Linear Structures of Correlation Function of Odd-Length Generalized Binary Barker Sequences

Abstract

Spread-spectrum, wideband and pulse compression are well-known currently in use technologies for radar, telecommunication, navigation and other radio systems. The main theoretical problem concerning these technologies boils down to a synthesis and study of signals with certain correlation properties. A collection of mathematical models, which are generation rules for binary sequences, and related sequences with regular structures, which generalize structural features of known binary Barker sequences, were suggested and described in literature and may be taken as a kind of generalized binary Barker sequences. In contrast with known binary Barker sequences, which are known up to length N = 13 and widely used in radar and other radio systems and techniques, the generalized binary Barker sequences can be synthesized by means of deterministic generation rules for lengths N = 4 k –1, N = 4 k , N = 4 k +1, where k is a positive integer, and contain binary Barker sequences of lengths N = 3, 4, 5, 7, 11, 13 as a particular case. However, generally, the correlation function of generalized binary Barker sequence has a high value of sidelobes and therefore they cannot be used separately for high quality signal detection. Because of that, multiplicative complementary sets of generalized binary Barker sequences with a mutual sidelobe reduction are usable, similarly to Golay additive complementary sequences. It is shown that a structuring of the correlation function of odd-length generalized binary Barker sequences lies in the fact that the correlation function may be presented by some finite number of linear structures and singular ones, which are exceptional points of the correlation function. It has been determined that the number of such structures for any odd-length generalized binary Barker sequence does not exceed 7 pieces, of which no more than 3 pieces are linear structures, and no more than 4 pieces are exceptional points of the correlation function. On the basis of the structural analysis a complete system of analytical expressions, which based on above-mentioned linear and singular structures, for the correlation function of off-length generalized binary Barker sequence of any type and subtype is synthesized and presented. Examples of a structuring of the correlation function, which based on linear and singular structures, for odd-length generalized binary Barker sequences of different types and subtypes at k = 8 (lengths N = 31 and N = 33) are done. The theoretical significance of presented results lies in the obtaining of a new additional feature of “generalization” of generalized binary Barker sequences, namely, the structuring of their correlation functions based on linear structures. The obtained results make possible a description of signal components, which are based on odd-length generalized binary Barker sequences, after their matched filtering by means of linear constituents. Bibl. 10, Fig. 3, Tables 3.