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.