is the complex reﬂectivity which

depends on the target back-scattering coefﬁcient and the atmospheric

attenuation, and w(t)∼ CN (0, σ2INt)denotes a stationary (ho-

moscedastic) additive white Gaussian noise (AWGN). In general,

the received signal may also have an additional inter-IRS interfer-

ence that should be included while accounting for the SNR. When

there is some blockage or obstruction between the radar and target,

we have αrtr '0,αritr,m '0and αrtir,m '0. We replace

αritir,m by αmfor notation brevity. The received signal becomes

y(t) =

m=1

αmHir,mΦmHti,mHit,mΦmHri,m

x(t−τritir,m) + w(t).(2)

Our goal is to design a radar system for inspecting a range cell

located at distance dtr with respect to (w.r.t.) the radar transmit-

ter/receiver for a potential target. Assume that the relative time

gaps between any two multipath signals are very small in compar-

ison to the actual roundtrip delays, i.e., τritir,m ≈τ0=2dtr

cfor

m∈ {1,...,M}, where cis the speed of light. We collect N

slow-time samples at the rate 1/Tsfrom the signal, at t=nTs,

n= 0,...,N −1. Hence, corresponding to the range-cell of inter-

est, the received signal vector is

y[n] =

m=1

αmHmx[n] + w[n],y[n]∈CNr×1,(3)

where x[n] = x(τ0+nTs)∈CNt×1,y[n]=[yi[n]]Nr

i=1, and we

deﬁne Hm=Hir,mΦmHti,mHit,mΦmHri,m ∈CNr×Nt. The

delay τ0is aligned on-the-grid so that n0=τ0/Tsis an integer [29].

Collecting all discrete-time samples for Nrreceiver antennas,

the received signal is the Nr×NmatrixY= [y[0],...,y[N−

1]] = PM

m=1 αmHmX+W,where X= [x[0],...,x[N−1]] ∈

CNt×Nand W= [w[0],...,w[N−1]] ∈CNr×N. Vectorizing as

y= vec (Y)yields

m=1

αmvec (HmX) + vec (W) = ˜

X˜

Hα+˜

w,(4)

where ˜

X=X>⊗INr,˜

H= [ ˜

H1,..., ˜

HM],˜

Hm= vec (Hm)for

m∈ {1,...,M},˜

w= vec (W)and α= [αm]M

m=1. Given that

w(t)is AWGN in (1), it is easily observed that y∼ CN (µ,R),

where µ=˜

X˜

Hαand R=σ2INrN. Note that, since w(n)is a

stationary process and i.i.d. with σ2variance, through vectorization

and stacking all ensembles as one vector, the resulting process is still

stationary and i.i.d with the same variance.

Our goal is to show the effectiveness of placing MIRS plat-

forms in estimating the DoA of the target in the LoS path, i.e. θtr .

For simplicity, we consider a two-dimensional (2-D) scenario, where

the radar, IRS platforms and the target are in the same plane. Our

analysis can be easily extended to 3-D scenarios. The following re-

mark states that the estimation of DoAs in the NLoS paths, θti,m ,

for m∈ {1,...,M}is equivalent to an estimation of θtr.

Remark 1. Estimation of the vector of target DoAs, ζ= [θti,1,

...,θti,M ]>is equivalent to estimating scalar DoA parameter, θtr .

This follows because, given the locations of radar, IRS platforms

and potential target range in the 2-D plane, we have ζ= [θtr +

θ1,...,θtr +θM]>, where θmfor m∈ {1,...,M}are known.

3. UQP-BASED CRLB OPTIMIZATION

For an unbiased estimator of a parameter θtr (θ, hereafter), the vari-

ance of ˆ

θis lower bounded as E{(ˆ

θ−θ)(ˆ

θ−θ)H} ≥ CRLB(θ)[30].

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

JOINTWAVEFORMANDPASSIVEBEAMFORMERDESIGNINMULTI-IRS-AIDEDRADARZahraEsmaeilbeig1?,ArianEamaz1?,KumarVijayMishray,andMojtabaSoltanalian??ECEDepartment,UniversityofIllinoisChicago,Chicago,IL60607,USAyUnitedStatesDEVCOMArmyResearchLaboratory,Adelphi,MD20783,USAABSTRACTIntelligentreectingsurface(IRS)tech...

展开>> 收起<<

JOINT WA VEFORM AND PASSIVE BEAMFORMER DESIGN IN MULTI-IRS-AIDED RADAR Zahra Esmaeilbeig1 Arian Eamaz1 Kumar Vijay Mishray and Mojtaba Soltanalian ECE Department University of Illinois Chicago Chicago IL 60607 USA.pdf

共6页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

JOINT WA VEFORM AND PASSIVE BEAMFORMER DESIGN IN MULTI-IRS-AIDED RADAR Zahra Esmaeilbeig1 Arian Eamaz1 Kumar Vijay Mishray and Mojtaba Soltanalian ECE Department University of Illinois Chicago Chicago IL 60607 USA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: