Double-comb absorption spectrum high-precision extraction method for gas parameter measurement

Abstract

The invention provides a double-comb absorption spectrum high-precision extraction method for gas parameter measurement, and belongs to the field of laser absorption spectrum measurement. A double-comb absorption spectrum is polluted by obvious multiplicative noise caused by optical power and coherent fluctuation of a laser source, and obvious standard deviation of the calculated gas temperature can be caused. The method provided by the invention can effectively realize separation of signals and noise, so that the influence of multiplicative noise on double-comb absorption spectrum gas parameter measurement is inhibited. According to the method, the correlation between noise and signals is suppressed by using operations such as logarithm operation, baseline offset, high-dimensional expansion and singular value decomposition, and effective separation of noise and signal subspaces is realized, so that noise-reduced absorption spectrum data is extracted. According to the invention, the dependence on a light source system and a self-adaptive acquisition system is reduced, and double-comb absorption spectrum measurement with high signal-to-noise ratio and high-precision gas parameter measurement can be realized.

Description

Technical Field

[0001] The present invention proposes a high-precision extraction method of dual-comb absorption spectra for gas parameter measurement, which belongs to the technical field of laser absorption spectroscopy. The method suppresses the influence of multiplicative noise on the high-precision extraction of dual-comb absorption spectra by effectively separating signal subspace and noise subspace. Background Art

[0002] Dual comb spectroscopy (DCS) has been widely used in the fields of gas parameter measurement, three-dimensional morphology detection and chemical reaction analysis due to its advantages of wide spectral coverage, fast response and high spectral resolution. However, the fluctuation of the optical power and coherence of the dual-comb laser source will cause serious multiplicative noise, which will lead to a significant decrease in the measurement accuracy of the absorption spectrum and gas temperature.

[0003] Traditionally, in order to suppress multiplicative noise and achieve dual-comb spectroscopy with high signal-to-noise ratio (SNR), two high-coherence optical frequency comb lasers with tight frequency locking are required. In 2010, I. Coddington et al. of the National Institute of Standards and Technology published a paper titled "Coherent dual-comb spectroscopy at high signal-to-noise ratio" in Physical Review A, Vol. 82, p. 043817, proposing a novel spectroscopy measurement technique. This technique uses two coherent optical frequency comb lasers to achieve precise measurement of the doubled frequency vibration of hydrogen cyanide (HCN) molecules. Optical frequency comb lasers are able to generate equally spaced frequency components over a wide range, which is essential for capturing the details of the molecular absorption spectrum. In 2021, Daniel MB Lesko et al. of the National Institute of Standards and Technology published a paper titled "Six-octave optical frequency comb from a scalable few-cycleerbium fibre laser" in Nature Photonics, Volume 15, Pages 281-286, which achieved six-octave coverage from the ultraviolet (UV) to the mid-infrared (MIR) spectral region, providing new possibilities for the expanded application of optical frequency combs. Due to its wide spectral coverage capability, DCS has been applied to measure uncommon absorption regions that are difficult to measure using traditional spectral measurement methods. In addition, in the fields of chemistry, quantum physics, atmospheric chemistry, combustion science, and biology, the high signal-to-noise ratio absorption spectrum measured by DCS provides a fast measurement method that does not require calibration and has a response speed of microseconds or even nanoseconds. In 2022, David Yun and others from the University of Colorado Boulder published a paper titled "Supersonic combustion diagnostics with dual comb spectroscopy" in the Proceedings of the Combustion Institute, Volume 39, pages 1299-1306. The paper introduced an innovative dual-frequency comb spectroscopy technology based on a mode-locked frequency comb that can accurately measure key fluid and thermodynamic parameters in supersonic environments, including fluid velocity, temperature, pressure, water vapor mole fraction, and air mass flux.This technology has been successfully applied to the diagnostic process of dual-mode ramjet engines during the ground test phase, providing a new measurement method for supersonic combustion research. In 2023, David A. Long and others from the National Institute of Standards and Technology published a paper titled "Nanosecond time-resolved dual-comb absorption spectroscopy" in Nature Photonics, Volume 18, Pages 127-131. They developed an innovative spectroscopic technique that can achieve high-precision spectral analysis of molecules on a nanosecond time scale, providing a new way to quickly measure molecular properties. However, in order to maintain the level of coherence and stability required for optical frequency comb lasers, complex light source systems and frequency locking systems are required, which seriously hinders the development of DCS.

