Exploiting Known Interference as Green Signal Power for Downlink Beamforming Optimization

C. Masouros; G. Zheng

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Exploiting Known Interference as Green Signal Power for Downlink Beamforming Optimization

C. Masouros, G. Zheng

arXiv:1505.01354v1May 6, 2015cs.IT

TL;DR

The paper addresses the inefficiency of treating known interference as harmful in multi-user MISO downlink beamforming. It designs data-aided PSK precoders and related SINR-balancing, multicast, and robust formulations that exploit constructive interference. The reported results show lower transmit power or higher SINR than conventional optimization, including power savings up to 50% and SINR gains of about 2–3 dB.

Problem
Conventional optimization suppresses interference to meet QoS, although known CSI and data can make some instantaneous interference constructive.
Method
The paper designs PSK beamforming constraints that treat constructive interference as useful signal power, then derives virtual multicast, gradient-projection, SINR-balancing, and bounded-CSI-error formulations.
Results
The proposed schemes outperform conventional precoding, delivering reduced transmit power for QoS or increased minimum SNR for a fixed power budget; reported gains include up to 50% power savings and about 2–3 dB SNR gains.
Takeaways & Limitations
Exploiting constructive interference improves fully optimized downlink precoding across power minimization, SINR balancing, multicast reformulations, and robust imperfect-CSI designs.

Abstract

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We propose a data-aided transmit beamforming scheme for the multi-user multiple-input-single-output (MISO) downlink channel. While conventional beamforming schemes aim at the minimization of the transmit power subject to suppressing interference to guarantee quality of service (QoS) constraints, here we use the knowledge of both data and channel state information (CSI) at the transmitter to exploit, rather than suppress, constructive interference. More specifically, we design a new precoding scheme for the MISO downlink that minimizes the transmit power for generic phase shift keying (PSK) modulated signals. The proposed precoder reduces the transmit power compared to conventional schemes, by adapting the QoS constraints to accommodate constructive interference as a source of useful signal power. By exploiting the power of constructively interfering symbols, the proposed scheme achieves the required QoS at lower transmit power. We extend this concept to the signal to interference plus noise ratio (SINR) balancing problem, where higher SINR values compared to the conventional SINR balancing optimization are achieved for given transmit power budgets. In addition, we derive equivalent virtual multicast formulations for both optimizations, both of which provide insights of the optimal solution and facilitate the design of a more efficient solver. Finally, we propose a robust beamforming technique to deal with imperfect CSI, that also reduces the transmit power over conventional techniques, while guaranteeing the required QoS. Our simulation and analysis show significant power savings for small scale MISO downlink channels with the proposed data-aided optimization compared to conventional beamforming optimization.

I. INTRODUCTION

The paper develops data-aided beamforming that exploits known constructive interference instead of suppressing all interference, targeting lower transmit power and higher SINR in multi-user MISO downlinks.

Motivation: Conventional downlink precoding methods trade performance against complexity, from sophisticated dirty paper coding and vector perturbation to lower-complexity channel inversion.Vector perturbation improves performance but requires solving an NP-hard perturbation search problem.
Motivation: Knowledge of both CSI and transmitted data lets the base station classify interference as constructive or destructive and exploit constructive components as signal power.Earlier closed-form precoders showed that constructive interference can increase receive SNR without increasing per-user transmit power.
Contributions: The proposed PSK precoder reduces transmit power for given QoS, supports cases with more users than transmit antennas, and extends constructive-interference optimization to SINR balancing.For fixed transmit power budgets, the SINR-balancing formulation targets higher minimum SINR values than conventional optimization.
Contributions: The paper recasts both optimizations as virtual multicast problems, derives optimal-solution structure, develops an efficient solver, and extends the approach to bounded CSI errors.The robust formulation is designed for imperfect CSI while retaining QoS guarantees.
Scope: The approach focuses on PSK modulation and is especially useful in high-interference settings using reliable low-order formats such as BPSK and QPSK.The concept can also be extended to QAM by adapting constellation decision thresholds.

