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Foundations of MIMO Radar Detection Aided byReconfigurable Intelligent Surfaces

Stefano Buzzi, Senior Member, IEEE, Emanuele Grossi, Senior Member, IEEE, Marco Lops, Fellow, IEEE,Luca Venturino, Senior Member, IEEE

Abstract—A reconfigurable intelligent surface (RIS) is a flatlayer made of sub-wavelength-sized reflective elements capableof adding a tunable phase shift to the impinging electromagneticwave. This paper considers the fundamental problem of targetdetection in a RIS-aided multiple-input multiple-output (MIMO)radar system. At first, a general signal model is introduced,which includes the possibility of using up to two RISs (one closeto the transmitter and one close to the receiver) and subsumesboth a mono-static and a bi-static radar configuration with orwithout a line-of-sight (LOS) view of the prospective target. Uponresorting to a generalized likelihood ratio test (GLRT), the designof the RIS phase shifts is formulated as the maximization of theprobability of detection in the resolution cell under inspectionfor a fixed probability of false alarm, and suitable optimizationalgorithms are proposed and discussed. Both the theoretical andthe numerical analysis clearly show the benefits, in terms ofthe signal-to-noise ratio (SNR) at the radar receiver, granted bythe use of the RISs and shed light on the interplay among thekey system parameters, such as the radar-RIS distance, the RISsize, and location of the prospective target. A major finding isthat the RISs should be deployed in the near-field of the radartransmit/receive array. The paper is then concluded by discussingsome open problems and foreseen applications.

Index Terms—MIMO Radar, LOS/NLOS, Bi/Mono-Static, Tar-get Detection, Reconfigurable Intelligent Surfaces, Metasurfaces.

I. INTRODUCTION

A reconfigurable intelligent surface (RIS) is an emergingtechnology that has recently attracted a huge interest from bothacademic and industry researchers in the wireless communi-cations and sensing community [1]–[3]. An RIS is a planarstructure that is made of engineered material with tunableelectromagnetic characteristics that can be controlled to alterthe wireless propagation environment [4]–[6]. Over the lastfew years, the RISs have been widely investigated in wirelesscommunications, with the aim of understanding how and towhat extent they could enhance the networks performance.In [7], the RIS is modeled as an array of reflecting elementswith tunable phase shifts and, with reference to a base station(BS) serving several downlink devices, the transmit power and

The authors are planning to submit part of this work at the 25th InternationalITG Workshop on Smart Antennas (WSA 2021).

S. Buzzi, E. Grossi and L. Venturino are with the Department of Electricaland Information Engineering (DIEI), University of Cassino and SouthernLazio, 03043 Cassino, Italy, and with Consorzio Nazionale Interuniversi-tario per le Telecomunicazioni, 43124 Parma, Italy (e-mail: buzzi@unicas.it;e.grossi@unicas.it; l.venturino@unicas.it).

M. Lops is with the Department of Electrical and Information Technology(DIETI), University of Naples Federico II, 80138 Naples, Italy, and with Con-sorzio Nazionale Interuniversitario per le Telecomunicazioni, 43124 Parma,Italy (e-mail: lops@unina.it).

the phase shifts are jointly designed to maximize the energyefficiency. In [8], an indoor environment is considered and adeep neural network able to learn the mapping between theposition of the users and the RIS phase shifts is designed,so as to maximize the signal strength at the user’s location.The work in [9] assumes that the phase shifts of the RISelements can take value in a finite set and investigates therelation between the set cardinality and the rate degradation.In [10], the RIS phase shifts are exploited in conjunction withthe communication transmitter to encode a message, and itis shown that this scheme achieves a larger capacity thanthat obtained when the RIS is designed only to maximize thesignal-to-noise ratio (SNR) at the receiver. The work in [11],instead, considers a point-to-point multiple-input multiple-output (MIMO) link and, accounting for the finite alphabetinput, performs an alternating optimization of the transmitprecoder and RIS phase shifts to minimize the symbol errorrate. The problem of channel estimation in RIS-aided wirelesscommunications has been tackled in [12]–[14], while otherstudies have assessed the RIS benefits in the context of wire-less power transfer [15], non-orthogonal multiple access [16],design of innovative antenna configurations [17], and coverageextension [18]. The reader may refer to [1]–[3] and thereferences therein for a more comprehensive overview.

The RIS potentials have been also investigated in position-ing and localization tasks, as discussed in [19]. In [20], ascenario with one BS, one mobile device, and one RIS isconsidered, and the problem of finding the device positionand orientation is discussed; the Cramer-Rao lower bounds tothe estimation error are computed and the RIS configurationmaximizing the sum of the SNRs at each BS antenna is given.As shown in [21], the use of an RIS allows to perform jointsynchronization and user localization without using two-waytransmission, even if the mobile user is equipped with oneantenna. In [22], a similar approach is pursued for the casethat the BS has only one antenna, and joint three-dimensionallocalization and synchronization in a single-user multi-carriersystem is addressed, under the assumption of line-of-sight(LOS) propagation and in the presence of a uniform planarreflecting array. Joint localization and communication is thenaddressed in [23], wherein a procedure to determine the RISphase shifts based on the use of hierarchical codebooks andfeedback from the mobile station is presented.

In the last few months, the possible benefits that the RISscan bring to radar systems and to the spectral coexistencebetween radar and communication have started being in-vestigated. The work in [24] exploits an RIS to enhance

3

the sensing and communication capabilities of a mmWavedual function transceiver; here, separate RIS elements areused for sensing and communication, and a two-dimensionalhierarchical codebook for the RIS configuration is presented,along with a method to choose the number of snapshots usedin each stage to get a desired target localization performance.In [25], a dual function system is again considered, and theradar detection probability is maximized by optimizing theBS transmit beamformer and the RIS phase shifts, subject tosignal-to-interference-plus-noise-ratio and power consumptionconstraints. A similar approach is pursued in [26], whereinthe design of the phase shifts and of the precoding matrix ofan RIS-aided BS is considered, which is carried out by max-imizing the SNR at the radar receiver subject to sensing andcommunication constraints. Finally, the transmitted waveformand the RIS configuration are chosen in [27] to mitigate themultiuser interference under a beam pattern constraint.

With regard to the target detection problem, in our earliercontribution [28] we assumed that the radar is capable of form-ing (a) two transmit beams pointing towards the prospectivetarget and the RIS and one receive beam pointing towardsthe prospective target only or, viceversa, (b) one transmitbeam pointing towards the prospective target only and tworeceive beams pointing towards the prospective target andthe RIS. Under such a scenario, a theoretical analysis iscarried out for closely- and widely-spaced (with respect to thelocation of the prospective target) radar and RIS deployments,showing that large gains can be provided by the nearby RIS,and initial hints on the optimal RIS placement are provided.In [29], the case where no direct path between the radarand the prospective target exists is considered, mimickingthe companion situation where the RIS provides an indirectlink between a communication source and a destination thatwould be otherwise not reachable. Finally, [30] represents afirst contribution to the scenario considered here and, althoughpassing over a number of relevant effects and situations, asdetailed in the sequel of this contribution, makes the basicpoint that an RIS can indeed aid the radar detection.

