[Paper Review] Hybrid Precoding For Millimeter Wave MIMO Systems: A Matrix Factorization Approach

Juening Jin, Yahong Rosa Zheng+2 more|arXiv (Cornell University)|Mar 26, 2020

Millimeter-Wave Propagation and Modeling21 references35 citations

TL;DR

This paper proposes a matrix factorization framework for hybrid precoding in mmWave MIMO with finite-alphabet inputs, including conditions for optimality and a BFGS-based algorithm.

ABSTRACT

This paper investigates the hybrid precoding design for millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems with finite-alphabet inputs. The precoding problem is a joint optimization of analog and digital precoders, and we treat it as a matrix factorization problem with power and constant modulus constraints. Our work presents three main contributions: First, we present a sufficient condition and a necessary condition for hybrid precoding schemes to realize unconstrained optimal precoders exactly when the number of data streams Ns satisfies Ns = minfrank(H);Nrfg, where H represents the channel matrix and Nrf is the number of radio frequency (RF) chains. Second, we show that the coupled power constraint in our matrix factorization problem can be removed without loss of optimality. Third, we propose a Broyden-Fletcher-Goldfarb-Shanno (BFGS)-based algorithm to solve our matrix factorization problem using gradient and Hessian information. Several numerical results are provided to show that our proposed algorithm outperforms existing hybrid precoding algorithms.

Motivation & Objective

• Investigate hybrid precoding design for mmWave MIMO with finite-alphabet inputs.
• Formulate hybrid precoding as a matrix factorization problem under power and constant modulus constraints.
• Derive conditions under which hybrid precoding can realize unconstrained optimal precoders.
• Show that the coupled power constraint can be removed without loss of optimality.
• Develop an efficient BFGS-based algorithm with gradient and Hessian information.

Proposed method

• Model the mmWave MIMO system with analog and digital precoders under constant modulus constraints for the analog part.
• Formulate the design as min ||F_opt − F_RF F_BB||_F^2 subject to a power constraint, where F_opt is the unconstrained optimum.
• Prove equivalence between the original problem and a relaxed formulation where the power constraint is removed at optimality.
• Reformulate the problem to optimize only the phase of F_RF via F_RF = (1/√Nt) exp(j Φ_RF) and reduce to a single-variable (phase) optimization problem.
• Derive closed-form expressions for the gradient and Hessian of the objective with respect to F_RF, enabling a BFGS-based algorithm.
• Address nonconvexity and saddle points by restricting the optimization to a class with a zero first row in the phase matrix.

Experimental results

Research questions

• RQ1 Under what conditions can hybrid precoding realize unconstrained optimal precoders exactly given the mmWave channel structure?
• RQ2 Can the power constraint in the matrix factorization formulation be removed without loss of optimality?
• RQ3 How can the phase of the analog precoder be optimized efficiently to approach the unconstrained optimum?
• RQ4 What are the gradient and Hessian needed to implement a BFGS-based solver for the phase-based reformulation?
• RQ5 How does the proposed method compare to existing hybrid precoding algorithms under finite-alphabet inputs?

Key findings

• A sufficient condition exists where hybrid precoding can realize unconstrained optimal precoders exactly when the number of data streams satisfies Ns = min{rank(H), Nrf}.
• A necessary condition is provided for the existence of an S making UF S lie in the feasible set; this condition uses a rank bound on a constructed matrix KF.
• Removing the power constraint does not affect optimality at KKT points, enabling a simpler formulation without loss of optimality.
• A BFGS-based algorithm is developed using gradient and Hessian information to solve the matrix factorization with constant modulus constraints.
• The approach outperforms existing hybrid precoding algorithms in numerical results (as claimed in abstract).