A Direct Data Approach to Joint 2-D DOA and Frequency Estimation With L-Shaped Array
Yanming Zhang, Lijun Jiang
Abstract
This article presents a joint two-dimensional dynamic mode decomposition (JTDDMD) approach to estimate the azimuth angle, elevation angle, and frequency for multiple narrowband sources using the L-shaped array. According to the placement position of different coordinate axes of the receiving antenna in the array, i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> - or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> -axis, the received signal is divided into two datasets. Two-dimensional dynamic mode decomposition (TDDMD), the subalgorithm of JTDDMD, is used twice to deal with these two datasets separately. Through the analysis of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> -axis subarray received signal using TDDMD, the electrical angle in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> -direction and frequency are obtained with automatic pairing. Likewise, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> -axis data analysis reveals the paired <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> -direction electrical angle and frequency. Then, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> - and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> -direction electrical angles are linked by the frequency information in the two sets of paired parameters. Finally, the paired elevation angle, azimuth angle, and frequency are extracted through the inversion relationship within the L-shaped array. In addition, because the proposed method can determine the number of signal sources in the estimation process, it is suitable for the estimation cases where the number of signal sources is not known in advance. Several numerical experiments benchmark the proposed method. It is found that compared with previous similar methods, JTDDMD can achieve similar results with less execution time. Consequently, this article can serve as an effective tool for jointly estimating the direction of arrivals and frequency.