A. Contribution and paper structureThe bottom line of the current studies is that suitably

deployed RISs modify the wireless channel response, thusproviding novel degrees of freedom to design wireless systems.In this context, we aim at investigating if and under whichconditions the RISs may boost the performance of a radar,by focusing on the classical target detection problem. Withreference to the scenarios outlined in Fig. 1, we first derive atruly general and novel signal model, wherein a MIMO radartransmits a set of orthogonal waveforms and, assisted by aforward and/or a backward RIS, verifies the presence/absenceof a prospective target in a given resolution cell underinspection. The considered model includes mono-static, bi-static, LOS, and non-line-of-sight (NLOS) radar configurationsand accounts for the presence of up to four paths from thetransmitter to the prospective target to the receiver. Also, wedistinguish among the two relevant situations that the forwardand backward RISs are in the near or far-field of the transmitterand receiver, respectively.

Next, we derive the generalized likelihood ratio test (GLRT)receiver with respect to the unknown target response andtackle the design of the phase shifts of the reflecting ele-ments by maximizing the target detection probability for afixed probability of false alarm: interestingly, the optimizationproblem turns out to be separable, i.e., the forward andbackward RISs can be designed independently. Also, if theforward (backward) RIS and radar transmitter (receiver) arein each other’s far-field the optimum phase shifts are found inclosed form; instead, in a near-field situation, the optimizationproblem is strongly NP-hard in general and is approximatelysolved through either an alternate maximization or by resortingto a convex relaxation. Interestingly, this latter approach pro-vides a randomized solution whose expected value is at leastπ/4 ≈ 0.785 times the optimum. The special case in whichthe radar is mono-static and there is only one RIS providinga forward and a backward indirect path is also examined, andan ad-hoc algorithm to select the phase shift of each reflectingelement is proposed.

The theoretical analysis shows that only a marginal gain(as compared to the case where the radar operates alone) canbe obtained if radar and RIS are in each other’s far-field.This confirms the intuition in [28] that an RIS should bebetter placed in proximity of the radar trasmitter or receiver,unless a far-field deployment is necessary to inspect a NLOSspot that would otherwise remain blind [29]. An extensivenumerical analysis is also provided, aimed at assessing theimpact of the radar-RIS distance, the RIS size, and the locationof the inspected cell. The results show that a large gaincan be obtained under a near-field radar-RIS deployment;interestingly, such gain almost doubles when the prospectivetarget is in the view of both the forward and the backwardRIS, as compared to the case where only a single surface canilluminate/observe it. The performance of a mono-static radarwith a single RIS (the simplest and perhaps most realistic andcost-effective configuration) is also contrasted to that of anideal system where the phase shifts of the reflecting elementsare changed in between the transmission and reception phases,showing that these configurations are substantially equivalent.

The paper is organized as follows. In Sec. II, the systemdescription and the modelling assumptions are presented. InSec. III, the design of the phase shifts introduced by thereflecting elements of the RISs is studied. In Sec. IV, somenumerical examples are given to illustrate the advantages ofusing an RIS-aided radar. Finally, concluding remarks andhints for future developments are provided in Sec. V.

B. Notation

Column vectors and matrices are denoted by lowercaseand uppercase boldface letters, respectively. The symbols (·)∗,(·)T and (·)H denote the conjugate, the transpose, and theconjugate-transpose operation, respectively. IM is a M ×Midentity matrix, while Aij and ai denote the (i, j)-th entry ofthe matrix A and the i-th entry of the vector a, respectively.A � 0 means that A is Hermitian positive semidefinite. tr{A}and ‖a‖ denote the trace of the square matrix A and theFrobenius norm of the vector a, respectively, while Rank{A}

4

target

forwardRIS backward

RIS

Radar TX Radar RX

(ρ, ϕt)

(ρ, ϕt)

(d, ωt)

(d, ωt)

target

forward/backwardRIS

RadarTX/RX

(ωr,n,j , δn,j , ϕs,n,j) (ϕs,q,n, δq,n, ωr,q,n)(ϕs,n,j , δn,j , ωr,n,j)

(ϕs,q,n, δq,n, ωr,q,n)

(b) LOS Monostatic configuration(a) LOS Bistatic configuration

RadarTX/RX

target

forward/backwardRIS

target

forwardRIS

backwardRIS

Radar TX Radar RX

(c) NLOS Bistatic configuration (d) NLOS Monostatic configuration

(ωr,n,j , δn,j , ϕs,n,j)

(d, ωt)

(ϕs,q,n, δq,n, ωr,q,n)

(ρ, ϕt) (d, ωt)

(ρ, ϕ)(ρ, ϕ)

(d, ωt)(d, ωt)

(ϕs,n,j , δn,j , ωr,n,j)

(ϕs,q,n, δq,n, ωr,q,n)

(d, ωt)(d, ωt)

Figure 1. Examples of system geometries.

is the rank of the matrix A. vec{a1, . . . ,aN} is the MN -dimensional vector obtained by piling up the M -dimensionalvectors {an}Nn=1. diag{a} is the M×M matrix containing theentries of M -dimensional vector a on the main diagonal andzero elsewhere. The symbol ⊗ denotes the Kronecker product,E[ · ] the statistical expectation, and i the imaginary unit.

II. SYSTEM DESCRIPTION

We consider a radar system aimed at detecting a prospectivetarget in a given resolution cell under inspection. The radar isequipped with a transmitter and a receiver with Nr ≥ 1 andNr ≥ 1 closely-spaced elements, respectively, arranged into aplanar array; the building element of each array can be a singleantenna or a subarray module composed itself of multipleantennas. The radar operates with a carrier wavelength λ andemits Nr orthogonal and equal-power waveforms, one fromeach radiating element.1 A nearby RIS (referred to as theforward RIS) helps the transmitter illuminate the prospectivetarget; this surface is composed of Ns ≥ 1 closely-spacedreflecting elements of area As, arranged into a planar array; wedenote by ϕn the phase shift introduced by the n-th element,by ϕ = (ϕ1 . . . ϕNs

)T the vectors containing all phase shifts,

1The possible optimization of the radiated waveforms is not consideredhere and left as a future development.

and by Is ∈ {1, 0} the indicator variable specifying if this RISis present or not. Similarly, a nearby RIS (referred to as thebackward RIS) helps the receiver capture the power scatteredby the prospective target; this surface is composed of Ns ≥ 1closely-spaced reflecting elements of area As, arranged into aplanar array; we denote by ϕn the phase shift introduced by then-th element, by ϕ = (ϕ1 . . . ϕNs

)T the vectors containingall phase shifts, and by Is ∈ {1, 0} the indicator variablespecifying if this RIS is present or not.

The RISs modify the response of the environment to thewaveforms emitted by the radar [31]; a prospective targetcan be illuminated by the transmitter (direct path) and/or bythe forward RIS (indirect path); likewise, the correspondingreflections originated by the target may reach the receiverthrough a direct path and/or a indirect path bouncing on thebackward RIS. As shown in Fig. 1, up to four echoes may beobserved, corresponding to a direct illumination and a directreflection (err), a indirect illumination and a direct reflection(esr), a direct illumination and a indirect reflection (ers), anda indirect illumination and a indirect reflection (ess). Theconsidered model encompasses both a bi-static and a mono-static radar; in this latter case, the forward and backward RISmay collapse into a single surface redirecting both incidentwaves from the transmitter towards the target and from the

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target towards the receiver, as shown in Figs. 1(b)-(d). Also,the model encompasses both a LOS and a NLOS radar; inthe former case, both the transmitter and the receiver have adirect view of the inspected resolution cell and the indirectpaths granted by the RIS can be exploited to improve thedetection performance that the radar would have alone; in thelatter case, either the transmitter or the receiver or both donot have a direct view of the inspected resolution cell and theindirect paths granted by the RISs extend the field of view ofthe radar, as shown in Figs. 1(c)-(d). We underline that theconfiguration in Fig. 1(d) may be of interest also to replace apossibly expensive radar transceiver that would be requiredto cover the region of interest with a low-cost feeder thatsends/receives signals via a re-configurable surface capable ofelectronically-tunable beamforming [28].

