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Article

Research on Fourier Transform Spectral Phase Correction Algorithm Based on CTKB-NCM

by
Xiong Wang
1,2,3,
Chunhui Yan
1,3,
Zimin Huo
4,
Pengzhang Dai
1,2,3 and
Dong Yao
1,2,3,*
1
Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China
2
University of Chinese Academy of Sciences, Beijing 100049, China
3
State Key Laboratory of Dynamic Optical Imaging and Measurement, Changchun 130033, China
4
School of Navigation and Internet of Things, Aerospace Information Technology University, Jinan 250299, China
*
Author to whom correspondence should be addressed.
Photonics 2025, 12(3), 219; https://doi.org/10.3390/photonics12030219
Submission received: 9 December 2024 / Revised: 12 February 2025 / Accepted: 24 February 2025 / Published: 28 February 2025
(This article belongs to the Special Issue Emerging Trends in Spectral Analysis with Optical Sensors: Modern Approaches and Applications)
Browse Figures
Figure 1
<p>The schematic of a Michelson-type time-modulated Fourier Transform Spectrometer.</p> "> Figure 2
<p>A block diagram of the CTKB-NCM phase correction algorithm structure.</p> "> Figure 3
<p>The linear phase error correction process of the CTKB method.</p> "> Figure 4
<p>A flowchart of the Rao algorithm.</p> "> Figure 5
<p>(<b>a</b>) A comparison of the normalized restored spectra after linear phase error correction using the CTKB method and the Mertz method; (<b>b</b>,<b>c</b>) comparisons of the local spectra after linear phase error correction by the two methods; (<b>d</b>) the difference between the normalized restored spectra of the two methods.</p> "> Figure 6
<p>The convergence curve of the objective function of the Rao algorithm.</p> "> Figure 7
<p>(<b>a</b>,<b>b</b>) Comparisons of the normalized restored spectra obtained from the SO<sub>2</sub> measurement spectrum after linear phase error correction using the CTKB method, with the instrument phase correction parameters calculated using the least squares method and the Rao algorithm, respectively. (<b>c</b>) A comparison of the normalized spectral reconstruction errors for the two methods mentioned above.</p> "> Figure 8
<p>The phase correction process of the CTKB-NMPRC algorithm.</p> "> Figure 9
<p>(<b>a</b>) Comparison of spectral restoration using CTKB-NCM, Mertz, and Forman Methods. (<b>b</b>,<b>c</b>) Local spectral restoration comparisons. (<b>d</b>) Comparison of restoration spectral errors among three methods.</p> ">
Versions Notes

Abstract

:
In Fourier Transform Spectrometers, phase errors in spectral measurement induce distortion in reconstructed spectra. Existing phase correction algorithms demonstrate insufficient precision in addressing both linear phase and instrumental phase components, resulting in limited applications for the restored spectra in the field of precision measurement. This paper proposes an algorithm called the Cross-Teager–Kaiser ψ B Energy Operator–Nonlinear Calibration Model (CTKB-NCM) for phase error correction. The algorithm first uses the cross-Teager–Kaiser ψ B energy operator (CTKB) method to correct linear phase errors, then applies the Nonlinear Calibration Model (NCM) to solve for the instrument phase correction parameters at each wavenumber, and finally uses the instrument phase correction parameters to correct the residual phase after the linear phase error has been corrected. The Rao algorithm is used to determine the optimal instrument phase correction parameters. Simulation experiments demonstrate that the CTKB-NCM method achieves an order-of-magnitude improvement in normalized reconstructed spectral accuracy for SO2 gas compared to the conventional Mertz method.

