MU-MIMO Communications with MIMO Radar: From Co-existence to Joint Transmission

I. INTRODUCTION

• The joint MIMO RadCom system simultaneously transmits radar probing signals and communication symbols to multiple downlink users.The design focuses on transmit beamforming rather than waveform redesign, preserving the original communication modulation scheme.Source: C. The Contribution of Our Work · p. 2, §II · p. 3
• The study identifies receiver techniques as outside its scope and leaves them for future work.The paper focuses on joint transmission of the RadCom system.Source: C. The Contribution of Our Work · p. 2
• Separated deployment partitions antennas between radar and communications, using zero-forcing to eliminate radar interference and SDR to solve the resulting non-convex design.The communication covariance matrix is shaped to match the radar beampattern while meeting communication requirements.Source: C. The Contribution of Our Work · p. 2, A. Separated Deployment · p. 3
• Shared deployment uses all antennas for both functions, treating communication signals as radar probing waveforms while imposing downlink power and SINR constraints.This formulation avoids radar interference to users in the shared signal model.Source: C. The Contribution of Our Work · p. 2
• Weighted shared-deployment optimizations incorporate SINR constraints as penalty terms and use manifold optimization under total-power or per-antenna constraints.The resulting feasible regions are Riemannian manifolds, enabling low-complexity manifold solvers.Source: C. The Contribution of Our Work · p. 2
• The paper derives computational complexity for the proposed weighted solvers and evaluates the beamforming approaches using numerical simulations under Rayleigh-fading channels.The simulations use a 20-antenna half-wavelength-spaced ULA and compare SDR-based constrained designs with RCG-based penalty designs.Source: C. The Contribution of Our Work · p. 2

B. Shared Deployment

• Shared deployment assigns all N antennas to both radar detection and downlink communications.The received-user model uses a beamforming vector over the full shared array.Source: B. Shared Deployment · p. 4
• The system employs the communication signal as the radar probing waveform, creating a dual-functional transmission.The channel is assumed to be flat Rayleigh fading and perfectly estimated.Source: B. Shared Deployment · p. 4
• The shared model's user SINR contains no radar interference term.The covariance matrix of the precoded symbols is used in the shared beamforming formulation.Source: B. Shared Deployment · p. 4

III. PROBLEM FORMULATION

• The beamformer must closely match the desired radar beampattern, guarantee downlink-user SINR, and satisfy the transmit-power budget.Source: §III · p. 4

A. MIMO Radar Beampattern Design

• MIMO radar beampattern design is equivalent to designing the covariance matrix of the probing waveforms.The cited passage states that convex optimization can be used for this covariance-matrix design.Source: A. MIMO Radar Beampattern Design · p. 4
• The steering vector is determined by antenna spacing, wavelength normalization, and the number of transmit antennas.The vector is defined over the transmit antenna array and depends on the normalized element spacing.Source: A. MIMO Radar Beampattern Design · p. 4
• The covariance matrix, power budget, scaling factor, and desired gain specify the beampattern optimization variables and target.Equal average power across antennas is enforced by an additional constraint.Source: A. MIMO Radar Beampattern Design · p. 4
• A second formulation targets a desired 3dB main-beam width while controlling the sidelobe region.The main-beam location is θ0, the width is determined by θ2 − θ1, and Ω denotes the sidelobe region.Source: A. MIMO Radar Beampattern Design · p. 4
• The radar beampattern optimization problems are convex and can therefore be solved efficiently with numerical tools.Source: A. MIMO Radar Beampattern Design · p. 4

