SYNTHESIS OF ANALYTICAL DESCRIPTION OF THE AUTOCORRELATION FUNCTION FOR GENERALIZED BINARY BARKER SEQUENCES OF TYPE 1 BASED ON ITS DECOMPOSITION USING LINEAR COMPONENTS

Abstract

Correlation functions (aperiodic and periodic auto- and cross-correlation) of sequences and codes are important for analysis and synthesis of signal-code constructions, because they define different characteristics (spectral, detection and others) in different radar, navigation, telecommunication and other kinds of signal processing and radiotechnical systems. The synthesis of mathematical expressions for the autocorrelation function for generalized binary Barker sequences of type 1, which based on the analysis and decomposition of a structure of the autocorrelation function using its linear components, is considered in the article. The hypothesis that the autocorrelation function for some kinds of sequences can be represented by some number of linear components and separated local points is possible due to an existence of periodical and recurrent constituents in structures of such sequences. The considered approach is characterized by a direct analysis of obtained structures of the autocorrelation function for described kind of sequences, separation from them related linear components, and generalization of the form of analytical description for the autocorrelation function of sequence of any possible length using elements of set theory. The approach substantially simplifies the process of solving a problem of synthesis of analytical description of the autocorrelation function. However, it requires a correct choice of the decomposition of structures of the autocorrelation function from all possible ones. The result of a synthesis shows that a general form of analytical description of the autocorrelation function, i.e. for any possible sequence length, for generalized binary Barker sequences of type 1 contains 19 components, of which 13 are linear components, and 6 are separated local points.