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

Author

Golomb, Solomon W.

Volume

Issue

fYear

1972

fDate

3/1/1972 12:00:00 AM

Firstpage

310

Lastpage

312

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

Keywords

Filtering; Sequences; Binary sequences; Computer aided software engineering; Delta modulation; Equations; Information filtering; Information filters; Intersymbol interference; Low pass filters; Polynomials; State-space methods;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.1972.1054762

Filename

1054762

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=916969