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Adaptive Training for Correlated Fading Channels With Feedback

机译：具有反馈的相关衰落信道的自适应训练

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摘要

We consider data transmission through a time-selective, correlated (first-order Markov) Rayleigh fading channel subject to an average power constraint. The channel is estimated at the receiver with a pilot signal, and the estimate is fed back to the transmitter. The estimate is used for coherent demodulation, and to adapt the data and pilot powers. We derive the Hamilton--Jacobi--Bellman (HJB) equation for the optimal policy in a continuous-time limit where the channel state evolves as an Ornstein--Uhlenbeck diffusion process, and is estimated by a Kalman filter at the receiver. Finding an explicit solution to the HJB equation, as well as proving that a (twice-differentiable) solution exists, appears to be quite challenging. However, assuming that such a solution does exist, we explicitly determine the optimal pilot and data power control policies. The optimal pilot policy switches between zero and the maximum (peak-constrained) value (“bang-bang” control), and approximates the optimal discrete-time policy at low signal-to-noise ratios (SNRs) (equivalently, large bandwidths). The switching boundary is defined in terms of the system state (estimated channel mean and associated error variance), and can be explicitly computed. Under the optimal policy, the transmitter conserves power by decreasing the training power when the channel is faded, thereby increasing the data rate. Numerical results show a significant increase in achievable rate due to the adaptive training scheme with feedback, relative to constant (nonadaptive) training, which does not require feedback. The gain is more pronounced at relatively low SNRs and with fast fading. Results are further verified through Monte Carlo simulations.
机译：我们考虑通过时间选择的，相关的（一阶马尔可夫）瑞利衰落信道进行数据传输，该信道受平均功率约束。在接收机处利用导频信号估计该信道，并将该估计反馈给发射机。该估计用于相干解调，并适应数据和导频功率。我们推导了在连续时间限制内最优策略的汉密尔顿-雅各比-贝尔曼（HJB）方程，其中通道状态随着Ornstein-Uhlenbeck扩散过程而演变，并由接收机处的卡尔曼滤波器进行估计。找到HJB方程的显式解，以及证明存在（两次可微分）解，似乎是非常具有挑战性的。但是，假设确实存在这样的解决方案，我们将明确确定最佳的导频和数据功率控制策略。最佳导频策略在零和最大（峰值约束）值之间切换（“砰砰”控制），并在低信噪比（SNR）（等效地，大带宽）下近似最佳离散时间策略。切换边界是根据系统状态（估计的通道平均值和相关的误差方差）定义的，并且可以明确地进行计算。在最佳策略下，当信道衰落时，发射机通过降低训练功率来节省功率，从而提高数据速率。数值结果表明，相对于不需要反馈的恒定（非自适应）训练，由于带有反馈的自适应训练方案的可实现率显着提高。在相对较低的SNR和快速衰落的情况下，增益更加明显。通过蒙特卡洛模拟进一步验证了结果。

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