Subspace Analysis Using PAST Algorithm with Frequency Estimation via MUSIC Method

In this research, we employ the Projection Approximation Subspace Tracking (PAST) algorithm for subspace analysis and the Multiple Signal Classification (MUSIC) algorithm for frequency estimation. The implementation involves several key computational steps: The PAST algorithm utilizes recursive updating of the covariance matrix through efficient rank-1 updates, enabling real-time tracking of signal subspace components. This is typically implemented using matrix decomposition techniques like QR factorization or SVD for numerical stability. The MUSIC algorithm performs eigenvalue decomposition on the covariance matrix to separate signal and noise subspaces, followed by spectral peak searching using the pseudospectrum function. Our implementation includes adaptive thresholding for peak detection and incorporates smoothing techniques to enhance frequency estimation accuracy. These algorithmic approaches enable precise signal frequency tracking with reduced computational complexity compared to batch processing methods. Specifically, the PAST algorithm efficiently decomposes signals into principal components through iterative subspace updates, while MUSIC provides high-resolution frequency estimation through eigen-decomposition-based spectral analysis. This paper details the mathematical foundations of both algorithms, presents implementation methodologies with code optimization strategies, and provides comprehensive experimental results with performance analysis. Through this study, we further demonstrate the significance and practical value of PAST and MUSIC algorithms in modern signal processing applications, particularly in real-time spectrum analysis and adaptive filtering scenarios.