It has been found that the quantization error e for a conventional oversampled analog-to-digital (A/D) conversion behaves like /spl par/e/spl par//sup 2/=O(/spl tau//sup 2/) with respect to the sampling rate /spl tau/. 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|/sup 2/=O(/spl tau//sup 2/) 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.

A conventional A/D converter prefilters a signal by an ideal lowpass filter and performs sampling for bandlimited signals by the Nyquist sampling rate. Recent research reveals that A/D conversion in a shift invariant space provides more flexible choices for designing a practical A/D conversion system of high accuracy. This paper focuses on the maximal gap of a sampling set for the iterative algorithm in shift invariant spaces, which provides an explicit formula to calculate the maximal gap of a sampling set in terms of a generator of the undertaken shift invariant spaces.