A. Geometric parameters

Let Or, Or, Os, and Os be a Cartesian right-handedlocal reference system (LRS) placed at the transmitter, thereceiver, the forward RIS, and the backward RIS, respectively.We assume that each array is on the (y, z)-plane of thecorresponding LRS, with the array center located at the originand each element oriented towards the positive x-axis. At thetransmitter, we define the following quantities:• ρ, θaz

t , and θelt are the range and the azimuth and elevation

angles of the target in Or, respectively;2

• δ, θazs , and θel

s are the range and the azimuth andelevation angles of the center of the forward RIS in Or,respectively;

• δnj , θazs,nj , and θel

s,nj are the distance and the azimuth andelevation angles of the n-th forward reflecting element inthe LRS obtained upon translating Or at the j-th transmitelement, respectively; clearly, we have δnj = δ, θaz

s,nj =θazs , and θel

s,nj = θels if the j-th transmit element and the

n-th forward reflecting element are at the center of thecorresponding array.

At the forward RIS, we define the following quantities:• d, ωaz

t , and ωelt are the range and the azimuth and

elevation angles of the target in Os, respectively;• δ, ωaz

r , and ωelr are the range and the azimuth and

elevation angles of the center of the transmit array inOs, respectively;

• δnj , ωazr,nj , and ωel

r,nj are the range and the azimuthand elevation angles of the j-th transmit element in theLRS obtained upon translating Os at the n-th forwardreflecting element.

Specular geometric parameters can be defined at the trans-mitter and the backward RIS with respect to Or and Os,respectively, with an obvious modification of the symbolnotation (i.e., the overline is replaced with two dots). For

2We follow the standard notation that the azimuth angle of a point p ∈ R3

is the angle between the x-axis and the orthogonal projection of the vectorpointing towards p onto the (x, y)-plane, which takes values in [−π, π) andis positive when going from the x-axis towards the y-axis; also, the elevationangle is the angle between the vector pointing towards p and its orthogonalprojection onto the (x, y)-plane, which takes values in [−π/2, π/2) and ispositive when going toward the positive z-axis from the (x, y)-plane.

brevity, we denote by ϕ = {ϕaz, ϕel} the pair of azimuthand elevation angles ϕaz and ϕel. For the reader’s sake, themain geometric parameters are summarized in Fig. 1.

B. Design assumptions

At the design stage, we make the following assumptions.• There is no coupling among the elements of each array.• There is LOS propagation in the radar-RIS, radar-target,

and RIS-target hops (whenever present).• The waveforms are narrowband, so that the delays of the

target echoes reaching the receiver are not resolvable.• When both the transmitter and the forward RIS illu-

minate the prospective target, they see the same aspectangle [32], i.e., λ/Dt � maxn,j ξnj , where Dt is theeffective size of the target as seen from the transmitterand ξnj is the angle formed by the line segment linkingthe target with the j-th transmit element and the linesegment linking the target with the n-th forward reflect-ing element; similarly, when both the receiver and thebackward RIS observe the prospective target, they seethe same aspect angle, i.e., λ/Dt � maxq,n ξqn, whereDt is the effective size of the target as seen from thereceiver and ξqn is the angle formed by the line segmentlinking the target with the q-th receive element and theline segment linking the target with the n-th backwardreflecting element.

• The target and the forward (backward) RIS are in eachother’s far-field, i.e., we have [33]

d ≥ max{2D2t /λ, 2D

2s/λ, 5Dt, 5Ds, 1.6λ} (1a)

d ≥ max{2D2t /λ, 2D

2s/λ, 5Dt, 5Ds, 1.6λ} (1b)

where Ds and Ds are the maximum size of the forwardand backward RISs, respectively; also, the target and eachradar array are in each other’s far-field, i.e., we have [33]

ρ ≥ max{2D2t /λ, 2D

2r/λ, 5Dt, 5Dr, 1.6λ} (2a)

ρ ≥ max{2D2t /λ, 2D

2r/λ, 5Dt, 5Dr, 1.6λ} (2b)

where Dr and Dr are the maximum size of the transmitand the receive array, respectively. Accordingly, the phasecurvature of the wavefront can be neglected along theradar-target and RIS-target hops [33], [34].

• Any pair of transmit and forward reflecting elements arein each other’s far-field; similarly, any pair of receive andbackward reflecting elements are in each other’s far-field.This requires that [33]

minn,j

δnj ≥ max{2∆2r/λ, 2∆2

s/λ, 5∆r, 5∆s, 1.6λ} (3a)

minq,n

δqn ≥ max{2∆2r/λ, 2∆2

s/λ, 5∆r, 5∆s, 1.6λ} (3b)

where ∆r, ∆s, ∆r, and ∆s are the maximum size of eachelement of the transmit, forward reflecting, receive, andbackward reflecting arrays, respectively. We underlinethat the transmitter and the forward RIS and, similarly,the receiver and the backward RIS are not required to bein each other’s far-field. Accordingly, the phase curvature

6

of the wavefront across the array elements will not beneglected in the radar-RIS hops [17], [35].

C. Received signal

As customary, a range gating operation is at first performedby projecting the signal impinging on each receive elementalong a delayed version of each transmit waveform (this istantamount to sampling the output of a filter matched tothe waveform), with the delay tied to the distance of theinspected resolution cell from the transmitter and the receiver.Accordingly, a Nr×Nr observation matrix, sayR, is obtained,whose rows contains the Nr samples (one for each transmitelement) taken at each of the Nr receive elements; in thepresence of a steady target, this matrix can be modeled as

R = α(γrvrv

Tr γr︸ ︷︷ ︸

Err

+ γrvrvTs X(ϕ)Gγs︸ ︷︷ ︸Esr(ϕ)

+ γsGX(ϕ)vsvTr γr︸ ︷︷ ︸

Ers(ϕ)

+ γrGX(ϕ)vsvTs X(ϕ)Gγs︸ ︷︷ ︸

Ess(ϕ,ϕ)

)+W (4)

where we have defined the following quantities.• α ∈ C is the unknown target response.• γr ∈ C is the known channel between the reference

transmit element (assumed to be located at the centerof the array) and the target, which accounts for theradiated power, say Pr, the transmit gain, the path-loss,and phase delay; γs ∈ C is the known channel fromthe reference transmit element to the reference forwardreflecting element to the target, which also accounts forthe bi-static radar cross-section (RCS) of the reflectingelement.

• γr ∈ C is the known channel between the target andthe reference receive element, which accounts for thereceive gain, the path-loss, and phase delay; γs ∈ C is theknown channel from the target to the reference backwardreflecting element to the reference receive element, whichalso accounts for the bi-static RCS of the reflectingelement.

