1 Cram erRao Lower Bound Optimization for Hidden Moving Target Sensing via Multi-IRS-Aided Radar

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Cram´

er–Rao Lower Bound Optimization for Hidden

Moving Target Sensing via Multi-IRS-Aided Radar

Zahra Esmaeilbeig, Kumar Vijay Mishra, Arian Eamaz and Mojtaba Soltanalian

Abstract—Intelligent reﬂecting surface (IRS) is a rapidly

emerging paradigm to enable non-line-of-sight (NLoS) wire-

less transmission. In this paper, we focus on IRS-aided radar

estimation performance of a moving hidden or NLoS target.

Unlike prior works that employ a single IRS, we investigate this

problem using multiple IRS platforms and assess the estimation

performance by deriving the associated Cram´

er-Rao lower bound

(CRLB). We then design Doppler-aware IRS phase shifts by min-

imizing the scalar A-optimality measure of the joint parameter

CRLB matrix. The resulting optimization problem is non-convex,

and is thus tackled via an alternating optimization framework.

Numerical results demonstrate that the deployment of multiple

IRS platforms with our proposed optimized phase shifts leads to a

higher estimation accuracy compared to non-IRS and single-IRS

alternatives.

Index Terms—A-optimality, hidden target sensing, intelligent

reﬂecting surfaces, parameter estimation, radar.

I. INTRODUCTION

In recent years, intelligent reﬂecting surfaces (IRS) have

emerged as a promising technology for smart wireless envi-

ronments [1, 2]. An IRS consists of low-cost passive meta-

material elements capable of varying the phase of the im-

pinging signal and hence shaping the radiation beampattern

to alter the radio propagation environment. Initial research

on IRS was limited to wireless communication applications

such as range extension to users with obstructed direct links

[3], joint wireless information and power transmission [4],

physical layer security [5], unmanned air vehicle (UAV) com-

munications [6], and shaping the wireless channel through

multi-beam design [7]. Recent works have also introduced IRS

to integrated communications and sensing systems [5, 8–10].

In this paper, we focus on IRS-aided sensing, following the

advances made in [8, 11].

The literature on IRS-aided radar [9, 12] is primarily

focused on the radar’s ability to sense objects that are hidden

from its line-of-sight (LoS). While there is a rich body of

research on non-IRS-based non-line-of-sight (NLoS) radars

(see, e.g., [13] and the references therein), the proposed

Zahra Esmaeilbeig, Arian Eamaz and Mojtaba Soltanalian are with the

ECE Departement, University of Illinois at Chicago, Chicago, IL 60607 USA.

Email: {zesmae2, aeamaz2, msol}@uic.edu.

Kumar Vijay Mishra is with the United States DEVCOM Army Research

Laboratory, Adelphi, MD 20783 USA. E-mail: kvm@ieee.org.

This work was sponsored in part by the National Science Foundation Grant

ECCS-1809225, and in part by the Army Research 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 expressed 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.

formulations require prior and rather accurate knowledge of

the geometry of propagation environment. In contrast, IRS-

aided radar utilizes the signals received from the NLoS paths

to compensate for the end-to-end transmitter-receiver or LoS

path loss [11].

The potential of IRS in enhancing the estimation perfor-

mance of radar systems has been recently investigated in [8,

14, 15]. Some recent studies such as [14], employ IRS to

correctly estimate the direction-of-arrival (DoA) of a stationary

target. Nearly all of the aforementioned works consider a

single-IRS aiding the radar for estimating the parameters of a

stationary target. The IRS-based radar-communications in [15]

included moving targets but did not examine parameter esti-

mation. In this paper, we focus on the estimation performance

of a multi-IRS-aided radar dealing with moving targets.

