Correlation Functions of GBOC Navigation Signals as the Inverse Fourier Transform of the Energy Spectra

The modulating functions (MFs) of GBOC signals (generalized BOC signals) of prospective satellite radio navigation systems, such as Galileo (EU), GPS (United States), and BeiDou (China), are considered. A method for calculating the correlation functions (CFs) of single elements of MFs of GBOC signals based on inverse Fourier transform (FT) of their energy spectra is proposed. The method is based on representation of the energy spectrum as a weighted algebraic sum of the cosines of angles determined by characteristic points of the sine symbol of MFs of GBOC signals (CF breakpoints). According to the method, analytical expressions for the CFs of single elements of MFs of GBOC signals are obtained as the inverse FT of energy spectra with a pulse multiplicity coefficient \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{N}_{{\text{p}}}}$$\end{document} = 2 and 4 for different duty cycle values rho, where \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\rho $$\end{document}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \in $$\end{document} [0, 1]. In some cases, calculation of the CFs of GBOC signals based on the inverse FT of energy spectra is preferable (in particular, in terms of labor-intensity) compared with the method of obtaining CFs based on its general definition.