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We study the problem of A/D conversion and error-rate dependence of a class of nonbandlimited signals with finite rate of innovation. In particular, we analyze a continuous periodic stream of Diracs, characterized by a finite set of time positions and weights. Previous research has only considered sampling of this type of signals, ignoring the presence of quantization, necessary for any practical implementation. To this end, we first define the concept of consistent reconstruction and introduce corresponding oversampling in both time and frequency. High accuracy in a consistent reconstruction is achieved by enforcing the reconstructed signal to satisfy three sets of constraints, related to low-pass filtering, quantization and the space of continuous periodic streams of Diracs. We provide two schemes to reconstruct the signal. For the first one, we prove that the estimation mean squared error of the time positions is O(1/Rt2Rf3), where Rt and Rf are the oversampling ratios in time and frequency, respectively. For the second scheme, it is experimentally observed that, at the cost of higher complexity, the estimation accuracy lowers to O(1/Rt2Rf5). Our experimental results show a clear advantage of consistent over nonconsistent reconstruction. Regarding the rate, we consider a threshold crossing based scheme where, as opposed to previous research, both oversampling in time and in frequency influence the coding rate. We compare the error-rate behavior resulting, on the one hand, from increasing the oversampling in time and/or frequency, and, on the other hand, from decreasing the quantization stepsize.