• vr ∈ CNr (vr ∈ CNr ) is the direct transmit (receive)steering vector of the radar towards the direction θt (θt) ofthe target; vs ∈ CNs (vs ∈ CNs ) is the transmit (receive)steering vector of the forward (backward) RIS towardsthe direction ωt (ωt) of the target; these vectors are tiedto the array geometry and have unit-magnitude entries.

• X(ϕ) = diag{

(eiϕ1 · · · eiϕNs )T}∈ CNs×Ns and

X(ϕ) = diag{

(eiϕ1 · · · eiϕNs )T}∈ CNs×Ns are the

matrices accounting for the phase response of the forwardand backward reflecting elements, respectively.

• G ∈ CNs×Nr and G ∈ CNr×Ns are the known channelmatrices between the transmitter and the forward RIS andbetween the backward RIS and the receiver, respectively;

• Finally, W ∈ CNr×Nr is a matrix accounting forthe additive disturbance, whose entries are modeled asindependent and identically distributed (i.i.d.) circularly-symmetric Gaussian random variables with variance σ2

w.Depending on the values of γr, γs, γr, and γs only some ofthe echoes err, esr, ers, and ess may be actually present:

Table IPOSSIBLE SYSTEM CONFIGURATIONS

Radar γr γr γs γsTarget

illuminated by observed by

LOS 6= 0 6= 0

6= 0 6= 0 Radar & RIS Radar & RIS= 0 6= 0 Radar Radar & RIS6= 0 = 0 Radar & RIS Radar= 0 = 0 Radar Radar

NLOS

6= 0 = 06= 0 6= 0 Radar & RIS RIS= 0 6= 0 Radar RIS

= 0 = 0 6= 0 6= 0 RIS RIS

= 0 6= 06= 0 6= 0 RIS Radar & RIS6= 0 = 0 RIS Radar

all possible system configurations are outlined in Table I.Finally, we point out that GTX(ϕ)vs and GX(ϕ)vs can beinterpreted as the indirect transmit (receive) steering vector ofthe radar towards the target via the RIS-assisted path,3 whichare controlled through the phases of the reflecting elements.

It is now convenient to reorganize the above model into avector form; upon defining r = vec {R} ∈ CNrNr and

err = vec{Err

}=(γrvr

)⊗(γrvr

)(5a)

esr(ϕ) = vec{Esr(ϕ)

}=(γsG

TX(ϕ)vs

)⊗(γrvr

)(5b)

ers(ϕ) = vec{Ers(ϕ)

}=(γrvr

)⊗(γsGX(ϕ)vs

)(5c)

ess(ϕ, ϕ) = vec{Ess(ϕ, ϕ)

}=(γsG

TX(ϕ)vs

)⊗(γsGX(ϕ)vs

)(5d)

we have

r = α(err + esr(ϕ) + ers(ϕ) + ess(ϕ, ϕ)

)︸ ︷︷ ︸e(ϕ,ϕ)

+w (6)

where w = vec{W } is a zero-mean circularity-symmetricGaussian vector with covariance matrix4 σ2

wINrNr. Notice that

e(ϕ, ϕ) is the spatial signature of the prospective target; re-markably, it is verified by inspection that this vector possessesthe following Kronecker structure

e(ϕ, ϕ) =(γrvr + γsG

TX(ϕ)vs

)︸ ︷︷ ︸

e(ϕ)

⊗(γrvr + γsGX(ϕ)vs

)︸ ︷︷ ︸

e(ϕ)

(7)

where e(ϕ) is the transmit signature, resulting from the su-perposition of the direct and indirect transmit steering vectorswith weights equal to the corresponding channels γr andγs, and e(ϕ) is the receive signature, resulting from the

3We use here the term steering vector with some abuse of notation as theentries of GTX(ϕ)vs and GX(ϕ)vs may not have a unit-magnitude.

4The following developments can be also extended to the case wherethe covariance matrix of w is full-rank and has the separable structureE[wwH ] = Cw ⊗ Cw , with Cw ∈ CNr×Nr and Cw ∈ CNr×Nr .

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superposition of the direct and indirect receive steering vectorswith weights equal to the corresponding channels γr and γs.

D. Channel model

While the design methodologies proposed next are generalenough to remain independent of the adopted channel model,we will make at the analysis stage the following assumptions.

According to the standard radar equation [36], [37], we willmodel the channels γr, γs, γr, and γs as follows

γr =

(PrNr

G(θt)

4πρ2Lr

)1/2

e−i2πρ/λ (8a)

γs =

(IsPrNr

G(θs)ζ(ωr, ωt)

(4π)2δ2d2Ls

)1/2

e−i2π(δ+d)/λ (8b)

γr =

(G(θt)λ

2

(4π)2ρ2Lr

)1/2

e−i2πρ/λ (8c)

γs =

(Isζ(ωt, ωr)G(θs)λ

2

(4π)3d2δ2Ls

)1/2

e−i2π(d+δ)/λ (8d)

where G(ϕ) and G(ϕ) are the gain of each transmit andreceive element in the direction ϕ = {ϕaz, ϕel}, respectively,ζ(ϕin, ϕout) and ζ(ϕin, ϕout) are the bi-static RCS of eachforward and backward reflecting element towards the directionϕout = {ϕaz

out, ϕelout} when illuminated from the direction

ϕin = {ϕazin , ϕ

elin}, respectively, and Lr, Ls, Lr, and Ls are

loss factors accounting for any additional attenuation along thecorresponding paths, respectively.

Following [38]–[40], we will assume that

ζ(ϕin, ϕout) = As cos(ϕazin ) cos(ϕel

in)︸ ︷︷ ︸ζa(ϕin)

× (4πAs/λ2) cos(ϕaz

out) cos(ϕelout)︸ ︷︷ ︸

ζg(ϕout)

. (9)

This is a simple yet realistic model, wherein the RCS of eachelement is regarded as the product of an effective receiveaperture ζa(ϕin) and a transmit gain ζg(ϕout), with a cosine-shaped scan loss in both azimuth and elevation. This modeltreats the RIS as a reciprocal surface, i.e., ζ(ϕin, ϕout) =ζ(ϕout, ϕin); also, at broadside, the entire reflecting area isequal to the sum of the effective apertures of the buildingelements, so that the overall RIS behaves as a flat plate ofarea NsAs [34]. The model in (9) will be also employed forζ(ϕin, ϕout) upon replacing As with As.

Finally, following [35], the entries of G will be modeledas5 Gnj = 0, if γs = 0, and

Gnj =

(G(θs,nj)ζa(ωr,nj)δ

2

G(θs)ζa(ωr)δ2nj

)1/2

e−i2π(δnj−δ)/λ (10)

otherwise. In this latter case, we have Gnj = 1, if the j-thtransmit and the n-th forward reflecting element are at the

5The path model in (8) and (10) can be farther generalized by considering adifferent bi-static RCS function for each element of the forward and backwardRIS, which may be the case when the possible mutual coupling among theelements of the surface is accounted for.

center of their arrays, in keeping with (8). Similarly, G willbe modeled as Gqn = 0, if γs = 0, and

Gqn =

(G(θs,qn)ζg(ωr,qn)δ2

G(θs)ζg(ωr)δ2qn

)1/2

e−i2π(δqn−δ)/λ (11)

otherwise.