[0004] In order to improve the signal-to-noise ratio of the spectrum and reduce the difficulty of designing optical frequency comb lasers, adaptive dual-comb spectroscopy technology has emerged as a promising high-coherence measurement method. In 2014, Takuro Ideguchi et al. from the Max Planck Institute of Quantum Optics published a paper titled "Adaptive real-time dual-comb spectroscopy" on page 3375 of Volume 5 of Nature Communications, proposing a new type of real-time dual-comb spectroscopy technology. This technology effectively compensates for the instability of the laser through electronic signal processing, and uses a free-running femtosecond laser to achieve molecular spectroscopy with Doppler resolution. Through this technology, the research team successfully recorded the full-bandwidth spectrum of acetylene in the near-infrared region, providing a new real-time measurement method for the field of spectroscopy. The adaptive dual-comb spectroscopy measurement system combines feedback loops and active stabilization technology to offset the impact of noise and maintain the accuracy of the spectrum. The combination of adaptive dual-comb spectroscopy technology and cavity-enhanced dual-comb spectroscopy technology can further improve the signal-to-noise ratio of dual-comb spectroscopy measurements. In 2019, Zhang et al. from Tsinghua University published a paper titled "Adaptive cavity-enhanced dual-comb spectroscopy" on page 883 of Volume 7 of Photonics Research. They proposed an innovative adaptive cavity-enhanced dual-comb spectroscopy technology, which effectively compensated for the relative phase jitter of the optical frequency comb by adaptively post-correcting the dual-comb interference pattern, achieving a high signal-to-noise ratio of 440:1 and mode-resolved spectral resolution within 1 second, providing a spectral measurement technology with high resolution, high sensitivity and wide bandwidth for precision spectroscopy, atmospheric tracking gas observation, fast breathing analysis and dynamic combustion diagnosis. However, the adaptive acquisition system used to synchronize signals and extract frequency jitter information is very complex, which limits the application of adaptive dual-comb spectroscopy technology.

[0005] Mode-locked dual-comb lasers provide a simpler means of measurement, as they can generate two sets of frequency combs in the same laser. In 2022, J. Pupeikis et al. from ETH Zurich published a paper titled "Spatially multiplexed single-cavity dual-comb laser" on page 713 of Volume 9 of Optica, successfully designing a new single-cavity dual-comb laser technology. This technology not only maintains laser performance and reduces relative timing noise, but also realizes the integration and miniaturization of the laser cavity. Through reasonable configuration, the common-mode noise between pulses is effectively suppressed, while the coherence of the laser is enhanced. Due to its excellent performance, this mode-locked dual-comb laser has been widely used in many fields such as ranging, spectral analysis, and combustion diagnosis. In 2020, Xu et al. from the Chinese University of Hong Kong published a paper titled "Multipass-assisted dual-comb gas sensor for multi-species detection using a free-running fiberlaser" in Applied Physics B, Vol. 126, p. 39, proposing an innovative dual-comb gas sensor technology. This technology combines mode-locked dual-comb laser technology with multi-channel assisted technology to achieve high-resolution molecular spectroscopy measurements of C2H2 (acetylene) and NH3 (ammonia), and can achieve real-time gas sensing capabilities without the need for complex optical and electronic feedback systems. In 2021, Jacob Nürnberg and others from the Swiss Federal Institute of Technology published a paper titled "Dual-comb ranging with frequency combs from single cavity free-running laser oscillators" in Volume 29, page 24910 of Optics Express, proposing the use of mode-locked dual-comb lasers to achieve the first attempt in the field of laser ranging, providing new possibilities for precise distance measurement.In 2021, Xu et al. from the Chinese University of Hong Kong published a paper titled "Dual-comb Spectroscopy for Laminar Premixed Flames with a Free-running Fiber Laser" in Combustion Science and Technology, Vol. 194, pp. 2523-2538, proposing a new dual-comb spectroscopy technique based on a mode-locked dual-comb laser. This technology breaks through the limitations of traditional methods and can achieve simultaneous and non-invasive measurement of water vapor (H2O) and acetylene (C2H2) in laminar premixed flames without relying on any phase-locking system, providing a new and efficient means for flame diagnosis. Generally, the measurement accuracy of mode-locked dual-comb spectroscopy measurement technology without frequency locking is far inferior to that of traditional DCS technology and adaptive dual-comb spectroscopy measurement technology, resulting in serious deviations in gas parameter measurement. In order to improve the measurement signal-to-noise ratio and parameter measurement accuracy of mode-locked dual-comb spectroscopy, a phase correction algorithm is needed to extract and compensate for the phase jitter in the dual-comb interferogram caused by coherent fluctuations. However, the phase correction algorithms reported so far can only correct the phase jitter between adjacent dual-comb interferograms, rather than the phase jitter of each point within the interferogram, which will lead to residual phase fluctuations. Under the influence of residual phase fluctuations and optical power fluctuations, the absorption spectrum will be contaminated by multiplicative noise.

