Tensor MIMO Perfect-Equalizer Design Using New Projection-Based Method and Approximation-Error Analysis

As real environments are quite complex, multiple-input-multiple-output (MIMO) channels should be characterized by the multidimensional-multirelational models. Tensor linear algebra has been emerging as a powerful and promising tool for such MIMO channel modeling and estimation. However, to the best of our knowledge, there hardly exists any algebraic framework of MIMO channel equalization when the channel is modeled as an arbitrary-dimensional transfer tensor (a tensor whose entries are all rational z-functions). In this work, we first propose a new method based on the projection, namely, 'projection-based method,' to determine whether a given transfer tensor is a full-rank tensor or not and carry out the inversion of an invertible transfer tensor. Our idea is employing the projection subspace of the system transfer tensor to simplify the pertinent inversion process. Second, we design the finite-impulse-response (FIR) realization to approximate the inverse of an arbitrary transfer tensor such that all entries of the approximated inverse transfer tensor are finite-degree polynomials of z. Third, we also analyze the computational and memory complexities required by our proposed new projection-based method for checking the full-rank condition of a given transfer tensor and for undertaking the tensor inversion. Our developed new theories and analysis in this work can be applied to determine the FIR model-orders of an approximated inverse MIMO transfer tensor often encountered in signal processing, control, communications, etc.