Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Tutorials Issues in Progress Current Issue All Issues Early Posting Feature Issues

Norton-Beer apodization and its Fourier transform

Konstantin F. F. Ntokas, Jörn Ungermann, and Martin Kaufmann

Open Access Open Access

Abstract

In Fourier transform spectroscopy, apodization is used to alter the instrument line shape, reducing the prominence of its side lobes. The Fourier transform of the apodization window is of great interest as it allows us to compute or optimize the line shape. In the last decades, many apodization windows have been proposed, from which the group of Norton-Beer apodization functions gained large popularity in Fourier transform spectroscopy. While for a small set of specific Norton-Beer apodization functions analytical solutions of the Fourier transform have been presented in the past, we present here a general method, which allows us to calculate the analytical solution of the Fourier transform for any Norton-Beer apodization function. This paper also documents the free Python library called norton_beer. It contains functions to generate apodization windows and their Fourier transform following the presented analytical solution. Furthermore, new Norton-Beer apodization functions can be generated for any desired spectral resolution.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. INTRODUCTION

Fourier transform spectroscopy (FTS) applications leverage the fundamental concept of Fourier transformation to convert an interferogram into a spectrum. Due to the nature of a finite optical path difference, a discontinuity at the edges of the interferogram cannot be avoided. One models a finite interferogram by multiplying an assumed infinite interferogram with a boxcar function of appropriate length. The convolution theorem states that the Fourier transform of a pointwise multiplication of two functions is a convolution of their Fourier transforms. The Fourier transform of a boxcar is a sinc-function, which is notorious for its large side lobes that decay slowly toward zero. The spectral resolution of an instrument is measured by the full width at half-maximum (FWHM) of the main lobe of this sinc, which is inversely proportional to the length of the boxcar. The measured spectrum can then be expressed in terms of an “ideal” spectrum convolved with the sinc-function, resulting in an oscillating output due to the large side lobes of the sinc-function. The idea of apodization can be seen from two viewpoints [1]. First, the interferogram is pointwise multiplied with a function that decreases toward the edges to reduce the ringing artifacts caused by the discontinuity at the edges. Second, a window function is chosen in a way so that its Fourier transform contributes only to the spectral region close to zero. In other words, it minimizes the side lobes and maintains the localization of the spectral information. Both strategies end up in spectra with decreased oscillations. Sadly, there is a fundamental limit to this process. According to the uncertainty principle [2], the finite support of the interferogram being nonzero prevents a finite support for the instrument line shape; still a faster decay toward zero is possible and advantageous. In general, stronger decaying window functions decrease the side lobes more at a cost of an increased FWHM of the main lobe. Apodization, therefore, can be considered as a trade-off between spectral resolution and the leakage of spectral information to neighboring spectral samples. Many different apodization functions have been proposed in the past for different criteria of optimality. Examples are the decrease of discontinuity at the edges of the interferogram used by the triangle window [3] and a modified version of the Norton-Beer apodization [4], the decrease of the maximum absolute value of the first side lobe of the Fourier transform used by the original and extend Norton-Beer apodization [5–7] and the Hamming window [1], the minimization of the main lobe’s width for a given side lobe level used by Dolph-Chebyshev window [8], and the maximization of the energy concentration in the main lobe for a fixed side lobe level used by the Kaiser window [9,10]. Various measures have been employed to evaluate apodization functions, with the resolution of the main lobe (measured by the FWHM) and the maximum absolute value of the first side lobe of the Fourier transform being the most appropriate criteria for assessing their performance [1,11,12]. These two measures are incorporated into the Filler’s diagram, which plots the FWHM against the absolute maximum of the first side lobe [13]. The Norton-Beer apodization is optimal in the sense of these two measures because it minimizes the maximal value of the first side lobe for a given FWHM of the main lobe. Norton-Beer apodization, therefore, aligns with the second viewpoint mentioned earlier, which entails selecting a window function in such a manner that its Fourier transform primarily affects the spectral region close to zero [1]. Three apodization functions called Norton-Beer weak, medium, and strong were proposed, corresponding to an increase of the FWHM relative to the sinc-function by 20%, 40%, and 60%, respectively [5]. An extended version of the Norton-Beer apodization with 10 functions has been presented, where the increase of the FWHM ranges from 10% to 100% [7]. Additionally, a further development of the Norton-Beer apodization method has been published, focusing on ensuring the continuity at the edges of the apodized interferogram [4]. This results in a faster decay of the side lobes but increases the absolute value of the first side lobe [14]. Because of its optimal properties with respect to these to measures, the Norton-Beer apodization has a large popularity in the FTS community [15,16]. Nevertheless, the decay of the subsequent side lobes in weaker Norton-Beer apodization functions is relatively slow compared to other apodization functions. This can be primarily attributed to the presence of discontinuities at the edges of the apodized interferogram [14]. It is concluded that the two above-mentioned measures can be used for gas-phase spectra with very narrow lines, but when applying it on condensed-phase spectra, the decay of the side lobes plays a considerable role, and other apodization functions are recommended [14].

