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Cell-Free ISAC MIMO Systems: Joint Sensing and Communication Beamforming

Umut Demirhan, Ahmed Alkhateeb

arXiv:2301.11328v3cs.ITeess.SP

TL;DR

Cell-free ISAC MIMO systems require coordinated beamforming so distributed access points can jointly serve communication users and sense targets. This paper develops separate baselines and joint beamforming and power-allocation designs, finding that joint beamforming nearly matches the communication and sensing performance of the respective prioritized baselines while outperforming regularized zero-forcing power allocation in sensing.

  • Problem

    The paper addresses coordinated sensing and communication beamforming in cell-free ISAC MIMO systems, where distributed access points jointly serve users and sense targets.

  • Method

    The paper develops communication- and sensing-prioritized baselines, formulates joint sensing and communication beamforming under communication SINR constraints, and applies semidefinite relaxation to beamforming and fixed-beam power allocation problems.

  • Results

    The developed JSC beamforming nearly achieves the SINR of communication-prioritized sensing beamforming and almost the sensing SNR of sensing-prioritized communication beamforming, while outperforming regularized zero-forcing power allocation in sensing SNR at the same communication rates.

  • Takeaways & Limitations

    Joint beamforming provides a favorable sensing-communication performance balance through codesign of the two functions in cell-free ISAC MIMO systems.

Abstract

from arXiv · show

This paper considers a cell-free integrated sensing and communication (ISAC) MIMO system, where distributed MIMO access points (APs) jointly serve the communication users and sense the target. For this setup, we derive a sensing SNR for multi-static sensing where both joint communication and sensing signals transmitted by different APs are utilized. With this sensing objective, we develop two baseline approaches that separately design the sensing and communication beamforming vectors, namely communication-prioritized sensing beamforming and sensing-prioritized communication beamforming. Then, we consider the joint sensing and communication (JSC) beamforming design and derive the optimal structure of these beamforming vectors based on a max-min fairness formulation. In addition, considering any pre-determined JSC beam design, we devise a power allocation approach. The results show that the developed JSC beamforming is capable of achieving nearly the same communication signal-to-interference-plus-noise ratio (SINR) of the communication-prioritized sensing beamforming solution with almost the same sensing SNR of the sensing-prioritized communication beamforming approach. The proposed JSC beamforming optimization also provides a noticeable gain over the power allocation with regularized zero-forcing beamforming, yielding a promising strategy for cell-free ISAC MIMO systems.

I. INTRODUCTION

The paper addresses the limited integration of sensing and communication in distributed cell-free MIMO by studying coordinated beamforming for APs that jointly serve users and sense targets. It introduces baseline, joint, and power-allocation designs for this setting.

  • Cell-free ISAC MIMO coordinates distributed APs to jointly serve communication users and sense common targets.
  • Prior Work: Prior JSC research mainly studied single-node systems, while distributed-node studies often served each user through only one AP.
  • Contributions: The paper develops communication-prioritized sensing and sensing-prioritized communication beamforming as separate baseline strategies.
  • Contributions: It formulates joint beamforming to maximize sensing SNR under communication SINR constraints and applies semidefinite relaxation to obtain beamforming structures.
  • Contributions: For fixed beamforming vectors, it reformulates the design as an SDP-based power-allocation problem and reports gains from joint optimization over regularized zero-forcing power allocation.

B. Communication Model

The system models distributed APs jointly transmitting communication and sensing signals, with communication SINR and multi-static sensing SNR as the design objectives. The sensing objective aggregates contributions from communication and sensing streams across receiving APs.

  • Communication channels are stacked across APs to model the received signal at each communication user.
  • The received communication signal contains desired signal, multi-user interference, sensing interference, and receiver noise.
  • The communication SINR is expressed using individual beamforming variables and stacked user-channel vectors.
  • The sensing model uses a single-point reflector and a transmit-to-receive path between transmitting and receiving APs.
  • The sensing SNR uses joint processing across receiving APs and includes contributions from all communication and sensing streams.
  • Beamforming is designed assuming that communication channels and sensing target angles are known at the transmitting APs.

III. COMMUNICATION-PRIORITIZED SENSING BEAMFORMING DESIGN

The paper develops separate baseline designs that prioritize either communication or sensing, then optimizes the lower-priority beamforming subject to preserving the higher-priority function.

