Beamforming Codebooks for Two Transmit Antenna Systems Based on Optimum Grassmannian Packings
Beamforming Codebooks for Two Transmit Antenna Systems Based on Optimum Grassmannian Packings
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DOI：
10.1109/tit.2011.2165820
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发表时间：
2011-10
影响因子：
2.5
通讯作者：
Renaud-Alexandre Pitaval;Helka-Liina Määttänen;Karol Schober;O. Tirkkonen;R. Wichman
中科院分区：
文献类型：
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作者：
Renaud-Alexandre Pitaval;Helka-Liina Määttänen;Karol Schober;O. Tirkkonen;R. Wichman
已知有限反馈MIMO系统的预编码码本设计可归结为格拉斯曼流形上的离散化问题。双天线波束成形的情况是特殊的，因为它等效于量化真实的球体。Grassmannian G2，1与真实的球面S2的等距性表明Grassmannian G2，1中的离散化问题可以直接用相应的球面码来解决.值得注意的是，图2中的格拉斯曼线填充问题，即最大化最小距离，等价于真实的球面上的Tammes问题，因此最优球面填充给出最优格拉斯曼填充。此外，G2、1和S2之间的简单同构使得能够分析地导出具有低实现复杂度的封闭形式的简单码本。使用简单的几何形状的一些这些码本，我们推导出封闭形式的概率密度函数的表达式，由于有限的反馈的相对SNR损失。我们还研究了基于其他球形排列的码本，例如最大化码字之间相互距离的调和平均值的解决方案，这被称为汤姆森问题。我们发现，在某些特殊情况下，格拉斯曼码书的基础上，这些其他的球形安排优于格拉斯曼包装的码书。
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.