Generalized subband analysis and signal synthesis

Currently, one of the main approaches used in the analysis of properties and synthesis of signals of various classes is the subband methodology, which is carried out from the position of the Fourier transform of signal counts (frequency representations) into subbands of the transform definition domain (transformants). At the same time, the main tool of subband analysis, including wavelet analysis, which is widely used for these purposes, is bandpass filters, and first of all, filters with finite impulse response (FIR filters). The paper presents the basics of building a theory of subband analysis / synthesis of signals of various classes using transformations based on any orthonormal basis with weight. The proposed approach is based on the concept of the fraction of the square of the Euclidean signal norm in a given subband of the domain of definition of the transformant. It is shown that the basis of the mathematical apparatus of subband analysis is a new class of matrices called subband ones. Some properties of the eigenvalues of these matrices are established, the problem of optimal selection of additive components of signals is formulated and solved.