# Improvement of Mathematical Models with Time-Shift of Two- and Tri-Fragment Signals with Non-Linear Frequency Modulation

## DOI:

https://doi.org/10.20535/RADAP.2023.93.22-30## Keywords:

signals with non-linear frequency modulation, a mathematical model, instantaneous phase jump, autocorrelation function, maximum side lobes level## Abstract

The use of signals with intra-pulse modulation in radar systems allows us to increase the duration of sounding pulses, and therefore the radiated energy with limitations on peak power. In the processing system, compression of the signal provides the necessary time difference, but leads to the appearance of side lobes. This, in turn, causes the ``stretching'' of the passive interference zone by range and worsens the potential for target detection. Therefore, reducing the level of the side lobes of the processed signal is an urgent task of radar. With regard to the maximum level of side lobes, signals with non-linear frequency modulation have advantages, but the models used for their mathematical description need to be clarified. The research carried out by the authors of the article and the obtained results of mathematical modelling explain the mechanism of frequency and phase jumps in signals with nonlinear frequency modulation, consisting of several linearly frequency modulated fragments. These results are obtained for the mathematical model of the current time, when the time of each subsequent fragment is counted from the end of the previous one. The article discusses mathematical models of two- and three-fragment signals with using different approach. The difference is that the start time of each successive linearly frequency modulated fragment is shifted to the origin, that is, the shifted time is used. The advantage of this approach is the absence of frequency jumps at the joints of fragments, but phase jumps at these moments of time still occur. Thus, there is a need to develop a mathematical apparatus for compensating of such jumps. An analysis of known publications conducted in the first section of the article shows that for mathematical models of shifted time, the issue of determining the magnitude of phase jumps at the joints of fragments and the mechanisms for their compensation were not considered. From this follows the task of research, which is formulated in the second section of the work. Mathematical calculations for determining the magnitude of phase jumps that occur in these mathematical models, as well as the results of checking the improved mathematical apparatus, are given in the third section of the work. Further research is planned to be directed at the features of using the developed mathematical models in solving applied problems in radar systems.

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*Visnyk NTUU KPI Seriia - Radiotekhnika Radioaparatobuduvannia*, (93), pp. 22-30. doi: 10.20535/RADAP.2023.93.22-30.

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