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MU-MIMO Communications with MIMO Radar: From Co-existence to Joint Transmission
Fan Liu, Christos Masouros, Ang Li, Huafei Sun, Lajos Hanzo
TL;DR
The paper addresses transmit-beamforming design for a dual-functional MIMO RadCom system that must serve downlink users while probing radar targets. It compares separated and shared antenna deployments, then replaces constrained SINR formulations with penalty-based manifold optimizations. Numerically, shared deployment performs significantly better than separated deployment, while weighted optimizations achieve similar performance at much lower computational complexity.
Problem
A joint MIMO RadCom system needs beamforming that supports simultaneous radar probing and multi-user downlink communication within the same frequency band.
Method
The paper designs separated and shared beamformers, then incorporates SINR constraints as penalties and solves the weighted problems with manifold-based algorithms.
Results
The shared deployment has significantly better beampattern–downlink trade-offs than separated deployment, while weighted optimizations achieve similar performance to the original designs with much lower computational complexity.
Takeaways & Limitations
Sharing all antennas and solving the resulting weighted problems with manifold algorithms provides an effective lower-complexity approach to joint RadCom beamforming.
Abstract
from arXiv · showhide
Beamforming techniques are proposed for a joint multi-input-multi-output (MIMO) radar-communication (RadCom) system, where a single device acts both as a radar and a communication base station (BS) by simultaneously communicating with downlink users and detecting radar targets. Two operational options are considered, where we first split the antennas into two groups, one for radar and the other for communication. Under this deployment, the radar signal is designed to fall into the null-space of the downlink channel. The communication beamformer is optimized such that the beampattern obtained matches the radar's beampattern while satisfying the communication performance requirements. To reduce the optimizations' constraints, we consider a second operational option, where all the antennas transmit a joint waveform that is shared by both radar and communications. In this case, we formulate an appropriate probing beampattern, while guaranteeing the performance of the downlink communications. By incorporating the SINR constraints into objective functions as penalty terms, we further simplify the original beamforming designs to weighted optimizations, and solve them by efficient manifold algorithms. Numerical results show that the shared deployment outperforms the separated case significantly, and the proposed weighted optimizations achieve a similar performance to the original optimizations, despite their significantly lower computational complexity.
I. INTRODUCTION
The paper develops transmit-beamforming designs for a dual-functional MIMO RadCom system and compares separated and shared antenna deployments. It introduces weighted manifold optimizations to reduce constraints and computational cost while preserving beamforming performance.
- 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.
- I. INTRODUCTION: 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.
- I. INTRODUCTION: 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.
- I. INTRODUCTION: 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.
- I. INTRODUCTION: 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.
- I. INTRODUCTION: 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.
B. Shared Deployment
In shared deployment, all antennas serve radar detection and downlink communication through a common dual-functional signal. The communication signal itself becomes the radar probing waveform, while the design retains downlink SINR and power requirements.
- 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.
- B. Shared Deployment: 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.
- B. Shared Deployment: 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.
III. PROBLEM FORMULATION
The system requires a beamformer that matches a desired radar beampattern, meets downlink SINR requirements, and respects transmit-power limits. MIMO radar beampattern design is formulated through waveform covariance optimization, including sidelobe and main-beam-width objectives.
- III. PROBLEM FORMULATION: The beamformer must closely match the desired radar beampattern, guarantee downlink-user SINR, and satisfy the transmit-power budget.
- 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.
- A. MIMO Radar Beampattern Design: 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.
- A. MIMO Radar Beampattern Design: 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.
- A. MIMO Radar Beampattern Design: 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.
- A. MIMO Radar Beampattern Design: The radar beampattern optimization problems are convex and can therefore be solved efficiently with numerical tools.
B. Zero-forcing Beamforming for Separated Deployment
In the separated deployment, radar and communication antennas are partitioned, and radar interference is eliminated by placing radar signals in the users’ channel null-space. Communication covariance design then matches the radar beampattern under communication constraints.
- 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.
- B. Zero-forcing Beamforming for Separated Deployment: The separated design uses an NR × NR radar covariance matrix obtained after imposing the zero-forcing constraint.
- B. Zero-forcing Beamforming for Separated Deployment: The overall covariance matrix is block-structured because radar and communication signals are assumed statistically independent.
- B. Zero-forcing Beamforming for Separated Deployment: The communication beamformer is designed so the overall beampattern matches the radar-only beampattern, subject to a nonnegative scaling factor.
- B. Zero-forcing Beamforming for Separated Deployment: The zero-forcing optimization imposes user SINR thresholds and a downlink communication power budget.
- B. Zero-forcing Beamforming for Separated Deployment: 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.
- B. Zero-forcing Beamforming for Separated Deployment: Algorithm 1 obtains the radar covariance, solves the relaxed SDP, and then constructs an approximate solution.
C. Beamforming for Shared Deployment
The shared deployment uses all antennas for radar and communications, treating radar targets as virtual downlink users while retaining communication constraints. Weighted penalty formulations and manifold algorithms address feasibility and computational limitations of the original optimization.
