  In this paper, we design polar codes and polar lattices for i.i.d. fading
channels when the channel state information is only available to the receiver.
For the binary input case, we propose a new design of polar codes through
single-stage polarization to achieve the ergodic capacity. For the non-binary
input case, polar codes are further extended to polar lattices to achieve the
egodic Poltyrev capacity, i.e., the capacity without power limit. When the
power constraint is taken into consideration, we show that polar lattices with
lattice Gaussian shaping achieve the egodic capacity of fading channels. The
coding and shaping are both explicit, and the overall complexity of encoding
and decoding is $O(N \log^2 N)$.
