  A polar coding scheme for fading channels is proposed in this paper. More
specifically, the focus is Gaussian fading channel with a BPSK modulation
technique, where the equivalent channel could be modeled as a binary symmetric
channel with varying cross-over probabilities. To deal with variable channel
states, a coding scheme of hierarchically utilizing polar codes is proposed. In
particular, by observing the polarization of different binary symmetric
channels over different fading blocks, each channel use corresponding to a
different polarization is modeled as a binary erasure channel such that polar
codes could be adopted to encode over blocks. It is shown that the proposed
coding scheme, without instantaneous channel state information at the
transmitter, achieves the capacity of the corresponding fading binary symmetric
channel, which is constructed from the underlying fading AWGN channel through
the modulation scheme.
