Negative Beta Encoder

A new class of analog-to-digital (A/D) and digitalto-analog (D/A) converters using a flaky quantiser, called the β-encoder, has been shown to have exponential bit rate accuracy while possessing a self-correction property for fluctuations of the amplifier factor β and the quantiser threshold ν. The probabilistic behavior of such a flaky quantiser is explained as the deterministic dynamics of the multi-valued Rényi map. That is, a sample x is always confined to a contracted subinterval while successive approximations of x are performed using β-expansion even if ν 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 β which improves the quantisation error by more than 3dB when β > 1.5. The invariant subinterval under the Rényi map shows that ν should be set to around the midpoint of its associated greedy and lazy values. Furthermore, a new A/D converter is introduced called the negative β-encoder, which further improves the quantisation error of the β-encoder. A twostate Markov chain describing the β-encoder suggests that a negative eigenvalue of its associated transition probability matrix reduces the quantisation error.