How Much Feedback Is Enough for MIMO? III Codebook Design

[How Much Feedback Is Enough for MIMO? I Introduction]
[How Much Feedback Is Enough for MIMO? II Channel Estimation]
[How Much Feedback Is Enough for MIMO? IV Channel Quantization]
[How Much Feedback Is Enough for MIMO? V Feedback Reliabilities]
[How Much Feedback Is Enough for MIMO? VI Rank Deficiency]

Figure 2. Voronoi cell and various bounds

MIMO beamforming mismatch upper bound depends on the codebook design. The maximum MIMO beamforming mismatch can be determined by the largest radius of the codebook’s Voronoi cell { Vi : 1   ≤ i ≤ 2R }, which in general is the solution to the disk-covering problem that still is open. Instead of finding the exact boundary for the Voronoi cell Vi, a heuristic approach using sphere-packing bound and sphere cap to approximate the actual polytope boundary can be used. The result is an approximate of the sphere packing solution, in which all spheres are supposed to be non-overlappedly placed. With this approach, sphere caps are overlapped with each other in space but the interior of them has the same area as the Voronoi cell. The border of this sphere cap is named sphere-packing boundary. The relationship between sphere-packing boundary and Voronoi cell is shown in Fig. 2. For an uniform random codebook of size 2R in M-dimensional Euclid space, the area of a Voronoi cell is given by

A( Vk ) = 2πM / [ 2R Γ(M) ]

where Γ(*)denotes the gamma function. A heuristic upper bound of MIMO beamforming mismatch is given by

Dθ ≥ { (M-1)/M [ 1 - ( 1 - 2-R )1/(M-1) ] }1/2

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