The optimum reconstruction of vector signals using multi-input multi-output filter banks

In this paper, we define a multi-input multi-output system composed of given analysis-filter matrices, given sampler matrices and interpolation-filter matrices to be optimized, respectively. It is assumed that input-signal vectors of this system are contained in a certain given set of input-signal vectors. Firstly, we define new notations which expresses a kind of product between two matrices or between a vector and a matrix. Using these new notations, we show that the presented approximation satisfies a certain two conditions and prove that the presented approximation minimizes any upperlimit measure of error compared to any other linear or nonlinear approximation with same sample values, simultaneously.