MATHEMATICAL MODEL OF CURRENT TIME OF SIGNAL FROM SERIAL COMBINATION LINEAR-FREQUENCY AND QUADRATICALLY MODULATED FRAGMENTS
Authors: Kostyria O. O. , Нryzo A. A. , Khudov H. V. , Dodukh O. M. , Solomonenko Y. S. 
Published: June 2024

Context. One of the methods of solving the actual scientific and technical problem of reducing the maximum level of side lobes of autocorrelation functions of radar signals is the use of nonlinear-frequency modulated signals. This rounds the signal spectrum, which is equivalent to the weight (window) processing of the signal in the time do-main and can be used in conjunction with it.
A number of studies of signals with non-linear frequency modulation, which include linearly-frequency modulated fragments, indicate that distortions of their frequency-phase structure occur at the junction of the fragments. These distortions, depending on the type of mathematical model of the signal – the current or shifted time, cause in the generated signal, respectively, a jump in the instantaneous frequency and the instantaneous phase or only the phase. The paper shows that jumps occur at the moments when the value of the derivative of the instantaneous phase changes at the end of the linearly-frequency modulated fragment. The instantaneous signal frequency, which is the first derivative of the instantaneous phase, has an interpretation of the rotation speed of the signal vector on the complex plane. The second derivative of the instantaneous phase of the signal is understood as the frequency modulation rate.
Distortion of these components leads to the appearance of an additional component in the linear term of the instantaneous phase, starting with the second fragment. Disregarding these frequency-phase (or only phase) distortions causes distortion of the spectrum of the resulting signal and, as a rule, leads to an increase in the maxi-mum level of the side lobes of its autocorrelation function. The features of using fragments with frequency modulation laws in complex signals, which have different numbers of derivatives of the instantaneous phase of the signal, were not considered in the known works, therefore this article is devoted to this issue.
Objective. The aim of the work is to develop a mathematical model of the current time of two-fragment nonlinear-frequency modulated signals with a sequential combination of linear-frequency and quadratically modulated fragments, which provides rounding of the signal spectrum in the region of high frequencies and reducing the maximum level of side lobes of the autocorrelation function and increasing the speed of its descent.
Method. Nonlinear-frequency modulated signals consisting of linearly-frequency and quadratically modulated fragments were studied in the work. Using differential analysis, the degree of influence of the highest derivative of the instantaneous phase on the frequency-phase structure of the signal was determined. Its changes were evaluated using time and spectral correlation analysis methods. The parameters of the resulting signal evaluated are phase and frequency jumps at the junction of fragments, the shape of the spectrum, the maximum level of the side lobes of the autocorrelation function and the speed of their descent.
Results. The article has further developed the theory of synthesis of nonlinear-frequency modulated signals. The theoretical contribution is to determine a new mechanism for the manifestation of frequency-phase distortion at the junction of fragments and its mathematical description. It was found that when switching from a linearly-frequency modulated fragment to a quadratically modulated frequency-phase distortion of the resulting signal, the third derivative of the instantaneous phase becomes, which, by analogy with the theory of motion of physical bodies, is an acceleration of frequency modulation. The presence of this derivative leads to the appearance of new components in the expression of the instantaneous frequency and phase of the signal. The compensation of these distortions provides a decrease in the maximum level of the side lobes by 5 dB and an increase in its descent rate by 8 dB/deck for the considered version of the non-linear-frequency modulated signal.
Conclusions. A new mathematical model of the current time has been developed for calculating the values of the instantaneous phase of a nonlinear-frequency modulated signal, the first fragment of which has linear, and the second – quadratic frequency modulation. The difference between this model and the known ones is the introduction of new components that provide compensation for frequency-phase distortions at the junction of fragments and in a fragment with quadratic frequency modulation. The obtained oscillogram, spectrum and autocorrelation function of one of the synthesized two-fragment signals correspond to the theoretical form, which indicates the adequacy and reliability of the proposed mathematical model.