A theory of the optimum approximation of multiple-input multiple-output filter banks and trans-multiplexers

This paper presents the optimum approximation theory of multiple-transmission systems expressed as matrix filter banks. With respect to any measure of error that is identical to an arbitrary operator, functional or function of elements in the corresponding error matrix of the matrix filter bank, the presented approximation is able to achieve the minimum upper limit of the measure of error (worst-case measure of error) among all the matrix filter banks using the same analysis filter matrices and the same sampler matrices. In this paper, without entering details, we assume that the ordinary inner product (a, b) between two functions a = a(x) and b = b(x) and the ordinary inner product (a, b) between two row-vectors a and b are defined already. Further, we define inner product between two matrices in the following discussion. Because we use these different types of inner products in this paper, to avoid confusion of notations, we define new notation of inner product between two matrices. For this purpose, in the first part of this paper, we borrow the well known notation of inner product <; a|b > between a row-vector ( bra-vector) <; a| and a column-vector (cket-vector) |b > in quantum mechanics and we extend this expression to an inner product <; A|B > between two matrices A and B. Associated with the above extended inner-product of matrices, we define norms of matrices. Based on this concept of norm, we define a set of signal matrices each of which has a norm smaller than a given positive number A. The proposed optimum approximation is defined on this set of signal matrices. Using these notations, we succeed to show that both the approximation matrix and the corresponding error matrix are expressed by spectrum matrix F(ω) that are derived from the signal matrix and some kernel matrices. Secondly, we present an upper limit of the above measures of error. By differentiating the core-matrix which is contained in this upper limit and cons- dering the obtained matrix formulas to be zero, we derive the optimum interpolation matrices which minimize the upper limit of the measures of error. We prove that these optimum interpolation matrices satisfy two conditions of the optimum approximation which are previously reported by the authors and we establish that the presented approximation has the above remarkable feature of the optimum approximation. Thirdly, we show the optimum matrix trans-multiplexer with given matrix filter bank at receiving user side and with the optimized matrix filter bank at transmission side.