  • Communication-prioritized sensing: With communication prioritized, sensing beams are designed to maximize sensing performance without interfering with communication users.
  • Communication-prioritized sensing: When no communication users are present, conjugate sensing beamforming is optimal for the single-target model.
  • Communication-prioritized sensing: With communication users present, sensing beams are projected into the null space of the communication channels.
  • Sensing-prioritized communication: With sensing prioritized, communication beams are optimized while minimizing the impact of sensing interference.
  • Sensing-prioritized communication: Regularized zero-forcing balances multi-user interference and noise through a regularization parameter when the sensing target is absent.
  • Sensing-prioritized communication: For predetermined sensing beams, the communication design maximizes the minimum user rate through a quasiconvex optimization solved via feasibility and second-order-cone reformulation.

V. JOINT SENSING AND COMMUNICATION: BEAMFORMING OPTIMIZATION

The joint design maximizes sensing SNR while satisfying communication SINR constraints, using semidefinite relaxation to handle the non-convex beamforming problem. The relaxed solution can be mapped back to beamforming vectors, with optimality depending on the resulting matrix ranks.

  • The joint objective maximizes sensing SNR together with communication SINR performance.
  • The beamforming problem is a non-convex quadratically constrained quadratic program because the sensing SNR objective is non-convex.
  • Beamforming vectors are lifted into positive semidefinite matrices, introducing rank constraints that preserve their vector structure.
  • The lifted problem is relaxed by removing rank constraints and solved as a convex semidefinite program using convex optimization solvers.
  • When the relaxed user matrices are rank one and the sensing matrix has rank at most Q, eigenvectors recover optimal communication and sensing beamformers.
  • If the sensing matrix has rank greater than Q, the constructed solution is not optimal and becomes an approximation whose performance is examined separately.

A. How Many Sensing Streams Do We Need?

The paper characterizes the structure and maximum number of sensing streams under optimality conditions. In Rayleigh channels, the stream limit is governed by the numbers of transmitting APs and communication users.

  • The optimal sensing beamforming matrix lies in the nullspace of A − P ν⋆ and of Q_u for users whose SINR constraints are active.These nullspace conditions follow from the KKT complementary-slackness relations.
  • Sensing-beam existence is determined independently at each AP: a component has a nullspace only when ν⋆_m = ζ̄_m; otherwise, it is full rank.When ν⋆_m > 0, no sensing stream is present for that AP component.
  • The maximum number of sensing streams is bounded by the available nullspace dimensions, with a general upper bound of M_t dimensions.This follows from the rank structure of the sensing SNR matrix and the associated nullspace.
  • For random Rayleigh channels, the number of sensing streams is limited to M_t − U, the difference between transmitting APs and communication users.Each UE reduces the available sensing-stream dimensions with probability 1.
  • When transmitting APs outnumber users, cell-free massive MIMO may require almost one sensing stream per AP; when APs are fewer than users, no sensing streams are required.The latter case is stated as a consequence of the Rayleigh-channel stream limit.

VI. JOINT SENSING AND COMMUNICATION: POWER ALLOCATION WITH FIXED BEAMS

The paper formulates power allocation for predetermined communication and sensing beams. It converts the nonconvex problem into an SDR-based semidefinite formulation, while recognizing the relaxation and reconstruction limitations.

  • The proposed formulation allocates power over fixed beams while jointly accounting for communication and sensing objectives under per-AP power constraints.The fixed beams are represented by unit-power vectors and power coefficients.
  • The power-allocation problem is nonconvex because it contains square-root powers and coherent sums inside absolute-value terms.Stacked per-stream power vectors are introduced to rewrite these terms.
  • The reformulated problem is a non-convex QCQP because the sensing SNR is quadratic in the power variables.The formulation uses stacked power variables and AP selection matrices.
  • Semidefinite relaxation removes the rank-1 constraint and yields a convex SDP solvable by convex optimization methods.The lifted variables must satisfy positive semidefiniteness, while rank one remains nonconvex before relaxation.
  • The SDR solution may require rank-1 reconstruction; the paper uses the most significant eigenvector as a heuristic and retains the relaxed solution as an upper bound.The relaxation does not guarantee optimality for the original power-allocation problem.

VII. RESULTS

The evaluation compares separate sensing and communication designs with joint beam and power optimization. The compared methods differ in whether beams are fixed or jointly optimized and whether SDR upper bounds are used.