The Fourier transform of the apodization function is of large interest as this allows direct calculation in the spectral domain, which is mostly used in solving inverse problems. Retrieval including apodization induces correlations between the spectral samples. Accounting for them, the outcome of trace gas retrievals remains unchanged [17,18]. However, even ignoring these correlations in the retrieval causes only negligible differences [15]. There exists an analytical solution for the Fourier transform of the Norton-Beer weak, medium, and strong apodizations [5,6]. This is not provided for the extended apodization functions. We, therefore, present a general analytical solution of the Fourier transform corresponding to the proposed generic form. This paper is structured as follows. In Section 2.A, we give a short background on the mathematics and Norton-Beer apodization functions. The mathematical derivation of the analytical Fourier transform of the generic form of Norton-Beer apodization is presented in Section 2.B. In Section 3, we give an overview of developed Python toolbox called norton_beer [19]. It contains the generation of the apodization window in spatial domain and its analytical Fourier transform for a given set of parameters. Furthermore, new apodization functions can be generated, where the parameters are optimized for a fixed FWHM so that the absolute maximal value of the side lobes is minimized. Subsequently, we compare the newly generated apodization functions using the above-mentioned optimization algorithm with the extended Norton-Beer apodization and other apodization functions commonly used in FTS. The comparison shows that new apodization functions with a relative FWHM between one and two can be generated with the Python toolbox.

2. THEORETICAL FOUNDATION

A. Background

An infinite interferogram of an ideal instrument measuring poly-chromatic radiance consists of multiple overlapping sinusoidal waves, denoted by

$$I(x) = \int_{- \inf}^{{\inf}} B(\sigma){\left[{1 + \cos (2\pi \sigma x)} \right]}{\rm d}\sigma ,$$
where $x$ is the optical path difference, $\sigma$ the frequency, and $B$ is the poly-chromatic radiance. It is common to remove the constant term in Eq. (1), isolating the modulated part of the interferogram. Further, a finite detector cuts the interferogram off. The finite modulated part, thus, is described by
$$I(x) = \Pi (x)\int_{- \inf}^{{\inf}} B(\sigma)\cos (2\pi \sigma x){\rm d}\sigma ,$$
where $\Pi$ is a boxcar function denoted by
$$\Pi (x) = \left\{{\begin{array}{*{20}{c}}1&{{\rm if}\; x \in [- L,L]}\\0 &{{\rm else}}\end{array}} \right.,$$
where $L$ is the maximal optical path difference measured by the instrument. Taking the Fourier transformation of the finite modulated part defined in Eq. (2) gives
$${\cal F}(I)(\sigma) = {\cal F}(\Pi)(\sigma) * \frac{1}{2}B(|\sigma |)$$
$$= 2L\,{\rm sinc}(2\pi \sigma L) * \frac{1}{2}B(|\sigma |),$$
where we used the convolution theorem in Eq. (4) and the sinc-function is defined by ${\rm sinc}(x) = \frac{{\sin (x)}}{x}$. The convolution of the spectrum with the sinc-functions results in an unsmooth spectrum due to the large side lobes of the sinc-function. The side lobes are ringing artifacts caused by the discontinuity of the interferogram at the edges. The concept of apodization involves replacing the boxcar function $\Pi$ with another function whose Fourier transform exhibits reduced side lobes. Hereby, the main lobe will be kept as narrow as possible to maintain a low spectral resolution. Frequently, two criteria are used to assess the apodization functions [1,11–13]:
  • • ($W/{W_{{\rm sinc}}}$) the FWHM relative to the FWHM of the sinc function
  • • ($| {h/{h_{{\rm sinc}}}} |$) the height of the second absolute maximum relative to the height of the second absolute maximum of the sinc function

The Filler’s diagram then illustrates $| {h/{h_{{\rm sinc}}}} |$ against $W/{W_{{\rm sinc}}}$. Through empirical testing of various apodization classes, a mutually dependent boundary between $| {h/{h_{{\rm sinc}}}} |$ and $W/{W_{{\rm sinc}}}$ was found [13] and is defined by

$${{\rm log}_{10}}\left| {\frac{h}{{{h_{{\rm sinc}}}}}} \right| \approx 1.939 - 1.401\left({\frac{W}{{{W_{{\rm sinc}}}}}} \right) - 0.597{\left({\frac{W}{{{W_{{\rm sinc}}}}}} \right)^2}.$$

The Fourier transforms of the three Norton-Beer apodization functions lie on this boundary line [5]. All three functions follow the generic form,

$${A_{{\rm NB}}}(x) = \left\{{\begin{array}{ll}{\sum\nolimits_{k = 0}^{K - 1} {{c_k}} {{\left[{1 - {{\left({\frac{x}{L}} \right)}^2}} \right]}^k}}&{{\rm if}\; x \in [- L,L]}\\[4pt]0&{{\rm else},}\end{array}} \right.$$
where $K$ represents the number of parameters used to shape the apodization window to gain different degrees of apodization strength. They can be understood as even polynomials that strictly decrease monotonically (for $k \ge 1$) from zero toward the interval boundaries. For the extended form, the FWHM of its Fourier transform is set, and the parameters are optimized so that the second absolute maximum is minimized [7]. The parameters are presented in Table 1. Stronger damping of the first side lobe needs stronger apodization, which is achieved by coefficients of higher degrees as presented in Fig. 1.
Tables Icon

Table 1. Parameters of the Extended Norton-Beer Apodization Given by [7]a

View Table | View all tables in this article
 figure: Fig. 1.

