Precoding for Multi-Cell ISAC: From Coordinated Beamforming to Coordinated Multipoint and Bi-Static Sensing

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 23, NO. 10, OCTOBER 2024 14637 Precoding for Multi-Cell ISAC: From Coordinated Beamforming to Coordinated Multipoint and Bi-Static Sensing Nithin Babu , Member, IEEE, Christos Masouros , Fellow, IEEE, Constantinos B. Papadias , Fellow, IEEE, and Yonina C. Eldar , Fellow, IEEE Abstract— This paper proposes a framework for designing robust precoders for a multi-input single-output (MISO) system that performs integrated sensing and communication (ISAC) across multiple cells and users. We use Cramer-Rao-Bound (CRB) to measure the sensing performance and derive its expressions for two multi-cell scenarios, namely coordinated beamforming (CBF) and coordinated multi-point (CoMP). In the CBF scheme, a BS shares channel state information (CSI) and estimates target parameters using monostatic sensing. In contrast, a BS in the CoMP scheme shares the CSI and data, allowing bistatic sensing through inter-cell reflection. We consider both block-level (BL) and symbol-level (SL) precoding schemes for both the multi-cell scenarios that are robust to channel state estimation errors. The formulated optimization problems to minimize the CRB in estimating the parameters of a target and maximize the minimum communication signal-to-interferenceplus-noise-ratio (SINR) while satisfying a given total transmit power budget are non-convex. We tackle the non-convexity using a combination of semidefinite relaxation (SDR) and alternating optimization (AO) techniques. Simulations suggest that neglecting the inter-cell reflection and communication links degrades the performance of an ISAC system. The CoMP scenario employing SL precoding performs the best, whereas the BL precoding applied in the CBF scenario produces relatively high estimation error for a given minimum SINR value. Index Terms— ISAC, precoder, Cramer-Rao bound, CoMP, CBF. I. INTRODUCTION INTEGRATED sensing and communication (ISAC) has been identified as an enabler for next-generation wireless Manuscript received 19 December 2023; revised 16 April 2024; accepted 10 June 2024. Date of publication 28 June 2024; date of current version 11 October 2024. This work was supported by Project 6GMUSICAL part of the Smart Networks and Services Joint Undertaking (SNS JU) under the European Union’s Horizon Europe Research and Innovation Programme under Grant 101139176. The associate editor coordinating the review of this article and approving it for publication was L. Liu. (Corresponding author: Nithin Babu.) Nithin Babu and Christos Masouros are with the Department of EE, University College London (UCL), WC1E 7JE London, U.K. (e-mail: [email protected]; [email protected]). Constantinos B. Papadias is with the Research, Technology and Innovation Network (RTIN), Alba, The American College of Greece, 115 28 Athens, Greece (e-mail: [email protected]). Yonina C. Eldar is with the Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot 7632706, Israel (e-mail: [email protected]). Color versions of one or more figures in this article are available at https://doi.org/10.1109/TWC.2024.3417713. Digital Object Identifier 10.1109/TWC.2024.3417713 networks to augment communication services with sensing for emerging applications spanning connected vehicles, remote healthcare, smart homes, and more [1],[2]. The availability of large bandwidth, multiple antennas, and dense deployment of 5G-Advanced and 6G networks enable high-resolution radio sensing capability [3]. ISAC has been recognized as one of the key enablers for 6G by the standardization bodies such as the International Telecommunication Union (ITU) and Third Generation Partnership Project (3GPP) [4],[5]. Having sensing and communication capabilities mutually benefits both systems: sensing can be used by the communication system to understand the environment better and enhance, for instance, interference management and beamforming, whereas the connected network infrastructure enables coordinated sensing at an unprecedented scale. Numerous works have considered designing optimal transmit waveforms/precoders to improve a single-cell ISAC system’s sensing and communication performance [6],[7], [8],[9],[10],[11],[12],[13],[14],[15],[16],[17],[18], [19],[20]. However, the dense deployment of next-generation small-cell BSs causes the signals transmitted from a BS to its users to affect the sensing and communication performance of neighbouring BSs. When the BSs use the same time and frequency resources to serve users, a user experiences intra-cell interference by the signals intended for other users in the same cell and inter-cell interference (ICI) due to co-channel signal leakage from the neighbouring BSs, reducing the received SINR. The work in [21] discusses various resource allocation schemes and precoder designs to improve the received SINR for a communication-only multi-cell system by suppressing interference. In multi-cell ISAC, a notable difference lies in the fact that while the communication aspect is often limited by interference, multi-static sensing is not necessarily interference-limited. For ISAC, in addition to this, inter-cell reflections (ICR) will be received by a BS from its target due to the signal transmitted from the neighbouring BSs. The received power through ICR can degrade target parameter estimation if the BS is unaware of the data transmitted from the neighbouring BS. This emphasizes the need for coordination among BSs to effectively manage inter-cell communication and sensing links. The ISAC BSs can coordinate at different levels through coordinated beamforming (CBF) and coordinated multipoint (CoMP) schemes, similar to the communication-only 1536-1276 © 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: University College London. Downloaded on January 16,2025 at 10:30:53 UTC from IEEE Xplore. Restrictions apply. 