Performance Analysis of the MUSIC Algorithm with Implementation Insights

The MUSIC (Multiple Signal Classification) algorithm is a high-resolution Direction of Arrival (DOA) estimation technique based on subspace decomposition. Its performance analysis primarily focuses on success probability, estimation variance, and estimation errors. The algorithm operates by performing eigenvalue decomposition on the received signal covariance matrix to separate the signal subspace from the noise subspace, ultimately constructing a spatial spectrum function by leveraging the orthogonality between the noise subspace and steering vectors. In practical implementations, this involves computing the covariance matrix using MATLAB's cov() function and performing eigenvalue decomposition with eig() or svd().

Regarding success probability, the MUSIC algorithm requires that the number of signal sources must be smaller than the number of array elements, and the correlation coefficient between sources cannot be too high. When the Signal-to-Noise Ratio (SNR) exceeds a threshold value, the algorithm can correctly resolve sources with nearly 100% probability. As SNR decreases or the angular separation between sources narrows, the success probability shows a nonlinear decreasing trend. Code implementation typically involves threshold detection and source number estimation using information criteria like AIC or MDL.

In terms of estimation variance, theoretical Cramér-Rao Bound (CRB) analysis demonstrates that MUSIC's variance performance surpasses traditional beamforming methods. Under moderate to high SNR conditions, estimation variance inversely correlates with SNR and decreases with increasing snapshot numbers. Both array geometry and source angular separation significantly impact variance performance. Implementation-wise, variance analysis often requires Monte Carlo simulations with loop structures to statistically evaluate performance across multiple trials.

Estimation errors primarily stem from three sources: covariance matrix estimation errors due to finite snapshots, subspace leakage phenomena under low SNR conditions, and subspace rotation caused by source coherence. Particularly in low SNR regions, noise subspace contamination of the signal subspace leads to spectral peak shifts and spurious peak generation. In code implementations, these errors can be mitigated through forward-backward averaging and spatial smoothing techniques for coherent sources.

Practical performance analysis must consider computational complexity, where the eigenvalue decomposition step exhibits complexity proportional to the cube of the number of array elements. For large-scale arrays, fast subspace tracking algorithms like PAST or Oja's rule can reduce computational burden. Hardware implementations must also address numerical stability issues arising from finite word length effects, often requiring careful selection of data types and precision in FPGA or DSP implementations.