Data Synchronization With Timing: The Variable-Rate Case

Abstract

This paper extends the theory of data synchronization with timing for fixed-rate codes, previously developed by the authors, to the variable-rate case. Given a source code, a class of sync-timing codes called variable-rate cascaded (VRC) codes is considered that ldquowrap aroundrdquo the source code in such a way as to enable the decoder to not only resynchronize rapidly when the encoded bits are corrupted by insertion, deletion, or substitution errors, but also produce estimates of the time indices of the data symbols encoded by the source code. The estimates of the time indices are modulo- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> reductions of the actual time indices, for some integer <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> called the timing span of the code. These sync-timing codes are analyzed on the basis of the maximum timing span achievable for a given coding rate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> and permissible resynchronization delay <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> . It is shown that the timing span of VRC codes is upper-bounded by 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D(1-R)</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o(D)</sup> , and that this upper bound is achievable asymptotically in D. This exponential rate of growth of timing span with delay is the same as that found previously for certain fixed-rate sync-timing codes, e.g., (fixed-rate) cascaded codes.