14638 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 23, NO. 10, OCTOBER 2024 multi-cell system. In an ISAC CBF scheme, a single BS serves a disjoint set of users and estimates its target’s parameters through mono-static sensing. Here, the signal power received through the inter-cell links negatively affects the sensing and communication performances. Hence, each BS selects a transmit strategy jointly with other BSs to minimize the ICI and ICR using the globally shared CSI obtained from the users via feedback channels. Conversely, a user is served by all the BSs in an ISAC CoMP scenario, thereby improving the received communication SINR at the user. Since the CoMP scheme shares the user data to be transmitted and the CSI globally among the BSs through a high bandwidth backhaul network, the additional power received through ICR and ICI links enhances estimation accuracy and communication SINR at the expense of increased coordination overhead. In particular, ICR enables bistatic sensing. Each of these multi-cell scenarios can employ either of two existing precoding strategies that differ in how they deal with the co-channel interference experienced by a user: a) blocklevel precoding (BLP) [21] and b) symbol-level precoding (SLP) [22]. Conventional precoding, also called BLP, designs precoders for a set of users to transmit a given block of symbols. It treats the interference experienced by a user as a harmful element, whereas constructive interference (CI)-based precoding, also called SLP, uses the signal power received through the instantaneous interference to aid the received communication SINR. The SLP technique exploits knowledge of both CSI and downlink data to be transmitted at the BS to ensure the received symbol at a user falls within the constructive region of the signal constellation [23]. Considering the SLP in the CoMP scheme, each BS can utilize both the intra-cell and inter-cell interference to enhance the received signal power. Conversely, the SLP in the CBF scheme can only utilize the intra-cell interference due to the non-availability of the user data from the neighbouring BSs. The design of the optimal precoders for both coordination schemes depends on the available CSI at the BSs and the users. In practice, CSI is prone to errors necessitating that the designed precoders exhibit robustness to potential CSI uncertainties. Multi-cell ISAC setups with various combinations of CBF and CoMP schemes have been considered in [24],[25], [26],[27],[28],[29],[30],[31],[32], and [33]. The work in [24] and [25] consider orthogonal transmission among BSs, whereas [26] uses an additional BS as the receiver to enable bi-static sensing and hence does not consider the inter-cell interference links. Reference [27] considers a multi-static scenario where one BS transmits signals to one vehicle, and the echoes are captured by several other BSs for sensing purposes. The paper [28] presents a resource allocation strategy for ISAC in distributed antenna networks, demonstrating significant energy savings and operational efficiency through a novel alternating optimization algorithm. The authors of [29] propose a multistatic architecture in coordinated multi-cell networks for ISAC, featuring an alternating optimization algorithm to optimize transmit and receive beamforming, thereby reducing self-interference and enhancing quality of service. The study in [30] considers CoMP mode to serve a set of users while detecting a target. Here, the transmitter employs distinct sensing and communication signals, allowing for modeling the interference between them. The authors of [31] model multi-cell ISAC interference considering 3 BSs operating in CBF mode where the users are considered as targets. Reference [32] models the inter-user and inter-subsystem interference. In [33], the BSs serve the users in the CBF mode and neglect the inter-cell reflection links. The work in [26], [28],[29], and [30] consider BLP design to achieve various objectives spanning minimizing energy consumption [28], beampattern mismatch error [29] or total transmit power [30] to maximizing the minimum communication SINR [26] subject to radar and or communication constraints. Reference [31] employs a collaborative SLP to mitigate mutual interference in a 3-cell ISAC system and minimize the total transmit power. A. Main Contributions and Paper Organization The existing multi-cell ISAC works consider no uncertainty in CSI. Moreover, the work in [24],[25],[26],[27],[28],[29], [30],[31],[32], and [33] employ either orthogonal schemes or CoMP for serving the users, and the sensing model follows the properties of CBF in which the reflections from other targets are considered interference. The proposed sensing models