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

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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 .

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Information Theory, IEEE Transactions on  (Volume:18 ,  Issue: 2 )