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MISO Capacity with Per-Antenna Power Constraint

Mai Vu

arXiv:1003.1738v2cs.IT

TL;DR

The paper studies how realistic per-antenna power limits affect capacity and optimal signaling in single-user MISO channels. It derives closed-form solutions for constant channels known to both terminals and Rayleigh fading channels known only to the receiver. The constant-channel solution uses phase-matched beamforming with power-dependent amplitudes, while the fading solution uses independent antenna signals, and both capacities are usually below the sum-power case.

  • Problem

    Per-antenna-constrained MISO capacity and optimal signaling are less well understood than under sum power, despite practical RF-chain and distributed-antenna constraints.

  • Method

    The paper solves the capacity optimization in closed form, using a relaxed-problem argument for constant channels and Rayleigh-distribution symmetry for fading channels.

  • Results

    Constant channels use beamforming with channel-phase-matched, power-dependent amplitudes; Rayleigh fading channels use independent signals with constrained powers, and per-antenna capacity is usually below sum-power capacity.

  • Takeaways & Limitations

    Per-antenna constraints lead to signaling rules distinct from sum-power beamforming: amplitudes need not track channel amplitudes, while fading favors independent antenna signals.

  • Takeaways & Limitations

    The capacity of a constant MIMO channel with per-antenna constraint remains an open problem, although the Rayleigh-fading proof generalizes directly to MIMO fading.

Abstract

from arXiv · show

We establish in closed-form the capacity and the optimal signaling scheme for a MISO channel with per-antenna power constraint. Two cases of channel state information are considered: constant channel known at both the transmitter and receiver, and Rayleigh fading channel known only at the receiver. For the first case, the optimal signaling scheme is beamforming with the phases of the beam weights matched to the phases of the channel coefficients, but the amplitudes independent of the channel coefficients and dependent only on the constrained powers. For the second case, the optimal scheme is to send independent signals from the antennas with the constrained powers. In both cases, the capacity with per-antenna power constraint is usually less than that with sum power constraint.

I. INTRODUCTION

The paper addresses the less-understood MISO capacity problem under realistic per-antenna power constraints and derives closed-form capacity and signaling results for constant and Rayleigh fading channels.

  • Motivation: Per-antenna constraints reflect individual RF-chain limits and distributed antennas that cannot share power, making their capacity and signaling schemes practically relevant.Unlike sum power, these constraints prevent arbitrary power allocation across antennas.
  • Contribution: The paper establishes closed-form capacity and optimal signaling for single-user MISO channels under per-antenna power constraints.It considers constant channels known to both terminals and Rayleigh fading channels known only to the receiver.
  • Contribution: For constant channels, the optimal scheme beamforms with beam-weight phases matched to channel phases, while amplitudes depend only on constrained powers.The amplitudes are not matched to channel amplitudes.
  • Contribution: For Rayleigh fading known only at the receiver, the optimal scheme sends independent signals from the transmit antennas using their constrained powers.The proof uses symmetry of the Rayleigh fading distribution.
  • Problem formulation: The paper formulates capacity by optimizing the covariance of a zero-mean Gaussian input subject to the applicable power constraint.The remaining task is selecting the covariance that maximizes the achievable rate.

B. Power constraints

The paper contrasts sum, independent multiple-access, and per-antenna power constraints for MISO channels, then develops the constant-channel analysis under these alternatives.

  • Power constraints: Under a sum power constraint, total power P can be shared arbitrarily across antennas, represented by tr(Q) ≤ P.This is the standard comparison constraint used in the paper.
  • Power constraints: Under independent multiple-access constraints, each antenna operates independently with its own power budget, modeled by Q = diag{P1, . . . , Pn}.This models distributed antennas without explicit cooperation in coding and signal design.
  • Power constraints: Under per-antenna constraints, each antenna has a separate budget Pi while the antennas can fully cooperate in a centralized or cooperative distributed MISO system.The constraint arises from the individual limits of each transmit RF chain.
  • Constant channel: The constant-channel analysis optimizes the transmit covariance for a channel known at both transmitter and receiver, contrasting per-antenna signaling with established alternatives.The covariance optimization is convex, although the supplied passage states that no closed-form solution was previously available for the per-antenna problem.
  • Constant channel: For the sum-power constant-channel problem, the optimal covariance allocates all transmit power along the channel-conjugate eigenvector, yielding single-mode beamforming.The beamforming vector is h*/∥h∥.

2) Independent multiple-access capacity:

For the per-antenna constrained MISO channel, the optimal signaling depends on channel knowledge: constant known channels use beamforming, while receiver-only fading uses independent antenna signals. In the constant-channel case, closed-form analysis shows that beamforming matches channel phases while fixing amplitudes from the antenna powers.

  • Fading channel: For receiver-only fading, the covariance is fixed as Q = diag{P_1, …, P_n}, so antennas transmit different independent symbols with their constrained powers.The section identifies this case as requiring no covariance optimization.
  • Optimization: The capacity problem is convex because each constraint q_ii ≤ P_i is linear in the covariance matrix Q, although no closed-form solution was previously available.The paper obtains a closed-form solution by applying a matrix-minor relaxation and showing its optimum also solves the original problem.
  • Constant channel: Closed-form analysis shows that the optimal covariance has rank one, so the MISO channel's optimal signaling is beamforming.The solution is obtained by relaxing the positive-semidefinite constraint and proving the relaxed optimum remains feasible; rank(Q⋆)=1 for MISO.
  • Constant channel: For constant channels known at both ends, beamforming matches each channel coefficient's phase, while its amplitude is independent of the channel and fixed by P_i.There is no power allocation among antennas: antenna i transmits with power P_i.
  • Constant channel: Beamforming capacity increases as the angle between the beam weight and channel decreases; complete phase-and-amplitude matching achieves the maximum under sum power.The per-antenna solution is constrained by fixed antenna powers, unlike the fully matched sum-power beamformer.
  • Constant channel: The constrained beamformer generally does not fully align with the channel, except when √P_k is proportional to |h_k| across antennas.In other cases, the angle between the beamforming vector and channel is positive, but the solution still gives the largest rate without power allocation.

