# Mathematical Model of Two-Fragment Signal with Non-Linear Frequency Modulation in Current Period of Time

## DOI:

https://doi.org/10.20535/RADAP.2023.92.60-67## Keywords:

non-linear frequency modulation, mathematical model, autocorrelation function, sidelobe level, frequency and phase jumps## Abstract

Advantages of using frequency-modulated signals for locating objects are the possibility of using long-term probing pulses. Such signals provide the required radiated power while maintaining the desired discriminating power from range. One such signal that has found wide application is a signal with linear frequency modulation. An undesirable effect of the matched filtering of such a radio pulse is a sufficiently large level of side lobes of the compressed signal at the output of the processing device, the maximum level of which is approximately minus 13 dB. Such an effect may lead to an increase in the probability of false detection or masking of less powerful signals by side lobes of signals with greater power. One method of reducing the level of side lobes is the use of signals with non-linear frequency modulation. An example of such signals is a known two-fragment signal consisting of linearly-frequency modulated fragments in time. However, the mathematical models used to describe such a signal do not fully reflect the effects that occur at the moment of transition from one fragment of the signal to the second. These effects are manifested in a sudden change in frequency and phase, which leads to distortion of the signal spectrum, an increase in the level of the side lobes of the autocorrelation function and sharp changes in their level. Such effects have not been studied in known works, as evidenced by the results of the analysis of studies and publications given in the first section of the article. In the second section of the work, the research task is formulated. The third section of the work is devoted to the development of a mechanism for compensating for the manifestation of detected effects and its mathematical description, which is verified by modeling. Taking into account the detected effects, a new mathematical model of a non-linear frequency modulated signal has been developed. In contrast to those known in the proposed model, instantaneous frequency and phase jumps are compensated for, which occur at the moments when the frequency modulation rate changes during the transition from one signal fragment to another. Further studies should be focused on the peculiarities of compensation for the manifestation of detected effects for signals with a large number of fragments, as well as combinations of fragments with different types of modulation, as indicated in the conclusions on the work.

## References

**References**

Skolnik M. I. (1980). *Introduction to Radar Systems*. McGraw Hill, New York, 590 p.

Barton D. K. (2004). *Radar System Analysis and Modeling*. Artech House Publishers, 535 p.

Cook C. E. and Bernfeld M. (1967). *Radar Signals: An Introduction to Theory and Application*. Academic Press, 531 p.

Cook C. E, Paolillo J. (1964). A pulse compression predistortion function for efficient sidelobe reduction in a high-power radar. *Proceedings of the IEEE*, Vol. 52, Iss. 4, pp. 377–389. doi:10.1109/proc.1964.2927.

Alphonse S. and Williamson G. A. (2019). Evaluation of a class of NLFM radar signals. *EURASIP Journal on Advances in Signal Processing*, Article number: 62, 12 p. doi:10.1186/s13634-019-0658-9.

Kurdzo J. M., et al. (2014). A Pulse Compression Waveform for Improved-Sensitivity Weather Radar Observations. *Journal of Atmospheric and Oceanic Technology*, Vol. 31, Iss. 12, pp. 2713-2731. doi:10.1175/JTECH-D-13-00021.1.

Kurdzo J. M., Cheong B. L., Palmer R. D. and Zhang G. (2014). Optimized NLFM Pulse Compression Waveforms for High-Sensitivity Radar Observations. *International Radar Conference*, pp. 1-6. doi:10.1109/RADAR.2014.7060249.

Gomi K., et al. (2017). Pulse Compression Weather Radar with Improved Sensitivity, Range Resolution, and Range Sidelobe. *38th Conference on Radar Meteorology*, Poster Session 131, pp. 1-7.

Bharadwaj N. and Chandrasekar V. (2009). Frequency Diversity Wideband Waveforms for Dual-Polarization Weather Radars. *34th Conference on Radar Meteorology*, Poster Session P5.12.

Arnab Das, Venugopalan Pallayil. (2016). Analysis of Effective Signal Design for Active Sensing of Undersea Objects/Bottoms in Tropical Shallow Waters. *Conference OCEANS*. doi:10.1109/OCEANSAP.2016.7485558.

Jin G., et al. (2019). Nonlinear Frequency Modulation Signal Generator in LT-1. *Engineering IEEE Geoscience and Remote Sensing Letters*, Vol. 16, Iss. 10, pp. 1570–1574. doi: 10.1109/LGRS.2019.2905359.