B. SINR Balancing

SINR balancing maximizes the minimum achievable SINR under a total transmit-power budget. The paper adapts this objective to exploit constructive PSK interference through relaxed received-symbol constraints.

B. SINR Balancing: SINR balancing maximizes the minimum achievable SINR subject to a total transmit power budget P.The conventional formulation is non-convex and requires more complex iterative solutions.
Constructive Interference: Constructive interference is identified using known CSI and data, with received symbols judged by their distance from PSK decision thresholds rather than nominal constellation points alone.Interference is constructive when it moves received symbols away from decision thresholds.
Proposed Optimization: For constructive interference, instantaneous interfering symbols are optimized to contribute to received signal power instead of being constrained as harmful stochastic interference.The formulation uses the known phases and amplitudes of the users’ symbols to impose constructive alignment and satisfy SNR thresholds.
Constructive Interference: The proposed constraints allow nonzero phase shifts within the constructive region, whose maximum angle shift for M-PSK is θ = ±π/M.The relaxed region extends away from decision thresholds and enlarges the feasible optimization region.
Proposed Optimization: The relaxed optimization is a standard SOCP with a smaller minimum transmit power and an increased feasibility region compared with zero-angle-shift optimization.The paper also derives a more computationally efficient algorithm for solving the problem.

A. A Virtual Multicast Formulation of (11)

The constructive-interference broadcast problem is equivalent to a virtual multicast problem with a modified channel. This reformulation captures the correlation among data streams and yields a convex problem that is efficiently solvable.

The optimization in (11) is re-cast as a virtual multicasting problem through Theorem 1.
The modified channel is defined as ˜h_i = h_i e^{j(φ_1−φ_i)}.
The composite precoding term can be treated as a single vector w, producing the multicast reformulation.
Unlike classical multicast beamforming, the resulting problem is convex with a quadratic objective and 2K linear constraints.
The reformulation interprets transmission as sending a common data stream over reshaped channels to deliver intended data to each receiver.

B. Real Representation of the Problem

The problem is converted into a real-valued representation using modified channel and precoder components. Its dual is a non-negative least-squares problem whose optimal solution has a channel-based linear-combination structure.

The complex modified channel and precoder are decomposed into real and imaginary parts.
Real-valued vectors f_i, w_1, and w_2 are defined from those components to represent the optimization in real form.
The dual formulation introduces nonnegative variables μ and ν for the two constraint sets and combines them into λ.
The optimal precoding vector has a linear-combination structure based on channel coefficients.
The resulting dual problem is a non-negative least-squares problem involving the real matrix A = [f − g f + g].
When λ* is positive, Corollary 1 characterizes the condition for achieving the optimal solution of the original constructive-interference problem.

D. A Gradient Projection Algorithm to Solve (29)

Because the general solution of problem (29) is difficult to derive, the paper develops a gradient projection algorithm after rewriting it as a standard convex problem.

The general solution to problem (29) is difficult to derive, motivating an iterative solver.
The problem is rewritten as a standard convex problem before applying gradient projection.
Iterations continue until convergence, after which the algorithm outputs λ_opt.
Algorithm 1 initializes a nonnegative dual vector λ(0) and updates it iteratively.
The step size a_n can be selected using the Armijo rule or another line-search scheme.

V. CONSTRUCTIVE INTERFERENCE OPTIMIZATION FOR SINR BALANCING

Constructive interference is extended to SINR balancing by reformulating the problem with an auxiliary variable. The resulting formulation is convex and has an equivalent multicast representation.

The constructive-interference concept is applied to the SINR balancing problem.
Replacing √Γ_t with Γ_t^2 = √Γ_t resolves the nonconvexity caused by the concavity of √Γ_t.
The resulting formulation in (38) is convex and can be solved using standard convex optimization techniques.
The optimization is re-cast as a multicasting problem through an equivalent theorem.
Theorem 2 states that problem (38) is equivalent to the stated multicast problem.
The equivalent formulation is a standard SOCP that can be efficiently solved using known approaches.