[0006] In general, the multiplicative noise of DCS can be effectively suppressed by using two high-coherence optical frequency comb lasers with tight frequency locking, but the application of this method is limited by the complexity of the laser source system. Adaptive dual-comb spectroscopy reduces the requirements on laser source performance and achieves high-precision dual-comb spectroscopy measurement and high-precision gas parameter measurement by extracting and compensating the optical power and coherence fluctuations of the dual-comb laser source in real time. However, the adaptive dual-comb spectroscopy requires a large and complex post-processing system to adaptively extract frequency jitter information and synchronization signals, which limits the application of this technology. Mode-locked dual-comb spectroscopy provides a miniaturized and compact solution that eliminates common-mode noise between pulses by sharing the same laser and experiencing the same environment. Although the phase jitter between adjacent dual-comb interferograms can be corrected by a phase correction algorithm, there are still residual phase fluctuations, which causes the accuracy of the absorption spectrum to be reduced due to multiplicative noise. Therefore, it is crucial to propose an effective denoising technique to maintain the integrity of the absorption spectrum and ensure accurate gas parameter measurement.

[0007] Based on the above background, the present invention proposes a high-precision extraction method of dual-comb absorption spectra for gas parameter measurement, which can realize high-precision measurement of dual-comb spectra and gas parameters. The principal component analysis denoising method can effectively separate the signal and noise, thereby reducing the influence of multiplicative noise. This method can suppress the correlation between the signal and the multiplicative noise through baseline shift and logarithmic operation, preparing for the subsequent principal component analysis; through high-dimensional expansion and singular value decomposition, the signal and noise subspaces can be effectively separated, thereby effectively removing noise. A high-precision extraction method of dual-comb absorption spectra for gas parameter measurement proposed in the present invention can significantly reduce the multiplicative noise carried in the measured absorption spectrum, and significantly improve the measurement accuracy of gas parameters. Summary of the invention

[0008] The present invention proposes a high-precision extraction method of dual-comb absorption spectra for gas parameter measurement, which mainly includes the following three steps. Step 1, calculate the free induction decay signal sequence: first, it is necessary to perform a logarithmic operation on the measured absorption spectrum data to suppress the correlation between the signal and the noise, so as to facilitate the subsequent separation of the noise and signal subspaces. Then, the absorption spectrum data after the logarithmic operation is Fourier inverse transformed to the time domain to obtain the free induction decay signal; Step 2, separate the signal and noise subspaces: arrange the free induction decay signal sequence into a Hankel matrix, so as to achieve high-dimensional expansion of the signal. Perform singular value decomposition on the obtained Hankel matrix to obtain the corresponding singular value matrix. By distinguishing the size of the singular values, the effective separation of the noise and signal subspaces can be achieved; Step 3, extract the absorption spectrum data after noise reduction: retain the signal subspace obtained in step 2, and arrange the free induction decay signal after noise reduction. Perform Fourier transform and antilogarithmic operation on the free induction decay signal after noise reduction to obtain the absorption spectrum data after noise reduction.