III. SYSTEM DESIGN

Let f ∈ CNrNr be the unit-norm filter employed by thereceiver to focus the radar towards the specific direction ofthe inspected resolution cell, which is under the designer’scontrol; then, its output is

r = αfHe(ϕ, ϕ) + fHw (12)

resulting in the following SNR

SNR =σ2α

σ2w

∣∣fHe(ϕ, ϕ)∣∣2 (13)

where σ2α = E[|α|2]. Also, the GLRT discriminating between

the target presence (hypothesis H1) and absence (hypothesisH0) is [41]

|r|2

σ2w

H1

≷H0

η (14)

where η > 0 is the detection threshold, to be set according tothe desired probability of false alarm Pfa = e−η; assuming that|α|2 is non fluctuating, exponentially distributed, or gammadistributed with variance σ2

α/2, the corresponding detectionprobabilities are [36], [37]

Pd =

Q1

(√2SNR,

√2η)

(non fluctuating)

e−γ/(1+SNR) (exponential)(1 + γSNR/2

(1+SNR/2)2

)e−γ/(1+SNR/2) (gamma).

(15)where Q1(·, ·) is the Marcum’s Q−function. The receive filterand the RIS phases can be jointly chosen to maximize theSNR at the cell under inspection and, as a by product, thedetection probability in (15). For any ϕ and ϕ, the optimalfilter is the one matched to the signal to be detected, i.e.,

f =e(ϕ, ϕ)∥∥e(ϕ, ϕ)

∥∥ . (16)

Hence, upon plugging (16) into (13), the optimal phases areobtained as the solution to

maxϕ, ϕ

‖e(ϕ, ϕ)‖2. (17)

We underline here that the above design is independent ofthe target and noise strength (namely, of σ2

α and σ2w); also,

since the objective function ‖e(ϕ, ϕ)‖2 = ‖e(ϕ)‖2‖e(ϕ)‖2and the constraint set in (17) are separable, the forward andbackward phases can be optimized independently. Specifically,the problems to be solved are

maxϕ‖e(ϕ)‖2 (18)

when the forward RIS is present, and

maxϕ‖e(ϕ)‖2 (19)

when the backward RIS is present.

8

A. Phase optimization

In this section we discuss the solution to (18); the followingdevelopments are easily adapted to also tackle the problemin (19) which indeed presents the same structure of (18). Toproceed, let y = γrvr, Q = γsG

Tdiag (vs), and x(ϕ) =(eiϕ1 · · · eiϕNs )T; then, (18) can be rewritten as

maxϕ

∥∥y + Qx(ϕ)∥∥2. (20)

When Rank(Q) = 1, which occurs if the transmitter andthe forward RIS are in each other’s far-field (more on this inSec. III-B) or Nr = 1 or Ns = 1, then the solution to (20) isderived in closed form. Indeed, upon factorizing Q as qrqTs ,with qr ∈ CNr and qs ∈ CNs , we have that

∥∥y + qrqTs x(ϕ)

∥∥2=

∥∥∥∥∥y + qr

Ns∑n=1

|qs,n|ei(ϕn+∠qs,n)

∥∥∥∥∥2

= ‖y‖2 + ‖qr‖2∣∣∣∣∣Ns∑n=1

|qs,n|ei(ϕn+∠qs,n)

∣∣∣∣∣2

+ 2<

{yHqr

Ns∑n=1

|qs,n|ei(ϕn+∠qs,n)

}

≤ ‖y‖2 + ‖qr‖2(Ns∑n=1

|qs,n|

)2

+ 2|yHqr|Ns∑n=1

|qs,n| (21)

where the last upper bound is achieved when

ϕn = −∠qs,n − ∠(yHqr), n = 1, . . . , Ns. (22)

When Rank(Q) > 1, instead, we proceed as follows. Uponintroducing the positive semi-definite matrix6

B =

(QHQ QHyyHQ ‖y‖2

)∈ C(Ns+1)×(Ns+1) (23)

the problem in (20) can be equivalently recast as

maxϕ,φ

(e−iφxH(φ) e−iφ

)B(eiφxT(φ) eiφ

)T(24)

which in turn is equivalent to

maxz

zHBz

s.t. |zn| = 1, n = 1, . . . , Ns + 1.(25)

If z? solves (25), the solution to (20) can be recovered as ϕn =∠z?n−∠z?Ns+1

, n = 1, . . . , Ns. The problem (25) is a complexquadratic program, that has been shown to be strongly NP-hardin general [42]. A sub-optimal solution can be obtained via analternate maximization, which iteratively optimizes one entryof z at a time, as reported in Algorithm 1. Since zHBz =2<{z∗n∑j 6=n Bnj zj

}+Bnn|zn|2+

∑k 6=n

∑j 6=n z

∗kBkj zj , the

maximization over zn gives

zn =

∑j 6=n Bnj zj∣∣∣∑j 6=n Bnj zj

∣∣∣ (26)

if∑j 6=n Bnj zj 6= 0, and any unit-modulus complex number,

otherwise. Since the objective function is bounded above

6The positive-semidefiniteness is verified by exploiting the Schur comple-ment for block matrices.

Algorithm 1 Alternate maximization for Problem (25)

1: Choose ε > 0, Kmax > 0, and z ∈ CNs+1: |zn| = 1, ∀n2: k = 0 and f0 = zHBz3: repeat4: for n = 1, . . . , Ns + 1 do

5: zn =

∑j 6=n Bnj zj

|∑j 6=n Bnj zj |

, if∑j 6=n Bnj zj 6= 0

1, otherwise6: end for7: k = k + 18: fk = zHBz9: until (fk − fk−1)/fk < ε or k = Kmax

and monotonically increased at each iteration, convergence isensured. The complexity per iteration is O(N2

s ), with an initialcost of O(N3

s ) to form the matrix B in (23).Alternatively, the problem in (25) can be reformulated as

maxZ,z

Tr(BZ)

s.t. Znn = 1, n = 1, . . . , Ns + 1

Z � 0 , Z = zzH

(27)

that admits the following (convex) relaxation

maxZ

Tr(BZ)

s.t. Znn = 1, n = 1, . . . , Ns + 1 , Z � 0.(28)

The above semi-definite program is well-studied in combina-torial optimization (see, e.g., the max-cut problem [43]) andcan be solved by standard techniques, such as interior pointmethods, first-order methods on the associated dual problem,block coordinate ascent, etc. Once the matrix Z? solving (28)is found, a (sub-optimum) solution to (25) must be extracted.As in [42], [44], [45], we can generate a sample vector from acomplex zero-mean circularly symmetric Gaussian distributionwith covariance matrix Z? and normalize its entries to havea unit-magnitude, so as to obtain a sub-optimum solutions for (25). This randomized algorithm satisfies a noticeableproperty [42], [46]. Indeed, let f? be the optimal value of theobjective function in (25) and f?sub = sHBs; then, since Z?

solves the relaxed problem in (28), we have that

Tr(BZ?) ≥ f? ≥ E[f?sub] = Tr(BE[ssH]

)= Tr

(BF (Z?)

)(29)

where F (Z) ∈ C(Ns+1)×(Ns+1) is the matrix with entry (i, j)defined as [42], [45]

1

2ei∠Zij

∫ π

0

cos(ϕ) arcsin(|Zij | cos(ϕ)

)dϕ. (30)

Therefore, since Tr(BF (Z?)

