Transmit-Receive Beamforming Techniques

• Transmit-receive beamforming jointly optimizes transmitter and receiver strategies to enhance spectral efficiency and reduce interference in wireless networks.
• Modern methods employ alternating optimization, convex relaxation, closed-form solutions, and greedy algorithms to solve nonconvex Tx/Rx design problems.
• These techniques drive significant performance gains in MU-MIMO, ISAC, federated learning, and sparse array systems while balancing complexity and adaptability.

Transmit-receive beamforming techniques jointly optimize the spatial processing strategies at both the transmitter and receiver sides of a wireless link, with the goal of maximizing spectral efficiency, mitigating interference, enhancing robustness, embedding sensing functionalities, or achieving other system objectives under practical constraints. Modern research covers these methods for a range of system models, including multi-user MIMO, cognitive and underlay networks, federated learning, integrated sensing and communications (ISAC), and distributed or sparse array architectures. Key developments address performance limits, complexity-reduction, robustness to channel modeling, adaptability to instantaneous network circumstances, and integration with machine learning-driven hardware or channel reconfigurability.

1. System Models and Problem Formulations

Transmit-receive beamforming arises in an array of architectures:

• Multi-User MIMO: Considered in coordinated downlink transmission, where a multi-antenna base station (BS) serves multiple users, each possibly with multiple antennas, subject to channel state information (CSI) at the BS (An et al., 2010).
• Cognitive/Underlay Spectrum Sharing: Secondary users (SUs) opportunistically access bands occupied by primary users (PUs), with protection constraints on PU interference (Denkovski et al., 2015).
• Full-Duplex ISAC: A unified BS simultaneously serves users and senses targets, requiring joint optimization to balance communication rates, radar beampatterns, and suppression of residual self-interference (SI) (Liu et al., 2022).
• Sparse/Massive MIMO & Sensor Arrays: Selection or reconfiguration of active transmit/receive elements to balance array cost, detection performance, and interference rejection (Hamza et al., 2021).
• Over-the-Air Federated Learning and Computation: Joint design of device transmit scalars and server receive beamformers to enable analog aggregation of gradients or statistics under power/convergence constraints (Kalarde et al., 29 Jan 2025, Fang et al., 2021).
• Near-Field/Non-RF Applications: Joint current-weighting at tri-directional coils and combining weights for magnetic induction communications (Li et al., 30 May 2025).

The canonical signal model is

$$\mathbf{y} = \mathbf{H}\mathbf{W}\mathbf{d} + \mathbf{n}$$

where $\mathbf{H}$ is the MIMO channel, $\mathbf{W}$ the transmit precoding matrix, $\mathbf{d}$ the data vector, and $\mathbf{n}$ the noise. The receiver applies a combining matrix $\mathbf{U}$, yielding processed estimates $\hat{\mathbf{d}} = \mathbf{U}^\dagger\mathbf{y}$.

Typical objectives include weighted sum-rate maximization, SINR fairness, aggregate MSE minimization, radar beampattern shaping, pathloss minimization, and strict power or interference constraints.

2. Joint Optimization Algorithms and Frameworks

Due to the nonconvexity inherent in bilinear Tx/Rx interactions, most practical approaches employ alternating optimization (block coordinate descent) or reformulate the problem to exploit structure:

• Alternating Optimization: Iteratively update the transmit beamformer(s) with fixed receiver(s), then update receiver(s) for the new transmit solution. This approach is used in fairness and sum-rate beamforming for spectrum sharing (Denkovski et al., 2015), robust MIMO radar (Zhang et al., 2015), federated learning (Kalarde et al., 29 Jan 2025), and ISAC (Liu et al., 2022).
• Convex Relaxation: In settings such as underlay spectrum sharing, the transmit subproblem is cast as a semidefinite program (SDP), exploiting the structure of secondary-to-secondary, secondary-to-primary, and primary-to-secondary channels (Denkovski et al., 2015).
• Closed-Form or Procrustes Solutions: In systems with transmit covariance constraints (e.g., DFRC radar-communications), subproblems admit closed-form SVD solvers (e.g., Cauchy's interlacing theorem for orthogonality under a Hermitian constraint, and the orthogonal Procrustes problem) (Yang et al., 2023).
• Majorization-Minimization and Fractional Programming: Address nonconvex constraints arising in MIMO-OFDM ISAC by iteratively constructing lower bounds and reformulating the sum-rate objective, combined with neADMM decomposition for high-dimensional block-variable updates (Xiao et al., 2023).
• Combinatorial/Greedy Grouping: In multi-relay MIMO-OFDMA, grouping orthogonal spatial multiplexing components via exhaustive (ESGA) or greedy (OCGA) algorithms enables scalable zero-forcing (Cheung et al., 2016).
• Compressing Over-the-Air Constraints: For aggregation and computation over noisy MACs, transmit and receive beamformers are optimized with explicit connections to convergence or estimation error bounds (Kalarde et al., 29 Jan 2025, Fang et al., 2021).

