Fast Polarization and Finite-Length Scaling for Non-Stationary Channels

We consider the problem of polar coding for transmission over a non-stationary sequence of independent binary-input memoryless symmetric (BMS) channels $\left\{W_i\right\}_{i=1}^{\infty}$, where the $i$-th encoded bit is transmitted over $W_i$. We show, for the first time, a polar coding scheme that achieves the effective average symmetric capacity $$ \overline{I}(\left\{W_i\right\}_{i=1}^{\infty}) \ := \lim_{N\rightarrow \infty} \frac{1}{N}\sum_{i=1}^N I(W_i), $$ assuming that the limit exists. The polar coding scheme is constructed using Ar{\i}kan's channel polarization transformation in combination with certain permutations at each polarization level and certain skipped operations. This guarantees a fast polarization process that results in polar coding schemes with block lengths upper bounded by a polynomial of $1/\epsilon$, where $\epsilon$ is the gap to the average capacity. More specifically, given an arbitrary sequence of BMS channels $\left\{W_i\right\}_{i=1}^{N}$ and $P_e$, where $0 < P_e <1$, we construct a polar code of length $N$ and rate $R$ guaranteeing a block error probability of at most $P_e$ for transmission over $\left\{W_i\right\}_{i=1}^{N}$ such that $$ N \leq \frac{\kappa}{(\overline{I}_N - R)^{\mu}} $$ where $\mu$ is a constant, $\kappa$ is a constant depending on $P_e$ and $\mu$, and $\overline{I}_N$ is the average of the symmetric capacities $I(W_i)$, for $i=1,2,,\dots,N$. We further show a numerical upper bound on $\mu$ that is: $\mu \leq 10.78$. The encoding and decoding complexities of the constructed polar code preserves $O(N \log N)$ complexity of Ar{\i}kan's polar codes.