[Paper Review] The Capacity of MIMO Channels with Per-Antenna Power Constraint

Mai Vu|arXiv (Cornell University)|Jun 24, 2011

Advanced MIMO Systems OptimizationEngineering4 references20 citations

TL;DR

This paper establishes the capacity and optimal input signaling for MIMO channels under per-antenna power constraints, formulating the problem as a semidefinite program (SDP) and deriving a closed-form solution for the optimal input covariance matrix in terms of dual variables. The proposed iterative algorithm efficiently computes the optimal beamforming strategy, showing that eigenbeamforming is suboptimal and independent signaling can outperform it; capacity under per-antenna constraints can approach sum-power capacity when constraints are balanced but diverges under skewness.

ABSTRACT

We establish the optimal input signaling and the capacity of MIMO channels under per-antenna power constraint. While admitting a linear eigenbeam structure, the optimal input is no longer diagonalizable by the channel right singular vectors as with sum power constraint. We formulate the capacity optimization as an SDP problem and solve in closed-form the optimal input covariance as a function of the dual variable. We then design an efficient algorithm to find this optimal input signaling for all channel sizes. The proposed algorithm allows for straightforward implementation in practical systems in real time. Simulation results show that with equal constraint per antenna, capacity with per-antenna power can be close to capacity with sum power, but as the constraint becomes more skew, the two capacities diverge. Forcing input eigenbeams to match the channel right singular vectors achieves no improvement over independent signaling and can even be detrimental to capacity.

Motivation & Objective

• To determine the capacity and optimal input signaling for MIMO channels under per-antenna power constraints, which are more realistic than sum power constraints due to individual RF chain power amplifiers.
• To address the lack of closed-form solutions for single-user MIMO capacity under per-antenna constraints, despite known solutions under sum power constraints.
• To develop an efficient, real-time implementable algorithm for computing the optimal input covariance matrix under per-antenna constraints.
• To evaluate the performance gap between per-antenna and sum power constraints, particularly under channel asymmetries and different signaling strategies.

Proposed method

• Formulates the MIMO capacity optimization problem under per-antenna power constraints as a semidefinite program (SDP) with a dual variable parameterization.
• Derives a closed-form expression for the optimal input covariance matrix as a function of the dual variable, enabling efficient computation.
• Proposes an iterative algorithm that updates the dual variable and computes the optimal input covariance in real time, suitable for practical implementation.
• Uses uplink-downlink duality and matrix identity expansions to analyze optimality conditions and derive the structure of the optimal solution.
• Applies singular value decomposition (SVD) and matrix partitioning techniques to decompose the problem into manageable subproblems for both n ≥ m and n < m cases.
• Employs convergence analysis and matrix inequalities to prove the monotonic decrease of the dual variable sequence, ensuring algorithm convergence.

Experimental results

Research questions

• RQ1 Is eigenbeamforming optimal for MIMO channels under per-antenna power constraints, or does it degrade performance compared to independent signaling?
• RQ2 Can a closed-form solution be derived for the optimal input covariance matrix under per-antenna constraints, and if so, how does it depend on the dual variable?
• RQ3 How does the capacity under per-antenna constraints compare to that under sum power constraints, especially under asymmetric power constraints?
• RQ4 What is the performance gap between optimal signaling and independent signaling (equal power per antenna) under per-antenna constraints?
• RQ5 Can an efficient, real-time algorithm be designed to compute the optimal input covariance for any MIMO size under per-antenna constraints?

Key findings

• Eigenbeamforming based on channel right singular vectors is suboptimal under per-antenna constraints and can even reduce capacity compared to independent signaling.
• The optimal input covariance matrix is not diagonalizable by the channel’s right singular vectors, unlike in the sum power case.
• The proposed iterative algorithm converges and allows real-time implementation of the optimal signaling strategy.
• With balanced per-antenna power constraints, the capacity under per-antenna constraints is close to that under sum power constraints.
• As the per-antenna power constraints become more skewed, the capacity gap between per-antenna and sum power constraints increases significantly.
• Independent signaling (equal power per antenna) can outperform eigenbeamforming under per-antenna constraints, especially in asymmetric scenarios.