Detection and Analysis of Signal Singularities Using Wavelet Analysis with DB Wavelet Decomposition and Reconstruction

In the implementation of wavelet analysis using MATLAB, we can utilize wavelet decomposition and reconstruction techniques to detect and analyze signal singularities. Wavelet decomposition is a method that breaks down a signal into multiple sub-signals at different frequency bands, while reconstruction combines these sub-signals back into the original signal. Through these techniques, we can gain comprehensive insights into signal characteristics and perform deeper analysis.

Key implementation approaches include using MATLAB's Wavelet Toolbox functions such as wavedec() for multi-level decomposition with Daubechies (db) wavelets, which are particularly effective for singularity detection due to their compact support and vanishing moments. The wenergy() function can calculate energy distribution across decomposition levels to identify abrupt changes. For reconstruction, waverec() or appcoef/detcoef functions allow selective reconstruction using approximation and detail coefficients. Critical algorithm considerations include proper wavelet selection (dbN where N indicates order), decomposition level optimization, and thresholding techniques for noise reduction while preserving singularities.

Sample workflow: 1) Perform multi-level decomposition using [C,L] = wavedec(signal,level,'db4'); 2) Analyze detail coefficients at different scales to locate singularities where coefficients exhibit local maxima; 3) Apply thresholding to detail coefficients using wthresh() for denoising; 4) Reconstruct signal using modified coefficients with waverec(). This approach enables precise identification of discontinuity points, abrupt changes, and transient features in signals across various applications including fault detection and biomedical signal processing.