Searching for Alternatives to the Savitzky–Golay Filter in the Spectral Processing Domain

Abstract

An elegant, well-established effective data filter concept, proposed originally by Abraham Savitzky and Marcel J.E. Golay, is undoubtedly a very effective tool, however not free from limitations and drawbacks. Despite the latter, over the years it has become a "monopolist” in many fields of spectra processing, claiming a "commercial" superiority over alternative approaches, which would potentially allow to obtain equivalent or in some cases even more reliable results. In order to show that basic operations performed on spectral datasets, like smoothing or differentiation, do not have to be equated to the application of the one particular single algorithm, several of such alternatives are briefly presented within this paper and discussed with regard to their practical realization. A special emphasis is put on the fast Fourier methodology (FFT), being widespread in the general domain of signal processing. Finally, a user-friendly Matlab routine, in which the outlined algorithms are implemented, is shared, so that one can select and apply the technique of spectral data processing more adequate for their individual requirements without the need to code it prior to use.

Keywords

Signal processing data smoothing noise reduction numerical differentiation Savitzky–Golay filter fast Fourier transform FFT

Introduction

Despite a continuous development of spectroscopy, it is not possible even now to experimentally acquire the "ideal" sets of spectral data. This regularity is inextricably linked to the primal imperfectness of the measured signal known as the noise. In order to preserve the original, significant information carried by the recorded signal, one would usually make the best possible efforts to reduce the level of the noise among the collected data. This can be generally done in two ways, at the stage of acquisition (optimization of an experiment) and right after it. The latter approach, involving mathematical and statistical apparatus is known under the name of filtering, smoothing, denoising, or more generally, signal (pre)processing.^1,2

The term processing, however, is not limited only to the problem of noise reduction. In fact, any modification applied to spectra supporting their interpretation may be included in this definition.² Particular attention is given in this paper to the group of rather basic transformations,³ enabling it to reveal additional information initially “hidden” within the originally recorded signal, namely, calculation of spectral derivatives. This is due to the fact that such straightforward at first sight operations can be effectively applied to a baseline correction^2,3 as well as overlapping band decomposition.^4–8

Since both smoothing and differentiation are rather standard and widely used operations performed in spectroscopic research,³ most of the computer packages related to the acquisition and/or analysis of spectra (data) are supplemented with the routines to allow them to automatically perform the proper calculations. While this arrangement in general tends to be highly convenient for the users, it has one potential pitfall. The majority of commercial user-friendly routines are based on (and thus supports) only one type of data processing algorithm, that is, the approach originally proposed by Savitzky and Golay.⁹

Although the Savitzky–Golay filter is undoubtedly an extremely useful tool (and thus has become a “monopolist” in the spectral processing domain), it has been reported to have some drawbacks, too.^10–15 Even though a variety of upgrades have been introduced to increase its efficiency,^14,16–18 yet, some limitations related to the very core of the algorithm still remain impossible to be fully resolved. For the above reasons, the Authors believe that alternative concepts of spectral processing should deserve more attention.

As an inspiring example of the substitute algorithm gaining popularity, a very intuitive concept of smoothing (denoising) the spectra proposed by Whittaker,¹⁹ and successfully implemented (and validated) by Eilers, can be given.^11,13,20 On the other hand, in various fields of signal processing, algorithms based on the Fourier transform methodology are commonly applied.²¹ Interestingly, in the spectroscopic research related to chemistry, these are used rather occasionally, in spite of the reports on their significant potential.^22–27 Perhaps this might result from seemingly complex Fourier methodology and/or from numerical difficulties related to its performance.^21,28

Therefore, trying to address the outlined issue, the Authors would like to present how to practically implement fast Fourier transform (FFT) and several other techniques (see Supplemental Material) for the processing of spectra. Within a few pages of this paper, the alternative algorithms to the commercialized Savitzky–Golay filter are concisely described and commented with regard to their practical realization. Finally, a functional, ready-to-use routine written in Matlab²⁹ is shared so that one does not have to individually code them prior to usage.

