On the survival of sequence information in filters (Corresp.)

Abstract

Given a binary data stream A = \{a_i\}_{i=o}^\infty and a filter F whose output at time n is f_n = \sum_{i=0}^{n} a_i \beta^{n-i} for some complex \beta \neq 0 , there are at most 2^{n +1) distinct values of f_n . These values are the sums of the subsets of \{1,\beta,\beta^2,\cdots,\beta^n\} . It is shown that all 2^{n+1} sums are distinct unless \beta is a unit in the ring of algebraic integers that satisfies a polynomial equation with coefficients restricted to +1, -1, and 0. Thus the size of the state space \{f_n\} is 2^{n+1} if \beta is transcendental, if \beta \neq \pm 1 is rational, and if \beta is irrational algebraic but not a unit of the type mentioned. For the exceptional values of \beta , it appears that the size of the state space \{f_n\} grows only as a polynomial in n if \mid\beta\mid = 1 , but as an exponential \alpha^n with 1 if \mid\beta\mid \neq 1 .