Wednesday, February 25, 2009

How Much Feedback Is Enough for MIMO? IV Channel Quantization

[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? III Codebook Design]
[How Much Feedback Is Enough for MIMO? V Feedback Reliabilities]
[How Much Feedback Is Enough for MIMO? VI Rank Deficiency]

Generally, MIMO channel quantizer or CQI generation, maps the input channel estimation vector to the index of a codeword in the codebook. The decode will do the reverse. It is similar to the vector coding in EVRC, AMR, MPEG-4, etc. The designing a best codebook as well as finding the general boundary of Voronoi cell is NP-hard.

Figure 1. MIMO Precoding Mismatching

With Figure 1, it is shown that there are multiple issues involving MIMO precoding mismatching. In most existing MIMO beamforming systems, the receiver tracks the channel norm information for link adaptation purpose and the phase information for beamforming precoding. In this case, Dh can be rewritten by

Dh = 2M σh2 [ 1 - ( 1- Dθ )1/2 ]

with Dθ denoting the phase quantization deviation. A lower bound for distortion rate of channel quantization for MIMO beamforming is

R( Dθ )   ≥ max{ - M log2[ 2 - 2( 1- Dθ )1/2 + σh2h2 ], 0 }

for each beam. With the feedback rate of R, it also tells us that the minimum precoding mismatch for forwardlink MIMO beamforming is

Dθ ≤ 1-[1-2-R/M-1 + σh2/(2σh2)]2

Figure 2. The rate-distortion region with M = 4

Tuesday, February 17, 2009

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 cellVi : 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( V) = 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