Multiple-input multiple-output sampling: necessary density conditions

We consider the problem of multiple-input multiple-output (MIMO) sampling of multiband signals. In this problem, a set of input signals is passed through a MIMO channel modeled as a known linear time-invariant system. The inputs are modeled as multiband signals whose spectral supports are sets of finite measure and the channel outputs are sampled on nonuniform sampling sets. The aim is to reconstruct the inputs from the output samples. This sampling scheme is quite general and it encompasses various others including Papoulis' generalized sampling and nonuniform sampling as special cases. We introduce notions of joint upper and lower densities for collections of sampling sets and then derive necessary conditions on these densities for stable sampling and consistent reconstruction of the channel inputs from the sampled outputs. These results generalize classical density results for stable sampling and interpolation due to Landau.