An alternative proof of channel polarization for channels with arbitrary input alphabets

We revisit channel polarization for arbitrary discrete memoryless channels. A closed-form expression is derived to characterize the difference between the mutual information of the original channel and the virtual channels after one step of channel transformation when the input alphabet and the operation used in the channel transformation form a monoid. We then provide an alternative proof to the one given in [4] for the channel polarization theorem for arbitrary DMCs when the input alphabet set forms a group. The results reveal the connections between channel polarization and zero-error capacity.