)≥ π

4 Tr(BZ?) [42], we have

Tr(BZ?) ≥ f? ≥ E[f?sub] ≥ π

4Tr(BZ?) (31)

which shows that this is a randomized π4 -approximation algo-

rithm.7 Clearly, multiple (independent) feasible points s can be

7A (randomized) α-approximation algorithm is an algorithm that runs inpolynomial time and produces a solution whose (expected) value is at least afraction α of the optimum value [47].

9

computed, and the one providing the largest objective functionin (25) can be chosen: in this case, the π

4 approximation ratiowill not only hold in mean but also with a probability thatgoes to 1 exponentially fast with the number of samples.

B. Far-field deployment of the RIS

We study here in more detail the case where the transmitarray and the forward RIS are in each other’s far-field andthe receive array and the backward RIS are in each other’sfar-field. In this situation, a plane wave approximation in theaforementioned radar-RIS hops can be made and, hence, thechannel matrices in (10) and (11) simplify to G = psp

Tr

and G = prpTs , respectively, where pr (pr) is the transmit

(receive) steering vector of the radar towards the directionθs (θs) of the forward (backward) RIS, and ps (ps) is thetransmit (receive) steering vectors of the forward (backward)RIS towards the direction ωr (ωr) of the radar transmitter(receiver); the vectors pr, ps, pr, and ps are tied to the arraygeometry and have unit-magnitude entries. Since the matrix

Q = pr︸︷︷︸qr

pTs diag(vs)γs︸ ︷︷ ︸qTs

(32)

has rank one, upon exploiting (21) and (22), we have that

maxϕ‖e(ϕ)‖2 = Nr|γr|2 + NrN

2s |γs|2 + 2Ns

∣∣γ∗r γsvHr pr∣∣(33)

that is achieved when

ϕn = −∠(γsps,nvs,n

)−∠(γ∗r v

Hr pr

), n = 1, . . . , Ns. (34)

Similarly, we have that

maxϕ‖e(ϕ)‖2 = Nr|γr|2 + NrN

2s |γs|2 + 2Ns

∣∣γ∗r γsvHr pr∣∣(35)

that is achieved when

ϕn = −∠(γsps,nvs,n

)−∠(γ∗r v

Hr pr

), n = 1, . . . , Ns. (36)

To get some insights into the achievable performance, wenow evaluate the optimal SNR under a LOS radar configura-tion (i.e., γrγr 6= 0); from (7), (13), (33), and (35), we obtain

SNR =σ2αNrNr|γrγr|2

σ2w︸ ︷︷ ︸

SNRr

×

(1 + N2

s

|γs|2

|γr|2+ 2Ns

|γs||γr|

∣∣vHr pr∣∣Nr

)︸ ︷︷ ︸

Γs

×

(1 + N2

s

|γs|2

|γr|2+ 2Ns

|γs||γr|

∣∣vHr pr∣∣Nr

)︸ ︷︷ ︸

Γs

(37)

where SNRr is the SNR when the radar operates alone, whileΓs > 1 and Γs > 1 are the gain granted by the RIS-aidedillumination and observation, respectively. Under the modelin (8), we have

Γs ≈ 1 + IsG(θs)LrG(θt)Ls

N2s ζ(ωr, ωt)

4πδ2

+ 2

(IsG(θs)LrG(θt)Ls

N2s ζ(ωr, ωt)

4πδ2

)1/2 |vHr pr|Nr

(38a)

Γs ≈ 1 + IsG(θs)Lr

G(θt)Ls

N2s ζ(ωt, ωr)

4πδ2

+ 2

(IsG(θs)Lr

G(θt)Ls

N2s ζ(ωt, ωr)

4πδ2

)1/2|vHr pr|Nr

(38b)

where the above approximations follow from the fact thatρ ≈ d and ρ ≈ d, respectively. Observe now that, to ensurefar-field operation, we must necessarily have δλ > 2D2

s andδλ > 2D2

s [33]; consequently, assuming that square RIS areemployed, so that NsAs = D2

s and NsAs = D2s , under the

model in (9) we have

N2s ζ(ωr, ωt)

4πδ2≤(NsAsδλ

)2

≤ 1

4(39a)

N2s ζ(ωt, ωr)

4πδ2≤

(NsAs

δλ

)2

≤ 1

4. (39b)

From (38) and (39) we conclude that, when γrγr 6= 0,the additional paths enabled by a far-field RIS deploymentonly provides a marginal SNR gain: the reason is that thereflecting area offered by the RIS cannot be sufficiently largeto compensate for the signal attenuation experienced in theradar-RIS hop. As also confirmed by the analysis in Sec. IV,the forward (backward) RIS should rather be placed in thenear-field of the radar transmit (receive) array, as close aspossible, to significantly improve the system performance.

As a final remark, we point out that a far-field RIS deploy-ment remains a competitive option whenever the indirect linksallow to reach a spot that would otherwise remain blind [29].

C. Mono-static radar with a bi-directional RIS

We study here in more detail a mono-static radar whereinthe same RIS helps both the transmitter and the receiver: inthis case, we have Ns = Ns, (d, ωt) = (d, ωt), ζ(·, ·) =ζ(·, ·), vs = vs, and γsγs 6= 0. The disjoint design discussedin Section III-A can be directly employed if the phase shiftintroduced by each reflecting element can be reprogrammed inbetween the arrival of the forward (from the radar transmitter)and the backward (from the target) incident wave. To simplifythe hardware, we also discuss next the relevant case wherethe additional constraint ϕ = ϕ = ϕ is enforced to (17);accordingly, the phase design is recast as

maxϕ‖e(ϕ,ϕ)‖2. (40)

Notice that forcing ϕ = ϕ is optimal if e(·) = e(·): this occurswhen the RIS is reciprocal and the radar employs the samereciprocal array for both transmission and reception; instead,it is inevitably sub-optimal if the system present an asymmetrybetween the transmit and the receive side. In this latter case,we propose to compute a sub-optimal solution to (40) byiteratively optimizing one phase shift at a time. To proceed,the objective function is expanded as

‖e(ϕ,ϕ)‖2 =∥∥tn + qne

iϕn∥∥2 ∥∥tn + qne

iϕn∥∥2

10

Algorithm 2 Alternate maximization for Problem (40)

1: Choose ε > 0, Kmax > 0, and {ϕn}Nsn=1

2: k = 0 and f0 = 03: repeat4: k = k + 15: for n = 1, . . . , Ns do6: ϕn = arg max

ϕ<{Ane

iϕ +Bnei2ϕ}

7: end for8: fk = ‖e(ϕ,ϕ)‖29: until (fk − fk−1)/fk < ε or k = Kmax

=(‖tn‖2 + ‖qn‖2

)(‖tn‖2 + ‖qn‖2

)+ 2(‖tn‖2 + ‖qn‖2

)<{tHnqne

iϕn}

+ 2(‖tn‖2 + ‖qn‖2

)<{tHnqne

iϕn}

+ 4<{tHnqnt

Hnqne

i2ϕn}

(41)

where qn is the n-th column of the matrix γsGTdiag (vs),

and tn = γrvr +∑Ns

j=1, j 6=n qjeiϕj at the transmit side, and

qn is the n-th column of the matrix γsGdiag (vs) and tn =γrvr +

∑Ns

j=1, j 6=n qjeiϕj at the receive side. Neglecting the

irrelevant terms in (41), the problem to be solved becomes

maxϕn

<{Ane

iϕn +Bnei2ϕn

}(42)

where An = 2(‖tn‖2 +‖qn‖2

)tHnqn+2

(‖tn‖2 +‖qn‖2

)tHnqn

and Bn = 4tHnqntHnqn. The overall procedure is summarized

in Algorithm 2: since the value of the objective function isbounded above and monotonically increased at each iteration,convergence is ensured. Finally, notice that the solution to (42)can be found among the zeros of the derivative of the objectivefunction, i.e., |An| sin (ϕn + ∠An)+2|Bn| sin (2ϕn + ∠Bn),which can be obtained numerically.

