Sub-Nyquist multicoset and MIMO sampling: Perfect reconstruction, performance analysis, and necessary density conditions

We study two sub-Nyquist sampling schemes for multiband signals known as multicoset sampling and multiple-input, multiple-output (MIMO) sampling. A multiband signal is one whose Fourier transform is supported on a set $F\subseteq R$ consisting of a finite union of intervals. Unlike uniform sampling, multicoset sampling allows perfect reconstruction of a multiband input at sampling rates arbitrarily close to its Landau minimum rate equal to the Lebesgue measure of $F$. We derive perfect reconstruction conditions, an explicit interpolation formula, and bounds on the aliasing error for signals not spectrally supported on $F$. We also examine the performance of the reconstruction system when the input contains additive sample noise. Using these measures of performance, we optimize the reconstruction system. We find that optimizing these parameters improves the performance significantly. There is an increased sensitivity to errors associated with nonuniform sampling, as opposed to uniform sampling. However, these errors can be controlled by optimal design, demonstrating the potential for practical multifold reductions in sampling rate. Multicoset sampling is applicable to Fourier imaging problems like synthetic aperture radar and magnetic resonance imaging, where the objective is to image a sparse object from limited Fourier data.; We then study the MIMO sampling problem, where a set of multiband input signals is passed through a MIMO channel and the outputs are sampled nonuniformly. MIMO sampling is motivated from applications like multiuser communications and multiple source separation. MIMO sampling encompasses several sampling strategies as special cases, including multicoset sampling and Papoulis's generalized sampling. We derive necessary density conditions for stable reconstruction of the channel inputs from the output. These results generalize Landau's sampling density results to the MIMO problem. We then investigate a special case of MIMO sampling called commensurate periodic nonuniform MIMO sampling , for which we present reconstruction conditions. Finally, we address the problem of reconstruction FIR filter design, formulating it as a minimization and recasting as a standard semi-infinite linear program. Owing to the generality of the MIMO sampling scheme, the design algorithm readily applies to several sampling schemes for multiband signals.