**V. A. Sergeev**1

1 National Research Lobachevsky State University of Nizhny Novgorod (Nizhny Novgorod, Russia)

Uniform circular array (UCA) antennas are widely used in radar and sonar systems to determine the angle-of-arrival (AoA) of a useful signal from an arbitrary direction. A fairly large number of papers have been devoted to the study of the characteristics of UCA antennas. In these papers, among other things, the maximum achievable accuracy of the signal AoA estimates has been obtained for the signal-to-noise ratios (SNR) values exceeding certain thresholds. At the same time, the relationship between these SNR threshold values and the maximum achievable accuracy of the signal AoA estimates, as well as the dependence of these two key metrics on the number of UCA elements, antenna diameter and signal wavelength has not been studied in known works. However, these characteristics of the UCA antenna are very important for optimal selection of the antenna geometry.

In this paper, we carefully study the problem of optimizing the parameters of the uniform circular antenna arrays taking into account two main metrics that define the system performance: the theoretical boundary for the accuracy of the useful signal AoA estimates and the threshold values of the signal-to-noise ratio at which this boundary is achieved.

We have developed a three-step algorithm for finding the maximum likelihood (ML) estimates of the angle-of-arrival of a useful signal coming from a source located in the far-field zone of the UCA antenna. Based on the detailed numerical simulations of the signal processing in the UCA antenna with different numbers of elements and the ratio of the antenna aperture to the signal wavelength, the dependences of the root mean-square errors (MSE) of ML estimates in a wide range of values of the signal-to-noise ratio have been obtained. Threshold signal-to-noise ratios values have been found. When SNR values exceed these thresholds, the accuracy of the maximum likelihood AoA estimates reaches the theoretical lower Cramer–Rao bounds (CRB) and does not depend on the signal radiation source direction.

Recommendations on the choice of optimal sizes of circular antenna arrays for specific practical problems have been given. Detailed study of the characteristics of the UCA with a different number of elements and different apertures allows to make practical recommendations for choosing the geometry of the UCA antenna systems.

*Sergeev V.A.* Optimization of the size of the circular antenna array in the problem of determining the direction to the radiation source in the far-field zone. Antennas. 2023. № 5. P. 13–22. DOI: https://doi.org/10.18127/j03209601-202305-02 (in Russian)

*Van Trees H.L.*Detection, estimation, and modulation theory. Part IV. Optimum array processing. N.Y.: Wiley. 2002.*Zolotarev I.D., Berezovskij V.A.*Pelengatsiya v dekametrovom diapazone pri mnogotselevoj situatsii. M.: Radiotekhnika. 2013. (in Russian)*Turchin V.I.*Vvedenie v sovremennuyu teoriyu otsenki parametrov signalov. N. Novgorod: IPF RAN. 2015. (in Russian)*Stoica P., Nehorai A.*MUSIC, maximum likelihood and Cramer–Rao bound. IEEE Transactions on Acoustics, Speech, and Signal Processing. 1989. V. 37. № 5. R. 720–741. DOI: 10.1109/29.17564.*Ottersten B., Viberg M., Stoica P., Nehorai A.*Exact and large sample ML techniques for parameter estimation and detection in array processing. In:*Haykin S., Litva J., Shepherd T.J.*(eds) Radar array processing. Springer Series in Information Sciences. V. 25. Springer, Berlin, Heidelberg. 1993. DOI: 10.1007/978-3-642-77347-1_4.*Vu D.T., Renaux A., Boyer R., Marcos S.A.*Cramer Rao bounds based analysis of 3D antenna array geometries made from ULA branches. Multidimensional Systems and Signal Processing. 2013. V. 24. P. 121–155. DOI: 10.1107/s11045-011-0160-5.*Vinogradov A.D.*Algoritm analiticheskogo razdeleniya radioluchej dvukhluchevogo elektromagnitnogo polya malobazovym radiopelengatorom s chetyrekhelementnoj ekvidistantnoj kol'tsevoj antennoj reshetkoj. Antenny. 2022. № 5. S. 49–55. (in Russian)*Gavrilov V.M., Glukhov R.N., Dement'ev V.K., Korneeva N.N.*Kommutiruemaya kol'tsevaya antennaya reshetka s elektronnym skanirovaniem. Antenny. 2021. № 3. S. 5–10. DOI: https://doi.org/10.18127/j03209601-202103-01. (in Russian)*Bolkhovskaya O.V., Maltsev A.A., Sergeev V.A.*The wavefront estimation and signal detection in multi-element antenna arrays at low SNR. 2nd European Conference on Electrical Engineering and Computer Science (EECS). Bern, Switzerland. 2018. P. 497–501. DOI: 10.1109/EECS.2018.00097.*Bolkhovskaya O.V., Maltsev A.A., Sergeev V.A., Keusgen W., Peter M.*Accurate iterative algorithm for detection and the signal AoA estimation in low SNR region. 2019 4th International Conference on Computing, Communications and Security (ICCCS). Rome, Italy. 2019. DOI: 10.1109/CCCS.2019.8888112.*Bolkhovskaya O.V., Maltsev A.A., Sergeev V.A.*Investigation of theoretical limits for unconditional AoA estimations in multi-element antenna arrays by simulations. IEEE International Conference on Industrial Technology (ICIT). Buenos Aires, Argentina. 2020. P. 663–668. DOI: 10.1109/ICIT45562.2020.9067150.*Sergeev V.A., Bolkhovskaya O.V., Maltsev A.A.*Testing the hypothesis of a plane wavefront of a signal received by a multi-element antenna array. Wave Electronics and its Application in Information and Telecommunication Systems (WECONF). 2022. P. 1–5. DOI: 10.1109/WECONF55058.2022.9803550.*Bolkhovskaya O.V., Sergeev V.A., Maltsev A.A.*A passive system for source detection and distance measurement based on signal wavefront estimation. Journal Radioengineering. 2022. V. 86. № 9. P. 98–112.*Bolkhovskaya O.V., Sergeev V.A., Maltsev A.A.*Cramer-Rao lower bounds for the task of joint estimation of signal initial phase and AoA in multi-element antenna arrays. Wave Electronics and its Application in Information and Telecommunication Systems (WECONF). St. Petersburg, Russian Federation. 2023. P. 1–6. DOI: 10.1109/WECONF57201.2023.10148023.*Turchin V.I., Rodionov A.A.*Obnaruzhenie i pelengatsiya istochnikov s ispol'zovaniem razrezhennykh antennykh reshetok. Izvestiya vuzov. Radiofizika. 2018. T. 61. № 2. C. 122–140. (in Russian)*Athley F.*Threshold region performance of maximum likelihood direction of arrival estimators. IEEE Transactions on Signal Processing. 2005. V. 53. № 4. P. 1359–1373.