do not capture the property that the reflections from other targets can aid or degrade the sensing depending on the coordination level among the BSs. Furthermore, the sensing performance is quantified using radar SINR value. Explicit optimization of estimation performance metrics has not been considered in [24],[25],[26],[27],[28],[29],[30],[31],[32], and [33]. We considered this aspect in our previous work [34] while deriving the CRB expression for estimating the azimuth angle of a target in a multi-cell ISAC setup. Subsequently, this expression is utilized to design block-level precoders that optimize a weighted combination of sensing and communication metrics without considering channel state uncertainty. It is important to note that the derived CRB expression becomes invalid when dealing with scenarios involving more than one unknown parameter per target or multiple targets per cell. Recall that the precoding design should be robust to CSI uncertainties. The work in [35] designed robust symbol-level precoders for a multi-cell communication system that minimizes the total transmission power without considering the sensing capability. Even though the problem formulations in the aforementioned works consider multi-cell ISAC scenarios, a relative system performance analysis under different levels of coordination amongst BSs spanning from CBF to CoMP in the presence of channel state estimation error has not been investigated. To the best of our knowledge, explicit optimization of a weighted combination of the estimation error and communication performance metrics under CSI uncertainty has not been considered in the context of a multi-cell ISAC network. Our main contributions are: •We propose a robust precoder design framework to minimize the target parameter estimation error variance and maximize the minimum communication SINR experienced by a user in a multi-cell multi-user MIMO ISAC system under CSI uncertainty. •We consider two multi-cell scenarios, CBF and CoMP, and extend the corresponding CRB expressions in [34] to Authorized licensed use limited to: University College London. Downloaded on January 16,2025 at 10:30:53 UTC from IEEE Xplore. Restrictions apply. BABU et al.: PRECODING FOR MULTI-CELL ISAC: FROM CBF TO CoMP AND Bi-STATIC SENSING 14639 Fig. 1. Diagram of a multi-cell ISAC system, illustrating intra-cell and inter-cell communication links and reflections between BSs, communication users, and targets. multiple parameters (location and complex amplitudes) per target. The important distinction between the two cooperation modes in terms of sensing is that in the CRB, the adjacent BSs’ signals act as interference to the monostatic sensing of the serving BS, while in CoMP, the same signals can be used as an additional source of bi-static sensing. •The derived CRB expressions are then utilized to formulate optimization problems that jointly minimize the CRB value and maximize the minimum communication SINR value subject to a total transmit power budget. The formulations consider block-level and symbol-level precoding techniques for the CBF and CoMP scenarios. •We solve all the non-convex optimization problems using a combination of the semidefinite relaxation (SDR) and alternating optimization (AO) methods. •Finally, we compare the sensing and communication performances of all the considered precoder design schemes through simulations. The remaining content of the paper is organized as follows: in Section II, we explain the system model and derive the CRB expressions for the CBF and CoMP multi-cell scenarios. Section III and Section IV discuss the block-level and symbollevel robust precoding schemes for the considered multi-cell scenarios. Finally, in Section VI, we present our main findings through numerical evaluation, and the paper is concluded in Section VII. Notations: Matrices, vectors, and scalars are denoted by bold uppercase, lowercase, and normal font letters, respectively. We use tr(),()T,()H, and ()∗to represent trace operation, transpose, Hermitian transpose, and the complex conjugate of the matrices or vectors. The real and imaginary parts of xare represented as xRand xI, respectively, and ∥∥ denotes the l2norm. We represent an N×Nnull matrix and an N×Nidentity matrix as 0Nand IN, respectively. Additionally,  0Ndenotes an N×1null vector. II. SYSTEM MODEL We consider a multi-cell multi-input single-output (MISO) ISAC system with Jcells and Ksingle-antenna users per cell. As shown in Fig. 1, in addition to the users, each cell has a target and a BS with a uniform linear array (ULA) of Ntx transmit antennas spaced at λ/2distance, where λ is the wavelength. The BS is also equipped with a λ/2-spaced receive ULA of Nrx elements for sensing and isolated from the transmit antenna elements. Moreover, the BSs are interconnected through optical cables to synchronize and exchange information. A. Communication Model and Performance Metric For the communication service, we aim to maximize the minimum SINR value experienced by a user in a cell. Recall that the corresponding SINR expressions vary depending on the BSs’ coordination level (CBF or CoMP) and precoding mode, BLP or SLP, and are detailed in the following. The proposed algorithm requires the channel state information (CSI) to be available at the BS and the users. We consider the system operating in time division duplexing (TDD) mode so that the downlink CSI can be derived from the uplink channel observations based on the received uplink sounding reference signal (SRS) transmitted by the users. Additionally, each user employs a simple equalizer for the composite channel hTw, where his the communication channel from the serving BS and wis the precoding vector. 