Objective Maximum likelihood (ML) algorithms can achieve high-resolution direction-of-arrival (DOA) estimation, yet they suffer from excessive computational complexity. As a simplified version of ML, the alternating projection (AP) method effectively transforms the multidimensional optimization problem into a sequence of one-dimensional subproblems; however, its complexity remains relatively high. To address the two drawbacks of the AP method—namely, \textO (n^3 ) -level matrix operations and a complicated search process—an efficient AP method is proposed.
Method By exploiting the highly structured information of orthogonal projection matrix, an iterative update algorithm is designed to overcome the former issue. It is well known that, given a steering matrix and its corresponding orthogonal projection matrix, the updated orthogonal projection matrix after appending a column to the steering matrix can be computed directly and efficiently. In contrast, when a column is removed from the steering matrix, the resulting orthogonal projection matrix is generally not unique, making it difficult to obtain through straightforward computation. To address this issue in column-removal scenarios, an effective strategy is to pre-store certain auxiliary information. By storing a normal matrix and computing the Schur complement together with the Sherman–Woodbury formula, the computational problems associated with column removal and updating are both mitigated efficiently. For Uniform Linear Array (ULA) scenarios, a Fast Fourier Transform (FFT)-based method can be employed instead of direct spectral search to obtain DOA information by exploiting the Toeplitz structure of the covariance matrix. However, under high-precision FFT settings, the advantage in computational complexity is significantly reduced due to the increased number of frequency bins. Inspired by the recent FFT without zero-padding scheme, this approach is integrated into the efficient AP framework. Furthermore, a corresponding iterative update algorithm is also constructed for the projection congruence transformation required in adapting the FFT-based procedure with an acceptable computational burden. In summary, apart from the FFT without zero-padding scheme, the proposed method substantially reduces the computational burden at the cost of modest storage overhead.
Results Simulation results confirm the effectiveness of the proposed scheme. Compared with the widely adopted subspace-based MUltiple Signal Classification (MUSIC) algorithm, the proposed method achieves gains of 3.2 dB and 14.9 dB at a Root Mean Square Error (RMSE) of 2°, respectively, for three equal-power uncorrelated sources and for three equal-power sources with a correlation coefficient of 0.95.
Conclusion The efficient AP method reduces the computational complexity by an order of magnitude and can also serve as a general scheme for the successive update of orthogonal projection matrices.
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