Beamforming Codebooks for Two Transmit Antenna Systems Based on Optimum Grassmannian Packings

Precoding codebook design for limited feedback MIMO systems is known to reduce to a discretization problem on a Grassmann manifold. The case of two-antenna beamforming is special in that it is equivalent to quantizing the real sphere. The isometry between the Grassmannian G2,1ℂ and the real sphere S2 shows that discretization problems in the Grassmannian G2,1ℂ are directly solved by corresponding spherical codes. Notably, the Grassmannian line packing problem in ℂ2, namely maximizing the minimum distance, is equivalent to the Tammes problem on the real sphere, so that optimum spherical packings give optimum Grassmannian packings. Moreover, a simple isomorphism between G2,1ℂ and S2 enables to analytically derive simple codebooks in closed-form having low implementation complexity. Using the simple geometry of some of these codebooks, we derive closed-form expressions of the probability density function of the relative SNR loss due to limited feedback. We also investigate codebooks based on other spherical arrangements, such as solutions maximizing the harmonic mean of the mutual distances among the codewords, which is known as the Thomson problem. We find that in some special cases, Grassmannian codebooks based on these other spherical arrangements outperform codebooks from Grassmannian packing.