Polar Coding for Processes With Memory

We study polar coding for stochastic processes with memory. For example, a process may be defined by the joint distribution of the input and output of a channel. The memory may be present in the channel, the input, or both. We show that the ψ-mixing processes polarize under the standard Arikan transform, under a mild condition. We further show that the rate of polarization of the low-entropy synthetic channels is roughly O(2^-√N), where {N} is the blocklength. That is, essentially, the same rate as in the memoryless case.