On simple oversampled A/D conversion in L2(R)

Accuracy of oversampled analog-to-digital (A/D) conversion, the dependence of accuracy on the sampling interval, t, and on the bit rate are characteristics fundamental to A/D conversion but not completely understood. These characteristics are studied in this paper for oversampled A/D conversion of band-limited signals in L^2(R). We show that the digital sequence obtained in the process of oversampled A/D conversion describes the corresponding analog signal with an error which tends to zero as t^2 in energy, provided that the quantization threshold crossings of the signal constitute a sequence of stable sampling in the respective space of band-limited functions. Further, we show that the sequence of quantized samples can be represented in a manner which requires only a logarithmic increase in the bit rate with the sampling frequency, R=0(|logt|), and hence that the error of oversampled A/D conversion actually exhibits an exponential decay in the bit rate as the sampling interval tends to zero.