DOA Estimation Using Forward Spatial Smoothing, Classical MUSIC Algorithm, and Toeplitz Algorithm

DOA (Direction of Arrival) estimation is a core technology in array signal processing, used to determine the azimuth angles of signal sources. For the challenging problem of DOA estimation with coherent signal sources, forward spatial smoothing technology effectively decorrelates coherent signals; the classical MUSIC algorithm achieves super-resolution estimation through subspace decomposition; while the Toeplitz algorithm enhances estimation accuracy using matrix reconstruction techniques.

The forward spatial smoothing technique divides the array into overlapping subarrays and performs smoothing processing on the covariance matrix to break the correlation between coherent signal sources. The core concept involves partitioning a uniform linear array into multiple overlapping subarrays, averaging the covariance matrices of these subarrays to reconstruct a full-rank covariance matrix. In code implementation, this typically requires calculating subarray covariance matrices using slicing operations and applying matrix averaging. This method significantly improves estimation performance in coherent signal scenarios.

The classical MUSIC algorithm operates based on the orthogonality principle between signal and noise subspaces, constructing a spatial spectrum function to identify angles corresponding to spectral peaks. The algorithm first performs eigenvalue decomposition on the received data covariance matrix, then uses noise eigenvectors to construct the spectrum function, and finally achieves DOA estimation through spectral peak search. Key implementation steps include eigendecomposition using functions like 'eig()', noise subspace identification, and peak finding algorithms. This algorithm demonstrates excellent super-resolution characteristics under non-coherent signal conditions.

The Toeplitz algorithm addresses estimation errors in covariance matrices caused by limited snapshots by enforcing the Toeplitz property through matrix reconstruction. The algorithm first computes the sample covariance matrix, then reconstructs the Toeplitz matrix by averaging its diagonal elements, and finally performs subspace decomposition based on the reconstructed matrix. Implementation typically involves matrix diagonal averaging operations and Toeplitz matrix reconstruction using dedicated functions. This method effectively improves estimation stability under low signal-to-noise ratio conditions.

These three algorithms each have distinct characteristics: forward spatial smoothing excels at handling coherent signals, MUSIC algorithm achieves the highest accuracy under ideal conditions, while the Toeplitz algorithm demonstrates better robustness in engineering practice. In practical applications, appropriate algorithms can be selected based on signal environment characteristics, or combined usage can be implemented to achieve optimal performance through algorithm fusion techniques.