Vector sampling expansion in Riesz bases setting and its aliasing error

We consider the multi-input multi-output (MIMO) system of vector valued band-limited signals, which are transferred through an M×N LTI (linear time invariant) system of channels. D. Seidner and M. Feder considered this vector sampling expansion (VSE) with the formal delta-method, which is a natural generalization of Papoulis' generalized sampling expansion (GSE) for single input multi-output systems. Later R. Venkataramani and Y. Bresler considered a more general MIMO sampling for multiband signals in the frame setting. In this work, we consider the VSE problem by using the Riesz basis method, which enables us to clarify obscurities in some of the previous works related to GSE and VSE problem. We obtain necessary and sufficient conditions for a VSE formula to hold as a Riesz basis expansion in the MIMO system. We also obtain an upper bound of the aliasing error which occurs when we apply the VSE formula to non-band-limited vector signals.