Robust and practical analog-to-digital conversion with exponential precision
Beta-encoders with error correction were introduced by Daubechies, DeVore, Gunturk and Vaishampayan as an alternative to pulse-code modulation (PCM) for analog-to-digital conversion. An N-bit beta-encoder quantizes a real number by computing one of its N-bit truncated beta-expansions where beta is an element of (1, 2) determines the base of expansion. These encoders have (almost) optimal rate-distortion properties like PCM; furthermore, they exploit the redundancy of beta-expansions and thus they are robust with respect to quantizer imperfections. However, these encoders have the shortcoming that the decoder needs to know the value of the base of expansion beta, a gain factor in the circuit used by the encoder, which is an impractical constraint. We present a method to implement beta-encoders so that they are also robust with respect to uncertainties of the value of beta. The method relies upon embedding the value of beta in the encoded bitstream. We show that this can be done without a priori knowledge of beta by the transmitting party. Moreover the algorithm still works if the value of beta changes (slowly) during the implementation.
beta-encoders quantization beta-expansions analog-to-digital (a/d) conversion sampling sigma-delta