Experimental

Theoretical Background

Differentiation of Spectra with Fast Fourier Transform. In the classical terms, the first derivative dy of the spectrum y can be estimated simply as a difference quotient (see Supplemental Material):^3,30

d y (x_{n}) \approx \frac{y (x_{n}) - y (x_{n - 1})}{x_{n} - x_{n - 1}} = \frac{y (x_{n}) - y (x_{n - 1})}{Δ x_{n}}

(1)

calculated for each data point x_n. Calculation of the derivatives of an order δ, that is, d^δy, is done just by analogy and can be illustrated as differentiation of the previously determined derivative of lower order (δ – 1).

At the same time, in terms of the Fourier transform methodology, a recipe for estimation of the derivative d^δy of the spectrum is expressed with a possibly less intuitive formula:^5,6,21,28

d^{δ} y = F^{- 1} {{(2 π i v)}^{δ} \cdot F y}

(2)

resulting from the mathematical relationship between a function f (x) and its derivative of the order δ in the inverse domain ν (F, Fourier transform):

F \frac{d^{δ} f (x)}{d x^{δ}} = {(2 π i v)}^{δ} \cdot F f (x)

(3)

Hence, the differentiation procedure comes down to three consecutive steps—calculation of the Fourier transform F of the spectrum y, its multiplication by a term (2πiν)^δ and computation of the inverse Fourier transform (F⁻¹) of the resulting product (see Figure 1).

Figure 1.

Schematic depiction of the relationship between spectra and their derivatives (Eq. 2) in both real (left) and the Fourier (right) domains, shown on the example of two overlapping Gaussian curves (dotted lines). The phase shift observed for the FFT of the first derivative dy, relative to both original spectrum y and its (negative) second derivative d²y, results from multiplication with the imaginary term 2πiν.

This very elegant theoretical concept has, however, one major pitfall in the exact form of the Fourier transform of the spectrum F y:

F y = \int_{- \infty}^{- \infty} y (x) \cdot e^{- 2 π i x v} d x

(4)

remains, in general, unknown, and thus cannot be determined analytically. Still, a very good approximation of the latter can be obtained through the numerical technique of the discrete fast Fourier transform (FFT).³¹

F y \approx \sum_{x_{n}}^{N} y (x_{n}) \cdot e^{- \frac{2 π}{N} i x_{n} v}

(5)

As the derivatives of any order are computed directly according to the exact formula (Eq. 2), no additional approximations are needed, which is an unquestionable advantage of the Fourier concept of differentiation. Hence, in this particular field, the FFT approach takes advantage over the Savitzky–Golay filter, as the latter assumes that the spectrum (signal) in a relatively small range of x_n–w – x_n+w (known as a data window), can be effectively reproduced with a polynomial curve of the order p:^3,9,32

(x_{n}) \approx a_{0} + a_{1} x_{n} + a_{2} x_{n}^{2} + a_{3} x_{n}^{3} + \dots + a_{p} x_{n}^{p}, x_{n - w} \leq x_{n} \leq x_{n + w}

(6)

and is thus differentiated analytically as

\begin{array}{l} d y (x_{n}) \approx a_{1} + 2 \cdot a_{2} x_{n} + 3 \cdot a_{3} x_{n}^{2} + a_{3} x_{n}^{3} + \dots + p \cdot a_{p} x_{n}^{p - 1} \\ d^{2} y (x_{n}) \approx 2 \cdot a_{2} + 6 \cdot a_{3} x_{n} + \dots + p \cdot (p - 1) \cdot a_{p} x_{n}^{p - 2} \end{array}

(7)

While such approximation tends to be fully reasonable, its implementation has (at least) one significant pitfall. The final output is strongly dependent on the choice of the parameters p (order of the fitted polynomial) and w (size of the data window), which in practice have to be adjusted individually to the particular dataset,^13,15 ensuring the effective performance of the algorithm (robustness versus oversimplification; see Supplemental Material). This is not the case for the derivative computation in terms of the Fourier methodology, for which no parametrization is conceptually introduced.