1) Coordinated Beamforming: Let Umk represent the kth user of the mth cell. In practice, the CSI is imperfectly known at the BS. Let the CSI uncertainty be limited within the spherical set Emk ={emk :∥emk∥2≤δ2}[22]. Then, the actual CSI, ˜ hn,mk, from the nth BS to Umk is the sum of the observed CSI, denoted as {hn,mk}, and emk: ˜ hn,mk =hn,mk +emk,∀Umk.(1) Let the mth BS transmits the symbol matrix Xm= [xm1,xm2,...,xmL] = WmSm∈ CNtx×L, where Wm= [wm1,wm2,...,wmK ]∀m∈ {1,2, .., J}are the dual-functional beamforming matrices to be designed, with L > Ntx being the length of the radar pulse/ communication frame. Here, Sm∈ CK×Lis the orthogonal data stream transmitted to Kusers of the mth BS: (1/L)SmSH m=IK. Then the received signal at Umk is expressed as yC mk =˜ hT m,mkXm+ J X n=m ˜ hT n,mkXn+zC mk,(2) where zC mk ∈ C1×Lis an AWGN noise vector with variance of each entry being σ2 C. Since no data is shared among the BSs in CBF mode, the first term in the RHS of (2) contains the useful signal and intra-cell interference from the users of the same cell, whereas the second term represents the inter-cell interference from the neighbouring cells. In a multi-carrier system, for example, based on orthogonal frequency-division multiplexing (OFDM), the input–output model (2) could describe one of the subcarriers. 2) Coordinated Multipoint: Since the BSs share the user data and the CSI in the CoMP mode, the J·Kusers in the multi-cell ISAC system can be considered to be served by a virtual single-cell BS with N=J·Ntx antennas. In this scenario, the precoder design occurs at a designated base station serving as the central node. The resulting precoder matrices are then shared with the respective base stations Authorized licensed use limited to: University College London. Downloaded on January 16,2025 at 10:30:53 UTC from IEEE Xplore. Restrictions apply. 14640 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 23, NO. 10, OCTOBER 2024 through optical cables. Let Ukfor k∈ {1,2, . . . , J ·K} represent the kth user of the virtual single-cell system. Note that Ukfor k∈ {K·(m−1) + 1, . . . , K ·m}represents K users of the mth BS. Let ˜ hk∈ CN×1be the actual channel vector from all the BSs to Ukrepresented as ˜ hk=hk+ek,(3) where ek∈ Ek={ek:∥ek∥2≤Jδ2}. The received signal at Ukis expressed as yC k=˜ hT kX+zC k,(4) where X= [X1;X2, . . . ;XJ]∈ CN×Lis the concatenated symbol matrix available at each BS and zC k∈ C1×Lis an AWGN noise vector with variance of each entry being σ2 C. Since intra-cell and inter-cell interference are treated differently by BLP and SLP, we will present the corresponding SINR expressions in the respective sections. B. Sensing Model and Performance Metric As shown in Fig. 1, the BSs transmit simultaneously, and each BS receives its echo signal and multiple echo signals from the neighbouring BSs due to ICR; we consider the dominant path among the ICR paths and the signal power received through the weaker paths is included in the noise term. Hence, the resulting echo signal received by the mth BS from its target is given as YI m=GmXm+ J X n=m GnXn | {z } ICR +ZR m,(5) where Gn=αnamvT n∀m, n ={1,2, .., J}≡J, is the target response matrix at the mth BS due to the transmission from the nth BS in which amand vnare the array response vectors in the directions at the angle-of-arrival θmand the angle of departure θn, respectively. Here, αnrepresents the complex amplitude of the received signal due to pathloss and the radar cross section of the target and ZR m∈CNrx×Lis an additive white Gaussian noise (AWGN) vector with the variance of each entry being σ2 R. Equation (5) assumes that all the neighbouring BSs have a line-of-sight (LoS) link to the mth BS’s target. The first term on the right-hand side (RHS) of (5) is the mono-static intra-cell reflection due to the signal vector from the same BS, whereas the second term represents ICR due to the signals from the remaining BSs. Note that when J= 1,(5) reduces to the sensing model proposed in [11]. Furthermore, the transmit antennas at the BS are conventionally down-tilted, resulting in negligible BS-BS interference received through the side lobes of the transmit beampattern [36], and hence in (5), we neglect the direct interference link between the BSs. Note that (5) applies to scenarios where users necessitate simultaneous sensing and communication services. In such instances, {Gn}include LoS channels to the users. 