Systematic LDPC Convolutional Codes: Asymptotic and Finite-Length Anytime Properties

Here we propose an ensemble of non-terminated systematic LDPC convolutional codes with increasing memory, and show that, over the binary erasure channel (BEC), these codes achieve anytime reliability asymptotically when decoded with an expanding-window message-passing decoder. The corresponding anytime exponents are determined through protograph-based extrinsic information transfer charts. Fundamental complications arising when transmitting with finite block lengths are identified and a combinatorial performance analysis, when transmitting over a static BEC with a fixed number of erasures per code-word block, is developed. Based on the performance analysis, we explore the use of feedback for achieving anytime behavior with constraints on block length. To meet complexity constraints, with or without feedback, the code memory can be limited at the cost of an error floor emerging with a delay proportional to the memory constraint. Although the analysis is developed for a static BEC we show numerically that we can design efficient low-complexity finite-length codes with anytime properties even for the conventional BEC.