Filtering of Spectra

In contrast to the outlined finite differences methodology (Eq. 1), both Savitzky–Golay model and Fourier transform allow to simultaneously reduce the (instrumental) noise, which can be represented as a set r of randomly distributed values, disturbing the ideal spectrum ŷ (see Figure 2).^1,32

y = \hat{y} + r

(8)

Figure 2.

Procedure of Fourier spectra filtering, schematically presented on the data taken from Figure 1. The crude spectrum (gray solid line) can be represented as the ideal spectrum (black solid line) disturbed by the noise (jagged horizontal line). The FFT of the former is apodized (zeroed) with a low-pass filter function (dashed line), cutting off the chaotic, “high-frequency” fragment of the computed Fourier transform (in the figure its real part is shown), dominated by the noise. In the result, the recovered spectrum (black dotted line) is characterized by a less “jagged” structure.

This is crucial especially for the estimation of derivatives of the higher order (see Figure S1, Supplemental Material).^3,5,6,27 The concept of the Savitzky–Golay filtering is rather intuitive, that is, “crude,” “jagged” spectrum y is point-by-point idealized with the fragments of the smooth polynomial curves ŷ, fitted to the dataset (Eq. 6) (iterative procedure).^3,9,14,32 The remaining residuals r are believed to express the refused, unwanted discrepancy of data. Instead of fitting the polynomials, a more robust method of moving average computation ((Eq. 6) reduced to a constant term a₀ see Supplemental Material)^1,33 can also be effectively utilized,³⁴ as it was shown, for example, by Norris and Williams.^3,35 On the other hand, reproducing the whole fragments of spectra with a series of model Gaussian and/or Lorentzian curves is also a frequently used practice.^4,36

Another ingenious, model-free concept on how the spectra may alternatively be preprocessed in terms of the classic methodology has been proposed and implemented by Eilers.¹¹ The ideal ŷ spectrum could be recovered from the noisy dataset according to the basic principle, derived from the original idea by Whitaker,¹⁹ that it should remain similar to the original crude spectrum y at the same time being less jagged. Practical realization of those two criteria comes down to a relatively simple-structured objective function given in a vector notation as

{| y - \hat{y} |}^{2} = λ \cdot {| d \hat{y} |}^{2} \to 0

(9)

solutions of which can be found in simple, direct matrix calculations (see Supplemental Material).^11,13 Thus, the outlined algorithm is free from pitfalls related to the iterative fitting of polynomials. The smoothness of the output signal is neatly expressed by its first (or second) derivative (computed as the difference quotient, Eq. 1), and can be freely modified by adjusting (only one) scaling parameter λ (the higher the value of λ is, the more “blurred” the reproduced spectrum becomes).^11,13 In consequence, the method of Eilers as suggested by the author to be a “perfect smoother”,¹¹ indeed takes advantage over the Savitzky–Golay filter in (at least) two fields outlined above.¹³ Yet in contrast to the latter, it cannot be applied alone for determination of spectral derivatives, which may be classified as its (only) potential drawback.

Finally, the issue of the (instrumental) noise reduction can be effectively addressed using the Fourier related methodology.^22,27,33 In principle, the representation of the ideal spectrum ŷ, being the combination of the bell-shaped bands (Voigt profiles), in the inverse domain ν can be illustratively compared to the damped harmonic oscillator (Figures 1 and 2).^{1,21,22,25,27} Thus, the signal in the “lower frequency” range is intense and well structured, while shifting towards the “high frequencies” it exponentially decays, eventually approaching zero. On the other hand, the randomly distributed noise maintains its structureless, chaotic form in the whole range of the Fourier domain and in consequence at some point it starts to dominate over the signal.²⁷ Hence, the main idea of the FFT filtering is to apodize (to zero) the Fourier transform of the spectrum (Eq. 4) above the certain value of ν₀, so that the noise component F r (Eq. 8) is effectively reduced (Figure 2).^4–6,27

F y = F \hat{y} + F r \approx F \hat{y}, v \geq v_{0}

(10)

