Methods for the solution of systems of nonlinear algebraic equations and functions’ minimization tasks: elements of theory and applications

Abstract

Due to the rapid development of modern telecommunication and information technologies, there is a great need to develop new and improve existing methods for signal demodulation, information encoding and decoding, automatic objects’ recognition etc. A large number of these tasks is reduced to solving systems of nonlinear algebraic equations and problems of minimizing functions. The monograph presents the methods of localization of all the roots of nonlinear algebraic equations systems based on Jacobian’s non-degeneracy, Kantorovich’s theorem, and Krawchyk’s operator are developed. Based on the method of searching all the roots of systems of nonlinear algebraic equations, the method of global minimization of functions of many variables is developed. Also, based on the developed methods and autoregressive model of speech, the new efficient high-speed methods of calculating the immittance spectral frequencies of speech signals have been developed. It is shown that the method has higher computational efficiency over the methods used in existing communication systems. Conditions for orthogonality and minimization of the Riesz ratio for wavelets based on Jacobi polynomials are established. Basing on maximizing a posterior probability function, the method of demodulation and estimation of channel parameters on the basis of particle filtering is developed, the method of demodulating signals with a continuous phase is developed, which is based on the use of signal phase values only and the method of blind separation of signals with amplitude-phase modulation. It is shown that the proposed particle filtering demodulator provides an advantage over standard demodulators based on the use of Gardner and Costas loops. The developed method of demodulation of signals with a continuous phase has shown greater resistance to phase and frequency distortions of the signal in the communication channel as compared to the traditional demodulator. The method of semi-automatic data classification based on the minimization of Tikhonov functional using discrete Laplacian and the quasi-optimal choice of the regularization parameter has been developed and verified for different data sources. Also a new computationally efficient method for automatic identification of the speaker’s gender based on the Gaussian mixture models was developed based on the maximization of likelihood function. This method is based on the use of information vectors consisting of the pitch period of the speech signal and the set of cepstral coefficients.