Signal Denoising via Singular Value Decomposition and Reconstruction

In signal processing, a widely adopted technique involves performing singular value decomposition (SVD) on signals to extract their most significant components. This approach typically begins with constructing a Hankel matrix from the original signal, followed by computing its SVD using matrix factorization algorithms. The decomposition yields three matrices: U (left singular vectors), S (diagonal matrix of singular values), and V (right singular vectors). The key implementation step involves analyzing the differential spectrum - the differences between consecutive singular values - to identify the maximum inflection point. This critical point indicates the optimal threshold for separating meaningful signal components from noise. In practical implementations, this can be automated using peak detection algorithms applied to the differential spectrum array. Based on this identified threshold point, signal reconstruction is performed by retaining only the significant singular values and their corresponding singular vectors. This selective reconstruction effectively filters out noise components while preserving the essential signal features. The reconstruction process involves matrix multiplication of the truncated matrices: U reduced × S truncated × V reduced transposed. This methodology not only accomplishes effective noise reduction but also enhances the clarity of underlying signal characteristics. The implementation typically requires linear algebra libraries for matrix operations and may incorporate additional optimization through techniques like optimal hard thresholding for singular value selection.