MIMO Capacity Characterization for Movable Antenna Systems

Motivation

• Fixed-position antennas cannot fully utilize spatial channel variation, especially when only a limited number of antennas are available.Antenna selection also increases hardware, channel-estimation, and computational costs as the candidate set grows.Source: §I · p. 2

Related approaches

• Fluid antenna systems improve channel exploitation by switching a receive antenna among ports, but liquid-material constraints limit them to one antenna moving along a 1D line.Prior results report lower outage probability than multi-antenna MRC and significant capacity gains from favorable, interference-mitigated channels.Source: §I · p. 2

Proposed system

• Movable antennas use flexible-cable connections and controllers to adjust transmit and receive positions, reshaping the MIMO channel matrix relative to fixed-position antennas.The objective is to exploit additional spatial degrees of freedom for higher capacity.Source: §I · p. 3

Problem formulation

• The capacity problem jointly optimizes transmit and receive MA positions and the transmit-signal covariance matrix for a point-to-point system with perfect CSI at both ends.MA positions must balance channel gains across parallel spatial data streams.Source: §I · p. 3

Algorithms

• An alternating optimization algorithm iteratively updates the covariance matrix and each MA position, using convex relaxation for the positioning subproblem and converging to at least a locally optimal solution.Lower-complexity alternatives are also developed for asymptotically low SNR and MISO/SIMO channels.Source: §I · pp. 3–4

Results

• Numerical results show that jointly positioning transmit and receive MAs significantly improves total channel power, decreases channel condition number, and yields higher capacity than FPA-based systems.The evaluation compares the proposed systems with conventional FPA systems and benchmark schemes.Source: §I · p. 4

A. MA-Enabled MIMO System

• The system has N transmit MAs and M receive MAs connected to RF chains, allowing their positions to be adjusted in real time.Source: A. MA-Enabled MIMO System · p. 5
• Each transmit and receive MA moves freely within a specified 2D region, modeled as a square of size A × A.Source: A. MA-Enabled MIMO System · p. 5

B. Field-Response Based Channel Model

• The model assumes quasi-static block fading, fixed multipath components over the regions, and a far-field channel in which path phases vary with antenna position.Source: A. MA-Enabled MIMO System · p. 6, B. Field-Response Based Channel Model · p. 6

A. MA-Enabled MIMO System

• The channel matrix H(˜t, ˜r) depends on the collections of transmit and receive MA coordinates, while the transmit covariance is Q = E{ssH} ⪰ 0.Source: A. MA-Enabled MIMO System · p. 6

B. Field-Response Based Channel Model

• Transmit field responses are built from path-dependent propagation differences and stacked into G(˜t) ∈ C^Lt×N.Source: B. Field-Response Based Channel Model · p. 7
• Receive field responses are defined analogously from receive-path angles and propagation differences, forming F(˜r) ∈ C^Lr×M.Source: B. Field-Response Based Channel Model · pp. 7–8
• The path response matrix Σ ∈ C^Lr×Lt links each transmit path to each receive path and completes the position-dependent channel model.Source: B. Field-Response Based Channel Model · p. 8

C. Problem Formulation

• The objective is to characterize the capacity limit of an MA-enabled MIMO communication system.Source: C. Problem Formulation · p. 8
• Perfect CSI is assumed at both transmitter and receiver.Source: C. Problem Formulation · p. 8
• Unlike FPA MIMO, capacity depends on transmit and receive MA positions through H(˜t, ˜r) and the corresponding optimal covariance Q.Source: C. Problem Formulation · p. 8
• The optimization jointly selects ˜t, ˜r, and Q subject to transmit sum power and minimum-distance constraints between antennas.Source: C. Problem Formulation · pp. 8–9
• The problem is non-convex because the capacity objective is non-concave over MA positions and the separation constraints are non-convex.Source: C. Problem Formulation · p. 9

III. PROPOSED ALGORITHM

• The proposed solution uses alternating optimization, with additional lower-complexity treatments for low SNR and single-antenna MISO or SIMO cases.Source: §III · p. 9

