The Boolean averaging procedure, common to three systems in statistical mechanics, control and signal processing

Abstract

We investigate the Boolean averaging procedure (BAP) u(n) = sgn(/spl psi/ /spl Sigma//sub i=1/ /sup /spl infin// /spl gamma//sub i/u(n-i)), where /spl psi//spl isin//spl Ropf/ (/spl gamma//sub i/) is a weight function (i /spl isin/ /spl Nopf/), (u(n)) is a two-sided sequence (n /spl isin/ /spl Zopf/). It describes three systems: 1) the Hubbard model in statistical mechanics, 2) sampled-data control system, and 3) analog-to-digital converter with sigma-delta modulation and leaky integration. The periodic modes in the BAP generate so-called Hubbard configurations, in which (+1)-s is evenly distributed between (-1)-s (the phenomenon of even 2-colouring). Our new results shows that the trajectories (u(n)) become periodic for almost all /spl psi/ if the sequence (/spl gamma//sub n/) is convex.