Flexible Codebook Design for Limited Feedback Systems via Sequential Smooth Optimization on the Grassmannian Manifold

Grassmannian quantization codebooks play a central role in a number of limited feedback schemes for single and multi-user multiple-input multiple-output (MIMO) communication systems. In practice, it is often desirable that these codebooks possess additional properties that facilitate their implementation, beyond the provision of good quantization performance. Although some good codebooks exist, their design tends to be a rather intricate task. The goal of this paper is to suggest a flexible approach to the design of Grassmannian codebooks based on sequential smooth optimization on the Grassmannian manifold and the use of smooth penalty functions to obtain additional desirable properties. As one example, the proposed approach is used to design rank-2 codebooks that have a nested structure and elements from a phase-shift keying (PSK) alphabet. In some numerical comparisons, codebooks designed using the proposed methods have larger minimum distances than some existing codebooks, and provide tangible performance gains when applied to a simple MIMO downlink scenario with zero-forcing beamforming, per-user unitary beamforming and rate control (PU2RC), and block diagonalization signalling. Furthermore, the proposed approach yields codebooks that attain desirable additional properties without incurring a substantial degradation in performance.