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Ensemble Properties of RVQ-Based Limited-Feedback Beamforming Codebooks

机译：基于RVQ的有限反馈波束成形密码本的合奏特性

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The ensemble properties of random vector quantization (RVQ) codebooks for limited-feedback beamforming in multiinput multioutput (MIMO) systems are studied with the metrics of interest being the received $ {ssr SNR} $ loss and mutual information loss, both relative to a perfect channel state information (CSI) benchmark. The simplest case of unskewed codebooks is first studied in the correlated MIMO setting and these loss metrics are computed as a function of the number of bits of feedback ($B$), transmit antenna dimension ( $N_{t}$), and spatial correlation. In particular, it is established that: 1) the loss metrics are a product of two components—a quantization component and a channel-dependent component; 2) the quantization component, which is also common to analysis of channels with i.i.d. fading, decays as $B$ increases at the rate $2^{-B/(N_{t}-1)}$ ; 3) the channel-dependent component reflects the condition number of the channel. Further, the precise connection between the received ${ssr SNR}$ loss and the squared singular values of the channel is shown to be a Schur-convex majorization relationship. Finally, the ensemble properties of skewed codebooks that are generated by skewing RVQ codebooks with an appropriately designed fixed skewing matrix are studied. Based on an estimate of the loss expression for skewed codebooks, it is established that the structure of a good skewing matrix is critically dependent on the condition numbers of the effective channel (product of the true channel and the skewing matrix) and the skewing matrix.
机译：研究了多输入多输出（MIMO）系统中有限反馈波束形成的随机矢量量化（RVQ）码本的合奏特性，感兴趣的指标是相对于理想情况的接收$ {ssr SNR} $损耗和互信息损耗。通道状态信息（CSI）基准。首先在相关MIMO设置中研究未偏斜码本的最简单情况，并根据反馈位数（$ B $），发射天线尺寸（$ N_ {t} $）和空间来计算这些损耗度量相关性。特别地，可以确定：1）损耗度量是两个分量的产物-量化分量和信道相关分量； 2）量化分量，这对于使用i.d.衰落，随着$ B $以$ 2 ^ {-B /（N_ {t} -1）} $的速率增加而衰减； 3）通道相关组件反映了通道的条件编号。此外，所接收的$ {ssr SNR} $损失与信道的奇异值平方之间的精确连接被显示为舒尔凸主化关系。最后，研究了通过适当设计固定的偏斜矩阵来偏斜RVQ码本所产生的偏斜码本的整体性质。基于对偏斜码本的损失表达的估计，可以确定一个良好的偏斜矩阵的结构关键取决于有效通道（真实通道和偏斜矩阵的乘积）和偏斜矩阵的条件数。

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