Synthesis of Quasi-Harmonic Frames for Multi-Window Spectral Analysis

For spectral analysis of discrete real signals, it is proposed to use a new approach based on the use of a system of quasi-harmonic functions that form a frame. The advantages of this approach compared to the known ones are substantiated. An algebraic method for synthesizing an optimal multiwindow quasi-harmonic system that allows the best approximation of the signal is developed, and it is proved that it forms a tight frame. On its basis, strict analytical expressions describing the optimal quasi-harmonic frame are obtained in vector-matrix form. It is shown that the algorithm for constructing such a frame includes a product of sparse matrices and allows for a highly efficient computational implementation. The results of a computational experiment are presented, which show that all window functions of a quasi-harmonic frame have good time-frequency localization and can be used for spectral analysis of signals.