Negative Beta Encoder
  A new class of analog-to-digital (A/D) and digital-to-analog (D/A) converters using a flaky quantiser, called the $\beta$-encoder, has been shown to have exponential bit rate accuracy while possessing a self-correction property for fluctuations of the amplifier factor $\beta$ and the quantiser threshold $\nu$. The probabilistic behavior of such a flaky quantiser is explained as the deterministic dynamics of the multi-valued R\'enyi map. That is, a sample $x$ is always confined to a contracted subinterval while successive approximations of $x$ are performed using $\beta$-expansion even if $\nu$ may vary at each iteration. This viewpoint enables us to get the decoded sample, which is equal to the midpoint of the subinterval, and its associated characteristic equation for recovering $\beta$ which improves the quantisation error by more than $3{dB}$ when $\beta>1.5$. The invariant subinterval under the R\'enyi map shows that $\nu$ should be set to around the midpoint of its associated greedy and lazy values. %in terms of its quantisation MSE (mean square error). Furthermore, a new A/D converter is introduced called the negative $\beta$-encoder, which further improves the quantisation error of the $\beta$-encoder. A two-state Markov chain describing the $\beta$-encoder suggests that a negative eigenvalue of its associated transition probability matrix reduces the quantisation error. 
Tohru Kohda, Satoshi Hironaka, and Kazuyuki Aihara