B. Zero-forcing Beamforming for Separated Deployment

• Radar signals are forced into the null-space of the channel between radar antennas and downlink users to eliminate interference.Source: B. Zero-forcing Beamforming for… · p. 4
• The separated design uses an NR × NR radar covariance matrix obtained after imposing the zero-forcing constraint.Source: B. Zero-forcing Beamforming for… · p. 5
• The overall covariance matrix is block-structured because radar and communication signals are assumed statistically independent.Source: B. Zero-forcing Beamforming for… · p. 5
• The communication beamformer is designed so the overall beampattern matches the radar-only beampattern, subject to a nonnegative scaling factor.Source: B. Zero-forcing Beamforming for… · p. 5
• The zero-forcing optimization imposes user SINR thresholds and a downlink communication power budget.Source: B. Zero-forcing Beamforming for… · p. 5
• The non-convex problem is relaxed to an SDP by omitting rank-1 constraints, after which eigenvalue decomposition or Gaussian randomization approximates rank-1 solutions.Source: B. Zero-forcing Beamforming for… · p. 5
• Algorithm 1 obtains the radar covariance, solves the relaxed SDP, and then constructs an approximate solution.Source: B. Zero-forcing Beamforming for… · p. 5

C. Beamforming for Shared Deployment

• The separated deployment adds radar-interference-cancellation constraints that may cause poor radar-beampattern performance.Source: C. Beamforming for Shared Deployment · p. 5
• In the shared deployment, radar targets are treated as virtual downlink users in LoS channels alongside real users in fading channels.Source: C. Beamforming for Shared Deployment · p. 5
• The shared optimization formulates a radar beampattern and communication beamforming covariance while using all available transmit power.Source: C. Beamforming for Shared Deployment · pp. 5–6
• SDP relaxation of the shared problem may fail to produce solutions satisfying strict per-antenna power equalities, limiting use of the available power budget.Source: C. Beamforming for Shared Deployment · p. 6
• For K ∈ [N −2, N], the shared problem becomes infeasible with high probability because available degrees of freedom may be insufficient.Source: C. Beamforming for Shared Deployment · p. 6
• Weighted optimizations incorporate penalty terms and can be solved by manifold algorithms to avoid the drawbacks of the original formulation.Source: C. Beamforming for Shared Deployment · p. 6

Penalty formulation

• SINR constraints are incorporated into the objective as penalty terms to simplify beamforming optimization and improve feasibility probability.The same strategy can also be applied to the zero-forcing optimization.Source: D. SINR Penalty Terms · p. 6

Sum-Square Penalty

• The sum-square penalty minimizes the squared errors between actual SINR values and their thresholds.Source: D. SINR Penalty Terms · p. 6

Max Penalty

• The max penalty maximizes the minimum downlink-user SINR to promote fairness among users.Source: D. SINR Penalty Terms · p. 6

Penalty formulation

• Including both penalty terms in the objective makes the beamforming problem always feasible.Source: D. SINR Penalty Terms · p. 6

Manifold optimization

• Riemannian optimization searches directly on constrained manifolds, avoiding SDR rank-1 approximation and preserving equality constraints.The approach is described as more computationally efficient than SDR.Source: D. SINR Penalty Terms · p. 6, A. Preliminaries on Riemannian… · pp. 6–7

Problem Reformulation

• The weighted sum-square formulation uses a radar covariance matrix, SINR penalty, and weights for radar and communication objectives.Source: B. Beamforming under Total Power… · p. 7
• Replacing the rank-1 constraint with a total-power constraint makes the feasible set a complex hypersphere of dimension NK − 1.Source: B. Beamforming under Total Power… · p. 7
• Both weighted problems are solved with a Riemannian conjugate gradient algorithm instead of a conventional SDR solver.The method seeks near-global local minima of the non-convex objectives.Source: B. Beamforming under Total Power… · p. 7
• The max SINR penalty is smoothed with a log-sum-exp upper bound because the original max function lacks a gradient.Source: B. Beamforming under Total Power… · p. 8

Riemannian Conjugate Gradient Algorithm

• The RCG method combines current and previous descent directions, transports them between tangent spaces, and updates points through retraction.Source: B. Beamforming under Total Power… · p. 8
• Direct hypersphere search avoids SDR rank-1 approximation and guarantees satisfaction of the strict equality power constraint.Source: B. Beamforming under Total Power… · p. 8

