  We propose a novel scheme for rate-compatible arbitrary-length polar code
construction for the additive white Gaussian noise (AWGN) channel. The proposed
scheme is based on the concept of non-uniform channel polarization. The
original polar codes can only be designed with code lengths that are powers of
two. Puncturing, shortening and extension are three strategies to obtain
arbitrary code lengths and code rates for polar codes. There are other ways to
design codes with arbitrary length but which have encoding and decoding with
higher complexity such as multi-kernel, concatenated codes and specific
constructions for Belief propagation (BP) or Successive Cancellation (SC)
decoding. In general, the quality of the projected bit channels by these
arbitrary-length techniques differs from that of the original bit channels,
which can greatly affect the performance of the constructed polar codes. The
proposed Non-Uniform Polarization based on Gaussian Approximation (NUPGA) is an
efficient construction technique for rate-compatible arbitrary-length polar
codes, which chooses the best channels (i.e., selects the positions of the
information bits) by re-polarization of the codeword with desired length. A
generalization of the Gaussian Approximation is devised for both polarization
and re-polarization processes. We also present shortening and extension
techniques for design polar codes. Simulations verify the effectiveness of the
proposed NUPGA designs against existing rate-compatible techniques.