  • The compared methods are evaluated as distinct beamforming and power-allocation solutions for cell-free ISAC MIMO systems.The section defines the solution set before presenting the numerical results.
  • The evaluation compares nullspace-sensing designs paired with RZF or optimized communication beams, conjugate-beam sensing with optimized communication, and joint beam optimization.These baselines separate sensing and communication design, whereas JSC Beam Optimization jointly designs both functions.
  • JSC Power Optimization allocates powers for predetermined beams using the SDR formulation and a rank-1 eigenvector heuristic.Its predetermined beams are taken from the nullspace-sensing and RZF-communication approach.
  • JSC Beam SDR UB is an upper bound because its SDR solution has no rank constraint on the communication and sensing beams.JSC Power SDR UB similarly provides an upper bound because its power variables have no rank constraint.

A. LoS Channels

The LoS evaluation uses two distributed APs and randomly placed users and target in a one-dimensional geometry. It averages performance over 1000 realizations under fixed transmit and sensing-stream settings.

  • The setup uses M_t = M_r = 2 APs at (25, 0) and (75, 0), each with a 16-antenna ULA, and places one target and five users randomly along y = 50 m.The x-coordinates are drawn uniformly from [0, 100].
  • The AP transmit power is P_m = 0 dBW and the number of sensing streams is Q = 1.
  • The reported results are averaged over 1000 random realizations.

1) Providing NS Sensing - OPT Comm SINR for All UEs:

The evaluation compares beamforming and power-allocation solutions under communication-rate constraints, showing that joint beamforming preserves strong communication performance while substantially improving sensing.

  • The evaluation uses an equal communication SINR target for every UE, based on the minimum SINR obtained from solution (ii).
  • The proposed JSC optimization provides a significant sensing gain while satisfying the best communication SINR across power-allocation ratios.
  • The joint solution achieves communication performance matching the best separate solution and sensing performance close to MF sensing.
  • Across target distances, joint beamforming maintains almost constant sensing performance, approaches solution (i) at small separations, and exceeds the other solutions at larger separations.
  • With individual SINR targets taken from solution (i), beamforming optimization achieves the same SINR and a significant sensing-SNR gain over solutions without sensing beams.
  • In the realistic setup, Q = M − U sensing streams attain the beamforming upper bound while satisfying 10dB minimum communication SINR constraints.
  • The heuristic rank-1 optimization does not provide sufficient SNR, while power optimization can outperform a single sensing beam when U ≤ 2.
  • The paper concludes that JSC beamforming nearly matches communication-prioritized SINR and sensing-prioritized sensing SNR.

APPENDIX

The appendix derives the sensing-SNR expression and rewrites it into a semidefinite-program form using trace identities, expectation manipulations, and matrix definitions.

  • The sensing-SNR numerator is simplified by expanding the Frobenius norm, interchanging expectation and trace, and rearranging trace terms.
  • The denominator is derived separately before the resulting expressions are combined in the sensing-SNR formula.
  • The numerator is then expressed in SDP form using the definitions of a, D_mt, f_s, F_s, and A together with trace properties.

C. Proof of Proposition 1

The proof constructs rank-1 sensing solutions from an SDR solution while preserving the objective and satisfying the relevant constraints under a stream-rank condition.

  • The SDR is reformulated by eliminating the sensing variable F_Q and optimizing over user-stream variables and the aggregate matrix.
  • A candidate rank-1 solution is constructed from the SDR variables, and its optimality is checked through objective preservation and constraint satisfaction.
  • The construction preserves the objective because it depends on the unchanged summation variable, while the constraints follow from the matrix definitions and retained aggregate variable.
  • The sensing matrices are obtained from the largest Q eigenvectors of the SDR matrix decomposition.
  • The eigenvector construction is valid when rank(P_q) is no greater than the number of sensing streams Q.

D. Derivation of D2.1-SDR

The appendix derives the dual of the SDR by forming the Lagrangian and imposing matrix conditions that keep its supremum bounded.

  • The Lagrangian includes user and sensing matrices, semidefinite multipliers, SINR multipliers, and AP power-constraint multipliers.
  • Collecting terms yields compact trace expressions involving B_u + Z_u and B_Q + Z_Q.
  • The Lagrangian supremum is bounded only when the coefficient matrices satisfy B_u + Z_u = 0, leading to negative-semidefinite dual constraints.
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