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 $left{W_iright}_{i=1}^{infty}$, 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 $$ overline{I}(left{W_iright}_{i=1}^{infty}) := lim_{Nrightarrow infty} frac{1}{N}sum_{i=1}^N I(W_i), $$ assuming that the limit exists. The polar coding scheme is constructed using Ar{i}kanu0027s 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 $left{W_iright}_{i=1}^{N}$ and $P_e$, where $0 u003c P_e u003c1$, we construct a polar code of length $N$ and rate $R$ guaranteeing a block error probability of at most $P_e$ for transmission over $left{W_iright}_{i=1}^{N}$ 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}kanu0027s polar codes.