Two-Step Spherical Harmonics ESPRIT-Type Algorithms and Performance Analysis

Spherical arrays have been widely used in direction-of-arrival (DOA) estimation in recent years, and the high-resolution estimation of signal parameter via rotational invariance technique (ESPRIT) was developed in the spherical harmonics domain. However, the spherical harmonics ESPRIT (SHESPRIT) cannot estimate the DOA when the elevation approaches 90°. To solve this problem, we present a two-step SHESPRIT (TS-SHESPRIT) based on two new recurrence relations of complex spherical harmonics. Furthermore, we develop a real-valued two-step SHESPRIT (RTS-SHESPRIT) that exploits a unitary matrix to obtain a real-valued relation between the signal subspace and the steering matrix to further reduce the computational complexity. However, the number of sources that are estimated by RTS-SHESPRIT is limited. Therefore, we propose the semi-RTS-SHESPRIT method, which reduces the computational complexity associated with eigenvalue decomposition (EVD) and avoids the limitations of RTS-SHESPRIT. Relative to SHESPRIT and TS-SHESPRIT, RTS-SHESPRIT and semi-RTS-SHESPRIT reduce the computational burden by 75% during EVD. Furthermore, we derive the mean square errors (MSEs) of the above algorithms and significantly simplify the MSE expressions. Different expressions for the MSEs are due to different recurrence relations used by different SHESPRIT-type algorithms. All proposed two-step SHESPRIT-type algorithms have higher accuracy than traditional SHESPRIT. The simulation results demonstrate the satisfactory performance of our methods.