1) Sensing Performance Metric: We assume the target parameter estimation happens locally in the CBF and CoMP modes. The sensing process aims to estimate the target parameters using the received echo signal samples YI m. We aim to minimize the variance of the error in the parameter estimation. For an unbiased estimator, the error variance is lower bounded by the CRB given by the inverse of the Fisher information matrix (FIM). In general, not all the unknowns need to be estimated. Let θm,∗be the set of parameters of interest, and ηm,∗represents the nuisance parameters which are of no interest but present in the received echo signal. Hence, the FIM can be represented as Fm,∗="Fθm,∗θm,∗Fθm,∗ηm,∗ FT θm,∗ηm,∗Fηm,∗ηm,∗#,(6) where Fθm,∗θm,∗,Fθm,∗ηm,∗and Fηm,∗ηm,∗are the block matrices. The element at the lth row and the pth column of Fm,∗can be calculated by [37], Flp = tr dµH m,∗ dζml C−1 m,∗ dµm,∗ dζmp !,(7) where ζml, ζmp ∈ {θm,∗,ηm,∗}. Equation (7) is derived from the observation that the received echo signal at the mth BS is a multi-variate Gaussian random variable with mean µm,∗and covariance matrix Cm,∗. The entries of θm,∗,ηm,∗µm,∗and Cm,∗depend on whether the BSs are operating in the CBF or the CoMP mode, whose corresponding expressions are derived in the following propositions by extending the CRB derivation in [37] to a multi-cell ISAC scenario. 2) Coordinated Beamforming for Mono-Static Sensing: In the case of CBF, ICR acts as an interference term to the serving BS’s mono-static sensing since {Xn}is unknown to BS m. Therefore, from Eq. (5), µm,cbf =GmXm,(8) Cm,cbf =L J X n=m GnWnWH nGH n+σ2 RINrx .(9) Here, the set of unknown parameters is {θm, αR m, αI m}. In this work, we consider each BS needs to estimate its’ target’s azimuth angle. Hence θm,cbf =θmand ηm,cbf ={αR m, αI m}. Consequently, the FIM will be a 3×3matrix given by 1 Fm,cbf = 2 "FR θmθmfθmηm,cbf fT θmηm,cbf Fηm,cbf ηm,cbf #,(10) whose elements can be calculated using Proposition 1. Additionally, when ζml ={θm},(7) represents the Fisher information value derived in [34]. Proposition 1: The entries of Fm,cbf are computed as fθmηm,cbf = [FR θmαm−FI θmαm],(11) Fηm,cbf ηm,cbf =FR αmαm−FI αmαm −FI αmαmFR αmαm,(12) 1The extension to multiple targets and multiple parameters can be done using (10) of [37]. Authorized licensed use limited to: University College London. Downloaded on January 16,2025 at 10:30:53 UTC from IEEE Xplore. Restrictions apply. BABU et al.: PRECODING FOR MULTI-CELL ISAC: FROM CBF TO CoMP AND Bi-STATIC SENSING 14641 with Fθmθmand the elements of fθmηm,cbf and Fηm,cbf ηm,cbf determined using the following equations: 1 Lα2 m Fθmθm=˙ aH mC−1 m,cbf ˙ am·vH mR∗ Xmvm +˙ aH mC−1 m,cbf am·vH mR∗ Xm˙ vm +aH mC−1 m,cbf ˙ am·˙ vH mR∗ Xmvm +aH mC−1 m,cbf am·˙ vH mR∗ Xm˙ vm,(13) 1 Lα∗ m Fθmαm=˙ aH mC−1 m,cbf am·vH mR∗ Xmvm +aH mC−1 m,cbf am·˙ vH mR∗ Xmvm,(14) 1 LFαmαm=aH mC−1 m,cbf am·vH mR∗ Xmvm,(15) where, RXm=1 LXmXH m=WmWH m=PK k=1 wmkwH mk = PK k=1 Wmk. Here, the derivatives of amand vmwith respect to θmare denoted by ˙ amand ˙ vm, respectively. Proof: In CBF, as no data is shared between the BSs, the mth BS knows only the data symbol matrix Xm. Using the definitions of Gm, we have dµm,cbf dθm =αm˙ amvT m+αmam˙ vT mXm,(16) dµm,cbf dαR m =amvT mXm.(17) We get (13)-(15) using (16) and (17) and the cyclic property of the trace operation: tr(ABC) = tr(BCA)in (7).□ 3) Coordinated Multipoint Enabling Bi-Static Sensing: In the case of CoMP, ICR becomes useful as an additional source of bi-static sensing, in addition to the serving BS’s mono-static sensing. However, the ICR-related parameters {{θn},{αn}} △ ={θn,αn}are unknown to the mth Bs. The set of unknown parameters associated with BS min the CoMP mode is represented as ηm,comp ={θn, αR m, αI m,αR n,αI n}. The FIM for the CoMP case Fm,comp is similar to (10) with the entries computed using Proposition 2. Proposition 2: For the mth BS configured in CoMP mode, the FIM is expressed as Fm,comp = 2 "FR θmθmfθmηm,comp fT θmηm,comp Fηm,compηm,comp #.(18) The entries of the FIM are computed using using (19)- (26), shown at the bottom of the next page, where Dm= diag(0Ntx , .., INtx , .., 0Ntx )∈ CN×N; v′ m={ 0Ntx , .., vm, .., 0Ntx } ∈ CN×1, and ˆ RXop = DoRXDH p∀o, p ∈ {m, n}. Proof: Since each BS knows the transmitted symbol matrix of the other BSs, the ICR of (5) aids the target’s angle estimation in a bi-static manner. Consequently, the received echo at the mth BS is a multi-variate Gaussian random variable with mean µm,cmp =G′ mDmX+PJ n=mG′ nDnX where G′ n=αnamv′T nand the covariance matrix Cm,cmp = σ2 RINrx . Hence, dµm,cmp dθm =αmh˙ amv ′T m+am˙ v ′T mi | {z } ˙ Gm,θm DmX + J X n=m αn˙ amv ′T n |{z} ˙ Gn,θm DnX,(27) dµm,cmp dθn =αnam˙ v ′T n |{z} ˙ Gn,θn DnX,(28) dµm,cmp dαo =hamv ′T oi | {z } ˙ Go,αo DoX,∀o∈ {m, n}.(29) Equations (19),(23)-(26) are obtained using (27) -(29) in (7). □ The corresponding Fisher Information matrices, Fm,cbf and Fm,comp, are obtained using (13)-(15) and (19)-(21) in (10). The following problem formulations optimise the precoders towards a specific direction of interest [37]. III. ROBUST BLOCK LEVEL PRECODING The BLP constrains the co-channel interference experienced by a user so that the received symbol is within a certain distance from the nominal constellation symbol. In this section, we explain the BLP design framework for CBF and CoMP BS coordination schemes. A. BLP: Coordinated Beamforming We take a worst-case approach for the transmit precoding design to guarantee the resulting solution is robust to all possible channel uncertainties within Emk. Hence, in the CBF mode, for the mth BS, we aim to solve the following optimization problem: (P1) : maximize {wmk,tm,cbf ,γ} ρ·tm,cbf NFblp R,cbf +(1 −ρ)·γ NFblp C,cbf , s.t."FR θmθm−tm,cbf fθmηm,cbf fT θmηm,cbf Fηm,cbf ηm,cbf #⪰0, (30a) min emk (γmk)≥γ∀Umk,(30b) K X k=1 tr wmkwH mk≤Pt,(30c) where γmk =|˜ hT m,mkwmk|2 K X l=k|˜ hT m,mkwml|2 | {z } InCImk +PJ n=m K X l=1 |˜ hT n,mkwnl|2 | {z } ICIn,mk +σ2 C , (31) is the received SINR at Umk obtained using (2). The objective function of (P1) is the weighted sum of two components: minimizing the CRB in estimating the target’s angle and maximizing the minimum SINR value among the users of the mth cell. The weighting factor, ρ∈[0,1], determines the balance between the communication and sensing performance matrices. The normalization factors NFblp R,cbf and NFblp C,cbf are Authorized licensed use limited to: University College London. Downloaded on January 16,2025 at 10:30:53 UTC from IEEE Xplore. Restrictions apply. 