• The algorithm alternates among optimizing Q, one transmit MA position, and one receive MA position while fixing the remaining variables.Source: A. Alternating Optimization for (P1) · p. 10, 1) Optimization of Q with given {tn}N… · p. 10
• With fixed MA positions, Q is obtained by eigenmode transmission over the nonzero singular modes of H(˜t, ˜r).Source: 1) Optimization of Q with given {tn}N… · p. 10
• The receive-position objective can be rewritten using a sum of M rank-one matrices, decoupling the receive-MA position variables.Source: 2) Optimization of rm with given Q,… · pp. 10–11
• Each receive-position update remains non-convex because its objective and minimum-distance constraints are non-convex.Source: 2) Optimization of rm with given Q,… · p. 12
• Transmit-position optimization has the same structure as receive-position optimization and can be solved similarly.Source: 3) Optimization of tn with given Q,… · pp. 13–14
• Channel reciprocity permits the transmit-covariance treatment to be applied to H(˜t, ˜r)H using an equivalent covariance S.Source: 3) Optimization of tn with given Q,… · p. 13

• SCA constructs a quadratic surrogate for the non-concave receive-position objective using gradient and Hessian information.Source: B. Solution for Problem (P2-m) · p. 14
• The surrogate globally lower-bounds the objective, so maximizing the original objective is replaced by maximizing the surrogate.Source: B. Solution for Problem (P2-m) · pp. 14–15
• Without violated region or separation constraints, the concave quadratic surrogate has a global optimum for the receive-MA position.Source: B. Solution for Problem (P2-m) · p. 15
• When the unconstrained update violates a non-convex constraint, first-order Taylor expansion linearizes the distance constraints.Source: B. Solution for Problem (P2-m) · p. 15
• The resulting problem is a quadratic program because its objective is quadratic and its constraints are linear in the receive position.Source: B. Solution for Problem (P2-m) · pp. 15–16
• The algorithm iterates until the objective increase falls below ǫ1, then outputs the updated receive-MA position.Source: B. Solution for Problem (P2-m) · p. 16
• The objective sequence is non-decreasing and converges to a maximum value, while the stated complexity is polynomial in N, Lr, and M.Source: B. Solution for Problem (P2-m) · pp. 16–17

• Algorithm 1 is modified to optimize each transmit MA position t_n by solving problem (P3-n).Source: C. Solution for Problem (P3-n) · pp. 17–18
• The position-update procedure guarantees monotonic convergence when solving problem (P3-n).Source: C. Solution for Problem (P3-n) · p. 17
• Algorithm 2 initializes parameters including the transmit and receive regions, distance constraints, and convergence tolerances before iterating.Source: C. Solution for Problem (P3-n) · p. 18
• The overall alternating-optimization procedure outputs the optimized transmit positions, receive positions, and covariance matrix Q.Source: C. Solution for Problem (P3-n) · p. 18

• Algorithm 2 alternates among three subproblems: optimizing Q, sequentially updating receive-MA positions, and sequentially updating transmit-MA positions.Source: D. Overall Algorithm · p. 18
• The iterations stop when the channel-capacity increase falls below the predefined tolerance ǫ2.Source: D. Overall Algorithm · p. 18
• The objective value is non-decreasing during iterations and is upper-bounded by a finite channel capacity.Source: D. Overall Algorithm · p. 18
• Any limit point generated by Algorithm 2 satisfies the Karush–Kuhn–Tucker condition of problem (P1).Source: D. Overall Algorithm · p. 18
• When no variable can further increase the objective, Algorithm 2 converges to an at least locally optimal solution of problem (P1).Source: D. Overall Algorithm · pp. 18–19
• Water-filling computations in steps 4 and 9 have complexity O(MN min(M, N)).Source: D. Overall Algorithm · p. 19

Initialization

• The initialization scheme arranges equal-radius, non-overlapping circles inside the transmit or receive region and uses their centers as initial MA positions.Source: E. Initialization · p. 19
• Circle packing is used because transmit and receive MAs should be sufficiently separated during initialization.Source: E. Initialization · p. 19
• Increasing MA separation can reduce coupling effects and make feasible high-capacity position updates more likely during SCA.Source: E. Initialization · p. 19

Low-SNR alternative

• In the asymptotically low-SNR regime, capacity maximization can be reformulated in terms of transmit and receive MA positions for lower-complexity optimization.Source: F. Alternative Solution for Problem… · p. 20
• At low SNR, the optimal transmission strategy allocates all transmit power to the strongest eigenchannel through single-stream beamforming.Source: F. Alternative Solution for Problem… · p. 20

Low-SNR case

• In the low-SNR regime, transmit and receive position updates optimize strongest-singular-vector-based channel expressions instead of the full capacity objective.Source: §2 · p. 20
• The low-SNR alternating procedure obtains a locally optimal capacity solution by iteratively optimizing the three variable sets.Source: §2 · p. 20

