Author et al., International Journal of Emerging Technologies in Computational and Applied Sciences, 8(1), March-May, 2043, pp. xx-xx Critical Analysis of Spectral Deconvolution Methods J. Dubrovkin Computer Department, Western Galilee College 2421 Acre, Israel Abstract: Spectral deconvolution methods are critically reviewed using simple mathematical description and graphical illustration. It is shown that all methods are based on the compromise between the maximum resolution enhancement and the minimum intensity of the side lobes. It is emphasized that there exists no general solution ("panacea") for all deconvolution problems and the optimal deconvolution algorithm should be chosen with regard to the particular analytical problem under study. The general approach to selecting the quasi optimal algorithm is suggested. Keywords: Spectral deconvolution; Tikhonov regularization; self-deconvolution; artificial improvement of the spectral resolution; inverse filtering. I. Introduction Spectrometry is one of the most widespread theoretical and experimental physical methods, which has been employed in various fields of science and technology for more than a century [1]. Computer-enhanced industrial and laboratory spectral instruments are powerful tools used by researchers and practitioners in physics, chemistry, biology and medicine, pharmacology, metallurgy, food industry, astronomy, archaeology, criminalistics, as well as in the environment control. Using smart mathematical software algorithms, it is possible to automatically obtain correct information about the samples under study. In general, spectral data processing can be divided into two sequential steps: pre-processing of raw data and extracting the necessary information [2]. The main goal of the first stage is reducing the noise, removing the non-zero baseline and resolving overlapped spectral components (lines and bands). Overlapping of spectral components is accounted for their broadening due to the instrumental distortions (e.g., limited maximum delay time in the interferogram obtained using Fourier-transform infrared spectrometers and non-zero width of the instrumental function of monochromators) and inter- and/or intra-atomic and molecular interactions (e.g., Stark broadening in dense plasma [3], spin-spin interactions in NMR-spectroscopy [4], inter- and intra-molecular associations in IR-spectroscopy of liquids [5]) in the samples under study. Overlapping spectral components can be only partly separated by spectral data processing usually referred to as deconvolution. The artificial resolution improvement (spectral peak sharpening) and decomposition of a composite spectrum into individual pure-component spectra are sometimes viewed as a deconvolution process in a more general sense. From the theoretical point of view, deconvolution is generally considered as an inverse problem, which is commonly encountered in practical applications [6]. Since the 1930s, deconvolution has been the subject of numerous studies in optics, spectroscopy, analytical chemistry, and chemometrics [7-21]. Derivative spectroscopy (DS), which is one of the first deconvolution methods, is now widely used for pre-processing [11, 12]. Since, in this method, the low-order polynomial background is suppressed and spectral resolution is enhanced, the first- and second-order spectrum derivatives are commonly used for qualitative and quantitative analysis in many practical applications. Another method widely used in spectroscopic practice is Fourier self-deconvolution (FSD) [7-10, 13, 14], which was developed on the basis of Fourier transform spectroscopy [9]. The mature period of deconvolution started with the approach of viewing deconvolution as part of Digital Signal Processing (DSP) of spectral data in the framework of the functional engineering model of a spectral instrument (see review [15]). A set of effective deconvolution algorithms based on a complex of mathematical tools, such as digital filters, spline approximation, and Tikhonov regularization (Ridge regression), is described in [15-21]. General mathematical theory of deconvolution has its foundation in the integral equation theory [23]. Unfortunately, the researchers who are engaged in the deconvolution problem often present similar results using different terminology and mathematical tools. For example, in NMR spectroscopy, mathematical correction of instrumental imperfections is referred to as reference deconvolution (see references in [24]). Moreover, the use of formal mathematical approach makes it difficult for the reader who was brought up on classical concepts of optics and spectroscopy [7-10, 25] and is unfamiliar with DSP to gain detailed understanding of deconvolution and viewing the problem as a whole. In the present spectroscopy-oriented analytical review, the most common methods of deconvolution of overlapping symmetrical spectral components are critically discussed. The advantages and disadvantages of these methods are demonstrated using sufficiently simple mathematical tools which should be clear to wide audience. In what follows, for the sake of simplicity, term “line” is used for short of phrase "line and band". Standard algebraic notations are used throughout the article. All calculations were performed and the plots were built using the MATLAB program.