Joint Grassmann-Stiefel Quantization for MIMO Product Codebooks

We consider product codebook strategy where a single small codebook is implemented at the receiver to quantize larger multi-input multi-output (MIMO) channels, e.g. aggregate channels of cooperative MIMO base stations. The present work focuses on the codebook design, codebook construction, and codeword selection under this scenario, for single- or multistream MIMO transmission. Designing point-to-point unitary precoding codebook is related to a discretization problem on the Grassmann manifold, where a Grassmannian codeword is an equivalence class of rectangular-unitary/Stiefel matrices. For practical needs, one has to choose the rectangular unitary matrix to represent each Grassmann codeword. In this paper, we choose appropriate representatives so that product codebook quantization becomes competitive with global Grassmannian quantization. For this, we propose a novel joint Grassmann-Stiefel codebook design aiming at good quantization/discretization of Grassmann and Stiefel manifolds with a single codebook. To find low-distortion codebooks, we present a vector quantizer generating a Stiefel codebook conditioned on a fixed Grassmann codebook. For this purpose, we provide an exact solution for computing centroids in the Stiefel manifold with chordal distance. Furthermore, concrete examples of analytical joint Grassmann-Stiefel packings are given. Finally, we discuss low-complexity codeword selection methods.