Fig. 1. Summands of Eq. (7) denoted by ${[{1 - {{({\frac{x}{L}})}^2}}]^k}$ presented for different $k$.

Download Full Size | PDF

B. Analytical Solution of the Fourier Transform of the Norton-Beer Apodization

So far, there exists only an analytical solution for the Fourier transform of the Norton-Beer weak, medium, and strong apodizations [5,6]. For other Norton-Beer apodizations, it would be possible to generate the instrument line shape numerically by constructing a window function in the spatial domain and calculate the discrete Fourier transform numerically. However, in order to mitigate aliasing and other undersampling effects, it is necessary to have a fine sampling of the window function. A detailed discussion including a comparison between the numerical and analytical solution is given in Appendix C. The degree of fineness required depends on the specific application and needs to be evaluated in advance. The analytical solution, thus, gives a more user-friendly solution. In this section, we, therefore, present the mathematical derivation of the Fourier transform of the generic form given in Eq. (7). Utilizing the linearity property of the Fourier transform, it is given by

$${\cal F}({A_{{\rm NB}}})(\sigma) = \sum\limits_{k = 0}^K {c_k}\underbrace {{\cal F}\left({{{\left[{1 - {{\left({\frac{x}{L}} \right)}^2}} \right]}^k}} \right)}_{= :{Q_k}},$$
and thus it can be applied to any set of parameters. ${Q_k}$ can be calculated by
$${Q_k} = \int_{- L}^L {\left[{1 - {{\left({\frac{x}{L}} \right)}^2}} \right]^k}{e^{- i2\pi \sigma x}}{\rm d}x$$
$$= \sum\limits_{m = 0}^k \left({\begin{array}{*{20}{c}}k\\m\end{array}} \right){(- 1)^m}\underbrace {\int_{- L}^L {{\left({\frac{x}{L}} \right)}^{2m}}{e^{- i2\pi \sigma x}}{\rm d}x}_{= :{{\tilde Q}_m}},$$
where the binomial theorem is used in Eq. (10). ${\tilde Q_m}$ can be calculated recursively by using integration of parts twice. Appendix A gives a detailed calculation. The final solution is given by
$${\tilde Q_0} = 2L\;{\rm sinc}(a),$$
$${\tilde Q_m} = 2L\;\left({{\rm sinc}(a) + \frac{{2m}}{{{a^2}}}\cos (a)} \right) - \frac{{2m(2m - 1)}}{{{a^2}}}{\tilde Q_{m - 1}},$$
where $a = 2\pi \sigma L$. An explicit evaluation of ${Q_0}$ until ${Q_8}$ is given in Appendix B.

3. IMPLEMENTATION AND RESULTS

A Python toolbox called norton_beer is developed in accordance with this research [19]. It allows us to generate the apodization window in the spatial domain and its analytical Fourier transform for a given set of parameters. The generation of the apodization window is straightforward and will not be discussed in detail. The generation of its Fourier transform calculates the ${Q_k}$ presented in Eq. (10) using the recursive form for ${\tilde Q_m}$ shown in Eqs. (11) and (12). For $|a| \lt 1$, the analytical solution becomes numerically unstable due to the ${a^{2n}}$ for $n = 1,2,\ldots $ in the divisor. The evaluated ${Q_1}$ to ${Q_8}$ in Appendix B show examples of the divisors ${a^{2n}}$ for $n$ up to 8. Taylor expansion is used for the cos and sinc, denoted by

 figure: Fig. 2.

Fig. 2. Norton-Beer apodization in the (a) spatial domain and (b) spectral domain.

Download Full Size | PDF

$$\cos (a) = \sum\limits_{n = 0}^\infty \frac{{{{(- 1)}^n}}}{{(2n)!}}{a^{2n}},$$
$${\rm sinc}(a) = \sum\limits_{n = 0}^\infty \frac{{{{(- 1)}^n}}}{{(2n + 1)!}}{a^{2n}},$$
which eliminate the ${a^{2n}}$. Note that at least the first $K$ summands in the Taylor series need to be taken to eliminate all denominators, where $K$ is the number of parameters used in Eq (7); to ensure a high level of accuracy, the toolbox calculates the first $2(K + 1)$ summands. This effectively reduces the maximal approximation error below $1 \times {10^{- 10}}$. To reduce the computational cost, the scalar numerators are calculated first, resulting in the numerators in Eqs. (B2)–(B9). Subsequently, the sinc- and cosine-functions are evaluated on the abscissa $a$ and summed up with the denominators ${a^{2n}}$. This allows us to calculate ${\rm sinc}(a)$, $\cos (a)$, and $\frac{1}{{{a^{2n}}}}$ for each $n$ only once. The window functions of the extended Norton-Beer apodization following Eq. (7) with the parameters given in Table 1 are shown in Fig. 2(a). The corresponding Fourier transform using the analytical derivation presented in this section is shown in Fig. 2(b).