1. Introduction

After a century of evolutionary development, spectroscopic technology has become a critical methodology for the qualitative and quantitative analysis of molecular structures and compositional characteristics. Modern spectrometers serve as indispensable precision analytical instruments in industrial detection systems, enabling the determination of molecular composition through characteristic parameters including the position, intensity, and line profile of infrared absorption features in target substances. As a significant branch of spectroscopic instrumentation, Fourier Transform Spectrometers (FTSs) derive spectral information through the Fourier transformation of interferometric fringe patterns [1]. These instruments feature advantages such as a high optical throughput, high signal-to-noise ratio, and simultaneous multi-channel measurement [2], showcasing significant application value in scientific research [3], industrial production [4], healthcare [5], and military aerospace [6]. Particularly in gas detection applications, infrared FTSs enable simultaneous species identification and concentration quantification, exhibiting substantial utility in environmental monitoring [7] and gas-phase chemical process research [8].
In FTSs, spectral radiometric phase errors arise during measurement processes due to factors including modulation component imperfections and sampling interval non-uniformity. These errors essentially impose a phase modulation component onto the incident spectrum, causing spectral distortion in instrument-reconstructed spectra compared to the target spectra [9]. To obtain more-accurate spectral information from measured objects, precise phase correction of the instrument-reconstructed spectra becomes necessary, thereby achieving higher-fidelity target spectral reconstructions.
Regarding the phase correction issue in Fourier Transform Spectrometers, some studies have been conducted. Traditional spectral radiation phase correction methods include the Mertz method, the Forman method, and the absolute value method, among others. Mertz et al. proposed a frequency-domain multiplication method [10], which achieves the phase correction of the spectra in the frequency domain. This algorithm is simple and performs well in correcting linear phase errors, but is less effective for nonlinear phase errors. Forman et al. introduced an interferogram convolution method [11], which performs phase correction in the time domain. This method demonstrates good correction effects for both linear and nonlinear phase errors and allows for the convenient integration of digital filters to suppress noise. However, the convolution operation is computationally intensive and challenging to implement in hardware. Cheng et al. developed a phase correction model called the phase correlation method–all-pass filter (PCM-APF) [12]. This model utilizes phase correlation and all-pass filters for phase correction, demonstrating relatively high accuracy. However, the algorithm is complex in design and highly susceptible to noise interference.
The aforementioned methods are pure signal processing approaches for spectral phase correction based on the characteristics of interferometric data signals, entirely detached from the specific characteristics of actual instruments. They use interferometric data collected under certain conditions to apply phase correction to the original phase in the data according to specific rules, resulting in phase errors being a smooth function of the wavenumber. However, due to the truncation effects of interferometric data and the influence of the physical characteristics of the instruments themselves, their phase correction accuracy is insufficient, and errors such as negative spectra may occur. Therefore, purely signal processing-based methods cannot effectively achieve the high-precision reconstruction of Fourier transform spectra.
To address the limitations of pure signal processing-based spectral phase correction methods, recent research has increasingly considered the influence of instrument characteristics on phase correction, introducing the concept of an “instrumental phase” to describe phase errors induced by these characteristics. Saggin et al. proposed a temperature-dependent instrumental phase correction method [13], which accounts for the impact of significant temperature variations and mechanical vibrations during spectral measurements on the spectral reconstruction results. This method employs least squares and complex spectral radiation calibration techniques to correct linear and instrumental phase errors, respectively. However, the use of a linear model in the instrumental phase correction stage introduces certain deviations between the measured and calibrated spectra. Turbide et al. developed an algorithm for phase correction in spectrometers with instrument characteristic variations lacking temperature stability [14]. This algorithm corrects phase errors by calibrating the zero optical path difference offset between the scene measurement spectrum and the target emission spectrum, achieving spectral reconstruction errors of less than 1% within one minute of data acquisition. Despite its effectiveness, the algorithm is computationally intensive and can produce significant spectral errors under conditions of rapid environmental temperature changes.
This study focuses on phase correction algorithms for spectral reconstruction in Fourier Transform Spectroscopy. Recognizing the limitations of existing approaches—where pure signal processing-based methods exhibit insufficient correction accuracy while instrument-dependent algorithms are significantly influenced by instrumental characteristics—we propose a Cross-Teager–Kaiser ψ B Energy Operator–Nonlinear Calibration Model (CTKB-NCM) algorithm through the comprehensive analysis of phase error sources and the systematic integration of interferometric signal characteristics with instrumental properties. The CTKB-NCM algorithm first employs the cross-Teager–Kaiser ψ B energy operator method (CTKB) to calculate the zero optical path difference sampling offset, effectively correcting linear phase errors. It then applies the Nonlinear Calibration Model (NCM) for complex spectra to correct instrument phase errors. This study also conducts simulation experiments on spectral radiation phase error correction for gas target detection. A comparative analysis of spectral reconstruction errors among different phase correction algorithms is performed, verifying the superiority of the proposed algorithm in achieving high–precision spectral reconstruction.

