Sub-NYQUIST Multichannel Blind Deconvolution

Abstract

We consider a continuous-time sparse multichannel blind deconvolution problem. The signal at each channel is expressed as the convolution of a common source signal and its impulse response given as a sparse filter. The objective is to identify these sparse filters from sub-Nyquist samples of channel outputs by leveraging the correlation across channels. We present necessary and sufficient conditions for the unique identification. In particular, the sparse filters should not share a common sparse convolution factor and it is necessary to have 2L or more samples per channel from at least two distinct channels. We also show that L-sparse filters are uniquely identifiable from two channels provided that there are 2L2 Fourier measurements per channel, which can be computed from sub-Nyquist samples. Additionally, in the asymptotic of the number of channels, 2L Fourier measurements per channel are sufficient. The results are applicable to the design of multi-receiver, low-rate, sensors in applications such as radar, sonar, ultrasound, and seismic exploration.