Furthermore, new apodization functions can be generated, where the parameters are optimized for a given FWHM so that the absolute maximal value of the side lobes is minimized. Hereby, the objective function takes a set of parameters, calculates the Fourier transform of the apodization window for the given parameter set, and derives the FWHM of the main lobe and the absolute maximum of the first side lobe. These two measures are used in the minimization process to optimize the parameter set, denoted by

$$\mathop {\min}\limits_p \;c{({W_{{{{want}}}}} - W)^2} + h,$$
where $p$ refers to the parameter set, ${W_{{\rm want}}}$ and $W$ denote the wanted and gained FWHM of the main lobe, $h$ is the absolute maximum of the first side lobe, and $c$ is a weighting factor. The weighting factor is set to 500 by default to ensure a main lobe’s FWHM being close to the aimed FWHM. However, the weighting factor can be adjusted by the user. This can be done for a fixed number of parameters or for a range of numbers, where the minimum of that range is returned. The latter is used to generate new apodization functions for multiple relative FWHMs. Their FWHMs and absolute maxima of the side lobes relative to sinc-function, denoted by $W/{W_{{\rm sinc}}}$ and $| {h/{h_{{\rm sinc}}}} |$, are presented within the Filler’s diagram, which is introduced in Section 2.A. Further, the results are compared to commonly used apodization functions. For this, we calculate the $| {h/{h_{{\rm sinc}}}} |$ and $W/{W_{{\rm sinc}}}$ for each apodization window presented in Table 2. It should be noted that, if possible, we also adjust a parameter $p$ to achieve varying degrees of apodization strength.
Tables Icon

Table 2. Apodization Functions and Parameters Used in Fig. 3a

View Table | View all tables in this article

The outcome is displayed in Fig. 3, where the parameters are arranged in ascending order, aligning with markers representing a transition from small to large FWHM. However, for the Hamming and Gauss windows, the order of parameters corresponds to markers representing a transition from large to small FWHM. The extended Norton-Beer apodization functions follow mainly the solid boundary line and show the overall best result [7]. The newly generated apodization functions using the optimization algorithm from Section 3 consistently lie below the solid boundary line for relative FWHM values ranging from 1 to 2. This demonstrates that the Python toolbox is capable of generating novel apodization functions within that specific FWHM range. The optimized apodization windows and their Fourier transform are presented in Fig. 4. It is remarkable that, for very low FWHM below 1.2, the apodization functions increase again toward the edges indicating inadequate suppression of discontinuity. This phenomenon occurs because the procedure focuses solely on the minimization of the maximum absolute values of the first side lobe, neglecting the decay of the side lobes. However, the decay of the side lobes is closely connected to the suppression of the discontinuity [14]. The Kaiser apodization shows similar results in Fig. 3, but with higher side lobes. This comes from the fact that Norton-Beer and Kaiser windows are designed in a similar fashion. As mentioned, the Norton-Beer apodization minimizes the side lobes for a fixed FWHM [7]. The Kaiser window maximizes the energy concentration in the main lobe for a fixed maximal side lobe [11]. The Gauss window follows the Kaiser window but drifts away for large relative FWHM. The Hamming windows exhibit favorable characteristics as they closely align with the boundary line for smaller relative FWHM values up to 1.5. However, for larger FWHM values, they gradually deviate from the boundary line. The Blackman apodization shows an overall poor performance. The Hanning window can be considered as a special case of the Hamming window, where the parameter is set to 0.5. Similarly, it can also be seen as a special case of the Blackman apodization with the parameter set to 0. However, the Hanning window, like the Blackman window, deviates significantly from the boundary line.

 figure: Fig. 3.

Fig. 3. Filler’s diagram for commonly used apodization functions with different parameters. The functions and parameters used for each apodization window are presented in Table 2. The parameters in increasing order correspond to the markers from small to large FWHM, except for Hamming and Gauss window, where it corresponds to the markers from large to small FWHM. The solid line represents the boundary defined by Eq. (6). “opt_norton_beer” shows the result for the newly generated apodization function using the optimization algorithm presented in Section 3.

Download Full Size | PDF

 figure: Fig. 4.

Fig. 4. Newly generated Norton-Beer apodization functions using the optimization algorithm from Section 3 in the (a) spatial domain and (b) spectral domain. Every second $x$-marker from Fig. 3 is presented. The legend shows the relative FWHM set at the beginning of the optimization.

Download Full Size | PDF

4. CONCLUSION

This paper provides the analytical solution for the Fourier transformation of the generalized class of Norton-Beer apodization functions. This analytical solution can be applied to any given parameter set used to shape the Norton-Beer apodization in the spatial domain. For the extended Norton-Beer apodization [7], the evaluated analytical solution is presented. A Python toolbox has been developed [19], offering the capability of generating apodization windows in the spatial domain and their corresponding analytical Fourier transforms based on user-defined parameters. Additionally, the toolbox allows for the generation of new apodization functions by optimizing parameters for a fixed full width at half-maximum (FWHM) to minimize the maximum value of side lobes. By employing this algorithm, novel Norton-Beer apodization functions have been created, demonstrating comparable or superior performance compared to the extended Norton-Beer. Consequently, the Python toolbox proves its usefulness in generating Norton-Beer apodizations for any relative FWHM within the range of 1–2. To provide a comprehensive assessment, a comparison is made with widely employed apodization windows, revealing that Norton-Beer apodization functions strike the optimal balance between enhancing spectral resolution and minimizing the impact of side lobes.

