Method of Local Optimization of Complex Signal Ensembles Based on the Algorithms of Gradient Descent and Levenberg-Marquardt

Abstract

The article proposes a method of local nonlinear optimization of complex signal ensemble parameters based on the integration of the Gradient Descent and Levenberg-Marquardt algorithms. The proposed approach provides a two-stage optimization of complex signal ensembles after their initial formation: in the first stage, a rapid global reduction of the error function and elimination of major correlation outliers are achieved using Gradient Descent, while in the second stage, stable local convergence and improved accuracy in the nonlinear correlation-function space are ensured by the Levenberg-Marquardt method. Experimental modeling was carried out for five types of sequences (M-sequences, Kasami, Gold, Fibonacci, and exponential), as well as for LTE (15 kHz SCS) and 5G NR (30 kHz SCS) standards. The optimization included 30 iterations, with 20 corresponding to the Gradient Descent stage and 10 to the Levenberg-Marquardt phase, providing an optimal balance between convergence speed and computational stability. The results show that during the Gradient Descent stage, the average and maximum mutual correlation coefficients decrease by 30–35 %, ensuring rapid ensemble alignment and suppression of correlation outliers. The subsequent local optimization using the Levenberg-Marquardt method refines parameters within local minima, providing an additional 10–15 % reduction in correlation and stabilizing the ensemble’s energy characteristics. As a result, the crest factor (CF) decreases by 8–12 %, while the effective signal base increases by 7–9 %. It has been demonstrated that the proposed method ensures stable convergence in multimodal spaces and enhances the noise immunity of complex signal ensembles