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# Joint Frequency and DOA Estimation with Automatic Pairing Using the Rayleigh–Ritz Theorem

Haiming Du1,*, Han Gao1, Wenjing Jia2
1 College of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou, 450003, China
2 Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW, 2007, Australia
* Corresponding Author: Haiming Du. Email:

Computers, Materials & Continua 2021, 67(3), 3907-3919. https://doi.org/10.32604/cmc.2021.015969

Received 16 December 2020; Accepted 23 January 2021; Issue published 01 March 2021

### Abstract

This paper presents a novel scheme for joint frequency and direction of arrival (DOA) estimation, that pairs frequencies and DOAs automatically without additional computations. First, when the property of the Kronecker product is used in the received array signal of the multiple-delay output model, the frequency-angle steering vector can be reconstructed as the product of the frequency steering vector and the angle steering vector. The frequency of the incoming signal is then obtained by searching for the minimal eigenvalue among the smallest eigenvalues that depend on the frequency parameters but are irrelevant to the DOAs. Subsequently, the DOA related to the selected frequency is acquired through some operations on the minimal eigenvector according to the Rayleigh–Ritz theorem, which realizes the natural pairing of frequencies and DOAs. Furthermore, the proposed method can not only distinguish multiple sources, but also effectively deal with other arrays. The effectiveness and superiority of the proposed algorithm are further analyzed by simulations.

### Keywords

Array signal processing; DOA estimation; automatic pairing; Rayleigh–Ritz theorem

H. Du, H. Gao and W. Jia, "Joint frequency and doa estimation with automatic pairing using the rayleigh–ritz theorem," Computers, Materials & Continua, vol. 67, no.3, pp. 3907–3919, 2021.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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