Xampling of Unknown Pulses

We develop sub-Nyquist sampling systems for signals comprised of several, possibly overlapping, finite duration pulses with unknown shapes and time positions. To the best of our knowledge, stable and low-rate sampling strategies for a superposition of unknown pulses without knowledge of the pulse locations have not been derived. We propose a multichannel scheme based on Gabor frames that exploits the sparsity of signals in time and enables sampling at sub-Nyquist rates. Moreover, if the signal is additionally essentially multiband, then the sampling scheme can be adapted to lower the sampling rate without knowing the band positions. Our approach is based on modulating the input signal in each channel with a properly chosen waveform, followed by integration.