Universal Polarization for Processes with Memory

TL;DR

This work develops universal polarization for processes with memory by modeling channels/sources via FAIM processes and a forgetfulness property. It introduces a two-stage universal transform: a slow basic slow transform (BST) that creates polarized medial index sets, followed by a fast Arıkan-stage polarization, enabling a single universal code to achieve vanishing error probability while approaching the infimal information rate $I^*$ over a family of memoryful channels. The analysis hinges on a block-based decomposition (BI-process) and forgetfulness-based contraction results, extended to cascades of BSTs to tune rate via a cascade threshold entropy $h(oldsymbol{c})$. Decoding leverages successive-cancellation trellis methods that account for memory, with complexities of $O(| ext{S}|^3 ext{Λ} ext{log} ext{Λ})$ and achievable rates approaching $I^*$. Numerical results on Gilbert-Elliott and memoryless channels illustrate practical performance and robustness of the proposed universal polar codes with list decoding enhancements. The framework provides a principled path to universal coding for channels with memory, bridging memoryless universal polar code results with memory-containing processes through forgetful FAIM modeling and contraction-based analyses.

Abstract

A transform that is universally polarizing over a set of channels with memory is presented. Memory may be present in both the input to the channel and the channel itself. Both the encoder and the decoder are aware of the input distribution, which is fixed. However, only the decoder is aware of the actual channel being used. The transform can be used to design a universal code for this scenario. The code is to have vanishing error probability when used over any channel in the set, and achieve the infimal information rate over the set. The setting considered is, in fact, more general: we consider a set of processes with memory. Universal polarization is established for the case where each process in the set: (a) has memory in the form of an underlying hidden Markov state sequence that is aperiodic and irreducible, and (b) satisfies a `forgetfulness' property. Forgetfulness, which we believe to be of independent interest, occurs when two hidden Markov states become approximately independent of each other given a sufficiently long sequence of observations between them. We show that aperiodicity and irreducibility of the underlying Markov chain is not sufficient for forgetfulness, and develop a sufficient condition for a hidden Markov process to be forgetful.

Figures (16)

• Figure 1: Roadmap of the various ways to read this paper. All paths start at \ref{['sec_notation conventions and reminders']} and end at \ref{['sec_numerical results']}.
• Figure 2: Illustration of an Arıkan transform. It transforms two input symbols, $U$ (input-$\textrm{I}\xspace$) and $V$ (input-$\textrm{II}\xspace$) to two output symbols, $F_1$ (output '$-$') and $F_2$ (output '$+$').
• Figure 3: Index sets in level $n$ of the basic slow transform. A Level-$n$ block comprises $N_n = 2L_n + M_n$ s/o-pairs. The first $L_n$ and the last $L_n$ s/o-pairs are lateral s/o-pairs and the remaining $M_n$ s/o-pairs are medial s/o-pairs.
• Figure 4: A schematic description of forming lateral s/o-pairs of a level-$(n+1)$ block from two level-$n$ blocks.
• Figure 5: Forming the medial symbols of level $n+1$ of the basic slow transform. Arıkan transforms are used with a symbol from $[\textup{med}_{+}(n)]$ of one block as their input-$\textrm{I}\xspace$ and a symbol from $[\textup{med}_{-}(n)]$ of the other block as their input-$\textrm{II}\xspace$. One Arıkan transform is highlighted using thicker edges.

Theorems & Definitions (98)

• Theorem 1
• Definition 1: s/o-pair
• Definition 2: s/o-process
• Definition 3: s/o-block
• Definition 4: H-transform
• Example 1
• Definition 5: Monopolarizing H-transform
• Remark 1
• Definition 6: Ancestors and Base-ancestors
• Lemma 2