Guodong Jin, et al. (2019). An Advanced Nonlinear Frequensy Modulation Waveform for Radar Imaging With Low Sidelobe. *IEEE Transactions on Geoscience and Remote Sensing*, Vol. 57, No. 8, pp. 6155-6168. doi: 10.1109/TGRS.2019.2904627.

Zhao Y., et al. (2020). Non-continuous piecewise nonlinear frequency modulation pulse with variable sub-pulse duration in a MIMO SAR Radar System. *Remote Sensing Letters*, Vol. 11, Iss. 3, pp. 283-292. doi:10.1080/2150704X.2019.1711237.

Xu W., et al. (2021). Staring Spotlight SAR with Nonlinear Frequency Modulation Signal and Azimuth Non-Uniform Sampling for Low Sidelobe Imaging. *Sensors*, Vol. 21, Iss. 19, 6487. doi:10.3390/s21196487.

Hosseini N., Matolak D. W. (2021). Nonlinear Quasi-Synchronous Multi User Chirp Spread Spectrum Signaling. *IEEE Transactions on Communications*, Vol. 69, Iss. 5, pp. 3079 - 3090. doi:10.1109/TCOMM.2021.3055508.

Fan Z., Meng H. (2021). Coded excitation with Nonlinear Frequency Modulation Carrier in Ultrasound Imaging System. *2020 IEEE Far East NDT New Technology & Application Forum (FENDT)*, pp. 31-35. doi:10.1109/FENDT50467.2020.9337517.

Sira S. P., et al. (2009). Waveform-Agile Sensing for Tracking. *IEEE Signal Processing Magazine*, Vol. 26, Iss. 1, pp. 53–64. doi:10.1109/MSP.2008.930418.

Swiercz E., Janczak D., Konopko K. (2022). Estimation and Classification of NLFM Signals Based on the Time–Chirp Representation. *Sensors*, Vol. 22, Iss. 21, 8104. doi: 10.3390/s22218104.

Milczarek H., Leśnik C., Djurovi´c I., Kawalec A. (2021). Estimating the Instantaneous Frequency of Linear and Nonlinear Frequency Modulated Radar Signals—A Comparative Study. *Sensors*, Vol. 21, Iss. 8, 2840. doi:10.3390/s21082840.

Shahrezaei H., Kazerooni M., Fallah M. (2016). A Robust SAR NLFM Waveform Selection Based on the Total Quality Assessment Techniques. *Journal of Communication Engineering*, Vol. 5, No. 2, pp. 116-135. doi:10.22070/jce.2017.1912.1018.

Song Chen, et al. (2022). A Novel Jamming Method against SAR Using Nonlinear Frequency Modulation Waveform with Very High Sidelobes. *Remote Sensing*, Vol. 14, Iss. 21, 5370. doi:10.3390/rs14215370.

Jeyanthi J. E., Shenbagavalli A., Mani V. R. S. (2017). Study of Different Radar Waveform Generation Techniques for Automatic Air Target Recognition. *International Journal of Engineering Technology Science and Research*, Vol. 4, Iss. 8, pp. 742-747.

Doerry A. W. (2006). Generating nonlinear FM chirp waveforms for radar. *Sandia Report*, SAND2006-5856, 34 p. doi:10.2172/894743.

Prakash B. L., Sajitha G. and Rajeswari K. R. (2016). Generation of Random NLFM Signals for Radars and Sonars and their Ambiguity Studies. *Indian Journal of Science and Technology*, Vol. 9, Iss. 29, pp. 1-7. doi:10.17485/ijst/2016/v9i29/93653.

Saleh M., Omar S.-M., Grivel E., Legrand P. (2021). A Variable Chirp Rate Stepped Frequency Linear Frequency Modulation Waveform Designed to Approximate Wideband Non-Linear Radar Waveforms. *Digital Signal Processing*, Vol. 109, 102884. doi:10.1016/j.dsp.2020.102884.

Kurdzo J. M., et al. (2019). A Neural Network Approach for Waveform Generation and Selection with Multi-Mission Radar. *IEEE Radar Conference*, pp. 1-6. doi:10.1109/RADAR.2019.8835803.

Anoosha Chukka and Krishna B. T. (2022). Peak Side Lobe Reduction analysis of NLFM and Improved NLFM Radar signal. *AIUB Journal of Science and Engineering (AJSE)*, Vol. 21, Iss. 2, pp. 125–131. doi:10.53799/ajse.v21i2.440.

