On Robustness Properties of Beta Encoders and Golden Ratio Encoders

Abstract

The beta encoder was recently proposed as a quantization scheme for analog-to-digital (A/D) conversion; in contrast to classical binary quantization, in which each analog sample <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> isin[-1, 1] is mapped to the first <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> bits of its base-2 expansion, beta encoders replace each sample <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> with its expansion in a base beta between 1<beta<2. This expansion is nonunique when 1<beta<2, and the beta encoder exploits this redundancy to correct inevitable errors made by the quantizer component of its circuit design. The multiplier element of the beta encoder will also be imprecise; effectively, the true value beta at any time can only be specified to within an interval [beta <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">low</sub> , beta <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">high</sub> ]. This problem was addressed by the golden ratio encoder (GRE), a close relative of the beta encoder that does not require a precise multiplier. However, the GRE is susceptible to integrator leak in the delay elements of its hardware design, and this has the same effect of changing beta to an unknown value. In this paper, we present a method whereby exponentially precise approximations to the value of beta in both GREs and beta encoders can be recovered amidst imprecise circuit components from the truncated beta expansions of a rdquotestrdquo number <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">test</sub> isin[-1, 1] and its negative counterpart - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">test</sub> . That is, beta encoders and GREs are robust with respect to unavoidable analog component imperfections that change the base beta needed for reconstruction.