Paradoxically, processing of the signal in such manner may be done without need to formally compute the FFT (see Supplemental Material). The spectrum y can be convoluted in its original domain with a proper filter function f (often bell-shaped Gaussian or Lorentzian profiles),^1,24 which remains a well-defined numerical operation (y ⊗ f).^1,21 According to the so-called convolution theorem

F y \otimes f = F y \cdot F f = (F \hat{y} + F r) \cdot F f

(11)

the related hypothetical Fourier transform of the spectrum (F y) would be then multiplied with an exponentially decaying curve (F f), effectively dumping its high-frequency part, and thus blurring the unwanted jagging of the signal.^1,4–6,27

Results and Discussion

Regretfully, so elegant in theory Fourier methodology of spectra processing faces several practical hindrances, resulting mostly from the numerical issues.^21,28 Two main of such difficulties are shortly reported below, followed by a brief commentary on how the Authors managed to (successfully) overcome them. Some minor remarks on the FFT implementation, on the other hand, can be found in the Supplemental Material.

Effect of the Edge

First and the major challenge to be coped with is a so-called “edge effect” or “edge problem”,^21,37,38 commonly encountered in signal and image processing. In principle, this results from a discrepancy between the discrete nature of the processed datasets and the continuous character of the Fourier operations performed on them. Due to the very core of the Fourier transform (Eq. 4), the input is expected to be a function defined in the infinite domain of x ∈ (–∞,∞). Yet, the spectra are recorded as the finite sequence of points in the limited range of x₁ ≤ x_n ≤ x_N. Consequently, the regions where the spectrum unexpectedly ends, are prone to a various types of numerical instabilities and for this reason are often being distorted during the data-processing procedure.^37,38

A neat solution to overcome this obstacle is to artificially model the near-edge fragments of the datasets (Figure 3).³⁷ In case of the numerical convolution procedure performed purely in the original (real) domain (Eq. 11), surprisingly effective is to expand the signal by duplicating its edge values (i.e., first and last elements) several times, so that the problematic point is simply moved away from the actual spectrum³⁸ (Data Filtering Through Numerical Convolution, Supplemental Material). However, when the Fourier transform has to be actually computed, more sophisticated type of the “buffer” should be applied.³⁷ As the highest stability of the FFT (Eq. 5) is expected for the functions that are continuous and absolutely integrable (Eq. 4) upon N-periodic self-replication,³⁹ the Authors suggest extrapolating the signal according to the following expression, empirically developed by them for this purpose.

y_{e x} (x_{N + m}) = \frac{a_{0} + a_{1} \cdot x_{N + m}}{1 + \exp {\frac{x_{n + m} - X}{σ}}}

(12)

Figure 3.

Practical implementation of the Fourier transform to differentiation of the ultraviolet–visible (UV–Vis) absorption spectrum of anthracene (data taken from the previous work by Kałka and Turek⁴¹). Prior to the calculation of the FFT, the original signal y (black solid line) has to be extrapolated y_ex in such a way that it reaches zero value at its edge values (black dotted line). The computed Fourier transform (in the figure its real part is shown) is next multiplied by a proper term (2πiν)^δ according to the formula (Eq. 2). The obtained product output is next damped with a low-pass (sigmoidal) filter function (dash-dotted line) and its inverse transform is determined; the searched derivative d²y (here of the second order) is then finally obtained (gray solid line).

The above formula is a neat combination of two types of functions: a polynomial function (here of the first degree, that is, a linear one), approximating the evolution of the signal beyond the known values (numerator), and the sigmoidal function, dumping its intensity (denominator). The “patched” fragment of dataset y_ex reproduced this way remains consistent with the original signal (determination of the coefficients a₀ and a₁ is performed by fitting the polynomial curve to near-edge points of the spectrum y) and eventually approaches zero (start and the ratio of the decay are controlled by the parameters x_σ and σ, respectively), which effectively mimics the continuity of the signal (function) in the remaining complementary domain (Figure 3; Processing of Data with FFT, Supplemental Material). With reference to the above, it should be noted that spectra in the absorbance rather than transmittance mode (zero baseline) are naturally preferred for the above-described approach to FFT processing.

