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

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 < \alpha < 2
if
\mid\beta\mid \neq 1
.