Source-linked AI summary
Multi-Beam Forming with Movable-Antenna Array
Wenyan Ma, Lipeng Zhu, Rui Zhang
TL;DR
Fixed-position arrays trade off beamforming gain in desired directions against interference suppression in undesired directions. This paper jointly optimizes antenna positions and weights for a linear movable-antenna array, and reports stronger beamforming results than benchmark schemes, with effective interference suppression compared with FPA arrays.
Problem
Fixed-position antenna arrays have fixed steering-vector correlations across angles, forcing a trade-off between desired-direction beamforming gain and undesired-direction interference suppression.
Source: §I · p. 1Method
The paper jointly optimizes the antenna position vector and antenna weight vector using alternating optimization, with convex relaxation for the subproblems, to maximize minimum desired-direction gain subject to an interference-power constraint.
Source: §I · p. 1Results
The proposed method achieves 99.7% of full beamforming gain, compared with 36.9% for FPA, 80.0% for APS, and 86.8% for AWI, and the conclusion reports effective interference suppression versus FPA arrays.
Source: §IV · p. 4, §V · p. 5Takeaways & Limitations
The results support using antenna-position optimization as an additional degree of freedom for multi-beam forming with movable-antenna arrays.
Source: §V · p. 5
Abstract
from arXiv · showhide
Conventional multi-beam forming with fixed-position antenna (FPA) arrays needs to trade-off between maximizing the beamforming gain over desired directions and minimizing the interference power over undesired directions. In this letter, we study the enhanced multi-beam forming with a linear movable-antenna (MA) array by exploiting the new degrees of freedom (DoFs) via antennas' position optimization. Specifically, we jointly optimize the antenna position vector (APV) and antenna weight vector (AWV) to maximize the minimum beamforming gain over multiple desired directions, subject to a given constraint on the maximum interference power over undesired directions. We propose an efficient alternating optimization algorithm to find a suboptimal solution by iteratively optimizing one of the APV and AWV with the other being fixed. Numerical results show that the proposed multi-beam forming design with MA arrays can significantly outperform that with the traditional FPA arrays and other benchmark schemes in terms of both beamforming gain and interference suppression.
I. INTRODUCTION
Fixed-position antenna arrays face a trade-off between gain in desired directions and interference suppression in undesired directions. The paper addresses the unstudied multi-beam case for movable-antenna arrays by jointly optimizing antenna positions and weights.
I. INTRODUCTION
- FPA arrays trade off maximizing beamforming gain in desired directions against minimizing interference in undesired directions, limiting multi-antenna system performance,.Multi-beam forming supports concurrent transmissions to multiple receivers or receptions from multiple transmitters, with applications including satellite coverage, broadcasting or multicasting, and radar tracking.Source: §I · p. 1
- Prior work found MA arrays can improve channel and communication performance, but their multi-beam-forming improvement had not been investigated –.Earlier work showed that joint APV and AWV optimization can enhance single-beam forming, and under certain conditions achieve full desired-direction gain with null steering.Source: §I · p. 1
- The paper jointly optimizes the APV and AWV for a linear MA array, using alternating optimization and convex relaxation to obtain a suboptimal solution.The objective is to maximize minimum gain over desired directions subject to a maximum-interference-power constraint; simulations report gains in beamforming and interference suppression over FPA arrays and other benchmarks.Source: §I · p. 1
II. SYSTEM MODEL AND PROBLEM FORMULATION
The system is a linear movable-antenna array whose steering response depends on antenna positions and steering angle. The formulation jointly chooses positions and weights to maximize the weakest desired-direction signal power while bounding interference.
II. SYSTEM MODEL AND PROBLEM FORMULATION
- The linear array has N antennas movable within a one-dimensional segment of length D, and its steering vector depends on the APV x and angle θ.The APV orders the antenna positions from x1 to xN within [0,D].Source: §II · p. 1, Figure 1 · p. 2
- The design maximizes minimum signal power over K desired directions by jointly choosing x and w, subject to a bound on maximum interference power over L undesired directions.The AWV power is at most one; antenna positions remain within the segment and adjacent antennas are separated by at least D0 to avoid coupling.Source: §II · p. 2
- The problem is non-convex because the gain constraints are non-convex in w and x, and the two variables are coupled.These properties make the joint optimization more challenging to solve.Source: §II · p. 2
III. PROPOSED ALGORITHM
The proposed algorithm addresses the coupling between antenna positions and weights through alternating optimization. The section also analyzes convergence and complexity and presents an initialization scheme.
III. PROPOSED ALGORITHM
- The algorithm alternately optimizes w and x, holding the other variable fixed in each subproblem.The two subproblems are repeated in an alternating manner to address their coupling.Source: §III · p. 2
- The section analyzes the algorithm’s convergence and computational complexity.The analysis follows the description of the alternating subproblems.Source: §III · p. 2
- A low-complexity initialization scheme is presented for the algorithm.Source: §III · p. 2
A. Optimization of w with Given x
With antenna positions fixed, the algorithm optimizes the antenna weight vector by successively approximating a non-convex desired-gain constraint. The resulting subproblem is convex and can be solved efficiently.
