RIS-assisted Integrated Sensing and Communications A Subspace Rotation Approach Invited Paper

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RIS-assisted Integrated Sensing and

Communications: A Subspace Rotation Approach

(Invited Paper)

Xiao Meng1,2, Fan Liu2, Shihang Lu2, Sundeep Prabhakar Chepuri3and Christos Masouros4

1Beijing Institute of Technology, Beijing, China

2Southern University of Science and Technology, Shenzhen, China

3Indian Institute of Science, Bangalore, India

4University College London, London, UK

Abstract—In this paper, we propose a novel joint active and

passive beamforming approach for integrated sensing and com-

munication (ISAC) transmission with assistance of reconﬁgurable

intelligent surfaces (RISs) to simultaneously detect a target and

communicate with a communication user. We ﬁrst show that

the sensing and communication (S&C) performance can be

jointly improved due to the capability of the RISs to control the

ISAC channel. In particular, we show that RISs can favourably

enhance both the channel gain and the coupling degree of S&C

channels by modifying the underlying subspaces. In light of

this, we develop a heuristic algorithm that expands and rotates

the S&C subspaces that is able to attain signiﬁcantly improved

ISAC performance. To verify the effectiveness of the subspace

rotation scheme, we further provide a benchmark scheme which

maximizes the signal-to-noise ratio (SNR) at the sensing receiver

while guaranteeing the SNR at the communication user. Finally,

numerical simulations are provided to validate the proposed

approaches.

Index Terms—ISAC, RIS, beamforming, subspace.

I. INTRODUCTION

Sensing has been regarded as an important function in the

next-generation wireless networks [1], [2]. Many emerging

mobile applications, such as smart manufacturing and vehicle

to everything, not only require high-quality communication

with low latency and high rate, but also require location infor-

mation with high precision [3]. To provide better performance

and to efﬁciently use the spectrum, energy, and hardware,

integrating the sensing functionality and communication into

a single system becomes a promising approach. By sharing

the hardware and wireless resources and jointly designing

the waveform and signal processing ﬂow between S&C, a

signiﬁcant performance gain can be obtained in integrated

sensing and communication (ISAC) systems [4], [5].

In parallel to the ISAC technology, reconﬁgurable intelli-

gent surfaces (RISs) or intelligent reﬂecting surfaces (IRSs),

which are well known for its ability to modify the wireless

propagation environment, has also drawn signiﬁcant attention

from both academia and industry [6]–[8]. By designing the

phase shift matrix, RIS is capable of simultaneously modifying

the communication channel and the sensing channel, which is

favorable for an ISAC system [4], [8]–[15]. In particular, RIS

can be designed to diminish interference between the radar

and communication system [11], and may also be designed to

ISAC BS

( )



( )



( )



RIS

UE Target

Weakly

Coupled

RIS

Strongly

Coupled

Fig. 1. RIS-assisted ISAC system model.

reduce the multi-user interference (MUI) [12], [13]. As a step

forward, jointly designing the RIS and transmit beamformer,

one may leverage the constructive interference to facilitate the

ISAC transmission [14].

Motivated by the above research, in this paper we investigate

the joint active and passive beamforming design for the RIS-

assisted ISAC system, where a multi-antenna base station

(BS) simultaneously serves a single antenna user and tracks

a target. We ﬁrst point out that compared with the individual

S&C systems, the additional performance gain provided by the

RIS mainly comes from the improvement of channel/subspace

correlation and the enhanced channel gain, by presenting a

brief analysis on the RIS-assisted channel as well as the

structure of the beamformer. Based on these ﬁndings, we

then develop a heuristic method to rotate and expand the

S&C subspaces. To provide a performance baseline, we also

introduce a benchmark beamforming technique to maximize

the sensing signal-to-noise ratio (SNR) while guaranteeing

the communication SNR. To solve the optimization problem,

we employ alternative optimization (AO) algorithm to itera-

tively optimize the active beamformer at the BS and passive

beamformer at the RIS. Finally, we provide numerical results

to verify the effectiveness of the proposed subspace rotation

approach.

II. SYSTEM MODEL

Let us consider an RIS-assisted ISAC system, where a

multi-antenna BS simultaneously serves a single-antenna user

arXiv:2210.13987v1 [eess.SP] 23 Oct 2022

equipment (UE) and tracks a single target. As shown in Fig.