APPENDIX A: DERIVATION OF THE RECURSIVE DEFINITION

${\tilde Q_0}$, defined in Eq. (11), can be calculated by applying the fundamental theorem of calculus. ${\tilde Q_m}$, denoted in Eq. (12), can be calculated recursively by applying integration of the part twice as shown in the following:

$${\tilde Q_m} = \int_{- L}^L {\left({\frac{x}{L}} \right)^{2m}}{e^{- i2\pi \sigma x}}{\rm d}x$$
$$\begin{split}&= \left[{\frac{1}{{- i2\pi \sigma}}{e^{- i2\pi \sigma x}}{{\left({\frac{x}{L}} \right)}^{2m}}} \right]_{- L}^L - \frac{{2m}}{{- i2\pi \sigma L}}\int_{- L}^L {\left({\frac{x}{L}} \right)^{2m - 1}}\\&\quad\times{e^{- i2\pi \sigma x}}{\rm d}x\end{split}$$
$$\begin{split}&= 2L\;{\rm sinc}(2\pi \sigma L) \\&\quad- \frac{{2m}}{{- i2\pi \sigma L}}\left({\left[{\frac{1}{{- i2\pi \sigma}}{e^{- i2\pi \sigma x}}{{\left({\frac{x}{L}} \right)}^{2m - 1}}} \right]_{- L}^L -} \right.\end{split}$$
$${{{\left. {\int_{- L}^L \frac{{2m - 1}}{L}{{\left({\frac{x}{L}} \right)}^{2m - 2}}\frac{1}{{- i2\pi \sigma}}{e^{- i2\pi \sigma x}}{\rm d}x} \right)}}}$$
$$\begin{split}&= 2L\left({{\rm sinc}(2\pi \sigma L) + \frac{{2m}}{{{{(2\pi \sigma L)}^2}}}\cos (2\pi \sigma L)} \right) \\&\quad- \frac{{2m(2m - 1)}}{{{{(2\pi \sigma L)}^2}}}{\tilde Q_{m - 1}} .\end{split}$$

Assigning $a = 2\pi \sigma L$ gives Eq. (12).

APPENDIX B: EVALUATION OF ${Q_0}$ UNTIL ${Q_8}$

Using the recursive form Eqs. (11) and (12), ${Q_k}$ for $k$ up to $8$ are given by

 figure: Fig. 5.

Fig. 5. (a) Fourier transform of boxcar window function generated numerically versus analytically. (b) Difference between numerical and analytical Fourier transform for different number of spatial samples. The bottom panel shows a zoom-in of the top panel along the $y$-axis, respectively, for (a) and (b).

Download Full Size | PDF

$$\begin{array}{*{20}{l}}{{Q_0}} &={{\rm sinc}(a)}\end{array},$$
$$\begin{array}{*{20}{l}}{{Q_1}} &={\frac{2}{{{a^2}}}({\rm sinc}(a) - \cos (a))}\end{array},$$
$$\begin{array}{*{20}{l}}{{Q_2}} &={8\left[{\left({- \frac{1}{{{a^2}}} + \frac{3}{{{a^4}}}} \right){\rm sinc}(a) - \frac{3}{{{a^4}}}\cos (a)} \right]}\end{array},$$
$$\begin{array}{*{20}{l}}{{Q_3}} &={48\left[{\left({- \frac{6}{{{a^4}}} + \frac{{15}}{{{a^6}}}} \right){\rm sinc}(a) + \left({\frac{1}{{{a^4}}} - \frac{{15}}{{{a^6}}}} \right)\cos (a)} \right]}\end{array},$$
$$\begin{array}{*{20}{l}}{{Q_4}} &={384\left[{\left({\frac{1}{{{a^4}}} - \frac{{45}}{{{a^6}}} + \frac{{105}}{{{a^8}}}} \right){\rm sinc}(a) + \left({\frac{{10}}{{{a^6}}} - \frac{{105}}{{{a^8}}}} \right)\cos (a)} \right]}\end{array},$$
$$\begin{split}{{Q_5} }& ={3 840\left[{\left({\frac{{15}}{{{a^6}}} - \frac{{420}}{{{a^8}}} + \frac{{945}}{{{a^{10}}}}} \right){\rm sinc}(a) } \right.}\\&\quad +{ \left. {\left({- \frac{1}{{{a^6}}} + \frac{{105}}{{{a^8}}} - \frac{{945}}{{{a^{10}}}}} \right)\cos (a)} \right]}\end{split},$$
$$\begin{split}{{Q_6} }& = {46 080\left[{\left({- \frac{1}{{{a^6}}} + \frac{{210}}{{{a^8}}} - \frac{{4 725}}{{{a^{10}}}} + \frac{{10 395}}{{{a^{12}}}}} \right){\rm sinc}(a) } \right.}\\&\quad +{\left. {\left({- \frac{{21}}{{{a^8}}} + \frac{{1 260}}{{{a^{10}}}} - \frac{{10 395}}{{{a^{12}}}}} \right)\cos (a)} \right]},\end{split}$$
$$\begin{split}{{Q_7} }& = {645 120\left[{\left({- \frac{{28}}{{{a^8}}} + \frac{{3 150}}{{{a^{10}}}} - \frac{{62 370}}{{{a^{12}}}} + \frac{{135 135}}{{{a^{14}}}}} \right){\rm sinc}(a)} \right.}\\&\quad +{\left. {\left({\frac{1}{{{a^8}}} - \frac{{378}}{{{a^{10}}}} + \frac{{17 325}}{{{a^{12}}}} - \frac{{135 135}}{{{a^{14}}}}} \right)\cos (a)} \right]},\end{split}$$
$$\begin{split}{{Q_8}}& =10 321 920\left[\left(\frac{1}{{{a^8}}} - \frac{{630}}{{{a^{10}}}} + \frac{{51 975}}{{{a^{12}}}}\right.\right.\\&\quad - \left.\frac{{945 945}}{{{a^{14}}}} + \frac{{2 027 025}}{{{a^{16}}}} \right){\rm sinc}(a) \\&\quad +{ \left. {\left({\frac{{36}}{{{a^{10}}}} - \frac{{6 930}}{{{a^{12}}}} + \frac{{270 270}}{{{a^{14}}}} - \frac{{2 027 025}}{{{a^{16}}}}} \right)\cos (a)} \right].}\end{split}$$

