| Summary: | Recently there has been a surge of interest in sampling theory in signal processing community. New efficient sampling techniques have been developed that allow sampling and perfectly reconstructing some classes of non-bandlimited signals at sub-Nyquist rates. Depending on the setup used and reconstruction method involved, these schemes go under different names such as compressed sensing (CS), compressive sampling or sampling signals with finite rate of innovation (FRI). In this thesis we focus on the theory of sampling non-bandlimited signals with parametric structure or specifically signals with finite rate of innovation. Most of the theory on sampling FRI signals is based on a single acquisition device with one-dimensional (1-D) signals. In this thesis, we extend these results to the case of 2-D signals and multichannel acquisition systems. The essential issue in multichannel systems is that while each channel receives the input signal, it may introduce different unknown delays, gains or affine transformations which need to be estimated from the samples together with the signal itself. We pose both the calibration of the channels and the signal reconstruction stage as a parametric estimation problem and demonstrate that a simultaneous exact synchronization of the channels and reconstruction of the FRI signal is possible. Furthermore, because in practice perfect noise-free channels do not exist, we consider the case of noisy measurements and show that by considering Cramer-Rao bounds as well as numerical simulations, the multichannel systems are more resilient to noise than the single-channel ones. Finally, we consider the problem of system identification based on the multichannel and finite rate of innovation sampling techniques. First, by employing our multichannel sampling setup, we propose a novel algorithm for system identification problem with known input signal, that is for the case when both the input signal and the samples are known. Then we consider the problem of blind system identification and propose a novel algorithm for simultaneously estimating the input FRI signal and also the unknown system using an iterative algorithm.
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