1, the S&C channels may be weakly coupled, resulting in

poor performance of the ISAC system (which will be detailed

later). An RIS with Melements is deployed to provide

additional strongly-coupled channels, thus improving the joint

performance. The BS is equipped with Nttransmit antennas

and Nrreceive antennas, which transmits an ISAC signal x(t)

to perform both S&C tasks. By denoting the communication

channel, the transmit sensing channel, and the receive sensing

channel as hc∈CNt,ht∈CNt, and hr∈CNr, respectively,

the signal model of this ISAC system can be expressed as

Sensing Model: ys(t) = h∗

rhH

tx(t) + zs,∀t, (1)

Comms Model: yc(t) = hH

cx(t) + zc,∀t, (2)

where x(t) = ws(t)with w∈CNtdenoting the ISAC

beamformer and s(t)denoting the communication signal with

unit power, ys∈CNrand ycrespectively denote the received

signal at the BS and the UE. Here, zs∈CNrdenotes the

additive white Gaussian noise (AWGN) vector at the BS with

the variance of each entry being σsand zcdenotes the AWGN

at the UE with the variance being σc. In particular, the channel

can be modeled as

ht=αtat(θ)+αgGtΦb(φ)≈αt(at(θ) + GtΦ˜

b(φ)),(3)

hr=αrar(θ)+αgGrΦb(φ)≈αr(ar(θ) + GrΦ¯

b(φ)),(4)

hc=hBU +GtΦhRU ,(5)

where αt,αr, and αgdenote the reﬂection coefﬁcient and

path-loss coefﬁcient from the transmit antenna to the target,

that from the target to the receive antenna and that from the

RIS to the target, respectively, at∈CNt,ar∈CNrand

b∈CMdenote the steering vector from the transmit antenna

and receive antenna to the target and the steering vector from

the RIS to the target, Gt∈CNt×Mand Gr∈CNr×Mdenote

the channel from the transmit antenna array and the receive

antenna array to the RIS, hBU ∈CNtand hRU ∈CMdenote

the channel from the BS to the UE and that from the RIS to the

UE and Φ∈CM×Mis the diagonal phase shift matrix of the

RIS. While αt,αr, and αgmay be challenging to be explicitly

obtained, their relationship can be approximately estimated by

leveraging the geometric relationship among the BS, the RIS,

and the target. For notational convenience, we normalize bto

band ¯

bin (3) and (4), respectively.

To provide better sensing performance while ensuring com-

munication quality, we maximize the sensing SNR with a

given communication SNR threshold by jointly optimizing the

phase shift matrix Φand beamformer w, in which case the

optimization problem can be formulated as

max

Φ,wSNRs(6a)

s.t. SNRc≥Γ0,(6b)

||w||2≤Pt,(6c)

|ϕm|= 1,∀m= 1,2, ..., M, (6d)

Φ=diag(ϕ1, ...., ϕM),(6e)

Communication

Subspace

Increased correlation

Channel Gain

RIS

Assisted

RIS

Assisted

ISAC signal

Sensing

subspace

Fig. 2. ISAC performance enhanced by RIS: subspace expansion & rotation.

where SNRs=kh∗

rhH

twk2/σ2

sdenotes the sensing SNR,

SNRc=|hH

cw|2/σ2

cdenotes the communication SNR, ϕm

denotes the m-th non-zero element of the diagonal phase shift

matrix Φ,Γ0denotes the communication SNR threshold, and

Ptdenotes the transmit power budget.

III. PROPOSED SUBSPACE ROTATION SCHEME

In this section, we show that by introducing RIS we can

improve the performance of an ISAC system over that of in-

dividual S&C systems from the channel subspace perspective.

Towards this end, we propose a novel algorithm by exploiting

the subspace correlation between S&C channels.

A. ISAC Beamforming Design Without RIS

To begin with, we ﬁrst investigate the optimal beamformer

without the assistance of RIS, in which case we have ht=

αtat(θ),hr=αrar(θ)and hc=hBU . To maximize the

sensing SNR while guaranteeing the communication quality,

the optimization problem can be formulated as

max

wSNRs(7a)

s.t. SNRc≥Γ0,(7b)

kwk2≤Pt.(7c)

Lemma 1. The optimal solution of (7) satisﬁes

w∈span{hc,ht}.(8)

Proof. See [16].

This indicates that the optimal solution always belongs to

the linear subspace spanned by the communication subspace

hcand the sensing subspace ht. Using Lemma 1, the optimal

solution for (7) is given in the following theorem.

Theorem 1. The optimal solution to (7) is

w=(√Pt

khtk,if Pt|hH

cht|2≥Γ0σ2

ckhtk2

x1u1+x2u2,otherwise,(9a)

where

u1=hc

khck,u2=ht−(uH

1ht)u1

kht−(uH

1ht)u1k,(9b)

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RIS-assistedIntegratedSensingandCommunications:ASubspaceRotationApproach(InvitedPaper)XiaoMeng1;2,FanLiu2,ShihangLu2,SundeepPrabhakarChepuri3andChristosMasouros41BeijingInstituteofTechnology,Beijing,China2SouthernUniversityofScienceandTechnology,Shenzhen,China3IndianInstituteofScience,Bangalore,Indi...

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