Galushko V. G. (2019). Performance Analysis of Using Tapered Windows for Sidelobe Reduction in Chirp-Pulse Compression. *Radio Physics and Radio Astronomy*, Vol. 24, Iss. 4, pp. 300-313. doi:10.15407/rpra24.04.300.

Swiercz E., Janczak D., Konopko K. (2021). Detection of LFM Radar Signals and Chirp Rate Estimation Based on Time-Frequency Rate Distribution. * Sensors*, Vol. 21, Iss. 16, 5415. doi:10.3390/s21165415.

Adithyavalli N., Rani D. E., Kavitha C. (2019). An Algorithm for Computing Side Lobe Values of a Designed NLFM function. *International Journal of Advanced Trends in Computer Science and Engineering*, Vol. 8, No. 4, pp. 1026-103. doi:10.30534/ijatcse/2019/07842019.

Xu Z., Wang X., Wang Y. (2022). Nonlinear Frequency-Modulated Waveforms Modeling and Optimization for Radar Applications. * Mathematics*, Vol. 10, Iss. 21, 3939. doi:10.3390/math10213939.

Valli N. A., Rani D. E., Kavitha C. (2019). Modified Radar Signal Model using NLFM. *International Journal of Recent Technology and Engineering (IJRTE)*, Vol. 8, Iss. 2S3, pp. 513-516. doi: 10.35940/ijrte.B1091.0782S319.

Jeevanmai R., Rani N. D. (2016). Side lobe Reduction using Frequency Modulated Pulse Compression Techniques in Radar. *International Journal of Latest Trends in Engineering and Technology*, Vol. 7, Iss. 3, pp. 171-179. doi:10.21172/1.73.524.

Chan Y. K., Yam C. M., Koo V. C. (2009). Side lobes reduction using simple two and tri-stages non linear frequency modulation (NLFM). *Progress in Electromagnetics Research*, Vol. 98, pp. 33-52. doi:10.2528/PIER09073004.

Valli N. A., Rani D. E., Kavitha C. (2019). Windows for Reduction of ACF Sidelobes of Pseudo-NLFM Signal. *International Journal of Scientific & Technology Research*, Vol. 8, Iss. 10, pp. 2155-2161.

Ghavamirad R., Sebt M. A. (2019). Side lobe Level Reduction in ACF of NLFM Waveform. *IET Radar, Sonar & Navigation*, Vol. 13, Iss. 1, pp. 74-80. doi:10.1049/iet-rsn.2018.5095.

Valli N. A., Rani D. E., Kavitha C. (2019). Doppler Effect Analysis of NLFM Signals. *International Journal of Scientific & Technology Research*, Vol. 8, Iss. 11, pp. 1817-1821.

Parwana S., Kumar S. (2015). Analysis of LFM and NLFM Radar Waveforms and their Performance Analysis. *International Research Journal of Engineering and Technology (IRJET)*, Vol. 02, Iss. 02, pp. 334-339.

Widyantara M. R., et al. (2018). Analysis of Non Linear Frequency Modulation (NLFM) Waveforms for Pulse Compression Radar. *Jurnal Elektronika dan Telekomunikasi (JET)*, Vol. 18, No. 1, pp. 27-34. doi: 10.14203/jet.v18.27-34.

Doerry A. W. (2006). Technical Report: SAR Processing with Non-Linear FM Chirp Waveforms. *Sandia National Laboratories*, 66 р. doi:10.2172/902597.

Alphonse S., Williamson G. A. (2014). Novel radar signal models using nonlinear frequency modulation. *22nd European Signal Processing Conference (EUSIPCO)*, pp. 1024-1028.

Bayındır C. (2015). A Novel Nonlinear Frequency-Modulated Chirp Signal for Synthetic Aperture Radar and Sonar Imaging. *Journal of Naval Science and Engineering*, Vol. 11, No. 1, pp.68-81.

Kavitha C., Valli N. A., Dasari M. (2020). Optimization of two-stage NLFM signal using Heuristic approach. *Indian Journal of Science and Technology*, Vol. 13, Iss. 44, pp. 4465-4473. doi:10.17485/IJST/v13i44.1841.

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*Visnyk NTUU KPI Seriia - Radiotekhnika Radioaparatobuduvannia*, (92), pp. 60-67. doi: 10.20535/RADAP.2023.92.60-67.

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