Problem formulation

• The original radar covariance matrix is obtained under diagonal per-antenna power constraints before formulating the weighted SINR-penalty problem.Source: C. Beamforming under Per-Antenna Power… · p. 9
• The feasible region under per-antenna constraints is modeled as a complex oblique manifold.Source: C. Beamforming under Per-Antenna Power… · p. 9

Riemannian gradients

• The weighted problems are treated as unconstrained optimization on a Riemannian manifold after projecting Euclidean gradients onto the tangent space.Source: C. Beamforming under Per-Antenna Power… · p. 9

RCG algorithm

• Algorithm 3 applies Armijo step sizes, retraction, vector transport, and descent-direction updates to solve the per-antenna problems.Source: C. Beamforming under Per-Antenna Power… · p. 9

Complexity comparison

• Because SDR lacks closed-form complexity expressions, the paper compares SDR and RCG complexity through simulations.Source: §V · p. 9

Flop accounting

• A flop is defined as one floating-point addition, subtraction, multiplication, or division.Source: §V · p. 9

Complexity comparison

• Table I lists the per-iteration complexity of Algorithms 2 and 3 by operation.Source: Table I · p. 10, §V · p. 10

Flop accounting

• The analysis focuses on complexity per iteration because the total number of iterations is difficult to predict.Source: §V · p. 10

• For problem (32), retraction requires 14NK flops, while gradient-related computations dominate with 23N^2K + 12NK^2 + 42NK flops.The total cost also includes vector transport, inner products, and subsequent optimization steps.Source: A. Complexity of Beamforming Problems… · p. 10
• Both total-power weighted formulations therefore have comparable asymptotic computational complexity.Source: A. Complexity of Beamforming Problems… · p. 10
• For problem (41), the log-sum-exp approximation preserves the same gradient-computation complexity order as problem (32).The exponential operation is treated as having constant cost, so lower-order terms are omitted.Source: A. Complexity of Beamforming Problems… · p. 10

• For problems (56) and (57), retraction costs 14NK flops and gradient, vector-transport, and inner-product costs have the same order as the total-power formulations.Source: B. Complexity of Beamforming Problems… · p. 10
• The computational complexities of both beamforming problems are O(n) according to the complexity discussion.Source: B. Complexity of Beamforming Problems… · p. 10
• Simulations use SDR with CVX for constrained optimizations and RCG for penalty problems under a 20-antenna half-wavelength ULA.The channels are modeled with i.i.d. standard complex Gaussian entries.Source: §VI · p. 10
• The separated and shared deployments are compared through multi-beam beampatterns, with shared deployment achieving higher peaks because it retains more degrees of freedom.Source: A. Comparison between the Separated… · p. 10, Figure 3 · p. 11
• At Γ = 10dB, feasibility decreases as users increase; when users equal BS antennas, problem (20) has feasibility below 5%, whereas weighted optimizations remain feasible.Separated zero-forcing becomes generally infeasible for large user counts.Source: B. Feasibility Comparison between… · p. 11

Performance trade-offs

• Under shared deployment, total-power methods provide a 1.7dB SINR gain over per-antenna-constrained methods in the PSLR–SINR trade-off.Source: C. Performance of Beamforming Methods… · p. 12
• Weighted optimizations achieve nearly the same PSLR–SINR performance as constrained optimizations, with negligible performance loss.Source: C. Performance of Beamforming Methods… · p. 12
• The max SINR penalty with total power can outperform the corresponding constrained optimization in the MSE–SINR trade-off.Source: C. Performance of Beamforming Methods… · p. 12

SINR variability

• Weighted optimizations have instantaneous SINR distributions generally farther from the mean than constrained optimizations, despite similar overall variation.Source: C. Performance of Beamforming Methods… · p. 13

Computational complexity

• Weighted optimizations reduce execution complexity to below 50% of constrained methods, while other comparisons report up to an order-of-magnitude reduction.Max-penalty methods converge fastest among the evaluated weighted formulations.Source: D. Computational Complexity · pp. 13–14

Conclusion

• The shared deployment has better beampattern–SINR trade-offs than the separated deployment, and weighted designs offer a favorable performance–complexity trade-off.Source: §VII · p. 14