D. More insights on the radar and RIS interplay

The radar can be focused on a different resolution cellin the illuminated region by only relying on receive signalprocessing; to be more specific, the range gating selectsa region (namely, a circular or elliptical arc for a mono-and bi-static configuration, respectively) corresponding to agiven propagation delay, while the subsequent spatial filter fpoints towards a specific direction. When required, multipleresolutions cells can be simultaneously inspected by adjustingthe gating time and the spatial filter. The achievable SNR ineach resolution cell is tied to the set of emitted waveforms, theradiation pattern of the transmit/receive elements, and the arraygeometry. An RIS-aided radar system can now benefit fromadditional degrees of freedom, namely, the choice of the phaseshifts of the reflecting elements. Since their response must beprogrammed before the transmission of the probing signal, thesystem engineer must decide in advance how the RISs shouldoperate. Among the possible options, the design proposed in(17) aims at choosing the phase shifts so as to maximize theSNR at one specific resolution cell.8 The cell of interest can

8More generally, the forward and backward RISs could be pointed towardsa different pair of resolution cells; also, they could be defocused to pointtowards multiple (widely-spaced or adjacent) resolution cells.

15 km

10 km 2 km 8 km

5 km 5 km

45◦ 45◦

δ δ

P3P2P1

Radar TX Radar RX

forwardRIS

backwardRIS

Figure 2. Considered system geometry.

be for example the one where an alert was previously found(if the radar is operating in an alert/confirm mode [48]) or theone where a previously-detected target is expected to be found(if the radar is operating in a tracking mode) or a particularspot that the system inspects on demand to prevent specifictreats. Even if the RISs are optimized to look at a specificlocation, the radar can still continue to simultaneously inspectother resolution cells; we anticipate here that an SNR gain canas a by-product be obtained also in a large area close to thecell towards which the RISs have nominally been focused, asindicated by our examples in Sec. IV.

IV. NUMERICAL ANALYSIS

We present here some examples to demonstrate the per-formance achievable by an RIS-aided radar. The analysis iscarried out under the model in Sec. II-D, with Lr = Ls =Lr = Ls in (8). Also, Algorithms 1 and 2 are implementedwith Kmax = 200, ε = 10−5, and a random initialization.

A. System geometry

We consider the geometry in Fig. 2, which features a bi-static radar operating at 5 GHz, aided by a forward and/orbackward RIS. The local x and y axes of the transmit,receive, and reflecting arrays lays on the same plane, whichalso contains the prospective target. The transmitter and thereceiver employ a linear array with a λ/2 element spacingalong the local y-axis, Nr = 3, and (unless otherwise stated)Nr = 8; each array element has a rectangular shape of sizeλ/2 and λ along the local y and z axes, respectively, anda power pattern with a 3-dB beamwidth of 120◦ in azimuthand of 60◦ in elevation. The RISs (if present) have a squareshape and are composed of Ns = Ns = Ns adjacent squareelements with area As = As = λ2/4; also the center of thesesurfaces are located at the same distance δ = δ = δ from thearray center of the transmitter and the receiver, respectively.According to (3), the minimum required separation betweeneach transmit (receive) element and each forward (backward)reflecting element is 0.5 m. Finally, the target has an effectivesize of Dt = Dt = 10λ. Unless otherwise stated, we reportnext the performance when the target is at P2 and the RISsare designed to maximize the SNR at this same location.

11

2 4 6 8 10 12 140

2

4

6

8

10

12

Figure 3. SNR gain (as compared to case where the radar operates alone)obtained by using a forward and/or a backward RIS versus δ, whenNs = 225.The system geometry in Fig. 2 is considered.

2 4 6 8 10 12 14-15

-10

-5

0

5

10

15

Figure 4. SNR gain (as compared to case where the radar operates alone) ofthe echoes err , ers, esr , and ess and of their superposition versus δ, whenboth RIS are employed (“Radar&RIS - Radar&RIS” case) and Ns = 225.The system geometry in Fig. 2 is considered.

B. Impact of the radar-RIS distance

We first study the system behavior when δ is varied andNs = 225. Fig. 3 reports the SNR gain obtained by using a for-ward and a backward RIS, a forward RIS only, and a backwardRIS only, as compared to the case where the radar operatesalone (i.e., only the direct echo err exists): the configurationsconsidered here correspond to the “Radar&RIS - Radar&RIS”,“Radar&RIS - Radar", and “Radar - Radar&RIS" cases re-ported in Table I. As a benchmark, we include the performanceobtained with the far-field design in (34) and (36). It is seenthat a larger SNR gain is obtained when the radar and RISget closer, and, in this regime, the far-field design turns outto be detrimental. The SNR gain becomes instead negligiblewhen the radar and RIS arrays are located in each other’s far-field at both the transmit and the receive side, which in thisexample occurs when δ ≥ 11.5 m (as indicated by the verticaldotted line). Simultaneously using a forward and a backward

2 4 6 8 10 12 140

5

10

15

20

25

30

Figure 5. Maximum bandwidth ensuring that the delays of all target echoesreaching the receiver are not resolvable versus δ. The system geometry inFig. 2 is considered.

RIS almost doubles the gain as compared to using a singlereflecting array. If only a single RIS has to be employed (e.g.,due to cost constraints), then, for the considered scenario, itshould be that at the transmit side: this is a consequence ofthe fact that the reflecting elements of the forward RIS offera larger bi-static RCS towards P2 than those of the forwardcounterpart. To get further insights, in Fig. 4 we considerthe case where both RISs are simultaneously employed andreport the SNR gain of the echoes err, ers, esr, and ess andof their superposition. The echo ers is stronger than esr, inkeeping with the fact that the elements of the forward RISoffer a larger bi-static RCS towards P2; also, the echo ess isdominant when the distance between the radar and RIS arraysis very small, but it more rapidly fades out when δ is increaseddue to the more severe path-loss. Finally, Fig. 5 reports themaximum system bandwidth compatible with the requirednarrow-band assumption; this value have been computed as(τmax − τmin)/10, where τmax and τmin are the maximumand the minimum propagation delays from the transmit to thereceive elements. Not surprisingly, the maximum bandwidthdecreases as δ and/or the number of hops are increased.

C. Impact of the RIS size

Next, we study the system behavior when the number ofreflecting elements along each dimension, i.e.,

√Ns, is varied.

