An Analytical Approach to Signal Denoising Based on Singular Value Decomposition (I)

Signal denoising is a fundamental task in signal processing that aims to extract the true underlying signal from noisy observations. Existing signal denoising methods, such as wavelet transform and Fourier-based filtering, suffer from low computational efficiency and potential loss of important signal components during reduction. Moreover, determining the optimal threshold for singular value selection remains a challenge in traditional techniques that are based on singular value decomposition. In this paper, we introduce an efficient, non-iterative algorithm for signal denoising that leverages two noisy observations. By constructing Hankel matrices from these observations, the proposed method establishes a threshold using the largest singular value of their difference, effectively separating true signal components from noise without the need for iterative optimization. We validate the approach on synthetic data and real-world measurements, including smartphone sensor readings and displacement data from a micro-positioning system with a piezoelectric actuator.