Vector sampling expansion

The vector sampling expansion (VSE) is an extension of Papoulis' generalized sampling expansion (GSE) to the vector case. In VSE, N bandlimited signals, all with the same bandwidth B, are passed through a multi-input-multi-output (MIMO) linear time invariant system that generates M (M > N) output signals. The goal is to reconstruct the input signals from the samples of the output signals at a total sampling rate of N times Nyquist rate, where the Nyquist rate is B/π samples per second. We find necessary and sufficient conditions for this reconstruction. A surprising necessary condition for the case where all output signals are uniformly sampled at the same rate (N/M times the Nyquist rate) is that the expansion factor M/N must be an integer. This condition is no longer necessary when each output signal is sampled at a different rate or sampled nonuniformly. This work also includes a noise sensitivity analysis of VSE systems. We define the noise amplification factor, which allows a quantitative comparison between VSE systems, and determine the optimal VSE systems.