Sampling Streams of Pulses With Unknown Shapes
This paper develops a theory for sampling and perfectly reconstructing streams of short pulses of unknown shapes in the continuous-time domain. The single pulse is modelled as the delayed version of a wavelet sparse signal, which is normally not band-limited. As the delay can be an arbitrary real number, it is difficult to develop an exact sampling result for this type of signals. We manage to achieve the exact reconstruction of the pulses by using only the knowledge of the Fourier transform of the signal at specific frequencies. We further introduce a multichannel acquisition system that uses a new family of compact-support sampling kernels for extracting the Fourier information from the samples. The shape of the kernel is independent of the wavelet basis in which the pulse is sparse, and hence the same acquisition system can be used with pulses that are sparse on different wavelet bases. By exploiting the fact that pulses have short duration and that the sampling kernels have compact support, we finally propose a local and sequential algorithm to reconstruct streaming pulses from the samples.