14642 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 23, NO. 10, OCTOBER 2024 determined by solving (P1) for ρ= 1 and ρ= 0, respectively. Using (30a), we limit Fisher information value to be less than or equal to tm,cbf using the Schur complement: FR θmθm− fθmηm,cbf Fηm,cbf ηm,cbf fT θmηm,cbf >=tm.cbf . Moreover, (30b) is the SINR constraint derived from (2), whereas (30c) is the total power constraint with Ptbeing the total available power at the BS. Problem (P1) is difficult to solve because of (a) the infinite possibilities of emk in (30b), (b) the non-convex multiplication between γand the interference terms in (30b) and (c) the non-convex form of Cm,cbf in (30a). We tackle the infinite possibilities of emk by representing it as the ratio of the minimum of the numerator to the maximum of the denominator: min emk (|˜ hT m,mkwmk|2) max emk (InCImk +PJ n=mICIn,mk +σ2 C)≥γ. (32) The minimum value of the numerator and the maximum value of the denominator of (32) can be determined using Proposition 3. Proposition 3: For a given CSI uncertainty set Emk = {emk :∥emk∥2≤δ2}, the following equalities hold for any Umk. min emk |˜ hT m,mkwmk|2=tr (Qm,mkWmk),(33) max emk |˜ hT m,mkwmk|2=tr (Qm,mkWmk)+δ2tr (Wmk) +δ∥hT m,mkWmk∥+δ∥Wmkh∗ m,mk∥, (34) where Qm,mk =h∗ m,mkhT m,mk and Wmk =wmkwH mk. Proof: Let, |˜ hT m,mkwmk|2 = tr ˜ Qm,mkWmk(35) = tr ˜ h∗ m,mk ˜ hT m,mkWmk(36) = tr ˜ hT m,mkWmk ˜ h∗ m,mk(37) = tr hT m,mk +eT mkWmk h∗ m,mk +e∗ mk (38) = tr hT m,mkWmkh∗ m,mk+ tr hT m,mkWmke∗ mk + tr eT mkWmkh∗ m,mk+ tr eT mkWmke∗ mk.(39) Now, max ∥emk∥2≤δ2tr ˜ Qm,mkWmk = tr (Qm,mkWmk) +∥hT m,mkWmke∗ mk∥+∥eT mkWmkh∗ m,mk∥ + tr e∗ mkeT mkWmk(40) Fθmθm= tr  |αm|2˙ Gm,θmˆ RXmm ˙ GH m,θm+ J X n=m α∗ nαm˙ Gm,θmˆ RXmn ˙ GH n,θm+α∗ mαn˙ Gn,θmˆ RXnm ˙ GH m,θm + J X n=m J X n=m|αn|2˙ Gn,θmˆ RXnn ˙ GH n,θm C1×1(19) fθmηm,comp =L σ2 R [fR θmθn, FR θmαm,−FI θmαm,fR θmαn,−fI θmαn]∈ C1×(3·J−1) (20) Fηm,compηm,comp =L σ2 R        FR θnθnfR θnαm−fI θnαmFR θnαn−FI θnαn fRT θnαmFR αmαm−FI αmαmfR αmαn−fI αmαn −fIT θnαm−FI αmαmFR αmαm−fI αmαnfR αmαn FRT θnαnfRT αmαn−fIT αmαnFR αnαn−FI αnαn −FIT θnαn−fIT αmαnfRT αmαn−FIT αnαnFRT αnαn        ,∈ C(3·J−1)×(3·J−1) (21) fθmθn= [Fθmθn]∈ C1×(J−1);Fθmαm∈ C1×1;fθmαn= [Fθmαn]∈ C1×(J−1); Fθnθn= [Fθnθn]∈ C(J−1)×(J−1);Fαmαm∈ C1×1; fθnαm= [Fθnαm]∈ C(J−1)×1;Fθnαn= [Fθnαn]∈ C(J−1)×(J−1);Fαnαn= [Fαnαn]∈ C(J−1)×(J−1); fαmαn= [Fαmαn]∈ C1×(J−1);(22) Fθmθn= tr α∗ mαn˙ Gn,θnˆ RXnm ˙ GH m,θm+|αn|2˙ Gn,θnˆ RXnn ˙ GH n,θm;Fθnθn= tr |αn|2˙ Gn,θnˆ RXnn ˙ GH n,θn; (23) Fθmαm= tr α∗ m˙ Gm,αmˆ RXmm ˙ GH m,θm+α∗ n˙ Gm,αmˆ RXmn ˙ GH n,θm;Fθnαm= tr α∗ n˙ Gm,αmˆ RXmn ˙ GH n,θn; (24) Fθmαn= tr α∗ m˙ Gn,αnˆ RXnm ˙ GH m,θm+α∗ n˙ Gn,αnˆ RXnn ˙ GH n,θm;Fθnαn=tr α∗ n˙ Gn,αnˆ RXnn ˙ GH n,θn;(25) Fαmαm= tr ˙ Gm,αmˆ RXmm ˙ GH m,αm;Fαmαn= tr ˙ Gn,αnˆ RXnm ˙ GH m,αm;Fαnαn= tr ˙ Gn,αnˆ RXnn ˙ GH n,αn. (26) Authorized licensed use limited to: University College London. Downloaded on January 16,2025 at 10:30:53 UTC from IEEE Xplore. Restrictions apply. BABU et al.: PRECODING FOR MULTI-CELL ISAC: FROM CBF TO CoMP AND Bi-STATIC SENSING 14643 ≤tr (Qm,mkWmk) + δ∥hT m,mkWmk∥ +δ∥Wmkh∗ m,mk∥+ tr e∗ mkeT mktr (Wmk) ≤tr (Qm,mkWmk) + δ∥hT m,mkWmk∥ +δ∥Wmkh∗ m,mk∥+δ2tr (Wmk).(41) Here, (41) is derived from (40) using Cauchy-Schwarz inequality. The minimum value of the numerator is obtained by substituting δ= 0 in (41).□ Consequently, max emk (InCImk) = K X l=k tr (Qm,mkWml) + δ∥hT m,mkWml∥ +δ∥Wmlh∗ m,mk∥+δ2tr (Wml),(42) max emk (ICIn,mk) = K X l=1 tr (Qn,mkWnl) + δ∥hT n,mkWnl∥ +δ∥Wnlh∗ n,mk∥+δ2tr (Wnl).(43) The non-convex multiplication between γand the interference terms in (30b) is addressed by representing (30b) through a set of intra-cell and inter-cell leakage constraints (44)-(46): max emk (InCImk)≤Iintra m,cbf ,∀Umk,(44) max enk (ICIm,nk)≤Iinter cbf J−1∀Unk n=m, (45) tr (Qm,mkWmk)−γ(Iintra m,cbf +Iinter cbf )≥γσ2 C∀Umk.(46) Equation (44) limits the intracell interference to any user served by mth BS to be less than Iintra m,cbf . The left-hand-side (LHS) of (45) gives the maximum ICI from mth BS to Unk. The constraint (45) satisfied by all the BSs restricts the ICI to any user in the system to Iinter cbf . Note that for given Iintra m,cbf and Iinter cbf values, the constraints (44)-(46) become convex. Lastly, for a given {Wnk},Cm,cbf can be estimated using (9), which makes the entries of Fm,cbf of (30a), an affine function function of RXm=PK kWmk. Hence, we solve (P1) by solving the following two optimization problems alternatively: (P1.A) : maximize {Wmk,tm,cbf ,γ} ρ·tm,cbf NFblp R,cbf +(1 −ρ)·γ NFblp C,cbf , s.t. K X k=1 tr (Wmk)≤Pt,(47a) rank(Wmk) = 1; Wmk ⪰0,(47b) (30a),(44) −(46).