APPENDIX C: ERROR SOURCES OF NUMERICAL FOURIER TRANSFORM

The numerical generation of the Fourier transform involves constructing a window function in the spatial domain and performing a discrete Fourier transform. However, achieving an accurate solution that closely aligns with the analytical Fourier transform requires specific considerations regarding the number of samples of the window function, which we will outline in this section. Furthermore, to obtain the desired spectral abscissa, precise attention must be paid to setting the correct parameter for zero padding.

C.1. Errors Due to Aliasing Effect

A sampled window function is represented as

$${A_s} = A{\delta _d},$$
where $A$ represents a continuous window function, and ${\delta _d}$ is an infinite impulse train with a spacing of $d$. The Fourier transform of that impulse train is given by another impulse train with a spacing of $\frac{1}{d}$. Therefore, the Fourier transform of a sampled window function can be expressed as
$${\cal F}({A_s}) = {\cal F}(A) * {\delta _{\frac{1}{d}}}.$$

It is worth noting that the Nyquist limit is defined as $\frac{1}{{2d}}$, which determines the resolvable spectral region to $[- \frac{1}{{2d}},\frac{1}{{2d}}]$. The convolution with the infinite impulse train ${\delta _{\frac{1}{d}}}$, thus, sums up frequencies beyond the Nyquist limit and folds them into the resolvable spectral area. This phenomenon is commonly known as aliasing [16].

To illustrate the aliasing effect, consider the boxcar function, whose Fourier transform is a sinc-function. In Fig. 5(a), we compare the sinc-function (the analytical Fourier transform of a boxcar) with the numerical Fourier transform of a discrete boxcar window function containing 101 samples. In the lower zoomed-in panel of Fig. 5(a), one can observe an overestimation of the side lobes in the numerical solution near the Nyquist limit (in this example, at ${\pm}50{L^{- 1}}$, where $L$ represents the maximal optical path difference). This overestimation leads to errors of up to 0.3% relative to the peak value of the sinc-function, as depicted by the gray line in Fig. 5(b). This discrepancy is a direct consequence of the aliasing phenomenon.

To mitigate this aliasing effect, one can increase the sampling density of the window function, which effectively raises the Nyquist limit, expanding the resolvable spectral area. When we limit the spectral axis to its original region (in this example to ${\pm}50{L^{- 1}}$), we consider only the folded-in frequencies from spectral areas further apart from the zero frequency. These folded-in frequencies typically carry less intensity due to the decreasing side lobes of the Fourier transform. This effect is demonstrated in Fig. 5(b), where the difference between the numerical and analytical solutions of the sinc-function decreases as the number of samples increases. Therefore, to achieve a more accurate numerical estimate of the analytical solution, it is advisable to use a finer sampling of the window function.

 figure: Fig. 6.

Fig. 6. All sub-figures correspond to Norton-Beer weak (1.2). (a) Apodization function for different number of samples. Note that a low number of spatial samples is presented to increase the visibility of the differences. (b) Fourier transform of Norton-Beer 1.2 function generated numerically versus analytically. (c) Difference between numerical and analytical Fourier transform for different number of spatial samples. The bottom panel in (b) and (c) shows a zoom-in of the top panel along the $y$-axis.

Download Full Size | PDF

C.2. Low Frequency Errors Due to Undersampling

Coarse sampling of a bell-shaped window function can introduce errors in its Fourier transform, particularly in lower frequencies. In Fig. 6(a), we illustrate the Norton-Beer 1.2 apodization with various sampling levels. It is important to note that we intentionally display a lower number of samples to enhance the visibility of the differences. The subsequent spectral difference analysis is carried out using 101 samples, depicted in Fig. 6(b) as it was done in Appendix C.1.

Subsequently, the deviation between the numerical and analytical Fourier transform at the zero frequency is explained. The zero frequency corresponds to the integral of the apodization function. Figure 6(a) highlights that a coarsely sampled window underestimates the integral of the continuous function, entailing an underestimation of the numerical solution at the zero frequency as displayed by the gray line in Fig. 6(c). The discrepancy at higher frequencies is attributed to the aliasing effect, as discussed in Appendix C.1. When we examine the differences without applying finer sampling [reference the gray line in Fig. 6(c)], we observe errors at the peak position of the instrument line shape, reaching up to 0.4% for Norton-Beer weak. This underestimation at the zero frequency is amplified for stronger Norton-Beer apodization, owing to an increased curvature of the bell shape. However, for stronger Norton-Beer apodization, the discrepancies resulting from aliasing near the Nyquist limit are reduced due to the diminishing side lobe.