In particular, we jointly estimate target reﬂectivity and

Doppler velocity with multiple IRS platforms in contrast to the

scalar parameter estimation via a single IRS in [14]. We derive

the Cram´

er-Rao lower bound (CRLB) for these parameter

estimates and then determine the optimal IRS phase shifts

using CRLB as a benchmark. Previously, maximization of

signal-to-noise ratio (SNR) or signal-to-interference-to-noise

ratio (SINR) was employed in [16] to determine the optimal

phase shifts for target detection. However, optimization of the

SNR or SINR does not guarantee an improvement in target

estimation accuracy. Our previous works [11] and [17] intro-

duced multi-IRS-aided radar for NLoS sensing of a stationary

target and derived the best linear unbiased estimator (BLUE)

focusing only on target reﬂectivity and the CRLB of DoA,

respectively.

The rest of the paper is organized as follows. In the next

section, we introduce the signal model for the multi-IRS-aided

radar. In section III, we derive the CRLB for joint parameter

estimation. section IV presents the algorithm to optimize the

IRS phase shifts. We evaluate our methods via numerical

experiments in section V and conclude the paper in section VI.

Throughout this paper, we use bold lowercase and bold

uppercase letters for vectors and matrices, respectively. Cand

Rrepresent the set of complex and real numbers, respectively.

(·)>and (·)Hdenote the vector/matrix transpose, and the Her-

mitian transpose, respectively. The trace of a matrix is denoted

by Tr(.). Diag(.)denotes the diagonalization operator that

produces a diagonal matrix with same diagonal entries as the

entries of its vector argument, while diag(.)outputs a vector

containing the diagonal entries of the input matrix. The mn-th

element of the matrix Bis [B]mn. The Hadamard (element-

wise) and Kronecker products are denoted by notations and

⊗, respectively. The element-wise matrix derivation operator

arXiv:2210.05812v3 [eess.SP] 27 Nov 2022

is [∂A

∂b ]ij =∂Aij

∂b . The s-dimensional all-one vector and

the identity matrix of size s×sare denoted as 1sand Is,

respectively. Finally, Re (·)and Im (·)return the the real part,

and the imaginary part of a complex input vector, respectively.

II. SYSTEM MODEL

Consider a single-antenna pulse-Doppler IRS-aided radar,

which transmits a train of Nuniformly-spaced pulses s(t),

each of which is nonzero over the support [0, τ], with the

pulse repetition interval (PRI) Tp; its reciprocal 1/Tpis the

pulse repetition frequency (PRF). The transmit signal is

x(t) =

N−1

n=0

s(t−nTp),0≤t≤(N−1)Tp.(1)

The entire duration of all Npulses is the coherent processing

interval (CPI) following a slow-time coding procedure [19].

Assume that the propagation environment has KIRS

platforms, each with Mreﬂecting elements, deployed on

stationary platforms at known locations (Fig. 1). Each of the

Mreﬂecting elements in the k-th IRS or IRSkreﬂects the

incident signal with a phase shift and amplitude change that

is conﬁgured via a smart controller [20]. Denote the phase

shift matrix of IRSkby

Φk=Diag βk,1ejφk,1,...,βk,M ejφk,M ,(2)

where φk,m ∈[0,2π], and βk,m ∈[0,1] are, respectively,

the phase shift and the amplitude reﬂection gain associated

with the m-th passive element of IRSk. In practical settings,

it usually sufﬁces to design only the phase shifts so that

βk,m = 1 for all (k, m). The radar-IRSk-target-IRSk-radar

channel coefﬁcient/gain is

hNLoS,k =b>(θir,k)Φkb(θti,k)b>(θti,k )Φkb(θir,k),(3)

where θir,k (θti,k) is the angle between the radar (target) and

IRSk, and each IRS is a uniform linear array with the inter-

element spacing d, with the steering vector

b(θ) = h1, ej2πd

λsin θ,...,ej2πd

λ(M−1) sin θi>,(4)

and λdenoting the carrier wavelength.

