Oversampled A/D Conversion and Error-Rate Dependence of Non-Bandlimited Signals with Finite Rate of Innovation
We study the problem of A/D conversion and error-rate dependence of a class of non-bandlimited signals which have a finite rate of innovation, particularly, 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, which is necessary for any practical application. We first define the concept of consistent reconstruction for these signals and introduce the operations of both: a) oversampling in frequency, determined by the bandwidth of the low pass filtering used in the signal acquisition, and b) oversampling in time, determined by the number of samples in time taken from the filtered signal. Accuracy in a consistent reconstruction is achieved by enforcing the reconstructed signal to satisfy three sets of constrains, defined by: the low-pass filtering operation, the quantization operation itself and the signal space of continuous periodic streams of Diracs. We provide two schemes to reconstruct the signal. For the first one, we prove that the mean squared error (MSE) of the time positions is of the order of O(1/R_t^2R_f^3), where R_t and R_f are the oversampling ratios in time and in frequency, respectively. For the second scheme, which has a higher complexity, it is experimentally observed that the MSE of the time positions is of the order of O(1/R_t^2R_f^5). Our experimental results show a clear advantage of consistent reconstruction over non-consistent reconstruction. Regarding the rate, we consider a threshold crossing based scheme where, as opposed to previous research, both oversampling in time and also in frequency influence the coding rate. We compare the error-rate dependence behavior that is obtained from both increasing the oversampling in time and in frequency, on the one hand, and on the other hand, from decreasing the quantization stepsize.