In this paper, we study a fading broadcast channel (BC) with perfect channel state information at the receiver (CSIR) and only a quantized version of it at the transmitter due to limited-rate links for channel feedback at each user. We find an achievable region for the fading BC under this condition using super-position coding and show that it is sum-rate optimal. We also derive a closed-form expression for finding channel partitioning, which turns out to be the same in form as that for water-filling of power over time in fading channels. Using the derived closed form expression with temporal water-filling of power at the transmitter in an iterative manner, we show numerically that a single iteration is adequate to achieve most of the capacity. Thus the complexity of finding the optimal (global maximum) or close to optimal (local maximum) channel partitioning is greatly reduced as compared to using a search-based k-mean clustering algorithm like Lloyd's algorithm that requires multiple iterations.
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