Method for Differentiating Complex Matrix Functions in Digital Antenna Array Response Models

The paper proposes a method for differentiating the Khatri–Rao products of several matrix functions used in digital antenna array (DAA) response models, which are matrix expressions depending on the same vector of unknowns. The product of block matrices of values of amplitude-frequency responses (AFR) of several series-connected digital filters (quadrature I/Q demodulators, decimating filters, fast Fourier transform (FFT) filters, or so-called frequency filter banks, etc.), as well as the AFR of antenna elements or several stages of DAA receiver channels.

The procedure for the factorization of matrix derivatives of the specified type is considered. For compact notation of differentiation results, it is proposed to use matrix operations of block sum, column, and row diagonalization. For example, we present the results of differentiation of the Khatri–Rao products consisting of two and three matrices depending on one vector of variables, and the variants of factorization of such derivatives.

The proposed differentiation method is generalized to the case of a combination in one expression of different types of derivatives with respect to the block vector of the Khatri–Rao product, when the elements of not all, but several block matrices depend on the same block of unknowns.

The factorization procedure allows us to simplify the differentiation process and reduce calculations when forming the Fisher information matrix to analyze the potential accuracy of signal parameter estimation in DAA.