In this work, a two-dimensional analytical model of a multi-position radar system with ambiguous range measurements is elaborated and tested. In the proposed analytical model, properties of ambiguous measurements of fractional parts of relative unambiguity intervals in each radar are determined. Theorems are formulated defining the conditions of the unambiguous mapping of a target’s coordinates onto the aforementioned fractional parts, as well as a reverse unambiguous mapping of those fractional parts onto a two-dimensional vector of integers. The vector contains ranges from the target to pairs of radars composing the considered system. The theorems are based on a principle of mapping the measurements of fractional parts onto a multidimensional unit cube, and the interpretation of the total set of measurements as a multilayer structure of this cube. Moreover, each layer is a multidimensional hypersurface bounded by the cube faces, and the unambiguity conditions are reduced to the conditions that these layers do not intersect with each other. Based on the developed model and the formulated theorems, an algorithm is proposed for disclosing the ambiguities of the fractional parts mentioned, as well as for obtaining unambiguous estimates of the target coordinates. An example of a multi-position radar system and results of modeling chosen elements of the algorithm for disclosing ambiguities are also presented. The aims of further research are formulated, particularly regarding the synthesis of multiposition radar systems and the elaboration of an analytical model for systems for the localization of emission sources of periodic radio signals.
The paper presents the results of the development of a method and an algorithm for the synthesis of optimal basic signalcode structures in the form of code binary sequences, with a minimum criterion for the side lobes of the periodic autocorrelation function of the indicated sequences. To develop this method, approaches based on set theory and number theory were used. The method is based on a discrete representation of the periodic autocorrelation function of sequences in the form of a system of equations defined on a set of integers, set-theoretic interpretation of the constituent parts of sequences, their integer transformations, mutual properties and relations. A number of transformations of the constituent parts of the sequences are developed, analytical expressions for the dependence of the sum modulus of the sequence elements on the sum of the side lobe levels of their periodic autocorrelation function are derived, and the necessary conditions for the existence of sequences are defined and formulated. The relationship between the parameters of the code binary sequence and the canonical representation of the Euler function on the dimension of the sequence is determined. Analytical relationships between the levels of the side lobes of the periodic autocorrelation function and the parameters of the transformed sequence structures are obtained. The criterion of the effectiveness of the developed method and the corresponding algorithm is the ratio of the number of all possible variants of code binary sequences of a given dimension to a quantity that is determined by the developed algorithm; an expression was obtained to estimate the indicated amount. This efficiency is confirmed by the results of simulation and experimental research. The developed method can be used for the creation of secretive noise-proof data transmission radio systems, remote control systems, radar, and communications.