Mathematical Model of Shifted Time of Combined Signal as Part of Fragments with Linear and Quadratic Frequency Modulation

The issues of synthesis of nonlinear-frequency modulated probing signals, which, in comparison with well-known linear-frequency modulated signals, have a lower maximum level of side lobes of the autocorrelation function, have a practical orientation, relate to actual problems of theory and practice of forming signals with intra-pulse modulation for radio electronic means for various purposes.The paper considers a combined nonlinear-frequency modulated signal consisting of linearly and quadratically-frequency modulated fragments. The peculiarity of the proposed approach to the description of its mathematical model is the introduction of frequency-phase compensation components, which reduces the maximum level of the side lobes of the autocorrelation function of the signal. Calculation of values of compensation components is based on consideration of the effect of derivatives of function of instantaneous phase of fragments up to the most significant order inclusive. The limitation of the method should include the requirement for the existence of their final quantity.In the first section of the article, an analysis of known studies and publications is carried out, which states that for the mathematical model of shifted time, considered in the work, the proposed method of compensating for frequency-phase distortions has not been previously considered. Therefore, in the second section of the work, the corresponding task of the study is formulated. In order to achieve the formulated task of research, the third section of the work develops a mathematical model of the shifted time of the combined signal, which contains the compensatory components of the specified distortions. The importance of their consideration in the resulting signal is theoretically substantiated and clearly demonstrated.As a result of the research, the theory of synthesis of combined signals has been developed, the composition and determination of the magnitude of frequency-phase distortions, which are caused by the appearance of the third derivative of the instantaneous phase function of the quadratically-frequency modulated fragment, have been established.As a direction for further research, it is planned to develop and study a mathematical model of shifted time of a three-fragment combined signal with a quadratically-frequency modulated fragment.