On Robustness Properties of Beta Encoders and Golden Ratio Encoders

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
x
isin[-1, 1] is mapped to the first
N
bits of its base-2 expansion, beta encoders replace each sample
x
with its expansion in a base beta between 1_low
, beta
_high
]. 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
x

_test
isin[-1, 1] and its negative counterpart -
x

_test
. That is, beta encoders and GREs are robust with respect to unavoidable analog component imperfections that change the base beta needed for reconstruction.