Peak deconvolution with significant noise suppression and stability using a facile numerical approach in Fourier space

Abstract

The deconvolution problem for resolution and feature enhancement of analytical signals is known to be a noise-enhancing procedure to the extent that some authors have called it “violently unstable.” The sharpened result is frequently buried in high-amplitude noise in direct deconvolution using the Fourier division method. With a Gaussian deconvolving function, one can get a “Not-a-Number (NaN) error” due to division by near zero (complex) numbers in Fourier space. Herein, a simple numerical method that stabilizes the calculation of direct Fourier deconvolution and ensures that the noise is controlled is proposed. In this calculation, a small positive constant or symmetric positive peak function is added to the denominator during division in Fourier space, resulting in stability and high-frequency noise suppression. The key advantages of this non-iterative method are speed, numerical stability during division, noise suppression, peak area and location invariance, and minimization of the edge effects after deconvolution. Also, ringing effects in the baseline and negative overshoots are significantly mitigated using the modified denominator method. The numerical approach can be applied to enhance the analytical signals consisting of peaks, e.g., in NMR, IR, Raman spectroscopy, or separation sciences. Both symmetric and one-sided (asymmetric) broadening functions can be used for deconvolution in this method. Applications are also demonstrated for various test functions (square pulse, Voigt, or highly overlapped Gaussian peaks) with clear advantages with respect to the standard Fourier division method. The new denominator addition method is compared with two constrained deconvolution procedures in matrix formalism and Fourier domain.