Both of these effects can be mitigated by employing finer sampling of the apodization function in the spatial domain as shown in Fig. 6(c), following the approach outlined in Appendix C.1.

Funding

European Metrology Programme for Innovation and Research (19ENV07 MetEOC-4).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

REFERENCES

1. F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978). [CrossRef]  

2. G. B. Folland and A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal. Appl. 3, 207–238 (1997). [CrossRef]  

3. R. B. Blackman and J. W. Tukey, “The measurement of power spectra from the point of view of communications engineering — Part I,” Bell Syst. Tech. J. 37, 185–282 (1958). [CrossRef]  

4. L. Lee, “Extension of the Norton-Beer apodizing functions in Fourier transform spectrometry,” Appl. Opt. 51, 4622–4666 (2012). [CrossRef]  

5. R. H. Norton and R. Beer, “New apodizing functions for Fourier spectrometry,” J. Opt. Soc. Am. 66, 259–264 (1976). [CrossRef]  

6. R. H. Norton and R. Beer, “Errata: new apodizing functions for Fourier spectrometry,” J. Opt. Soc. Am. 67, 419 (1977). [CrossRef]  

7. D. A. Naylor and M. K. Tahic, “Apodizing functions for Fourier transform spectroscopy,” J. Opt. Soc. Am. A 24, 3644–3648 (2007). [CrossRef]  

8. H. Helms, “Digital filters with equiripple or minimax responses,” IEEE Trans. Audio Electroacoust. 19, 87–93 (1971). [CrossRef]  

9. R. W. Hamming, Digital Filters, 2nd ed., Prentice-Hall Signal Processing Series (Prentice-Hall, 1983).

10. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, 1975).

11. J. Kaiser and R. Schafer, “On the use of the I0-Sinh window for spectrum analysis,” IEEE Trans. Acoust. Speech Signal Process. 28, 105–107 (1980). [CrossRef]  

12. C. Barnet, J. Blaisdell, and J. Susskind, “Practical methods for rapid and accurate computation of interferometric spectra for remote sensing applications,” IEEE Trans. Geosci. Remote Sens. 38, 169–183 (2000). [CrossRef]  

13. A. S. Filler, “Apodization and interpolation in Fourier-transform spectroscopy,” J. Opt. Soc. Am. 54, 762–767 (1964). [CrossRef]  

14. C. Zhu and P. R. Griffiths, “Extending the range of Beer’s law in FT-IR spectrometry. Part I: theoretical study of Norton-Beer apodization functions,” Appl. Spectrosc. 52, 1403–1408 (1998). [CrossRef]  

15. C. D. Boone, S. D. McLeod, and P. F. Bernath, “Apodization effects in the retrieval of volume mixing ratio profiles,” Appl. Opt. 41, 1029–1034 (2002). [CrossRef]  

16. P. R. Griffiths and J. A. De Haseth, Fourier Transform Infrared Spectrometry, 2nd ed. (Wiley-Interscience, 2007).

17. U. Amato, D. De Canditiis, and C. Serio, “Effect of apodization on the retrieval of geophysical parameters from Fourier-transform spectrometers,” Appl. Opt. 37, 6537–6543 (1998). [CrossRef]  

18. S. Ceccherini, C. Belotti, B. Carli, and P. Raspollini, “Use of apodization in quantitative spectroscopy,” Opt. Lett. 32, 1329–1331 (2007). [CrossRef]  

19. K. F. F. Ntokas, “Norton-Beer apodization python package,” Zenodo (2023), https://doi.org/10.5281/zenodo.8085999.

Data availability

No data were generated or analyzed in the presented research.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)
Fig. 1.
Fig. 1. Summands of Eq. (7) denoted by ${[{1 - {{({\frac{x}{L}})}^2}}]^k}$ presented for different $k$.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Norton-Beer apodization in the (a) spatial domain and (b) spectral domain.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Filler’s diagram for commonly used apodization functions with different parameters. The functions and parameters used for each apodization window are presented in Table 2. The parameters in increasing order correspond to the markers from small to large FWHM, except for Hamming and Gauss window, where it corresponds to the markers from large to small FWHM. The solid line represents the boundary defined by Eq. (6). “opt_norton_beer” shows the result for the newly generated apodization function using the optimization algorithm presented in Section 3.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Newly generated Norton-Beer apodization functions using the optimization algorithm from Section 3 in the (a) spatial domain and (b) spectral domain. Every second $x$-marker from Fig. 3 is presented. The legend shows the relative FWHM set at the beginning of the optimization.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. (a) Fourier transform of boxcar window function generated numerically versus analytically. (b) Difference between numerical and analytical Fourier transform for different number of spatial samples. The bottom panel shows a zoom-in of the top panel along the $y$-axis, respectively, for (a) and (b).

