Geodesical Codebook Design for Precoded MIMO Systems

Abstract

We propose a numerical method for finding packings of multiple-input and multiple-output (MIMO) semi-unitary precoding matrices in Grassmannian manifolds with different metrics. The proposed expansion-compression algorithm ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ECA</i> ) is practical, simple and produces efficient packings without the need for a sophisticated initialization. With chordal distance metric, the algorithm tends to converge into a degenerated point constellation, where two points contain identical as well as orthogonal columns and distance between them cannot increase further along geodesic. Therefore, we alternate between <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max-min<i/> and <i>min-max</i> clustering parts of <i>ECA</i> algorithm, where the latter prevents degenerated constellations. With Fubini-Study distance metric, the algorithm converges to best known packings without extra <i>min-max</i> processing.</i>