On simple oversampled A/D conversion in shift-invariant spaces

It has been found that the quantization error e for a conventional oversampled analog-to-digital (A/D) conversion behaves like ||e||²=O(τ²) with respect to the sampling rate τ. Recently, conventional A/D conversion has been extended to A/D conversion based on shift-invariant spaces. As consequences of such extension, it offers rich choices to build a nonideal A/D conversion system of high accuracy and low computational complexity, as well as reduces the noise sensitivity and computational complexity in digital-to-analog (D/A) conversion. Therefore, it is necessary to establish the estimate of quantization error for the extended A/D conversion based on shift-invariant spaces. In this paper, we introduce a constructive method to establish an estimate of the quantization error as |e|²=O(τ²) for oversampled A/D conversion in shift-invariant spaces. Meanwhile, we demonstrate that the bit rate required to encode the converted digital signal in such A/D conversion scheme only increases as the logarithm of the sampling ratio. Therefore, the quantization error is an exponentially decaying function of the bit rate. In order to establish such an estimate, we need the nonuniform sampling theorem for shift-invariant spaces, which, as the necessary preparation, is studied prior to introducing the constructive method.