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. All sub-figures correspond to Norton-Beer weak (1.2). (a) Apodization function for different number of samples. Note that a low number of spatial samples is presented to increase the visibility of the differences. (b) Fourier transform of Norton-Beer 1.2 function generated numerically versus analytically. (c) Difference between numerical and analytical Fourier transform for different number of spatial samples. The bottom panel in (b) and (c) shows a zoom-in of the top panel along the $y$-axis.

View in Article | Download Full Size | PDF

Tables (2)

Tables Icon

Table 1. Parameters of the Extended Norton-Beer Apodization Given by [7]a

View Table | View all tables in this article
Tables Icon

Table 2. Apodization Functions and Parameters Used in Fig. 3a

View Table | View all tables in this article

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

I ( x ) = ∫ − inf inf B ( σ ) [ 1 + cos ⁡ ( 2 π σ x ) ] d σ ,
I ( x ) = Π ( x ) ∫ − inf inf B ( σ ) cos ⁡ ( 2 π σ x ) d σ ,
Π ( x ) = { 1 i f x ∈ [ − L , L ] 0 e l s e ,
F ( I ) ( σ ) = F ( Π ) ( σ ) ∗ 1 2 B ( | σ | )
= 2 L s i n c ( 2 π σ L ) ∗ 1 2 B ( | σ | ) ,
l o g 10 | h h s i n c | ≈ 1.939 − 1.401 ( W W s i n c ) − 0.597 ( W W s i n c ) 2 .
A N B ( x ) = { ∑ k = 0 K − 1 c k [ 1 − ( x L ) 2 ] k i f x ∈ [ − L , L ] 0 e l s e ,
F ( A N B ) ( σ ) = ∑ k = 0 K c k F ( [ 1 − ( x L ) 2 ] k ) ⏟ =: Q k ,
Q k = ∫ − L L [ 1 − ( x L ) 2 ] k e − i 2 π σ x d x
= ∑ m = 0 k ( k m ) ( − 1 ) m ∫ − L L ( x L ) 2 m e − i 2 π σ x d x ⏟ =: Q ~ m ,
Q ~ 0 = 2 L s i n c ( a ) ,
Q ~ m = 2 L ( s i n c ( a ) + 2 m a 2 cos ⁡ ( a ) ) − 2 m ( 2 m − 1 ) a 2 Q ~ m − 1 ,
cos ⁡ ( a ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! a 2 n ,
s i n c ( a ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) ! a 2 n ,
min p c ( W w a n t − W ) 2 + h ,
Q ~ m = ∫ − L L ( x L ) 2 m e − i 2 π σ x d x
= [ 1 − i 2 π σ e − i 2 π σ x ( x L ) 2 m ] − L L − 2 m − i 2 π σ L ∫ − L L ( x L ) 2 m − 1 × e − i 2 π σ x d x
= 2 L s i n c ( 2 π σ L ) − 2 m − i 2 π σ L ( [ 1 − i 2 π σ e − i 2 π σ x ( x L ) 2 m − 1 ] − L L −
∫ − L L 2 m − 1 L ( x L ) 2 m − 2 1 − i 2 π σ e − i 2 π σ x d x )
= 2 L ( s i n c ( 2 π σ L ) + 2 m ( 2 π σ L ) 2 cos ⁡ ( 2 π σ L ) ) − 2 m ( 2 m − 1 ) ( 2 π σ L ) 2 Q ~ m − 1 .
Q 0 = s i n c ( a ) ,
Q 1 = 2 a 2 ( s i n c ( a ) − cos ⁡ ( a ) ) ,
Q 2 = 8 [ ( − 1 a 2 + 3 a 4 ) s i n c ( a ) − 3 a 4 cos ⁡ ( a ) ] ,
Q 3 = 48 [ ( − 6 a 4 + 15 a 6 ) s i n c ( a ) + ( 1 a 4 − 15 a 6 ) cos ⁡ ( a ) ] ,
Q 4 = 384 [ ( 1 a 4 − 45 a 6 + 105 a 8 ) s i n c ( a ) + ( 10 a 6 − 105 a 8 ) cos ⁡ ( a ) ] ,
Q 5 = 3 840 [ ( 15 a 6 − 420 a 8 + 945 a 10 ) s i n c ( a ) + ( − 1 a 6 + 105 a 8 − 945 a 10 ) cos ⁡ ( a ) ] ,
Q 6 = 46 080 [ ( − 1 a 6 + 210 a 8 − 4 725 a 10 + 10 395 a 12 ) s i n c ( a ) + ( − 21 a 8 + 1 260 a 10 − 10 395 a 12 ) cos ⁡ ( a ) ] ,
Q 7 = 645 120 [ ( − 28 a 8 + 3 150 a 10 − 62 370 a 12 + 135 135 a 14 ) s i n c ( a ) + ( 1 a 8 − 378 a 10 + 17 325 a 12 − 135 135 a 14 ) cos ⁡ ( a ) ] ,
Q 8 = 10 321 920 [ ( 1 a 8 − 630 a 10 + 51 975 a 12 − 945 945 a 14 + 2 027 025 a 16 ) s i n c ( a ) + ( 36 a 10 − 6 930 a 12 + 270 270 a 14 − 2 027 025 a 16 ) cos ⁡ ( a ) ] .
A s = A δ d ,
F ( A s ) = F ( A ) ∗ δ 1 d .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use