2. Fourier Transform Spectroscopy Related Knowledge

2.1. Introduction to Fourier Transform Spectroscopy

A Fourier Transform Spectrometer utilizes the continuous optical path difference introduced by the movement of a movable mirror while a detector records the corresponding interference light intensity as the optical path difference changes. A Michelson interferometer, which generates equal-angle interference through the reciprocal movement of the mirror, is the typical structure of a time-modulated Fourier Transform Spectrometer. The optical principle is illustrated in Figure 1.
In this system, a beamsplitter divides the incident light, containing the target information, into reflected and transmitted beams. These beams are reflected by the movable and stationary mirrors, respectively, and then recombined at the beamsplitter. The combined light is focused by an imaging optical mirror onto a detector, producing an interference signal [15].
The relationship between the interferometric signal and the spectral signal measured by the spectrometer is as follows:
B m σ = ∫ − ∞ + ∞ I x e − i 2 π σ x − θ σ d x
B m σ represents the measured spectral signal, I x denotes the interferometric signal, x is the optical path difference, σ is the spectral wavenumber, and θ ( σ ) is the phase error function.
From Equation (1), it can be observed that directly applying the Fourier transform to the interferometric signal introduces a phase error, θ ( σ ) , which adversely affects the final reconstructed spectrum. Therefore, it is essential to extract the phase error function θ ( σ ) using a phase correction algorithm.
After obtaining the phase error function, the reconstructed spectrum using the symmetrized calculation is given by
B s y m σ = B m σ e − i θ σ
B s y m σ represents the reconstructed spectrum with the phase error eliminated. It provides a more accurate representation of the true target spectral information compared to B m σ . This is the ultimate goal of phase correction for the measured spectrum.

2.2. Analysis of Phase Error Sources

In Fourier Transform Spectrometers, spectral distortion in measured interferograms primarily arises from phase errors introduced during the instrument’s modulation and sampling processes. These phase errors alter the spectral phase characteristics of the incident radiation, ultimately resulting in deviations between reconstructed spectra and their theoretical counterparts.
An analysis of the sources of phase errors in the measured spectrum output by the spectrometer is conducted as follows [16,17].
  • Phase errors induced by the dispersion of the beamsplitter crystal and the machining precision of the wedge-shaped substrate of the beamsplitter.
  • Phase errors caused by modulation non-uniformity in spectrometers with equal optical path difference sampling. This occurs because the wavenumbers of the reference laser and the incident radiation are not identical, leading to inconsistent phase delays when the detector receives the two signals.
  • Phase errors introduced during sampling due to optical misalignment, random initialization states of the electronic system, motor positioning inaccuracies, and non-uniform sampling intervals of the detector. These errors stem from imperfections in the spectrometer’s optical alignment, electronic system, mechanical components, and sampling process.
Among the three sources of error mentioned above, the first two types of phase errors are wavenumber-dependent and are referred to as instrument phase errors, caused by the characteristics of the instrument. The third type of error, which is wavenumber-independent, is a random linear phase error.
Based on the analysis of phase errors in the spectrometer, the phase error of the measured spectral radiation can be expressed as follows [13]:
φ σ , t , T , v = 2 π σ δ t + 2 π σ ∆ x σ , T + φ ( σ v )
δ t represents the offset between the sampled zero optical path difference position and the actual zero optical path difference position. ∆ x σ , T is the optical path difference introduced by the crystal dispersion and thickness non-uniformity of the substrate material. φ ( σ v ) is the phase lag error caused by the electronics system. t denotes time, T is the temperature of the spectrometer, and v is the moving speed of the mirror.
The phase error of the measured spectral radiation, φ σ , t , T , v , consists of two components:
2 π σ δ t : This term represents the linear phase error introduced by sampling inaccuracies in the interferogram data.
2 π σ ∆ x σ , T + φ ( σ v ) : This term represents the phase error component caused by the instrument’s modulation characteristics, referred to as the instrument phase.

3. The CTKB-NCM Phase Correction Algorithm

3.1. Algorithm Description

Based on the analysis of phase error sources, it is evident that the phase error in a measured spectrum consists of two components: linear phase errors and instrument phase errors. By symmetrically eliminating these two components of phase error according to Equation (2), high-precision reconstructed spectra can be obtained.
Following this principle, the proposed algorithm proceeds as follows: When the spectrometer operates under stable environmental conditions, interferometric signal data are collected. First, the zero optical path difference offset is calculated using the cross-Teager–Kaiser ψ B energy operator method (CTKB) to correct the linear phase error. Next, for groups of measured spectra with corrected linear phase errors under different radiation temperatures, the complex spectral Nonlinear Calibration Model (NCM) is used to calculate the instrument phase correction parameters for each wavenumber using the Rao optimization algorithm. Finally, for any measured spectrum, the CTKB method is first applied to eliminate the linear phase error. Then, the instrument phase correction parameters for each wavenumber are used to correct the instrument phase error, completing the phase correction process. This phase correction algorithm is referred to as CTKB-NCM in this paper. Figure 2 illustrates the flow structure of the algorithm, with the specific steps detailed in the following subsections.