Apodization of the Residuals

According to the formula (Eq. 2), the derivative of the spectrum is obtained by multiplying its Fourier transform with a polynomial term of (2πiν)^δ. Hence, the decay of the signal in the inverse domain should be maintained, although its rate would be slightly slower (Figure 1 and the Filtering of Spectra section above).^25,27 Yet, due to the numerical accuracy limitations, the FFT in the region of the higher frequencies is usually approximated with close to zero values of a constant, low order of magnitude. Unfortunately, being the result of the performed multiplication, those residuals are effectively magnified and eventually the intensity of the output signal, expected to approach zero, starts to increase to significant values (Figure 3).^6,27 In consequence, as its core-product (Eq. 2) is ill-defined, the differentiation procedure may end up with a failure.⁶ As the outlined effect is additionally enhanced by the noise, the Authors suggest to smooth the spectra prior to the derivative calculation.

Therefore, in order to overcome the indicated numerical problem, apodization of the high-frequency fragment of the FFT can be applied.^5,27 For this purpose, the Authors suggest using the already introduced sigmoidal function (cf. Eq. 12) as a low-pass filter f (ν) (Figure 3).

f (v) = \frac{1}{1 + \exp {\frac{V- V_{0}}{σ}}}

(13)

Such an ingenious solution has at least several advantages: the dumping of the signal is done in a continuous way, the core low-frequency part of the FFT remains practically unchanged, and finally the cutoff can be easily modified by adjusting parameters ν₀ and σ, respectively (Supplemental Material). Ultimately, it is relatively straightforward for practical implementation. For all the reasons mentioned above, the Authors encourage to use this type of function also as an effective noise filter, alternatively to the classic bell-shaped profiles.^1,24

Practical Realization

The Fourier approach to spectral processing, as well as several other concepts,^3,33 namely, by Norris–Williams,^3,35 Savitzky–Golay,^9,32 and Whittaker (Eilers),^11,19 have been successfully implemented by the Authors in a form of two shared functional routines (see Supplemental Material) coded in the well-known Matlab environment that is particularly suitable for scientific calculations (and is thus expected to be also compatible with the open source GNU Octave software⁴⁰) and validated on the actual spectral datasets (e.g., Kałka and Turek⁴¹). The first script is dedicated to smoothing of the signal (s_smoother.m), while the other has been programmed with regard to the numerical derivative calculation (s_deriv_calc.m). The developed algorithms, in principle, are not related to only one type of spectral technique and thus can be applied to any type of measured spectra—the users are then welcome to select the methodology which is the most adequate to their needs. Importantly, as the package supports the graphical user interface (GUI) mode, it can be, upon downloading, effectively handled also by the users less experienced in the Matlab software.

Conclusion

The main purpose of the paper was to show that processing of spectra does not have to be equivalent to the application of the Savitzky–Golay filter. By outlining at least a few alternative concepts, the Authors hope to encourage other researchers to seek for and to deliberately choose the most adequate technique due to their individual requirements. To initially facilitate this task, a functional, ready-to-use Matlab routine is provided to the readers, supporting several different methods of signal processing. Prominent amongst these is the less intuitive, but extremely potent Fourier transform, details of which are particularly highlighted and elaborated within the article, especially with regard to its practical implementation.

Supplemental Material

Supplemental Material - Searching for Alternatives to the Savitzky–Golay Filter in the Spectral Processing Domain

Supplemental Material for Searching for Alternatives to the Savitzky–Golay Filter in the Spectral Processing Domain by Andrzej J. Kałka, and Andrzej M. Turek in Applied Spectroscopy

Footnotes

Author Contributions

Andrzej J. Kałka: Concept and methodology of the study; software design, development, and validation; writing the original draft.

Andrzej M. Turek: Supervision of the study; writing the original draft.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The undertaken research and development of the computer routines was supported by the funds of the DigiWorld Research Support Module under the program “Excellence Initiative—Research University” at the Jagiellonian University (U1U/W20/NO/28.11).