- C. Beamforming for Shared Deployment: The separated deployment adds radar-interference-cancellation constraints that may cause poor radar-beampattern performance.
- C. Beamforming for Shared Deployment: In the shared deployment, radar targets are treated as virtual downlink users in LoS channels alongside real users in fading channels.
- C. Beamforming for Shared Deployment: The shared optimization formulates a radar beampattern and communication beamforming covariance while using all available transmit power.
- C. Beamforming for Shared Deployment: SDP relaxation of the shared problem may fail to produce solutions satisfying strict per-antenna power equalities, limiting use of the available power budget.
- C. Beamforming for Shared Deployment: For K ∈ [N −2, N], the shared problem becomes infeasible with high probability because available degrees of freedom may be insufficient.
- C. Beamforming for Shared Deployment: Weighted optimizations incorporate penalty terms and can be solved by manifold algorithms to avoid the drawbacks of the original formulation.
D. SINR Penalty Terms
The paper replaces SINR constraints with penalty terms in beamforming objectives, using sum-square and max penalties, then solves the resulting problems with manifold-based methods.
- 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.
- Sum-Square Penalty: The sum-square penalty minimizes the squared errors between actual SINR values and their thresholds.
- Max Penalty: The max penalty maximizes the minimum downlink-user SINR to promote fairness among users.
- Penalty formulation: Including both penalty terms in the objective makes the beamforming problem always feasible.
- 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.
B. Beamforming under Total Power Constraint
Under a total-power constraint, the weighted beamforming problems become unconstrained optimization over a complex hypersphere and are solved with Riemannian conjugate gradient methods.
- Problem Reformulation: The weighted sum-square formulation uses a radar covariance matrix, SINR penalty, and weights for radar and communication objectives.
- Problem Reformulation: Replacing the rank-1 constraint with a total-power constraint makes the feasible set a complex hypersphere of dimension NK − 1.
- Problem Reformulation: 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.
- Problem Reformulation: The max SINR penalty is smoothed with a log-sum-exp upper bound because the original max function lacks a gradient.
- Riemannian Conjugate Gradient Algorithm: The RCG method combines current and previous descent directions, transports them between tangent spaces, and updates points through retraction.
- Riemannian Conjugate Gradient Algorithm: Direct hypersphere search avoids SDR rank-1 approximation and guarantees satisfaction of the strict equality power constraint.
C. Beamforming under Per-Antenna Power Constraint
For per-antenna power constraints, the weighted problems are defined on a complex oblique manifold and solved with a corresponding RCG procedure.
- Problem formulation: The original radar covariance matrix is obtained under diagonal per-antenna power constraints before formulating the weighted SINR-penalty problem.
- Problem formulation: The feasible region under per-antenna constraints is modeled as a complex oblique manifold.
- Riemannian gradients: The weighted problems are treated as unconstrained optimization on a Riemannian manifold after projecting Euclidean gradients onto the tangent space.
- RCG algorithm: Algorithm 3 applies Armijo step sizes, retraction, vector transport, and descent-direction updates to solve the per-antenna problems.
V. COMPLEXITY ANALYSIS
The complexity analysis counts floating-point operations per iteration for the proposed RCG algorithms and compares their computational burden with SDR through numerical simulations.
- Complexity comparison: Because SDR lacks closed-form complexity expressions, the paper compares SDR and RCG complexity through simulations.
- Flop accounting: A flop is defined as one floating-point addition, subtraction, multiplication, or division.
- Complexity comparison: Table I lists the per-iteration complexity of Algorithms 2 and 3 by operation.
- Flop accounting: The analysis focuses on complexity per iteration because the total number of iterations is difficult to predict.
A. Complexity of Beamforming Problems under Total Power Constraint
The paper analyzes the computational cost of solving weighted beamforming problems under total power constraints, comparing gradient and manifold-optimization components.
- 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.
- Both total-power weighted formulations therefore have comparable asymptotic computational complexity.
- 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.
B. Complexity of Beamforming Problems under Per-Antenna Power Constraint
The per-antenna-constrained formulations retain the same dominant computational structure as the total-power problems, while simulations evaluate their feasibility and beamforming performance.
- 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.
- The computational complexities of both beamforming problems are O(n) according to the complexity discussion.
- 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.
- The separated and shared deployments are compared through multi-beam beampatterns, with shared deployment achieving higher peaks because it retains more degrees of freedom.
- 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.
C. Performance of Beamforming Methods for Shared Deployment
For shared deployment, the weighted optimizations closely reproduce constrained-design performance while substantially reducing computational cost, although their instantaneous SINR varies more around the mean.
- 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.
- Performance trade-offs: Weighted optimizations achieve nearly the same PSLR–SINR performance as constrained optimizations, with negligible performance loss.
- Performance trade-offs: The max SINR penalty with total power can outperform the corresponding constrained optimization in the MSE–SINR trade-off.
- SINR variability: Weighted optimizations have instantaneous SINR distributions generally farther from the mean than constrained optimizations, despite similar overall variation.
- 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.
- Conclusion: The shared deployment has better beampattern–SINR trade-offs than the separated deployment, and weighted designs offer a favorable performance–complexity trade-off.