3.2. Linear Phase Error Correction Using the CTKB Method

Linear phase errors primarily arise from the offset between the actual sampling positions and the ideal sampling positions. This offset causes a misalignment between the zero optical path difference (ZOPD) position of the sampled interferogram and the ideal ZOPD position. Additionally, the linear phase components of the instrument phase error (e.g., the linear portion of the phase error introduced by beamsplitter dispersion) also contribute to the overall linear phase error. For linear phase errors under a stable instrument temperature, the entire error can be extracted without needing to pinpoint each specific source. It is sufficient to differentiate the primary sources of contribution. Moreover, since instrument phase extraction will be performed subsequently, removing the linear components of the instrument phase at this stage does not lead to omissions in the overall phase error correction for the spectrometer. Thus, in the following discussion, the instrument phase can be approximated as the nonlinear phase component of the spectrometer.
There have been various studies on methods for correcting linear phase errors. Wang proposed a linear phase error detection method based on the phase correlation approach, which can detect the zero optical path difference (ZOPD) offset from interferometric data and subsequently compute the linear phase error [18]. However, in the presence of noise interference, the envelope of the phase correlation method tends to exhibit jitter and distortion, leading to significant errors in ZOPD offset calculation.
To address these issues, this study employs the cross-Teager–Kaiser ψ B energy operator, which is more robust to noise and better reflects the transient characteristics of the signal, for determining the ZOPD offset [19]. Subsequently, Newton’s interpolation method is used to refine the interferometric signal corresponding to the ZOPD, achieving the high-precision extraction of the ZOPD offset.
The cross-Teager–Kaiser ψ B energy operator uses the first and second derivatives of a signal to characterize its relative changes and can be utilized to measure interactions between two signals [20]. It has been applied in fields such as time series analysis [21], envelope detection [22], and time delay estimation [23]. In this study, we use the cross-Teager–Kaiser ψ B energy operator to more accurately calculate the ZOPD position of the interferometric signal, based on which the ZOPD offset is determined and linear phase correction is performed.
For time series signals x t and y t , the cross-Teager–Kaiser ψ B energy operator is defined as follows:
ψ B ( x , y ) = 0.5 [ ψ C x , y + ψ C ( y , x ) ]
ψ C x , y = 0.5 x ˙ * y ˙ + x ˙ y ˙ * − 0.5 ( x y ¨ * + x * x ¨ )
x ˙ represents the first derivative of x , x ¨ represents the second derivative of x , x * denotes the complex conjugate of x , and similarly for y . Please note that ψ B ( x , y ) is a complex number.
ψ B x , y = ψ B x r , y r + ψ B x i , y i
ψ B x l , y l = x ˙ l y ˙ l − 0.5 x l y ¨ l + x ¨ l y l ,           l ∈ { r , i }
Here, x t = x r t + j x i t and y t = y r t + j y i t , where r and i denote the real and imaginary parts, respectively. For the real signals x t and y t , their complex forms can be obtained using the Hilbert transform.
Based on the above method, this study applies it to the extraction of the zero optical path difference (ZOPD) offset from interferometric signals, and names this linear phase correction method the CTKB method. The flowchart of the CTKB algorithm is shown in Figure 3, and its detailed steps are as follows:
  • Initialization:
Denote the sampled interferometric signal as I l x , where x = 1,2 , 3 , … , N . Let δ m a x be the optical path difference corresponding to the maximum value in I l x . Extract small bilateral interferometric data near the ZOPD from I l and denote them as I s x , where x = 1,2 , 3 , … , M . Convert I l and I s into complex signals using the Hilbert transform, i.e., I l = I l r + I l i and I s = I s r + I s i , where I l r and I s r represent the real parts and I l i and I s i represent the imaginary parts. The subscripts r and i will denote the real and imaginary parts of the signals hereafter.
2.
Sliding Window:
Set M as the window length of the cross-Teager–Kaiser ψ B energy operator. Slide the window across the interferometric signal I l and calculate the interferometric signal within each cross-Teager–Kaiser ψ B energy operator window, denoted as I j x , where I j x ∈ I l ;   x = j , … , M + j − 1 ;   j = 1 ,   2 , … , N − M + 1 .
3.
Energy Operator Calculation:
Compute the cross-Teager–Kaiser ψ B energy operator for each windowed interferometric signal I j x and the small bilateral interferometric data I s as follows:
ψ B j I j x , I s = ψ B r j I r j ( x ) , I s r + ψ B i j I i j ( x ) , I s i
Here, ψ B r j I r j ( x ) , I s r denotes the real part of the calculation and ψ B i j I i j ( x ) , I s i denotes the imaginary part.
4.
Cumulative Sum:
Compute the cumulative sum of ψ B j I j x , I s within each cross-Teager–Kaiser ψ B energy operator window, denoted as I s u m j = s u m ( ψ B j I j x , I s ) .
5.
Peak Identification:
Find the maximum value among all I s u m j , denoted as I m a x j . Using Newton’s interpolation method, refine I m a x j along with its neighboring points I m a x j − 1 and I m a x j + 1 to calculate the precise peak point of the sampled interferometric signal, denoted as I p e a k . The corresponding optical path difference δ p e a k of the I p e a k is identified as the ZOPD position.
6.
ZOPD Offset Calculation:
Calculate the ZOPD offset as ∆ δ = δ m a x − δ p e a k .
7.
Linear Phase Calculation:
Substitute the ZOPD offset ∆ δ into the following linear phase calculation formula:
φ l i n = ( δ m a x j + ∆ δ ) ν
δ m a x j is the optical path difference value corresponding to the point I m a x j .
After extracting the linear phase φ l i n , linear phase correction can be performed using the following equation, thereby eliminating the influence of linear phase errors.
B i n s t σ = B m σ e − i φ l i n ( σ )
At this stage, a spectrum B i n s t σ containing only the instrument phase error is obtained. The following sections analyze the method to eliminate this phase error.