A. Optimization of w with Given x
- For fixed x, successive convex approximation uses a first-order Taylor expansion at the current weight vector to construct a linear minorizer of the desired-direction gain.This relaxes the non-convex gain constraint with respect to w.Source: A. Optimization of w with Given x · p. 2
- The relaxed weight-optimization problem imposes the surrogate gain bound alongside the interference and weight-power constraints.The desired-direction constraint uses the surrogate evaluated at the current iterate.Source: A. Optimization of w with Given x · p. 2
- The resulting problem is a convex QCQP that can be efficiently solved using CVX.Its constraints are linear or convex quadratic in w.Source: A. Optimization of w with Given x · p. 2
B. Optimization of x with Given w
With the AWV fixed, the APV subproblem uses successive convex approximation (SCA) to relax the desired-gain and interference constraints into convex quadratic constraints.
B. Optimization of x with Given w
- For fixed w, SCA constructs a global lower-bound surrogate for desired-direction gain, yielding a convex quadratic constraint over x.The construction uses a second-order Taylor approximation of the cosine terms; the resulting quadratic matrix is negative semidefinite.Source: B. Optimization of x with Given w · pp. 2–3
- The gain expression uses a cosine of an antenna-pair phase quantity, formed by subtracting the weights’ phase difference (∠w_n−∠w_m) from the position-dependent phase ϕ_l(x_n−x_m).Source: B. Optimization of x with Given w · pp. 2–3
- The interference constraint is relaxed with a convex quadratic majorizing surrogate, producing an upper-bound constraint over x for each undesired direction.The resulting quadratic matrix is positive semidefinite, and the constraint bounds interference by η.Source: B. Optimization of x with Given w · p. 3
- The desired-gain and interference surrogates, together with the feasible-region and minimum-spacing constraints, form the convex QCQP used to update x.The QCQP can be efficiently solved using CVX.Source: B. Optimization of x with Given w · p. 3
C. Overall Algorithm and Performance Analysis
The algorithm alternates between AWV and APV optimization, using their subproblems until convergence. Its objective is non-decreasing and bounded, and the paper analyzes its computational complexity.
C. Overall Algorithm and Performance Analysis
- Algorithm 1 alternates between solving P2 for w with x fixed and P3 for x with w fixed, repeating the updates until convergence.Each subproblem uses its own convergence threshold, alongside a threshold for the overall algorithm.Source: C. Overall Algorithm and Performance… · pp. 3–4
- The objective is non-decreasing across outer iterations and bounded by a finite beamforming gain, so Algorithm 1 is guaranteed to converge.The paper also establishes monotonic convergence for each subproblem.Source: C. Overall Algorithm and Performance… · pp. 3–4
- The total complexity is O((K + L + 2)^0.5 N(N^2 + K + L) ln(1/β)(Iw + Ix)I) for interior-point QCQP solves with accuracy β.I, Iw, and Ix are the maximum outer, P2, and P3 iteration counts, respectively.Source: C. Overall Algorithm and Performance… · p. 4
D. Initialization
The initialization scheme spaces antennas uniformly within the feasible region, then obtains an initial AWV through an SDP and spectral factorization.
D. Initialization
- The initial APV x0 is uniformly spaced over [0, D], with spacing D/(N+1), to keep movable antennas sufficiently separated.For that uniform linear array, the paper states that the globally optimal initial AWV can be obtained.Source: D. Initialization · p. 4
- An SDP produces a positive semidefinite matrix W0 subject to desired-gain, interference, and AWV-power constraints.The constraints use the initial APV x0 and threshold η.Source: D. Initialization · p. 4
- The method derives the autocorrelation vector rw from W0, then reconstructs w0 using spectral factorization.The paper identifies rw as the autocorrelation vector of w0.Source: D. Initialization · p. 4
IV. NUMERICAL RESULTS
Across the tested settings, the proposed MA design outperforms the benchmark schemes in max-min beamforming gain, including when the number of interference directions is large.
IV. NUMERICAL RESULTS
- The benchmarks are a half-wavelength-spaced FPA array, alternating position selection with discrete locations, and the proposed algorithm run without outer iterations.The simulations set the maximum interference-power threshold to η = 0.1 and the minimum adjacent-MA spacing to D0 = λ/2.Source: §IV · p. 4
- 99.7% of full beamforming gain is achieved by the proposed method, versus 36.9% for FPA, 80.0% for APS, and 86.8% for AWI.The comparison uses N = 8, K = 2, and L = 2.Source: §IV · p. 4, Figure 2 · p. 5
- At N = 4, the proposed method still improves performance over FPA by 64.1%, and its advantage over all benchmarks increases with N.The proposed method outperforms all benchmark schemes across the tested values of N.Source: §IV · p. 4, Figure 3 · p. 5
- At L = 6 interference directions, the proposed MA method achieves about 3 dB gain over FPA and outperforms the other schemes.The comparison evaluates max-min beamforming gain as L varies.Source: Figure 4 · p. 5, §IV · p. 5
V. CONCLUSIONS
The paper concludes that jointly optimizing MA positions and antenna weights can improve max-min beamforming gain while suppressing interference relative to conventional FPA arrays.
V. CONCLUSIONS
- Joint APV and AWV optimization with an alternating algorithm yields a suboptimal solution that enhances max-min beamforming gain with effective interference suppression versus conventional FPA arrays.The algorithm iteratively optimizes one vector while fixing the other.Source: §V · p. 5