ORCID iD

Andrzej M. Turek

Supplemental Material

All supplemental materials mentioned in the text is available in the online version of the journal.

References

Willson

P.D.

Edwards

T.H.

. “Sampling and Smoothing of Spectra”. Appl. Spectrosc. Rev. 1976. 12(1): 1–81.

Mobley

P.R.

Kowalski

B.R.

Workman

J.J.

Bro

. “Review of Chemometrics Applied to Spectroscopy: 1985–95, Part 2”. Appl. Spectrosc. Rev. 1996. 31(1–2): 347–368.

Rinnan

Å.

Van Den Berg

Engelsen

S.B.

. “Review of the Most Common Pre-Processing Techniques for Near-Infrared Spectra”. TrAC, Trends Anal. Chem. 2009. 28(10): 1201–1222.

Maddams

W.F.

. “The Scope and Limitations of Curve Fitting”. Appl. Spectrosc. 1980. 34(3): 245–267.

Kauppinen

J.K.

Moffatt

D.J.

Mantsch

H.H.

Cameron

D.G.

. “Fourier Transforms in the Computation of Self-Deconvoluted and First-Order Derivative Spectra of Overlapped Band Contours”. Anal. Chem. 1981. 53(9): 1454–1457.

Cameron

D.G.

Moffatt

D.J.

. “A Generalized Approach to Derivative Spectroscopy”. Appl. Spectrosc. 1987. 41(4): 539–544.

Antonov

Stoyanov

. “Approach for Increased Information from the Second-Derivative Spectra in UV–Vis Absorption Spectroscopy”. Appl. Spectrosc 1993. 47(10): 1712–1715.

Max

J.J.

Chapados

. “Determination of Spectroscopic Band Shapes Using Second Derivatives, Part I: Theory”. Appl. Spectrosc. 2015. 69(3): 348–362.

Savitzky

Golay

M.J.

. “Smoothing and Differentiation of Data by Simplified Least Squares Procedures”. Anal. Chem. 1964. 36(8): 1627–1639.

10.

Madden

H.H.

. “Comments on the Savitzky–Golay Convolution Method for Least-Squares-Fit Smoothing and Differentiation of Digital Data”. Anal. Chem. 1978. 50(9): 1383–1386.

11.

Eilers

P.H.

. “A Perfect Smoother”. Anal. Chem. 2003. 75(14): 3631–3636.

12.

Luo

Ying

Bai

. “Properties of Savitzky–Golay Digital Differentiators”. Digit. Signal Process 2005. 15(2): 122–136.

13.

Urbas

A.A.

Choquette

S.J.

. “Automated Spectral Smoothing with Spatially Adaptive Penalized Least Squares”. Appl. Spectrosc. 2011. 65(6): 665–677.

14.

Figueiredo

N.S.

Ferreira

L.H.

Dutra

O.O.

. “An Approach to Savitzky–Golay Differentiators”. Circuits, Syst. Signal Process. 2019. 38(9): 4369–4379.

15.

Schmid

Rath

Diebold

. “Why and How Savitzky–Golay Filters Should Be Replaced”. ACS Meas. Sci. Au. 2022. 2(2): 185–196.

16.

Gans

Gill

J.B.

. “Smoothing and Differentiation of Spectroscopic Curves Using Spline Functions”. Appl. Spectrosc. 1984. 38(3): 370–376.

17.

Turton

B.C.H.

. “A Novel Variant of the Savitzky–Golay Filter for Spectroscopic Applications”. Meas. Sci. Technol. 1992. 3(9): 858–863,

18.

Iwata

Koshoubu

. “Minimization of Noise in Spectral Data”. Appl. Spectrosc. 1996. 50(6): 747–752.

19.

Whittaker

E.T.

. “On a New Method of Graduation”. Proc. Edinburgh Math. Soc. 1922. 41: 63–75.

20.

Eigenvector ResearchInc . PLS_Toolbox. Manson, Wisconsin: Eigenvector Research, Inc., 2022.

21.

Jansson

P.A.

. Deconvolution of Images and Spectra. San Diego, California: Academic Press, 1997.

