MATLAB Implementation of MUSIC Algorithm with Code Descriptions

The MUSIC (Multiple Signal Classification) algorithm is a high-resolution Direction of Arrival (DOA) estimation technique widely used in radar, sonar, and wireless communication systems. This algorithm leverages the orthogonality principle between signal subspace and noise subspace to effectively estimate arrival angles of multiple signals.

### Algorithm Implementation Approach Data Acquisition and Covariance Matrix Calculation: The implementation begins with signal reception through antenna arrays, followed by computation of the received signal's covariance matrix using MATLAB's cov() function or manual matrix operations. The covariance matrix captures signal correlations and forms the foundation for MUSIC algorithm processing. Eigenvalue Decomposition: The covariance matrix undergoes eigenvalue decomposition using MATLAB's eig() function, separating it into signal subspace (corresponding to larger eigenvalues) and noise subspace (corresponding to smaller eigenvalues). Proper thresholding techniques are implemented to distinguish between signal and noise components. Spatial Spectrum Estimation: Utilizing the orthogonality between noise subspace and signal direction vectors, the algorithm constructs a spatial spectrum function. Code implementation involves scanning through angle ranges and computing spectrum peaks using matrix multiplication operations. Peak detection algorithms identify DOA estimates by locating maximum values in the spectrum.

### Root-MUSIC Algorithm Comparison Root-MUSIC represents an enhanced version of the MUSIC algorithm that employs polynomial root-solving instead of spectrum peak search, reducing computational complexity while improving estimation accuracy. Key implementation differences include: Computational Efficiency: Root-MUSIC eliminates the spectrum search process required in standard MUSIC, implementing polynomial rooting using MATLAB's roots() function for faster computation. Resolution Performance: Root-MUSIC demonstrates superior resolution under high SNR conditions, though it may exhibit reduced stability compared to standard MUSIC in low SNR scenarios. Code implementation requires careful polynomial coefficient calculation from the noise subspace matrix. Implementation Complexity: While standard MUSIC offers straightforward implementation with clear spectrum visualization, Root-MUSIC requires additional polynomial manipulation steps and root selection criteria in the code.

In practical applications, algorithm selection depends on specific requirements. MUSIC algorithm suits scenarios with sufficient computational resources needing intuitive spectrum displays, while Root-MUSIC is preferable for applications prioritizing computational efficiency and estimation precision. MATLAB implementations typically include parameter tuning options for both algorithms to accommodate different operational conditions.