(47c) (P1.B) : minimize {Wmk,Iintra m,cbf ,Iinter m,cbf } Iintra m,cbf Iintra max +Iinter m,cbf (J−1) ·Iinter max , s.t."FR θmθm−t∗ m,cbf fθmηm,cbf fT θmηm,cbf Fηm,cbf ηm,cbf #⪰0, (48a) tr (Qm,mkWmk)−γ∗Iintra m,cbf +Iinter cbf ≥γ∗σ2 R, (48b) max enk (ICIm,nk)≤Iinter m,cbf J−1;Iinter m,cbf ≤Iinter cbf ,(48c) (47a),(44),(47b).(48d) The first problem, (P1.A), minimizes the CRB and maximizes the minimum communication SINR for a given {Wnk} ∀n= m∈ J,Iintra m,cbf , and Iinter cbf values. Equation (47a) is the equivalent representation of (30c). Using the solution of (P1.A): γ∗, and t∗ m,cbf , (P1.B) minimizes the leakage power towards the neighbouring BSs’ users while guaranteeing given sensing and communication performance through (48a) and (48b). Here, Iintra max and Iintra max represent the maximum tolerable intra-cell and inter-cell interference values, respectively. Omitting the rank constraint, (P1.A) and (P1.B) are convex optimization problems solved using MATLAB’s CVX solver [38]. The overall procedure is given in Algorithm 1. In the CBF mode, initially, all BSs select beamforming vectors wmk randomly and estimate their covariance matrices Cm,cbf . Each BS then independently solves (P1.A) to optimize the weighted combination of sensing and communication performance in parallel. Subsequently, (P1.B) is solved to minimize interference to neighbouring BSs, using the solutions from (P1.A). A similar approach is used in the CoMP mode, as outlined in steps 7-9. We can obtain wmk from Wmk using Eigenvalue decomposition technique if the rank of Wmk >1[17]. B. BLP: Coordinated Multi-Point The corresponding optimization problem in the CoMP scenario can be formulated as, (P2) :maximize {Wkf,γ} f·ρ NFblp R,comp +(1 −ρ)·γ NFblp C,comp , s.t."FR θmθm−tm,comp fθmηm,comp fT θmηm,comp Fηm,compηm,comp #⪰0, (49a) tm,comp ≥f, ∀m, (49b) min ek |˜ hT kwk|2 PJK l=k|˜ hT kwl|2+σ2 C! | {z } γk ≥γ, ∀k, (49c) K X k=1 tr DmwkwH kDH m≤Pt,∀m. (49d) The objective function of (P2) is the weighted combination of minimizing the maximum CRB and maximizing the minimum SINR values. Using (49a), we limit the Fisher information value to be less than or equal to tm,comp using the Schur complement:FR θmθm− fT θmηm,comp Fηm,compηm,comp fθmηm,comp >=tm,comp.In(49b), fis defined as the minimum Fisher information value for estimating θm. Equation (49c) is the minimum SINR constraint written using (4) whereas (49d) is the per-BS power constraint. The constants NFblp R,comp and NFblp C,comp are the corresponding maximum values of f and γobtained by setting ρ= 1 and ρ= 0, respectively. Here, wk= [w1k;w2k;...;wJk ]CN×1represents the precoding Authorized licensed use limited to: University College London. Downloaded on January 16,2025 at 10:30:53 UTC from IEEE Xplore. Restrictions apply. 14644 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 23, NO. 10, OCTOBER 2024 vectors for user kfrom all the BSs stacked vertically. Problem (P2) is non-convex due to the infinite possibilities of ekand the non-convex SINR constraint (49c). Similar to the CBF case, we tackle the infinite possibilities of ekby adapting Proposition 3to the CoMP case. Using Proposition 3, we have min ek|˜ hT kwk|2= tr (QkWk),(50) max ek|˜ hT kwl|2= tr (QkWl) + Jδ2tr (Wl) +√Jδ∥hT kWl∥+√Jδ∥Wlh∗ k∥,(51) where Qk=hkhH k∈ CN×N,Wk=wkwH k∈ CN×N, and RX=P2K k=1 Wk. Then, (49c) is reformulated into the following pair of constraints. tr (QkWk)−γIcomp ≥γσ2 C,(52) JK X l=k max ek|˜ hT kwl|2≤Icomp.(53) Equation (53) limits the total interference experienced by any user to be less than Icomp. For a given Icomp value, (52) and (53) become convex constraints. Hence, we solve (P2) by solving the (P2.A) and (P2.B) alternately. (P2.A) : maximize {Wk,f,γ} f·ρ NFblp R,comp +(1 −ρ)·γ NFblp C,comp , s.t. K X k=1 tr DmWkDH m≤Pt,(54a) Wk⪰0∀k, rank(Wk)=1,(54b) (49a),(49b),(52),(53).(54c) (P2.B) : minimize {Wk,Icomp}Icomp, s.t."FR θmθm−t∗ m,comp fθmηm,comp fT θmηm,comp Fηm,compηm,comp #⪰0, (55a) tm,comp ≥f∗,∀m, (55b) tr (QkWk)−γ∗Icomp ≥γσ2 C∀k, (55c) (53),(54a),(54b).(55d) Problem (P2.A) maximizes the objective function of (P2) for a given maximum value of the co-channel interference Icomp. Equation (54a) is the equivalent representation of (49d). Problem (P2.B) minimizes the co-channel interference while guaranteeing a maximum CRB f∗and minimum SINR γ∗ through (55a)-(55c). Dropping the rank constraints, (P2.A) and (P2.B) are convex optimization problems. Algorithm 1 summarizes the robust block-level precoder design procedure. Note that, for a given γ,ρ= 1,δ= 0 J= 1 setting, (P1), and (P2) are equivalent to the BLP design formulation, Eq. (18),of[11]. We verify the equivalence of the corresponding CRB expressions in Fig. 3. Certainly, our investigation into the multi-cell version of the CRB minimization problem introduces substantial differences in modeling. These distinctions make it impractical to directly apply the solutions presented in [11]. Furthermore, when δ= 0, both (P1) and (P2) reduce to the respective CBF and CoMP BLP design problems considered in our previous work [34]. Algorithm 1 Robust Block Level Precoder Design 1Input:u,{hm,nk},{hk},Iintra m,cbf ,Iinter cbf ,{Wmk},Icomp; 2while no convergence do 3if BSs in CBF mode then 4Determine Cm,cbf using {Wmk}in (9); 5Solve (P1.A) for each BS to obtain γ∗and {t∗l m,cbf }; 6Solve (P1.B) to update Iintra m,cbf ,Iinter cbf ,{Wmk}: Iinter cbf = max {Iinter m,cbf } 7if BSs in CoMP mode then 8Solve (P2.A) to obtain γ∗and f∗; 9Solve (P2.B) to update Icomp: Icomp = max ({Im,comp}) 10 Output:{Wm,k},{Wk}. Fig. 2. SLP: the ⟨dmks rotated noiseless received signal ˆymks =˜ hT m,mksxmsdmks should fall in the CI region of the transmitted QPSK symbol dmks. IV. ROBUST SYMBOL LEVEL PRECODING In this section, we explain the SLP design framework for CBF and CoMP BS coordination schemes. A. Overview: SLP The main idea involves leveraging co-channel interference constructively to increase the received signal power. This is achieved by instantaneously aligning the interfering signals with the desired signal at each receive antenna, as shown in Fig. 2. For a single-cell system, if the sth transmitted symbol is M-PSK-modulated: dmks =dejϕmks , the received signal at Umk can be represented as yC mks =˜ hT m,mk K X l=1 wmldmls +zC mks, =˜ hT m,mk K X l=1 wmlej(ϕmls−ϕmks)dmks +zC mks, =˜ hT m,mksxmsdmks +zC mks,(56) where ˜ hT m,mks =˜ hT m,mkej(−ϕmks)and xms = PK l=1 wmlej(ϕmls)(for d= 1). Hence, from Fig. 2, the Authorized licensed use limited to: University College London. Downloaded on January 16,2025 at 10:30:53 UTC from IEEE Xplore. Restrictions apply. BABU et al.: PRECODING FOR MULTI-CELL ISAC: FROM CBF TO CoMP AND Bi-STATIC SENSING 14645 condition for yC mks lying in the CI region of dmks to achieve a SINR of γcan be expressed as, |Im ˆ hT m,mksxms| ≤ Re ˜ hT m,mksxms −σC√γtanψ, (57) where, ψ=π/Mpsk. For a detailed discussion about SLP, we refer the readers to [22] and [39]. B. SLP: Coordinated Beamforming In the CBF case, the sth received symbol at Umk is obtained by extending (56) to a multi-cell scenario as follows: yC mks =˜ hT m,mksxmsdmks + J X n=m ˜ hT n,mksxnsdnks +zC mks, (58) where ˜ hT n,mks =˜ hT n,mkej(−ϕnks)and xns = PK l=1 wnlej(ϕnls). Since the data symbols are not shared among the BSs in the CBF mode, the SLP design can only align the intra-cell interference to enhance the received signal power, whereas the inter-cell interference degrades the SINR value. Hence, the SINR constraint is equivalently written as a CI constraint given by (60a), shown at the bottom of the next page, where γ′=√γ. The corresponding problem formulation using SLP is given by (P3) : maximize {xms,RXms ,tm,cbf ,γ′} tm,cbf ·ρ NFslp R,cbf +(1 −ρ)·γ′ NFslp C,cbf , s.t."FR θmθm−tm,cbf fθmηm,cbf fT θmηm,cbf Fηm,cbf ηm,cbf #⪰0, (59a) 1 Ltr L X s=1 RXms !≤Pt,(59b) RXms xms xH ms 1⪰0;RXms ⪰0∀s, (59c) (60a).(59d) The objective function of (P3) is the weighted combination of minimizing the CRB and maximizing the minimum SINR values. Note that, unlike the BLP design problem (P1), the optimization variables in this context are the transmit symbol vector xms and its covariance matrix Rxms . Additionally, the entries of Fm,cbf value is a function Rxm= (1/L)PL s=1 xmsxH ms = (1/L)PL s=1 Rxms . Similar to (P1), NFslp R,cbf and NFslp C,cbf are determined by solving (P3) for ρ= 1 and ρ= 0, respectively. Equation (59a) limits the Fisher information value to be less than or equal to tm,cbf using the Schur complement: FR θmθm− fθmηm,cbf Fηm,cbf ηm,cbf fT θmηm,cbf >=tm.cbf . Equation (59b) is the average power constraint. Equation (59c) is the semi-definite relaxed (SDR) representation of the relation between Rxms and xms using the Schur complement. Solving (P3) poses challenges stemming from (a) the infinite possibilities of emks in (60a), (b) the non-convex multiplication between γ′and the interference terms in (60a) and (c) the non-convex form of Cm,cbf in (59a). To tackle (a) and (b), let ˜ hm,mks =˜ hR m,mks +j˜ hI m,mks = hR m,mks +jhI m,mks +eR mks +jeI mks, then Im n˜ hT m,mksxmso=ˆ fT m,mksˆxms +ˆeT mksˆxms,(61) Re n˜ hT m,mksxmso=ˆ fT m,mksˆx′ ms +ˆeT mksˆx′ ms,(62) where ˆ fm,mks =hhR m,mks;hI m,mksi,ˆemks =eR mks;eI mks, ˆxms =xI ms;xR ms, and ˆx′ ms =xR ms;−xI ms. Then (60a) can be equivalently expressed as (60b) and (60c), shown at the bottom of the next page. Taking the maximum value of the LHS, the constraints are equal to, ˆ fT m,mksˆxms −ˆ fT m,mksˆx′ mstanψ+γ′qσ2 C+I2inter sl,cbf tanψ +δ∥ˆxms −ˆx′ mstanψ∥ ≤ 0,(63) −ˆ fT m,mksˆxms −ˆ fT m,mksˆx′ mstanψ+γ′qσ2 C+I2inter sl,cbf tanψ +δ∥ˆxms +ˆx′ mstanψ∥ ≤ 0,(64) where I2inter sl,cbf is the maximum ICI that satisfies max ∥enks∥2≤δ2∥˜ hT m,nksxms∥2≤I2inter sl,cbf J−1.(65) Note that the squared representation of ICI: I2inter sl,cbf , is to ensure convexity for the constraints (63) and (64). Using CauchySchwarz inequality, we have ∥hT m,nks +eT nksxms∥2≤∥hT m,nksxms∥+∥eT nksxms∥2. (66) Hence, (65) is equivalently written as ∥hT m,nksxms∥+δ∥xms∥ ≤ Iinter sl,cbf √J−1.(67) Note that for a given Iinter sl,cbf , the constraints (63),(64), and (67) that combined represent the SINR constraint are convex. Finally, given {RXn}, the estimation of Cm,cbf can be achieved using (9). This makes the entries of Fm,cbf in (59a) an affine function of RXm. Hence, we solve (P3) by solving the following two optimization problems alternatively untill convergence: (P3.A) : maximize {xms,RXms ,tm,cbf ,γ′} tm,cbf ·ρ NFslp R,cbf +(1 −ρ)·γ′ NFslp C,cbf , s.t.(59a) −(59c),(63),(64),(67);(68a) (P3.B) : [4pt] minimize {xms,Rxms ,I2inter m,sl,cbf }I2inter m,sl,cbf , s.t."FR θmθm−t∗ m,cbf fθmηm,cbf fT θmηm,cbf Fηm,cbf ηm,cbf #⪰0,(69a) Authorized licensed use limited to: University College London. Downloaded on January 16,2025 at 10:30:53 UTC from IEEE Xplore. Restrictions apply.