3.3. Instrument Phase Error Correction Using the NCM

3.3.1. Nonlinear Calibration Model (NCM)

After correcting the linear phase error in the measured data, the resulting B i n s t σ is a complex spectral dataset that contains only the instrument phase error. This study employs a Nonlinear Calibration Model (NCM) to process M s y m σ and eliminate the influence of instrument phase errors [24]. When the instrument operates under stable temperature conditions but the temperature range of the measured target source is large, the instrument’s spectral response to the target source’s radiation is not strictly linear. Therefore, a nonlinear model is used to describe the instrument’s spectral response.
In this model, the spectral radiation containing only the instrument phase is calibrated radiometrically. The input is the ideal spectrum of the real signal, while the output is the absolute value of the complex spectrum. Using the nonlinear model, the instrument response function for each wavenumber, corresponding to the instrument phase correction parameters, is derived.
To improve the fitting accuracy of the instrument’s Nonlinear Calibration Model for spectral radiation at different temperatures, it is necessary to collect target radiation spectra at multiple temperatures for model fitting [25]. The Nonlinear Calibration Model can then be expressed as follows:
B i n s t 1 σ = P σ + K σ B i d e a l 1 σ + Q σ B i d e a l 1 σ 2 B i n s t 2 σ = P σ + K σ B i d e a l 2 σ + Q σ B i d e a l 2 σ 2 … B i n s t N σ = P σ + K σ B i d e a l N σ + Q ( σ ) B i d e a l N σ 2
B i d e a l σ represents the ideal spectrum of the target radiation at different temperatures, while B i n s t σ is the spectrum containing only the instrument phase error. Based on the above equation, it is necessary to calculate the optimal solutions for the instrument phase correction parameters P σ ,   K σ , and Q ( σ ) . This study employs the Rao algorithm to optimize the parameters P σ ,   K σ , and Q ( σ ) . The details of the Rao algorithm are provided in Section 3.3.2.
With these parameters optimized, the calibrated spectrum can be expressed as follows:
B s y m σ = − K σ + K 2 σ − 4 Q ( σ ) ( P σ − B i n s t σ ) 2 Q ( σ )
where B s y m σ denotes the spectrum after correcting for instrument phase errors.
After the aforementioned processing, this study successfully achieved the high-precision calibration of the phase error in the spectrometer at a specific operating temperature. Subsequent simulation experiments further validated the effectiveness of the algorithm.

3.3.2. Rao Algorithm

The parameter optimization process of the Rao algorithm is illustrated in Figure 4. The Rao algorithm is a non-metaphor-based optimization algorithm that updates the population through the optimal and worst individuals within the population, as well as random interactions among individuals during the optimization process [26]. This algorithm requires only generic control parameters for optimization algorithms, such as population size and the number of iterations, without the need to set additional algorithm-specific control parameters.
The Rao algorithm is computationally simple and exhibits excellent performance for constrained optimization problems [27]. Let F x denote the objective function. In each iteration, assume there are n candidate individuals (i.e., population size, i = 1, 2, …, n), and each individual has m variables (i.e., j = 1, 2, …, m). Suppose the best candidate individual achieves the optimal value of F x among all candidates (denoted as F b e s t x ), and the worst candidate achieves the worst value of F x (denoted as F w r o s t x ). If x i , j , k represents the value of the j-th variable of the i-th candidate during the k-th iteration, this value is updated based on the following equation:
x i , j , k ′ = x i , j , k + r i , j , k ( x b e s t , j , k − x w o r s t , j , k )
x b e s t , j , k represents the value of the j-th variable of the best candidate individual during the k-th iteration. x w o r s t , j , k represents the value of the j-th variable of the worst candidate individual during the k-th iteration. x i , j , k ′ is the updated value of x i , j , k . r i , j , k is a random number associated with the j-th variable of the i-th candidate during the k-th iteration, uniformly distributed within the range [0, 1].
When solving complex optimization problems, population diversity tends to decrease as the number of iterations increases, making the algorithm susceptible to local optima. To enhance population diversity, the chaotic operator, known for its combination of randomness and regularity [28], is utilized. This operator has the ability to traverse all states within a certain range without repetition, and thus, a cubic chaotic map is employed to initialize the population [29].
According to the Nonlinear Calibration Model (Equation (11)), the coefficients to be optimized are P σ , K σ , and Q ( σ ) . The objective function min ⁡ F x is set as follows:
min ⁡ F x = r 1 μ + r 2 σ
μ = ∑ i = 1 N f i x N
σ = ∑ i = 1 N ( f i x − μ ) 2 N − 1
f i x = M e m i s s i σ − [ P σ + K σ M s y m i σ + Q σ M s y m i σ 2 ]
Here, μ and σ represent the mean and standard deviation, respectively. r 1 and r 2 are proportional parameters set during optimization. f i x denotes the nonlinear model residual when the target radiation temperature is iK.

