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Movable-Antenna Array Enhanced Beamforming: Achieving Full Array Gain with Null Steering

Lipeng Zhu, Wenyan Ma, Rui Zhang

arXiv:2308.08787v2cs.ITeess.SP

TL;DR

Fixed-position arrays face a trade-off between desired-direction gain and interference nulling. The paper jointly optimizes movable-antenna positions and weights, showing that suitable arrays can achieve full gain with simultaneous null steering, using closed-form solutions and phase-only analog weights.

  • Problem

    Fixed-position arrays generally cannot simultaneously maximize desired-direction array gain and null undesired directions because their steering vectors are spatially correlated.

  • Method

    The paper jointly optimizes the antenna positions vector and antenna weights vector of a linear movable-antenna array, deriving closed-form solutions through steering-vector orthogonality.

  • Results

    Under sufficient conditions on the numbers of antennas and null directions, movable-antenna beamforming achieves full desired-direction array gain with null steering over all undesired directions.

  • Takeaways & Limitations

    The optimal movable-antenna weights require only phase adjustment with fixed amplitude, enabling analog beamforming and reduced implementation complexity.

Abstract

from arXiv · show

Conventional beamforming with fixed-position antenna (FPA) arrays has a fundamental trade-off between maximizing the signal power (array gain) over a desired direction and simultaneously minimizing the interference power over undesired directions. To overcome this limitation, this letter investigates the movable antenna (MA) array enhanced beamforming by exploiting the new degree of freedom (DoF) via antenna position optimization, in addition to the design of antenna weights. We show that by jointly optimizing the antenna positions vector (APV) and antenna weights vector (AWV) of a linear MA array, the full array gain can be achieved over the desired direction while null steering can be realized over all undesired directions, under certain numbers of MAs and null-steering directions. The optimal solutions for AWV and APV are derived in closed form, which reveal that the optimal AWV for MA arrays requires only the signal phase adjustment with a fixed amplitude. Numerical results validate our analytical solutions for MA array beamforming and show their superior performance to the conventional beamforming techniques with FPA arrays.

I. INTRODUCTION

Conventional fixed-position antenna arrays face a trade-off between amplifying desired signals and suppressing interference because their steering vectors are spatially correlated. This letter uses movable-antenna positions as an additional beamforming degree of freedom.

  • Fixed-position arrays generally cannot simultaneously achieve full desired-direction array gain and null steering over undesired directions.The trade-off arises from inherent spatial correlation among steering vectors at different angles.
  • Movable antennas enable local position optimization for more favorable channel conditions and improved communication performance compared with conventional fixed-position systems.
  • The proposed method jointly optimizes the antenna positions vector and antenna weights vector of a linear movable-antenna array.
  • Under certain numbers of movable antennas and null-steering directions, the method achieves full array gain while null steering all undesired directions.
  • Closed-form solutions show that the optimal antenna weights require only signal-phase adjustment with fixed amplitude, enabling analog beamforming with lower implementation complexity.
  • Numerical results validate the analytical solutions and show superior performance to conventional fixed-position-array beamforming techniques.

II. PROBLEM FORMULATION AND TRANSFORMATION

The paper formulates joint antenna-position and weight optimization for a linear movable-antenna array under null-steering and spacing constraints. It transforms full-gain attainment into a steering-vector orthogonality condition that antenna-position optimization can satisfy under suitable dimensions.

  • A linear movable-antenna array has N antennas, with antenna positions collected in the antenna positions vector x.
  • The array steering vector is determined by antenna positions, wavelength λ, and steering angle θ.
  • The optimization maximizes desired-direction beam gain while forcing zero gain in K undesired interference directions and maintaining minimum antenna spacing.
  • The normalized AWV power constraint and a tight upper bound establish N as the full array gain over the desired direction.
  • For fixed-position arrays, zero-forcing beamforming generally incurs array-gain loss because antenna geometry and the resulting loss term are fixed.
  • Movable-antenna position optimization adds degrees of freedom that can reduce the zero-forcing loss, making full gain feasible for suitable N and K.
  • Zero loss is equivalent to orthogonality between the desired steering vector and every undesired steering vector, called the steering-vector orthogonality condition.
  • Under this condition, the optimal AWV has constant-modulus elements and can be implemented using analog beamforming.

III. ANTENNA POSITION OPTIMIZATION

The section constructs antenna position vectors that satisfy the SVO condition while preserving minimum antenna spacing, enabling full array gain and null steering under specific N and K conditions.

  • For K = 1, a feasible APV exists for any N ≥2 while satisfying the SVO condition and minimum-distance constraint.
  • Lemma 2 extends feasibility from N = N1 and K ≤K1 to N = N1N2 and K ≤K1 + 1 for any N2 ≥2.
  • The construction represents each antenna index with a unique factorization vector based on N’s ordered prime factors.
  • If N has IN prime factors, an APV satisfying the SVO condition and minimum-distance constraint exists for all K ≤IN.
  • Theorem 1 concurrently achieves full array gain over θ0 and null steering over K undesired directions when K ≤IN.
  • For N = 8 and K = 3, antennas are placed incrementally to satisfy each new SVO condition while maintaining earlier conditions; construction complexity is O(IN).
  • The theorem gives sufficient, not generally necessary, conditions; for K > IN, the existence of an optimal APV remains open and ZF handles remaining directions.

