Tensor z-Transform

Abstract

The multi-input multioutput (MIMO) systems involving multirelational signals generated from distributed sources have been emerging as the most generalized model in practice. The existing work for characterizing such a MIMO system is to build a corresponding transform tensor, each of whose entries turns out to be the individual z-transform of a discrete-time impulse response sequence. However, when a MIMO system has a global feedback mechanism, which also involves multirelational signals, the aforementioned individual z-transforms of the overall transfer tensor are quite difficult to formulate. Therefore, a new mathematical framework to govern both feedforward and feedback MIMO systems is in crucial demand. In this work, we define the tensor z-transform to characterize a MIMO system involving multirelational signals as a whole rather than the individual entries separately, which is a novel concept for system analysis. To do so, we extend Cauchy’s integral formula and Cauchy’s residue theorem from scalars to arbitrary-dimensional tensors, and then, to apply these new mathematical tools, we establish to undertake the inverse tensor z-transform and approximate the corresponding discrete-time tensor sequences. Our proposed new tensor z-transform in this work can be applied to design digital tensor filters including infinite-impulse-response (IIR) tensor filters (involving global feedback mechanisms) and finite-impulse-response (FIR) tensor filters. Finally, numerical evaluations are presented to demonstrate certain interesting phenomena of the new tensor z-transform.