Engineering the optimal filter: Quantitative assessment of linear noise-reducing filters in spectroscopy

Abstract

We capitalize on the intrinsic separation of information and noise in reciprocal space to quantitatively investigate linear noise-removal filters in spectroscopy. This is accomplished with a cost function that factors the action of these filters into loss of information and leakage of noise. Not surprisingly, these two actions conflict, making the perfect filter impossible. In addition, optimization depends on the data being processed, making the universal filter impossible. However, using this capability, we find that the best practical linear filter is the Gauss–Hermite version introduced by Hoffman et al. in 2002. Examples are provided, and recommendations for next-generation improvements are discussed. Engineering the optimal noise-reduction filter for specific applications is a problem now much closer to being solved.