C. Numerical examples

The numerical example evaluates a two-antenna MISO channel with a fixed complex channel and matches total sum-constrained power to total per-antenna power for a fair comparison.

  • C. Numerical examples: The example uses h = [0.3 + 0.2i 0.4 − 0.7i]^T with two transmit antennas and total power P = 10.The per-antenna powers are selected so that P_1 + P_2 = P for the sum-power comparison.

1) With 2 transmit antennas:

For two transmit antennas, beamforming improves capacity over independent signaling, while per-antenna power is generally below sum-power performance except at one matching allocation.

  • Beamforming increases capacity over independent multiple-access signaling by correlating the transmit signals.
  • Figure 1 compares capacities under sum-power, independent multiple-access, and per-antenna constraints for a 2 × 1 constant channel.
  • At P1 = P2 = 5, per-antenna capacity is about 93% of sum-power capacity and almost 30% above multiple-access capacity.
  • For more than two antennas, the section considers capacity as a function of antenna count, with equal per-antenna power budgets in the symmetric comparison.

2) With n transmit antennas:

With n transmit antennas, symmetric channels make per-antenna and sum-power capacities equal, whereas the non-symmetric comparison shows per-antenna capacity nearly matching sum-power capacity and exceeding multiple-access capacity.

  • For hk = k, per-antenna capacity is almost as high as sum-power capacity, and both are significantly better than multiple-access capacity.
  • For Rayleigh fading, the transmitter lacks channel knowledge while the receiver knows the channel vector perfectly.
  • Under sum-power signaling, the optimal covariance is proportional to the identity, so antennas transmit independent signals with equal power.
  • Compared with the constant-channel expression, the fading instantaneous capacity includes a factor of n dividing the power because the transmitter lacks channel information.

B. MISO capacity with per-antenna power constraint

Under per-antenna constraints, the Rayleigh fading problem is solved using symmetry rather than eigenvalue equivalence, yielding full-power independent signaling and no average-capacity gain from transmitter cooperation.

  • The per-antenna problem imposes diagonal constraints qii ≤ Pi on a Hermitian covariance matrix Q.
  • Eigenvalue decomposition cannot reduce this problem to the standard fading analysis because eigenvalue constraints differ from diagonal-entry constraints.
  • The optimal Rayleigh-fading covariance is Q = diag{P1, . . . , Pn}, so every antenna sends an independent signal at full power.
  • Without transmitter channel information, cooperation among antennas does not increase average capacity under per-antenna constraints.
  • The proof uses permutations and concavity of the logarithm, with equality only when Pi = P/n for every antenna.
  • With full channel state information, the optimal strategy is beamforming, whereas without transmitter channel information it is independent signaling.

C. Numerical examples

The numerical example compares ergodic capacities for a two-antenna Rayleigh fading MISO channel under different power constraints. In fading channels, the gap between sum-power and per-antenna capacities is smaller, and the capacities coincide when P1 = P/2 = 5.

  • Figure 3 plots ergodic capacities for a 2×1 Rayleigh fading channel as functions of the first antenna's transmit-power constraint.
  • The capacity gap between sum-power and per-antenna constraints is smaller for fading channels than for constant channels.
  • The two capacities are equal when P1 = P/2 = 5.

APPENDIX

The appendix solves the constant-channel covariance optimization by relaxing the positive-semidefiniteness constraints to 2×2 principal-minor constraints. The resulting covariance is shown to satisfy the original constraints and to be rank one.

  • The relaxed problem replaces the full positive-semidefiniteness constraint with constraints on all 2×2 principal minors.The relaxation imposes qii = Pi and M(ij) ≽ 0, equivalently |qij|2 ≤ PiPj.
  • The optimal off-diagonal entries saturate their bounds, satisfying |qij|2 = PiPj.Increasing an unsaturated qij in the channel-aligned direction increases the objective.
  • The resulting covariance is positive semidefinite with one positive eigenvalue λ1 = P Pi and n−1 zero eigenvalues.
  • The corresponding eigenvector is specified by the elements given in equation (8).

B. The rank of the optimal transmit covariance for constant channels

For the constant-channel problem, the covariance optimization is convex and can be analyzed with Lagrangian conditions. Complementary slackness yields an upper bound on the optimal covariance rank.

  • The constant-channel covariance problem is convex, so the Lagrangian method obtains its exact solution.
  • The stationarity and complementary-slackness conditions imply DQ = h∗hT Q and rank(Q⋆) ≤ rank(h).
  • At optimum, the per-antenna power constraints are active, making the diagonal dual matrix full rank.The associated dual variables are strictly positive because unused antenna power could increase the rate.

C. Optimal transmit covariance for Rayleigh fading channels

For Rayleigh fading known only at the receiver, symmetry of the channel distribution forces the optimal transmit covariance to be diagonal. Thus each antenna sends an independent signal using its constrained power.

  • The diagonal entries meet their individual power constraints, while the off-diagonal correlation constraint is inactive.
  • The proof uses invariance under sign flips of individual Rayleigh fading coefficients.
  • For two antennas, the symmetry argument gives q = 0, so the covariance is diagonal.
  • The optimal covariance for any number of antennas is Q = diag{P1, . . . , Pn}.Sign symmetry of the i.i.d. zero-mean Gaussian coefficients makes all off-diagonal entries zero at the optimum.
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