Given a binary data streamA = \{a_i\}_{i=o}^\inftyand a filterFwhose output at timenisf_n = \sum_{i=0}^{n} a_i \beta^{n-i}for some complex\beta \neq 0, there are at most2^{n +1)distinct values off_n. These values are the sums of the subsets of\{1,\beta,\beta^2,\cdots,\beta^n\}. It is shown that all2^{n+1}sums are distinct unless\betais 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\}is2^{n+1}if\betais transcendental, if\beta \neq \pm 1is rational, and if\betais 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 innif\mid\beta\mid = 1, but as an exponential\alpha^nwith1 < \alpha < 2if\mid\beta\mid \neq 1.