Flexible codebook design for limited feedback downlink systems via 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 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 smooth optimization algorithms for the Grassmannian manifold and the use of smooth penalty functions to obtain additional desirable properties. As one example, rank-2 codebooks with a nested structure and elements from a discrete alphabet are designed. In some numerical comparisons, codebooks designed using the proposed approach have better Fubini-Study distance properties than some existing codebooks, and provide tangible performance gains when applied to a simple MIMO downlink scenario with zero-forcing beamforming, PU²RC, and block diagonalization signalling. Furthermore, the proposed approach yields codebooks that attain desirable additional properties without incurring a substantial degradation in performance.