On the Compound Capacity of a Class of MIMO Channels Subject to Normed Uncertainty
On the Compound Capacity of a Class of MIMO Channels Subject to Normed Uncertainty
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DOI：
10.1109/tit.2011.2173727
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发表时间：
2012-04
影响因子：
2.5
通讯作者：
S. Loyka;C. Charalambous
中科院分区：
文献类型：
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作者：
S. Loyka;C. Charalambous
当信道由（已知）标称信道和约束范数（未知）不确定性描述的类建模时，考虑不确定多输入多输出信道的复合容量。在此框架内，使用奇异值分解和相关奇异值不等式作为主要工具，研究了受谱范数约束的加性和乘性不确定性的两类类别。复合容量是最大最小互信息，表示类别的容量，其中最小化是在信道类别上完成的，而最大化是在传输协方差上完成的。获得了这些类的复合容量的封闭式解，并提出了与发射和接收特征向量相关的几个属性。结果表明，在某些条件下，该类的复合容量等于最坏情况下的信道容量，从而建立了鞍点特性。针对最坏情况的信道不确定性和实现容量的传输协方差矩阵给出了显式的封闭式解决方案：实现复合容量的最佳传输策略是在标称（已知）信道本征模式上进行多重波束成形，并通过在降低的 SNR 下注水进行波束功率分布。随着不确定性的增加，使用的本征模会越来越少，直到只有最强的本征模保持活动状态，因此，在给出了明确条件的大不确定性情况下，发射波束成形是一种最佳的鲁棒传输策略。利用这些结果，针对其他有界不确定性构建了复合容量的上限和下限，并指出了一些通用属性。结果扩展到复合多址和广播频道。在所有考虑的情况下，信道不确定性所付出的代价是与频谱范数测量的不确定性设置半径相称的 SNR 损失（或者等效地，标称信道退化），并且最佳信令策略是在退化的标称信道上传输。
The compound capacity of uncertain multiple-input multiple-output channels is considered, when the channel is modeled by a class described by a (known) nominal channel and a constrained-norm (unknown) uncertainty. Within this framework, two types of classes are investigated with additive and multiplicative uncertainties subject to a spectral norm constraint, using the singular value decomposition and related singular value inequalities as the main tools. The compound capacity is a maxmin mutual information, representing the capacity of the class, in which the minimization is done over the class of channels while the maximization is done over the transmit covariance. Closed-form solutions for the compound capacity of the classes are obtained and several properties related to transmit and receive eigenvectors are presented. It is shown that, under certain conditions, the compound capacity of the class is equal to the worst-case channel capacity, thus establishing a saddle-point property. Explicit closed-form solutions are given for the worst-case channel uncertainty and the capacity-achieving transmit covariance matrix: the best transmission strategy achieving the compound capacity is a multiple beamforming on the nominal (known) channel eigenmodes with the beam power distribution via the water filling at a degraded SNR. As the uncertainty increases, fewer eigenmodes are used until only the strongest one remains active so that transmit beamforming is an optimal robust transmission strategy in this large-uncertainty regime, for which explicit conditions are given. Using these results, upper and lower bounds of the compound capacity are constructed for other bounded uncertainties and some generic properties are pointed out. The results are extended to compound multiple-access and broadcast channels. In all considered cases, the price to pay for channel uncertainty is an SNR loss (or, equivalently, the nominal channel degradation) commensurate with the uncertainty set radius measured by the spectral norm and the optimal signaling strategy is the transmission on the degraded nominal channel.