Optimized Bit Mappings for Spatially Coupled LDPC Codes over Parallel Binary Erasure Channels

TL;DR

This work investigates spatially coupled LDPC (SC-LDPC) codes over parallel channels and aims to optimize the bit mapper to improve decoding performance. It models parallel bit channels as independent binary erasure channels (BECs), focusing on a two-channel, 4-PAM BRGC example to study threshold gains. An iterative differential-evolution-based optimization finds bit-mapper assignments that raise the decoding threshold or reduce the spatial chain length for a given gap to capacity, with notable gains at finite chain lengths. For circular (tail-biting) ensembles, an optimized mapper can induce wave-like decoding behavior, effectively creating a termination-like boundary and enhancing performance.

Abstract

In many practical communication systems, one binary encoder/decoder pair is used to communicate over a set of parallel channels. Examples of this setup include multi-carrier transmission, rate-compatible puncturing of turbo-like codes, and bit-interleaved coded modulation (BICM). A bit mapper is commonly employed to determine how the coded bits are allocated to the channels. In this paper, we study spatially coupled low-density parity check codes over parallel channels and optimize the bit mapper using BICM as the driving example. For simplicity, the parallel bit channels that arise in BICM are replaced by independent binary erasure channels (BECs). For two parallel BECs modeled according to a 4-PAM constellation labeled by the binary reflected Gray code, the optimization results show that the decoding threshold can be improved over a uniform random bit mapper, or, alternatively, the spatial chain length of the code can be reduced for a given gap to capacity. It is also shown that for rate-loss free, circular (tail-biting) ensembles, a decoding wave effect can be initiated using only an optimized bit mapper.

Figures (8)

• Figure 1: Block diagram of the considered system model.
• Figure 2: Two PAM constellations labeled by the .
• Figure 3: Individual erasure probabilities $\varepsilon_i$ plotted over $\bar{\varepsilon}$.
• Figure 4: Graphical representation of the Tanner graph for the two-sided $(3,6,5,2)$ spatially coupled ensemble. Known are shown in red.
• Figure 5: Optimization results for the two-sided $(4,8,L,{w})$ ensemble. Color is helpful.