3. Robustness, Adaptivity, and Sparsity

Several works target adaptation and robustness:

• Adaptive Multi-Stream Selection: Instead of fixing the per-user stream allocations, allocate streams dynamically per user according to instantaneous channel metrics that minimize noise amplification under linear detection (e.g., sum of inverses of QR diagonal terms), yielding dramatic BER improvements (e.g., 2.5 dB SNR gain at BER $=10^{-2}$ in "coordinated Tx-Rx" compared to fixed allocation) (An et al., 2010).
• Robust Optimization under Mismatch: Probability-constrained approaches address randomly distributed steering-vector errors at both transmitter and receiver in MIMO radar, splitting joint constraints into separate transmit and receive parts and deriving convex SOCP subproblems (Zhang et al., 2015).
• Sparse Array Selection: Dynamic selection of active antennas at both ends using mixed $\ell_{1,2}$ regularization under joint constraints enables cost-effective adaptive beamforming, leveraging SCA for iterative refinement (Hamza et al., 2021). Reweighting mechanisms sharpen selection for minimum spatial aperture under SINR guarantees.
• CSI Imperfection/Tolerant Designs: In federated learning aggregation, per-round constraints and objectives are modified to account for noisy channel estimates, yielding up to 3 dB device power savings over naive perfect-CSI prescription (Kalarde et al., 29 Jan 2025).

4. Implementation Guidelines and Complexity Trade-offs

Practical deployment considerations include:

• Complexity Reduction: Successive transmit- or receive-only iterations (e.g., updating only the transmit side for many rounds, then updating receivers) can halve or better the iteration count compared to full-Joint Looping (Denkovski et al., 2015). For moderate antenna/user counts and SDPs or QPs per update, per-iteration times can be $\sim$100 ms or less.
• Scalability: Distributed architectures leverage simple one-bit feedback loops for large relay clusters to achieve $N$-fold SNR gain with $O(1)$ per-device overhead (scalable distributed receive beamforming) (Quitin et al., 2015).
• Parallelism & Sparsity: Group-sparse transmit-receive beamforming reduces hardware cost and power, and MMSE/Procrustes steps in multi-user MIMO yield exceptionally rapid convergence ($O(\text{few})$ iterations) (Yang et al., 2023).
• Joint Power/Interference Control: Underlay spectrum sharing and OFDMA designs enforce per-node and sum-power constraints as well as interference-temperature (PU protection) limits, with implementation either via centralized optimization or distributed message-passing (Denkovski et al., 2015, Cheung et al., 2016).

5. Application Domains and Performance Outcomes

Beamforming techniques are tightly coupled with specific system targets:

