Polarization theorems for arbitrary DMCs

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

“…Later, it was generalized to prime-input discrete memoryless channels (DMCs) in [9], to prime power-input DMCs in [5], [6] and to arbitrary DMCs in [2]- [4], [7], [8], [10]. The proofs in [5]- [8], [10] all follow Arıkan's proof technique, which is based on the properties of the Battacharyya parameter and mutual information of virtual channels.…”

“…Instead, the proof is based on the entropy inequality of virtual channels, i.e., the mutual information of the virtual channels is strictly different from that of the original channel. As an extension of [9], the This work has been funded in part by the European Research Council under ERC grant agreement 259663, the Spanish Ministry of Economy and Competitiveness under grant TEC2012-38800-C03-03, and the FP7 Network of Excellence NEWCOM# under grant agreement 318306. channel polarization theorem is proved in [4] for arbitrary DMCs with input alphabet set forming a quasigroup.…”

“…An alternative proof for the channel polarization theorem for arbitrary DMCs is provided. Similarly to [4], our approach does not consider the Battacharyya parameter. There are two main differences between our proof technique and the one proposed in [4].…”

“…There are two main differences between our proof technique and the one proposed in [4]. First, while [4] proves the entropy inequality of virtual channels by lower bounding the mutual information difference between the original and the "worse" virtual channel, we consider the difference between the "better" virtual channel and the original channel, for which a simple expression is given and bounded away from zero when the input alphabet and the operation used in the channel transformation forms a monoid. Though these two ideas might seem similar, this leads to a new approach for proving the strict inequality.…”

“…Moreover, we show that the extremal channels to which the virtual channels converge have a zero-error capacity equal to their capacity. We note that our proof of channel polarization theorem is restricted to group operations for now, while the stronger results in [4] apply to the wider class of quasigroups.…”