Fig. 6 reports the SNR gain granted by the use of a forwardand/or backward RIS, as compared to the case where theradar operates alone; not surprisingly, the gain increases as thereflecting surfaces get larger; however, it is important to noticethat, as long as

√Ns < 8, the radar and RIS arrays remain

in each other’s far-field (as indicated by the vertical dottedline) and only marginal gains are obtained; when instead

√Ns

gets larger, the advantage of using one or two RISs becomesevident. To farther investigates this phenomenon, in Fig. 7we consider the case where both RISs are simultaneouslyemployed and report the SNR gain of the echoes err, ers,esr, and ess and of their superposition. It is seen that the

12

5 10 15 20 25 300

5

10

15

Figure 6. SNR gain (as compared to case where the radar operates alone)obtained by using a forward and/or a backward RIS versus

√Ns, when δ =

4 m. The system geometry in Fig. 2 is considered.

5 10 15 20 25 30-15

-10

-5

0

5

10

15

20

Figure 7. SNR gain (as compared to case where the radar operates alone) ofeach echoes err , ers, esr , and ess and of their superposition versus

√Ns,

when δ = 4 m, both RIS are employed (“Radar&RIS - Radar&RIS” case),and Ns = 225. The system geometry in Fig. 2 is considered.

indirect echoes become progressively stronger as the RIS sizeis increased and, eventually, dominate over the direct one;hence, upon fixing the distance from the radar, exploitinga reflecting surface becomes competitive only if its size issufficiently large so as to positively balance the more severeattenuation over this path. In practice, the RIS size is limitedby cost and installation constraints; also, it remains understoodthat complementing an existing radar with a one or more RISscan be attractive only if the cost is inferior to that of upgradingthe radar itself with a better performing transceiver.

D. Impact of the target position

We now investigate the system performance as a functionof the target position, which is moved along the line segmentconnecting P1 to P3 (from left to right), which has a lengthof 20 Km, when Ns = 225 and δ = 1 m. At first, weassume that the RISs are designed to maximize the SNR at the

0 5 10 15 20-2

0

2

4

6

8

10

12

14

16

18

Figure 8. SNR gain (as compared to case where the radar operates alone)obtained by using a forward and/or a backward RIS versus the target positionalong the line segment going from P1 (= 0 Km) to P3 (= 20 Km), whenNs = 225 and δ = 1 m. The system geometry in Fig. 2 is considered.

actual target location and, in Fig. 8, report the correspondingSNR gain. Notice that the forward RIS cannot illuminate thetarget in the interval 0÷5 Km; accordingly, the “Radar&RIS–Radar” configuration does not provide any gain here, whilethe “Radar&RIS–Radar&RIS” and “Radar-Radar&RIS” con-figurations present the same gain. Similarly, the backwardRIS cannot observe the target in the interval 15 ÷ 20 Km;accordingly, the “Radar–Radar&RIS” is not effective here,while the “Radar&RIS–Radar&RIS” and “Radar&RIS–Radar”are equivalent. In between, the system can greatly benefitfrom the simultaneous use of two RISs, in keeping withthe results shown in the previous examples: indeed, the RISsize is here sufficiently large with respect to the radar-RISdistance that the strength of the echo ess is significant. Theobserved asymmetry in the reported gains, with respect tothe middle point of the line segment, is due to the fact thatNr 6= Nr. Finally, Fig. 9 reports the performance when theRISs are designed to maximize the SNR at P2 (marked herewith a dotted vertical line); for comparison, it is also shownthe achievable performance when the RISs are designed tomaximize the SNR at the actual target location. Even undera mismatched design, a noticeable gain is observed acrossa large region around P2, which progressively reduces bymoving away; the observed ripple is due to the fact that thedirect and indirect echoes may not always add constructively atlocations other than P2 and, indeed, a small negative gain mayeven occur occasionally. Overall, the results in Fig. 9 reinforcethe intuition that the system engineer can opportunisticallyactivate the available reflecting surfaces to redirect part of theradiated/received power towards/from specific spots to locallyboost the detection performance, while the radar transceivercan still keep looking at other regions with acceptable loss,which appears to be a nice feature.

E. Mono-static radar configurationWe assume here that the receive array is co-located with

the transmit array and the same RIS is employed for both

13

0 5 10 15 20 0 5 10 15 20-2

0

2

4

6

8

10

12

14

0 5 10 15 20

Figure 9. SNR gain (as compared to case where the radar operates alone) when using a forward RIS (left) or a backward RISs (center) or both RIS (right)versus the target position along the line segment going from P1 (= 0 Km) to P3 (= 20 Km), when Ns = 225 and δ = 1 m. The RISs are designed tomaximize the SNR at P2 (= 8 Km) or at the actual target position. The system geometry in Fig. 2 is considered.

5 10 15 20 250

2

4

6

8

10

12

14

16

18

20

Figure 10. SNR gain (as compared to case where the radar operates alone)versus Nr , when the forward and backward phase shifts are independentlyoptimized or are forced to be equal,

√Ns = 15, 20, 25, 30 and δ = 4 m.

The system geometry in Fig. 2 is modified by moving the receiver to the samelocation of the transmitter and by using a single bi-directional RIS.

forward and backward reflection; all other parameters remaindefined as in Sec. IV-A. Fig. 10 reports the SNR gain obtainedwhen the forward and backward phases of each reflectingelement are independently optimized or forced to be equal, asa function of the number of receive elements and for differentRIS sizes, when d = 4 m. Remarkably, enforcing equalphase values only incurs marginal losses, thus simplifying therequired hardware. Also, the obtained SNR gain reduces asNr is increased; indeed, if the radar is equipped with a betterreceiver, the reward for using a nearby RIS will be less.

V. CONCLUSIONS

In this work we have started unveiling the benefits ofcomplementing a MIMO radar with RISs placed close to thetransmitter/receiver. We have derived a general signal model,which includes mono-static, bi-static, LOS, and NLOS radarconfigurations and accounts for the presence of up to four

paths from the transmitter to the prospective target to thereceiver, and have recognized that the reflecting surfaces canbe employed to enhance the achievable detection performanceby redirecting/concentrating the radiated power towards/fromspecific spots. Our analysis have confirmed a simple intuition:an RIS should be better placed as close as possible to the radartransmitter/receiver in order to limit the additional path loss ex-perienced by an indirect link, with a far-field deployment onlyproviding a marginal advantage when the radar already has adirect view of the target. Also, in a mono-static configuration,the same surface can be used for both forward and backwardreflection by simply maintaining the same phase shift, whichgreatly simplifies its construction and control.

Future works should study the effect of the clutter on thesystem design and might consider how to point the RISstowards multiple distinct spots (e.g, this might be usefulfor multi-target confirmation or tracking) or towards a wideregion (e.g., this might useful for target search). Also, theemitted waverforms and the RISs phase shifts could be jointlydesigned, so as to conveniently focus both the radar transceiverand the reflecting surfaces towards specific (not necessarilythe same) regions. Future developments might also inves-tigate applications involving a high-resolution radar wherethe direct and indirect echoes have resolvable delays or apassive/opportunistic radar exploiting an RIS to redirect thesignal emitted by another (possibly non directional) source.A farther research line is studying if and how the estimationof the target parameters and the resolution of closely-spacedobjects can improve in the presence of nearby RISs. Finally,the use of active RISs have recently been proposed for future6G wireless communications [49], and their potential in thecontext of radar systems remains to be investigated.

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