  Channel polarization, originally proposed for binary-input channels, is
generalized to arbitrary discrete memoryless channels. Specifically, it is
shown that when the input alphabet size is a prime number, a similar
construction to that for the binary case leads to polarization. This method can
be extended to channels of composite input alphabet sizes by decomposing such
channels into a set of channels with prime input alphabet sizes. It is also
shown that all discrete memoryless channels can be polarized by randomized
constructions. The introduction of randomness does not change the order of
complexity of polar code construction, encoding, and decoding. A previous
result on the error probability behavior of polar codes is also extended to the
case of arbitrary discrete memoryless channels. The generalization of
polarization to channels with arbitrary finite input alphabet sizes leads to
polar-coding methods for approaching the true (as opposed to symmetric) channel
capacity of arbitrary channels with discrete or continuous input alphabets.
