We consider the problem of A/D conversion for non-bandlimitedsignals that have a finite rate of innovation, in particular, theclass of continuous periodic stream of Diracs, characterized by aset of time positions and weights. Previous research has onlyconsidered the sampling of these signals, ignoring quantization,which is necessary for any practical application (e.g. UWB,CDMA). In order to achieve accuracy under quantization, weintroduce two types of oversampling, namely, oversampling infrequency and oversampling in time. High accuracy is achieved byenforcing the reconstruction to satisfy either three convex setsof constraints related to: 1) sampling kernel, 2) quantization and3) periodic streams of Diracs which is then said to provide {\itstrong} consistency or only the first two, providing {\it weak}consistency. We propose three reconstruction algorithms, thefirst two achieving {\it weak} consistency and the third oneachieving {\it strong} consistency. For these three algorithms,respectively, the experimental MSE performance for time positionsdecreases as $O(1/{R_t^2 R_f^3})$, $O(1/{R_t^2 R_f^4})$ and$O(1/{R_t^2 R_f^5})$, where $R_t$ and $R_f$ are the oversamplingratios in time and in frequency, respectively. It is also provedtheoretically that our reconstruction algorithms satisfying {\itweak} consistency achieve an MSE performance of at least$O(1/{R_t^2 R_f^3})$.