VI. ROBUST POWER MINIMIZATION WITH BOUNDED CSI ERRORS

The robust design models each user’s channel as an estimate plus bounded spherical uncertainty and minimizes total transmit power under worst-case SINR constraints. The resulting robust problem admits an SDP relaxation that is convex and provides a lower bound on conventional power minimization.

The actual channel is modeled as the estimated channel plus an error vector constrained to a spherical uncertainty set of radius δ_i^2.
The robust formulation minimizes total transmit power while guaranteeing each user’s SINR constraint for every channel uncertainty within the specified set.
The robust beamforming problem is relaxed to a semidefinite programming problem.
The relaxed problem is convex and can be solved optimally, with its objective value providing a lower bound for conventional power minimization.
When the SDP relaxation returns rank-1 solutions, the optimal beamforming vectors can be recovered by matrix decomposition; otherwise, the original problem requires higher power.

C. Robust Precoding based on Constructive Interference

The robust constructive-interference design converts worst-case interference constraints under bounded CSI errors into tractable convex formulations. Real-valued reformulation and second-order cone programming enable efficient solution of the resulting robust precoder.

The robust constructive-interference design starts from the multicast formulation of the power-minimization problem and imposes worst-case CSI uncertainty.
The intractable infinite-error constraint is replaced by a modified constraint and then separated into real and imaginary components.
The complex channel and error are rewritten using real-valued channel, beamforming, and error vectors bounded by the CSI uncertainty radius.
The constructive-interference constraint is expressed as two worst-case constraints over the uncertainty set.
The final robust problem is a standard SOCP and can be efficiently solved, after which the robust beamformer is recovered from the optimized auxiliary variables.

D. Robust SINR Balancing

Robust SINR balancing is formulated as a tractable SOCP under bounded CSI uncertainty, while simulations compare constructive-interference and conventional schemes across power, feasibility, complexity, and achievable SINR. Constructive interference provides reported power, feasibility, SINR, and robustness advantages, with symbol-level optimization adding complexity.

D. Robust SINR Balancing: Robust SINR balancing is formulated directly under imperfect CSI and can be solved as a typical SOCP.
VII. NUMERICAL RESULTS: Up to 50% power savings are observed for the proposed constructive-interference optimization in the 5 × 4 scenario.
VII. NUMERICAL RESULTS: 92.6% of 3 × 4 cases are feasible for the proposed optimization, whereas conventional optimization is infeasible when N < K.
VII. NUMERICAL RESULTS: The gradient projection solver reduces average execution time to less than 15% of the BC and MC solver complexity in the N = 5 QPSK system.
VII. NUMERICAL RESULTS: The proposed optimization must be performed symbol by symbol, which can create excess complexity compared with conventional channel-dependent precoding in slow-fading scenarios.
VII. NUMERICAL RESULTS: Achievable SINR gains of about 2 dB and 3 dB are reported for the compared SINR-balancing configurations, attributed to constructive interference.
VII. NUMERICAL RESULTS: Under CSI errors, the proposed robust CI method has a modest power increase and less than 1 dB loss relative to perfect CSI, while conventional precoding is more sensitive.

VIII. CONCLUSION

The proposed constructive-interference precoding designs improve power and SINR performance in multi-user MISO downlink optimization, including robust designs for imperfect CSI. Their scope includes PSK signaling, bounded CSI errors, and single-cell systems, with QAM extension, optimal robust design, and multi-cell environments left for future work.

Conclusion: Constructive-interference designs reduce transmit power for given QoS constraints or increase minimum SNR for a fixed transmit-power budget.The schemes exploit constructive interference rather than suppressing it.
Conclusion: Both optimization problems were reformulated as virtual multicast problems that were more efficiently solvable.
Conclusion: Robust designs extend the proposed concept to imperfect CSI with bounded CSI errors.
Future work: Extending the approach to QAM requires redesigning constructive-interference sectors to determine the relevant optimization constraints.The paper identifies this extension as non-trivial.
Future work: The robust design uses a conservative approach, while optimal robust precoder design remains challenging.
Future work: The study considers a single-cell system, leaving multi-cell environments as a future investigation area.