Resolving features and derivatives in data with noise

A frequently occurring challenge in experimental and numerical observation is how to resolve features, such as spectral peaks - with center, width, height - and derivatives from measured data with unavoidable noise. Therefore, we develop a modified Whittaker-Henderson smoothing procedure that balances the spectral features and the noise. In our procedure, we introduce adjustable weights that are optimized using cross-validation. When the measurement errors are known, a straightforward error analysis of the smoothed results is feasible. As an example, we calculate the optical group delay dispersion of a Bragg reflector from synthetic phase data with noise to illustrate the effectiveness of the smoothing algorithm. The smoother faithfully reconstructs the group delay dispersion, allowing to observe details that otherwise remain buried in noise. To further illustrate the power of our smoother, we study several commonly occurring difficulties in data and data analysis and show how to properly smoothen unequally sampled data, how to obtain discontinuities, including discontinuous derivatives or kinks, and how to properly smooth data in the vicinity of boundaries to the domains.