On the Capacity of Precision-Resolution Constrained Systems

Abstract

Arguably, the most famous constrained system is the (d, k)-RLL (Run-Length Limited), in which a stream of bits obeys the constraint that every two 1's are separated by at least d 0's, and there are no more than k consecutive 0's anywhere in the stream. The motivation for this scheme comes from the fact that certain sensor characteristics restrict the minimum time between adjacent 1's or else the two will be merged in the receiver, while a clock drift between transmitter and receiver may cause spurious 0's or missing 0's at the receiver if too many appear consecutively. The interval-modulation scheme introduced by Mukhtar and Bruck extends the RLL constraint and implicitly suggests a way of taking advantage of higher-precision clocks. Their work however, deals only with an encoder/decoder construction. In this work we introduce a more general framework which we call the precision-resolution (PR) constrained system. In PR systems, the encoder has precision constraints, while the decoder has resolution constraints. We examine the capacity of PR systems and show the gain in the presence of a high-precision encoder (thus, we place the PR system with integral encoder, (p=1,alpha,theta)-PR, which turns out to be a simple extension of RLL, and the PR system with infinite-precision encoder, (infinity,alpha,theta)-PR, on two ends of a continuum). We derive an exact expression for their capacity in terms of the precision p, the minimal resolvable measurement at the decoder alpha, and the decoder resolution factor theta. In an analogy to the RLL terminology these are the clock precision, the minimal time between peaks, and the clock drift. Surprisingly, even with an infinite-precision encoder, the capacity is finite.