• Wireless Access: Coordinated Tx-Rx joint precoding, adaptive multi-streaming, and dynamic user-grouping provide significant throughput and SNR/BER gains (e.g., 2.5 dB SNR reduction at $10^{-2}$ BER in MU-MIMO) (An et al., 2010, Cheung et al., 2016).
• Cognitive/Underlay Networks: SDPs with fairness/SR objectives deliver near-capacity sum-rates and superior resilience to interference from and to incumbent systems (Denkovski et al., 2015).
• ISAC and DFRC Systems: Joint tradeoff between spectral efficiency and radar beampattern quality via transmit covariance constraints achieves at least 40% higher sum-rate (multi-user, high SNR) compared to prior methods (Yang et al., 2023); full-duplex FD-ISAC designs deliver 60 dB digital-residual SI suppression and >10% sum-rate boosts over HD (Liu et al., 2022).
• Federated Edge Learning: Over-the-air aggregation with optimized beamforming reduces device transmit power by over 25 dB at a fixed ML model convergence rate (Kalarde et al., 29 Jan 2025).
• Magnetic Induction/Non-RF Links: Alternating optimization yields 54% pathloss reduction and $<2$ dB angular pathloss fluctuation in tri-coil MI links (Li et al., 30 May 2025).
• Distributed/Sparse Arrays: Receive diversity and over-the-air beamforming architectures double diversity gain (vs. single-antenna destinations) and scale linearly with the number of distributed partners (Quitin et al., 2015, Domanovitz et al., 2019).

6. Extensions: Joint Physical Reconfiguration and Learning-Driven Adaptation

Recent efforts extend transmit-receive beamforming into:

• Polarization Reconfigurable Massive MIMO: Joint optimization of polarization and spatial beamforming at both ends, under severe observation constraints, via transformer-based deep learning frameworks that learn adaptive pilot sequences and processing—reducing overhead while achieving up to 2 dB gain over GRU-based alternatives with minimal pilots (Oh et al., 17 Aug 2025).
• Compressive Sensing and Sparse Signal Processing: Joint sparse echo recovery and multiuser beamforming in MIMO-OFDM ISAC, efficiently exploiting target sparsity to free spatial DoF for improved communication performance (Xiao et al., 2023).
• Integration with Federated Learning/Edge Intelligence: Structurally tailored QP-based updates guarantee monotonic decrease (convergence) and respect FL-specific error control, achieving both communication and learning goals simultaneously (Kalarde et al., 29 Jan 2025).
• Hardware and Robustness Considerations: Emphasis on high-dynamic range ADCs, the necessity for analog/propagation SI suppression, and sensitivity to CSI quality (Liu et al., 2022).

7. Performance Summaries and Comparative Insights

The table below summarizes representative numerical improvements from recent works as documented in the data, providing reference values for system designers.

Reference	Scenario	Notable Quantitative Gain
(An et al., 2010)	MU-MIMO adaptive Tx/Rx	2.5 dB lower SNR at BER=10⁻² (8×(2,2,2,2) MIMO)
(Denkovski et al., 2015)	Underlay MIMO/IC/BC/MAC	Approaches interference-free bound for large arrays/users
(Liu et al., 2022)	Full-duplex ISAC FD penalty-based	Up to 60 dB SI cancel, 10–20% sum-rate over HD, accurate ISAC
(Yang et al., 2023)	DFRC, multi-user, multi-stream	≥40% higher sum-rate (high SNR, K=4, d=4, N_rx=4)
(Kalarde et al., 29 Jan 2025)	FL over-the-air aggregation	≥25 dB power savings at fixed FL accuracy
(Hamza et al., 2021)	Sparse MIMO radar, MaxSINR	3–7 dB SINR gain over random sparse array
(Li et al., 30 May 2025)	Tri-coil MI joint BF	Up to 54% pathloss reduction, ≈2 dB PL fluctuation
(Xiao et al., 2023)	MIMO-OFDM ISAC, CS-aided	50% sum-rate improvement at fixed accuracy (Γ₀ = –5 dB)
(Oh et al., 17 Aug 2025)	PR-MIMO, transformer learning	1–2 dB beamforming gain advantage, reduced pilot overhead

Conclusion

Transmit-receive beamforming continues to be a primary enabler across wireless, sensing, and computation-driven network systems. By leveraging advanced convex optimization, block coordinate descent, adaptive and robust formulations, and deep learning-based sequence design, contemporary techniques deliver significant gains in spectral efficiency, reliability, robustness to modeling error, and resource consumption. Matching a beamforming strategy to the underlying architectural, hardware, algorithmic, and environmental constraints is critical, with the best results obtained from schemes that adaptively exploit instantaneous system state, balance constraints across communication, sensing, and computation, and provide practical complexity for large-scale deployments.