4. Simulation Experiments

4.1. Experiment on Linear Phase Correction Using CTKB Method

To validate the effectiveness of the CTKB method for linear phase correction described in Section 3.2, we conducted a simulation experiment using spectral data from the HITRAN database [30]. In the simulation, the spectrometer’s spectral resolution was set to 2.27   c m − 1 (corresponding to a maximum optical path difference of 0.44 cm) and the reference laser sampling wavelength was set to 1064 nm.
The SO2 gas emission spectrum was downloaded from the HITRAN database via the HAPI interface. The relevant parameters included an atmospheric pressure of one standard atmosphere, an SO2 gas temperature of 268 K, and the use of the Voigt profile as the spectral line shape function. The wavenumber range was set to 1000–1380 c m − 1 .
Figure 5a presents the normalized spectra of SO2 gas after phase correction using the CTKB method and the Mertz method. Both algorithms effectively eliminate linear phase errors in the reconstructed spectra, achieving high spectral reconstruction accuracy. Figure 5b,c provide a detailed comparison of local regions in the spectra reconstructed by the two methods.
Figure 5d shows the difference between the normalized reconstructed spectra processed by the two methods. It can be observed that the difference between the normalized spectra obtained by the CTKB and Mertz methods is in the order of 10 − 16 , demonstrating that the CTKB method has comparable performance to the Mertz method in correcting linear phase errors.

4.2. Experiment on Instrument PHASE Correction Parameters Calculation Using Rao Algorithm

Section 4.1 demonstrated the excellent performance of the CTKB method in correcting linear phase errors. Next, a simulation experiment for instrument phase correction was conducted using spectral data from the HITRAN database. Spectral data for SO2 gas were downloaded with the same parameters as in Section 4.1, except for the gas temperature. Five sets of spectral data were downloaded corresponding to temperatures of 255 K, 265 K, 275 K, 285 K, and 295 K for the simulation experiment.
The theoretical values of the SO2 gas emission spectra at these five temperatures, along with the spectrometer’s measured values, were substituted into Equation (11) to form a system of nonlinear calibration equations. The Rao algorithm was then employed to optimize the instrument phase coefficients P σ , K σ , and Q ( σ ) . The Rao algorithm parameters were configured as follows:
  • Population size: 5
  • Number of variables per individual (number of instrument phase coefficients): 3
  • Number of iterations: 2000
  • Objective function output calculated using Equation (14).
Figure 6 shows the convergence curve of the objective function values during the optimization of instrument phase correction parameters across 3 wavenumbers in the spectral range.
The experiment demonstrated that the Rao algorithm, initialized with a cubic chaotic map, exhibited excellent global search capability. By the 2000th iteration, the objective function rapidly converged to 10 − 12 of its initial value, highlighting the algorithm’s efficiency in optimization.
After confirming the Rao algorithm’s excellent convergence speed, an experiment was conducted to evaluate its parameter optimization capability and to assess the impact of different methods for calculating instrument phase correction parameters on spectral reconstruction accuracy. First, the linear phase errors of the SO2 gas emission spectra at five temperatures were corrected using the CTKB method. Then, Equation (11) was solved using both the least squares method and the Rao optimization algorithm.
Figure 7a,b compare the instrument phase correction parameters obtained by the two methods and their normalized spectral reconstruction results for a local spectral range of the SO2 gas emission spectrum at 268 K. Figure 7c illustrates the distribution of reconstruction errors for the spectra obtained using the two methods.
The experimental results indicate that the instrument phase correction parameters calculated using the Rao algorithm provide a better fit to the model, yielding higher spectral reconstruction accuracy. Moreover, the Rao algorithm demonstrates greater robustness across spectral curves with varying waveforms, making it a good choice for precise spectral reconstruction.

