  This work presents a polar coding scheme for fading channels, focusing
primarily on fading binary symmetric and additive exponential noise channels.
For fading binary symmetric channels, a hierarchical coding scheme is
presented, utilizing polar coding both over channel uses and over fading
blocks. The receiver uses its channel state information (CSI) to distinguish
states, thus constructing an overlay erasure channel over the underlying fading
channels. By using this scheme, the capacity of a fading binary symmetric
channel is achieved without CSI at the transmitter. Noting that a fading AWGN
channel with BPSK modulation and demodulation corresponds to a fading binary
symmetric channel, this result covers a fairly large set of practically
relevant channel settings.
  For fading additive exponential noise channels, expansion coding is used in
conjunction to polar codes. Expansion coding transforms the continuous-valued
channel to multiple (independent) discrete-valued ones. For each level after
expansion, the approach described previously for fading binary symmetric
channels is used. Both theoretical analysis and numerical results are
presented, showing that the proposed coding scheme approaches the capacity in
the high SNR regime. Overall, utilizing polar codes in this (hierarchical)
fashion enables coding without CSI at the transmitter, while approaching the
capacity with low complexity.
