Polar Coding for Non-Stationary Channels

The problem of polar coding for an arbitrary sequence of independent binary-input memoryless symmetric (BMS) channels <inline-formula> <tex-math notation="LaTeX">$\left \{{W_{i}}\right \}_{i=1}^{N}$ </tex-math></inline-formula> is considered. Such a sequence of channels is referred to as a <italic>non-stationary</italic> sequence of channels and arises in applications where data symbols experience different and independent channel characteristics. The sequence of channels is assumed to be completely known to both the transmitter and the receiver (a coherent scenario). Also, at each code block transmission, each of the channels is used only once. In other words, a codeword of length <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> is constructed and then the <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>-th encoded bit is transmitted over <inline-formula> <tex-math notation="LaTeX">$W_{i}$ </tex-math></inline-formula>. The goal is to operate at a rate <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> close to the average of the symmetric capacities of <inline-formula> <tex-math notation="LaTeX">$W_{i}$ </tex-math></inline-formula>’s, denoted by <inline-formula> <tex-math notation="LaTeX">$\overline {I}_{N}$ </tex-math></inline-formula>. To this end, we construct a polar coding scheme using Arıkan’s channel polarization transform in combination with certain permutations at each polarization level and certain skipped operations. In particular, given a non-stationary sequence of BMS channels <inline-formula> <tex-math notation="LaTeX">$\left \{{W_{i}}\right \}_{i=1}^{N}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$P_{e}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$0 < P_{e} < 1$ </tex-math></inline-formula>, we construct a polar code of length <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> and rate <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> guaranteeing a block error probability of at most <inline-formula> <tex-math notation="LaTeX">$P_{e}$ </tex-math></inline-formula> for transmission over <inline-formula> <tex-math notation="LaTeX">$\left \{{W_{i}}\right \}_{i=1}^{N}$ </tex-math></inline-formula> such that <inline-formula> <tex-math notation="LaTeX">$N \leqslant \frac {\kappa }{(\overline {I}_{N}- R)^{\mu }}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> is a constant and <inline-formula> <tex-math notation="LaTeX">$\kappa $ </tex-math></inline-formula> is a constant depending on <inline-formula> <tex-math notation="LaTeX">$P_{e}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula>. We further show a numerical upper bound on <inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> that is: <inline-formula> <tex-math notation="LaTeX">$\mu \leqslant 7.34$ </tex-math></inline-formula> for non-stationary binary erasure channels and <inline-formula> <tex-math notation="LaTeX">$\mu \leqslant 8.54$ </tex-math></inline-formula> for general non-stationary BMS channels. The encoding and decoding complexities of the constructed polar code preserve <inline-formula> <tex-math notation="LaTeX">$O(N \log N)$ </tex-math></inline-formula> complexity of Arıkan’s polar codes. In an asymptotic sense, when coded bits are transmitted over a non-stationary sequence of BMS channels <inline-formula> <tex-math notation="LaTeX">$\left \{{W_{i}}\right \}_{i=1}^{\infty }$ </tex-math></inline-formula>, our proposed scheme achieves the average symmetric capacity <inline-formula> <tex-math notation="LaTeX">$\overline {I}(\left \{{W_{i}}\right \}_{i=1}^{\infty }) \,\,{\textstyle \stackrel {\mathrm{ def}}{=}}\,\,\lim _{N\rightarrow \infty } \frac {1}{N}\sum _{i=1}^{N} I(W_{i})$ </tex-math></inline-formula>, assuming that the limit exists.