4.3. Experiment on CTKB-NCM Algorithm

To validate the phase correction effectiveness of the CTKB-NCM algorithm, this section uses the SO2 gas emission spectrum at 268 K, consistent with the parameters in Section 4.1, and the instrument phase correction parameters obtained from the Rao algorithm in Section 4.2.
Figure 8 compares three normalized spectra: the original SO2 gas emission spectrum, the spectrum corrected for linear phase errors using the CTKB method, and the spectrum fully processed by the CTKB-NCM algorithm.
The experimental results demonstrate that the CTKB-NCM algorithm effectively removes both linear phase errors and instrument phase errors. Compared to the spectrum corrected for linear phase errors using only the CTKB method, the spectrum processed by the CTKB-NCM algorithm exhibits smaller reconstruction errors. These findings confirm the effectiveness of the two phase correction steps in the CTKB-NCM algorithm for improving spectral reconstruction accuracy.
Figure 9a compares the normalized spectral reconstruction results for the 268 K SO2 gas emission spectrum in the wavenumber range of 1000–1380 c m − 1 using the CTKB-NMPRC method, the Mertz method, and the Forman convolution method. Figure 9b,c provide a closer look at the normalized spectral reconstruction results in specific local spectral ranges. Figure 9d illustrates the error distribution of the normalized reconstructed spectra for these three methods.
The experimental results show that in regions where the spectral curve changes sharply, the Mertz and Forman convolution methods exhibit poorer phase correction performance, resulting in larger reconstruction errors. In regions with slowly varying spectral curves, both the Mertz and Forman convolution methods perform well in phase correction, yielding smaller and more uniformly distributed reconstruction errors. In contrast, the CTKB-NMPRC method maintains consistently low spectral errors across the entire spectral range. This demonstrates that the Mertz and Forman convolution methods cannot fully eliminate the impact of phase errors, leading to significant distortions in the final reconstructed spectra. In comparison, the proposed CTKB-NMPRC method effectively removes phase errors, enabling higher-accuracy spectral reconstruction. It is worth noting that the 268 K SO2 gas radiation spectrum had not undergone instrument phase calibration. Thus, the experiment also demonstrated the broad applicability of the algorithm in correcting phase errors for radiation spectra at different temperatures.
The performance metrics for spectral reconstruction using the three methods are listed in Table 1. The simulation experiments were conducted on a computer equipped with a 13th Gen Intel(R) Core(TM) i9-13900HX processor running at 2.20 GHz with 16.0 GB of RAM. Within the target spectral range, the CTKB-NCM method achieves the smallest maximum error in the normalized reconstructed spectra, with a value of only 0.0058. Similarly, the CTKB-NCM method yields the lowest root mean square (RMS) error of the normalized spectra, at 7.7155 × 10 − 4 . Under identical conditions, the computation times of the three methods were measured. The computation times for the CTKB–NCM method, the Mertz method, and the Forman method were 2.518 s, 2.484 s, and 2.527 s, respectively.
Compared to the Mertz method, the CTKB-NCM method improves the accuracy of the reconstructed spectrum by an order of magnitude, while maintaining a computational speed comparable to the other two methods. This demonstrates that the CTKB-NCM method efficiently performs phase correction, enabling high-quality spectral reconstruction.

5. Conclusions

This paper introduces the composition and working principles of Fourier Transform Spectrometers and analyzes the sources of phase errors. To correct both linear phase errors and instrument phase errors, a CTKB-NCM phase correction algorithm is proposed. The algorithm first employs the cross-Teager–Kaiser ψ B energy operator method to obtain a more accurate zero optical path difference (ZOPD) offset for correcting linear phase errors. It then uses a Nonlinear Calibration Model to correct the residual instrument phase errors. During the calculation of the instrument phase correction parameters for the complex spectra at each wavenumber, the Rao optimization algorithm is applied to obtain optimal parameter estimates, improving the accuracy of the final spectral reconstruction.
Theoretical derivations and simulation experiments were conducted to validate the algorithm. The results show that the maximum error of the normalized reconstructed spectrum for SO₂ gas processed using this algorithm is only 0.0058, with a root mean square (RMS) value of 7.7155 × 10 − 4 , representing an order of magnitude improvement in accuracy compared to the Mertz method. This demonstrates that the proposed algorithm offers high precision and robustness.

