Sampling theorems for uniform and periodic nonuniform MIMO sampling of multiband signals

We examine a multiple-input multiple-output (MIMO) sampling scheme for a linear time-invariant continuous-time MIMO channel. The input signals are modeled as multiband signals with different spectral supports, and the channel outputs are sampled on either uniform or periodic nonuniform sampling sets, with possibly different but commensurate intervals on the different outputs. This scheme encompasses Papoulis' generalized sampling and several nonuniform sampling schemes as special cases. We derive necessary and sufficient conditions on the channel and the sampling rate that allow stable perfect reconstruction of the inputs or, equivalently, perfect inversion of the channel. From an implementation viewpoint, we note that it is desirable that the reconstruction filters have continuous frequency responses. We derive necessary and sufficient conditions that guarantee this continuity property. The frequency responses of the reconstruction filters are specified as solutions to a system of linear equations. Finally, we demonstrate that perfect reconstruction may be possible, even when the channel outputs are sampled at an average rate that does not allow the reconstruction of any output from its samples alone. In certain instances, this average rate can achieve the recently presented fundamental bounds on MIMO sampling density.