Consider a single Swerling-0 model [21], moving target

with fD0and α0being, respectively, its Doppler frequency

and target back-scattering coefﬁcient as observed via the LoS

path between the target and radar. Denote the same parameters

by, respectively, fDkand αk, for k∈ {1, . . . , K}as observed

by the radar from the NLoS path via IRSk. All Doppler

frequencies lie in the unambiguous frequency region, i.e.,

up to PRF. The received signal in a known range-bin is a

superposition of echoes from the LoS and NLoS paths as

y(t) = α0hLoS

N−1

n=0

s(t−nTp)ej2πfD0t

k=1

N−1

n=0

αkhNLoS,k s(t−nTp)ej2πfDkt+w(t)

≈α0hLoS

N−1

n=0

s(t−nTp)ej2πfD0nTp

k=1

N−1

n=0

αkhNLoS,k s(t−nTp)ej2πfDknTp+w(t)(5)

where hLoS is the radar-target-radar LoS channel state infor-

mation (CSI), hNLoS,k is the NLoS channel through radar-

IRSk-target-IRSk-radar path, w(t)is the random additive

signal-independent interference, and the last approximation

assumes the target is moving with low speed fDk1/τ for

k∈ {0, . . . , K}and have low acceleration, so that the phase

rotation within the CPI could be approximated as a constant.

We design a radar system to sense a moving target in the

two-dimensional (2-D) Cartesian plane. Our proposed signal

model and methods are readily extendable to 3-D scenarios

by replacing the 1-D DoA with 2-D DoA in (3) and (4).

In particular, we consider a target tracking scenario where the

radar wishes to examine a range-cell for a potential target [22,

23]. Accordingly, we collect Nslow-time samples of (5)

corresponding to Npulses in the vector

y=α0hLoS (xp(ν0)) +

k=1

αkhNLoS,k (xp(νk)) + w,(6)

where νk= 2πfDkTp∈[−0.5,0.5) for k∈

{0, . . . , K}are the normalized Doppler shifts, p(ν) =

1, ejν, . . . , ej(N−1)ν>is a generic Doppler steering vec-

tor, and x= [x(0), x(Tp), . . . , x((N−1)Tp)]>and w=

[w(0), w(Tp), . . . , w((N−1)Tp)]>are the N-dimensional

column vectors of the transmit signal and the zero-mean

random noise vector with a covariance R, respectively [24].

Assume the complex target back-scattering coefﬁcients ob-

served from the LoS and NLoS paths are collected in the vec-

tor α= [α0, α1, α2, . . . , αK]>. The corresponding channel co-

efﬁcients are collected in h= [hLoS , hNLoS,1, . . . , hNLoS,K ]>. De-

note the sensing matrix by A= [a0,a1,...,aK]∈CN×K+1,

where a0=hLoS [xp(ν0)] and ak=hNLoS,k [xp(νk)].

This implies that A=xh>P(ν), where the Doppler shift

matrix is P(ν) = [p(ν0),p(ν1),...,p(νK)] ∈CN×K+1.

Then, the received signal (6) in the vector form is given as

y=Aα+w,(7)

In presence of multiple-IRS platforms (Fig.1), we de-

ﬁne the LoS-to-NLoS link strength ratio (LSR) as γ=

|α0hLoS |2

k=1 |αkhNLoS,k |2, which governs the relative strengths of the

signals received from both paths. In cases where the LoS path

is obstructed or weaker than the NLoS paths as illustrated in

Fig. 1, we have hLoS hNLoS,k .

III. CRLB FOR MULTI-IRS-AIDED RADAR

Denote the 3(K+ 1) ×1vector of unknown target pa-

rameters as ζ=h˜

α>,ν>i>

, where ˜

α= [α>

R,α>

I]>, with

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1Cram´erRaoLowerBoundOptimizationforHiddenMovingTargetSensingviaMulti-IRS-AidedRadarZahraEsmaeilbeig,KumarVijayMishra,ArianEamazandMojtabaSoltanalianAbstractIntelligentreectingsurface(IRS)isarapidlyemergingparadigmtoenablenon-line-of-sight(NLoS)wire-lesstransmission.Inthispaper,wefocusonIRS-aided...

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