MISO case

• For MISO, maximum ratio transmission gives the optimal covariance matrix Q⋆, eliminating covariance updates from the alternating optimization.Source: G. Alternative Solution for Problem… · pp. 21–22
• With a single receive MA, the receive-position subproblem removes minimum-distance constraints and becomes convex after dropping the corresponding non-convex constraints.Source: G. Alternative Solution for Problem… · pp. 21–22

Complexity

• The alternative MISO and SIMO algorithms require much lower complexity than Algorithm 2.Source: G. Alternative Solution for Problem… · pp. 21–22

SIMO case

• For SIMO, the single transmit antenna simplifies the transmit covariance to Q⋆=P, and capacity has a form similar to the MISO case.Source: G. Alternative Solution for Problem… · pp. 21–23

Region size and SNR

• In both low- and high-SNR regimes, MA, RMA, and APS schemes outperform FPA systems, with gains increasing as the region size grows.The proposed algorithm achieves the best performance among all schemes for every region size and both SNR regimes.Source: §IV · pp. 24–25
• MA-enabled capacity converges across schemes when the normalized region size exceeds 4, indicating that finite transmit and receive regions can attain maximum capacity.The convergence is reported for the capacity-versus-region-size comparisons.Source: §IV · pp. 24–25

Benchmark comparisons

• Jointly optimizing transmit and receive MAs outperforms receive-only MA optimization, demonstrating an additional gain from optimizing both sides.This comparison is made against the RMA benchmark.Source: §IV · p. 25

SNR-dependent behavior

• SEPM nearly matches the proposed algorithm at low SNR but performs worst at high SNR because it concentrates power on the strongest eigenchannel and sacrifices other eigenchannels.At high SNR, capacity also reflects total channel power and condition number, so balanced eigenchannel power matters.Source: §IV · pp. 25–26

Channel properties

• The proposed algorithm increases channel total power while decreasing condition number as region size grows, yielding more balanced eigenchannel power and higher capacity.The paper reports higher total power and lower condition number than FPA, AS, RMA, and APS.Source: §IV · p. 26

Channel-path effects

• Capacity gaps between MA-based and FPA-based schemes increase with the number of channel paths, as stronger fading creates more regional capacity fluctuation and spatial degrees of freedom.The comparison covers different path counts and reports larger capacity for the proposed, RMA, and APS schemes than for FPA and AS.Source: §IV · p. 27

Contribution

• The proposed MA-enabled MIMO system exploits antenna-position optimization to increase channel capacity.Movable antennas provide position flexibility at the transmitter and receiver.Source: §V · p. 27

Main algorithm

• Capacity maximization jointly optimizes transmit and receive MA positions together with the transmit covariance matrix.The alternating procedure updates each transmit/receive MA position and the covariance matrix while holding the other variables fixed.Source: §V · p. 27

Special cases

• Lower-complexity algorithms are proposed for the asymptotically low-SNR regime and for MISO and SIMO channels.These alternatives complement the general alternating optimization approach.Source: §V · p. 27

Results

• Numerical results show significantly higher capacity for the proposed MA system than conventional FPA-based MIMO systems with or without antenna selection, especially in sufficiently large multipath regions.The conclusion attributes the improvement to joint transmit/receive position optimization and more favorable channel realizations.Source: §V · p. 27

Channel shaping

• Joint position optimization significantly improves MIMO channel power while reducing its condition number, producing more favorable channel realizations for capacity enhancement.The reported effect concerns the channel properties resulting from jointly optimized MA positions.Source: §V · pp. 27–28

Notation

• The appendix writes each entry of vector b in amplitude-phase form, separating amplitude |b_q| from phase ∠b_q.This notation is used before expressing ḡ(r_m).Source: A. Derivations of ∇¯g(rm) and ∇2¯g(rm) · p. 28

Derivation

• The Hessian matrix of ḡ(r_m) over r_m is represented explicitly as part of the appendix derivation.The displayed representation is introduced after defining ḡ(r_m).Source: A. Derivations of ∇¯g(rm) and ∇2¯g(rm) · p. 28

Curvature bound

• A scalar δ_m satisfying δ_m ≥ ||∇^2ḡ(r_m)||_2 yields the matrix inequality δ_mI_2 ⪰ ∇^2ḡ(r_m).This bound constrains the Hessian through its spectral norm.Source: B. Construction of δm · p. 29