Author Contributions

Conceptualization, D.Y. and X.W.; methodology, D.Y. and X.W.; software, X.W.; validation, X.W., C.Y., Z.H. and P.D.; formal analysis, P.D.; writing—original draft preparation, X.W.; writing—review and editing, D.Y., C.Y., Z.H. and P.D.; visualization, X.W.; supervision, D.Y.; project administration, C.Y.; funding acquisition, D.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the Innovation Fund of the Key Laboratory of the Chinese Academy of Sciences (CXJJ-22S014) and the National Key R&D Program of China (2022YFC2807402).

Data Availability Statement

All relevant data to interpret the results of this study are included in the figures. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The schematic of a Michelson-type time-modulated Fourier Transform Spectrometer.
Figure 1. The schematic of a Michelson-type time-modulated Fourier Transform Spectrometer.
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Figure 2. A block diagram of the CTKB-NCM phase correction algorithm structure.
Figure 2. A block diagram of the CTKB-NCM phase correction algorithm structure.
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Figure 3. The linear phase error correction process of the CTKB method.
Figure 3. The linear phase error correction process of the CTKB method.
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Figure 4. A flowchart of the Rao algorithm.
Figure 4. A flowchart of the Rao algorithm.
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Figure 5. (a) A comparison of the normalized restored spectra after linear phase error correction using the CTKB method and the Mertz method; (b,c) comparisons of the local spectra after linear phase error correction by the two methods; (d) the difference between the normalized restored spectra of the two methods.
Figure 5. (a) A comparison of the normalized restored spectra after linear phase error correction using the CTKB method and the Mertz method; (b,c) comparisons of the local spectra after linear phase error correction by the two methods; (d) the difference between the normalized restored spectra of the two methods.
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Figure 6. The convergence curve of the objective function of the Rao algorithm.
Figure 6. The convergence curve of the objective function of the Rao algorithm.
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Figure 7. (a,b) Comparisons of the normalized restored spectra obtained from the SO2 measurement spectrum after linear phase error correction using the CTKB method, with the instrument phase correction parameters calculated using the least squares method and the Rao algorithm, respectively. (c) A comparison of the normalized spectral reconstruction errors for the two methods mentioned above.
Figure 7. (a,b) Comparisons of the normalized restored spectra obtained from the SO2 measurement spectrum after linear phase error correction using the CTKB method, with the instrument phase correction parameters calculated using the least squares method and the Rao algorithm, respectively. (c) A comparison of the normalized spectral reconstruction errors for the two methods mentioned above.
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Figure 8. The phase correction process of the CTKB-NMPRC algorithm.
Figure 8. The phase correction process of the CTKB-NMPRC algorithm.
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Figure 9. (a) Comparison of spectral restoration using CTKB-NCM, Mertz, and Forman Methods. (b,c) Local spectral restoration comparisons. (d) Comparison of restoration spectral errors among three methods.
Figure 9. (a) Comparison of spectral restoration using CTKB-NCM, Mertz, and Forman Methods. (b,c) Local spectral restoration comparisons. (d) Comparison of restoration spectral errors among three methods.
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Table 1. Comparison of normalized spectral metrics for CTKB-NCM method, Mertz method, and Forman method.
Table 1. Comparison of normalized spectral metrics for CTKB-NCM method, Mertz method, and Forman method.
Phase Correction MethodMaximum Error 1Root Mean Square 2Time
CTKB-NCM0.0058 7.7155 × 10 − 4 2.518 s
Mertz0.02690.00952.484 s
Forman0.04920.01812.527 s
1,2 The comparison parameters in the table are the indicators of the normalized spectra.
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Wang, X.; Yan, C.; Huo, Z.; Dai, P.; Yao, D. Research on Fourier Transform Spectral Phase Correction Algorithm Based on CTKB-NCM. Photonics 2025, 12, 219. https://doi.org/10.3390/photonics12030219

AMA Style

Wang X, Yan C, Huo Z, Dai P, Yao D. Research on Fourier Transform Spectral Phase Correction Algorithm Based on CTKB-NCM. Photonics. 2025; 12(3):219. https://doi.org/10.3390/photonics12030219

Chicago/Turabian Style

Wang, Xiong, Chunhui Yan, Zimin Huo, Pengzhang Dai, and Dong Yao. 2025. "Research on Fourier Transform Spectral Phase Correction Algorithm Based on CTKB-NCM" Photonics 12, no. 3: 219. https://doi.org/10.3390/photonics12030219

APA Style

Wang, X., Yan, C., Huo, Z., Dai, P., & Yao, D. (2025). Research on Fourier Transform Spectral Phase Correction Algorithm Based on CTKB-NCM. Photonics, 12(3), 219. https://doi.org/10.3390/photonics12030219

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