22.

Kauppinen

J.K.

Moffatt

D.J.

Mantsch

H.H.

Cameron

D.G.

. “Smoothing of Spectral Data in the Fourier Domain”. Appl. Opt. 1982. 21(10): 1866–1872.

23.

Jones

R.N.

Shimokoshi

. “Some Observations on the Resolution Enhancement of Spectral Data by the Method of Self-Deconvolution”. Appl. Spectrosc. 1983. 37(1): 59–67.

24.

Mantsch

H.H.

Moffatt

D.J.

Casal

H.L.

. “Fourier Transform Methods for Spectral Resolution Enhancement”. J. Mol. Struct. 1988. 173: 285–298.

25.

Tooke

P.B

. “Fourier Self-Deconvolution in IR Spectroscopy”. TrAC, Trends Anal. Chem. 1988. 7(4): 130–136.

26.

Antonov

Nedeltcheva

. “Resolution of Overlapping UV–Vis Absorption Bands and Quantitative Analysis”. Chem. Soc. Rev. 2000. 29(3): 217–227.

27.

Wahab

M.F.

Gritti

O'Haver

T.C.

. “Discrete Fourier Transform Techniques for Noise Reduction and Digital Enhancement of Analytical Signals”. TrAC, Trends Anal. Chem. 2021. 143: 116354.

28.

Butola

Khare

. “Calculating Numerical Derivatives Using Fourier Transform: Some Pitfalls and How to Avoid Them”. Eur. J. Phys. 2018. 39(6): 065806.

29.

The Mathworks, Inc . Matlab v.R2015a. Natick, Massachusetts: The Mathworks Inc., 2022.

30.

Yeow

Y.L.

Leong

Y.K.

. “A General Computational Method for Converting Normal Spectra into Derivative Spectra”. Appl. Spectrosc. 2005. 59(5): 584–592.

31.

Epstein

C.L.

. “How Well Does the Finite Fourier Transform Approximate the Fourier Transform?” Comm. Pure Appl. Math. 2005. 58(10): 1421–1435.

32.

Press

W.H.

Teukolsky

S.A.

. “Savitzky‐Golay Smoothing Filters”. Comput. Phys. 1990. 4(6): 669–672.

33.

Alsberg

B.K.

Woodward

A.M.

Winson

M.K.

Rowland

Kell

D.B.

. “Wavelet Denoising of Infrared Spectra”. Analyst. 1997. 122(7): 645–652.

34.

Al-Mbaideen

A.A.

. “Application of Moving Average Filter for the Quantitative Analysis of the NIR Spectra”. J. Anal. Chem. 2019. 74(7): 686–692.

35.

Norris

Williams

. “Optimisation of Mathematical Treatments of Raw Near-Infrared Signal in the Measurement of Protein in Hard Red Spring Wheat. I. Influence of Particle Size”. Cereal. Chem. 1984. 61(2): 158–165.

36.

Mayerhöfer

T.G.

Popp

. “Quantitative Evaluation of Infrared Absorbance Spectra–Lorentz Profile Versus Lorentz Oscillator”. ChemPhysChem. 2019. 20(1): 31–36.

37.

Janecki

. “Gaussian Filters with Profile Extrapolation”. Precis. Eng. 2011. 35(4): 602–606.

38.

Kałka

A.J.

Turek

A.M.

. “Compensation of Temperature Effects on Spectra Through Evolutionary Rank Analysis”. Spectrochim. Acta, Part A. 2021. 245: 118770.

39.

Harris

F.J.

. “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform”. Proc. IEEE. 1978. 66(1): 51–83.

40.

Free Software Foundation, Eaton

J.W.

Yap

F.F.

, et al. GNU Octave. www.octave.org [accessed Jan 12 2023].

41.

Kałka

A.J.

Turek

A.M.

. “Do Spectra Live in the Matrix? A Brief Tutorial on Applications of Factor Analysis to Resolving Spectral Datasets of Mixtures”. J. Fluoresc. 2021. 31(6): 1599–1616.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.73 MB