IV. NUMERICAL RESULTS

Numerical results show that movable-antenna arrays jointly optimize positions and weights to achieve full desired-direction gain with simultaneous null steering, outperforming fixed-position arrays.

  • Digital beamforming: The digital-beamforming MA solution has end-to-end length 28.33λ, about 8 times the FPA array length.The compared FPA array uses half-wavelength antenna spacing.
  • Analog beamforming: With analog beamforming and N = 8, K = 3, the MA array again achieves full gain and three nulls, whereas the FPA array loses L(x) = 7.0.The analog FPA benchmark uses a Kronecker decomposition-based approach.
  • Analog beamforming: The analog-beamforming MA solution has end-to-end length 10.29λ, about 3 times the FPA array length.The FPA comparison again uses half-wavelength antenna spacing.
  • Single-null evaluation: For N = 8 and K = 1, MA beamforming always achieves full desired-direction gain, while FPA loss increases as θ1 approaches θ0 = 90◦.The FPA degradation occurs with both digital and analog beamforming because steering-vector correlation increases.

V. CONCLUSION

The conclusion reports that antenna-position optimization adds a useful degree of freedom for beamforming. Joint APV and AWV design enables simultaneous full array gain and null steering under stated conditions, with closed-form solutions and phase-only analog weights.

  • Conclusion: Jointly designing the APV and AWV enables full desired-direction array gain and simultaneous null steering under certain numbers of MAs and null-steering directions.The conclusion identifies antenna-position optimization as the added degree of freedom.
  • Conclusion: Closed-form APV and AWV solutions show that MA arrays require only signal phase adjustment with fixed-amplitude analog beamforming.Numerical results validate the analytical solutions.
  • Conclusion: Numerical results show superior MA-array performance compared with conventional FPA arrays using digital or analog beamforming.This comparison is reported as part of the paper’s validation.

APPENDIX A PROOF OF LEMMA 2

The proof constructs a larger movable-antenna position vector by replicating a valid smaller array with fixed offsets, preserving existing nulls and adding one new null direction while maintaining minimum spacing.

  • APPENDIX A PROOF OF LEMMA 2: The constructed APV consists of N2 translated copies of the N1-antenna APV, separated by a distance d.The offsets are represented by y = [0, d, 2d, · · · , (N2 −1)d]^T.
  • APPENDIX A PROOF OF LEMMA 2: The distance d is selected separately for the cases K = K1 + 1 and K ≤K1.This choice determines whether the construction adds a null-steering direction or preserves the existing set.
  • APPENDIX A PROOF OF LEMMA 2: For the existing directions, the smaller APV’s orthogonality guarantees that the constructed APV retains null steering over directions 1 through K1.The proof explicitly transfers the zero inner products from the original APV to the replicated construction.
  • APPENDIX A PROOF OF LEMMA 2: For the added direction k = K1 + 1, the selected spacing d makes the constructed APV satisfy the required orthogonality condition.Thus, the new construction satisfies the SVO condition over all K1 + 1 undesired directions.
  • APPENDIX A PROOF OF LEMMA 2: The construction also preserves the minimum-distance constraint between every pair of antennas.Within each copy, spacing follows the original APV; between copies, the ordered offsets ensure separation is at least dmin.
  • APPENDIX A PROOF OF LEMMA 2: Therefore, an APV satisfying both the SVO condition and the spacing constraint exists for N = N1N2 and K ≤K1 + 1.This completes the inductive enlargement step used in the proof.

APPENDIX B PROOF OF THEOREM 1

The proof extends the APV construction through prime-factor decomposition, assigning distances that enforce each null condition while maintaining minimum antenna spacing and yielding an optimal APV.

  • APPENDIX B PROOF OF THEOREM 1: Prime-factor induction establishes that a valid APV exists for all K ≤IN.Lemma 1 supplies the base case, while Lemma 2 increases the supported null count through factorized array dimensions.
  • APPENDIX B PROOF OF THEOREM 1: For each null-steering direction θi, the constructed distance di is chosen using the angle difference | cos θ0−cos θi| and wavelength λ.The resulting distance also satisfies the minimum-spacing requirement between successive antennas.
  • APPENDIX B PROOF OF THEOREM 1: For indices without additional null-steering directions, di is selected to guarantee the minimum distance between consecutive antennas.The construction continues spacing the antennas after all K null directions have been assigned.
  • APPENDIX B PROOF OF THEOREM 1: The vector zn uniquely encodes the factorized antenna index n through the relation n = zT nd.Its components determine the sequence of distance increments used to locate the antennas.
  • APPENDIX B PROOF OF THEOREM 1: The resulting APV satisfies the SVO condition and spacing constraint, making it an optimal solution for achieving full array gain in the desired direction.The proof concludes optimality after verifying both conditions for the constructed positions.
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