[0009] f m (v) represents the mth group of noisy absorption spectrum signals, expressed as:

[0010] f m (v) = α m (v)+n m (v) (1)

[0011] Among them, α m (v) and n m (v) represent the real absorption spectrum signal and multiplicative noise of the mth group, and n m (v) = σ m (v)α m (v), so the above formula can be simplified to:

[0012] f m (v)=(1+σ m (v))αm (v) (2)

[0013] In order to effectively separate noise from signal through principal component analysis, the independence between noise and signal must be guaranteed. In order to eliminate the correlation between signal and noise, it is necessary to perform logarithmic operation on equation (2) to convert multiplicative noise into additive noise.

[0014] f′ m (v) = ln(f m )=ln(1+σ m (v))+lnα m (v) (3)

[0015] Usually, the spectral profile of the laser source used for measurement is not flat, that is, the baseline is inevitably present in the measured absorption spectrum. In this case, equations (2) and (3) are written as:

[0016] f m (v)=(1+σ m (v))α m (v)+β(v)+β0 (4)

[0017]

[0018] Where β(v) represents the baseline of the absorption spectrum. According to the second term on the right side of equation (5), when the measured absorption spectrum contains a baseline, the logarithmic operation will convert σ m (v) and β(v) / α m (v) are coupled together. In other words, the multiplicative noise factor σ m (v) and absorption signal α m There is still a correlation between (v), which will lead to the failure of noise reduction. In order to suppress the multiplicative noise coefficient σ m (v) and absorption signal α m (v) The present invention applies a baseline shift factor to the measured absorption spectrum, expressed as β0, to move the baseline position. Next, it is explained how β0 suppresses the multiplicative noise coefficient σ m (v) and absorption signal α m (v) The correlation between them.

[0019] Let σ m (v)+[β(v)+β0] / α m (v)∈(ξ,η), we can get:

[0020]

[0021] Due to α m (v)>0, so as long as β0 is large enough, 0<<ξ<η. So:

[0022]

[0023] According to Equation (7), it can be obtained that:

[0024]

[0025] At this time, the third term on the right side of Equation (8) can be Taylor-expanded:

[0026]

[0027] where k = 1 / [1+(ξ+η) / 2], A m (v)=[β(v)+β0] / α m (v)-(ξ+η) / 2. When β0 is large enough, 0<<ξ<η, and at this time 0<k<<1. Only the first term on the right side of Equation (9) is taken, and the remaining terms are ignored. Equation (9) can be simplified to:

[0028]

[0029] The above formula mainly includes four terms, namely the signal term, the interference term, the constant term and the noise term. Since the signals between different periods are highly correlated, the singular value of the signal term is relatively large. The interference term is negatively correlated with the signal term, which means that the singular value of the interference term is also large. Since the constant term is correlated everywhere, its singular value is usually larger than the singular values of the signal term and the interference term. In contrast, due to the linear independence between different periods, the singular value of the noise term is significantly smaller than the other three terms.

[0030] As the value of β0 increases, the value of k decreases, the signal term remains unchanged, the constant term increases, and the noise term decreases. As long as β0 is large enough, the multiplicative noise is completely degraded into additive noise, and the correlation between the noise and the signal will decrease, which effectively promotes the separation of the noise and the signal. In the present invention, the value of β0 satisfies the relationship: min{f m (v)+β0}≥10.

[0031] Since the present invention uses the method of principal component analysis to separate the signal and the noise, when the amount of data is large, the calculation time will increase sharply. Considering that the free induction decay signal caused by absorption has a very short duration in the time domain. Therefore, the present invention performs principal component analysis on the measurement data in the time domain, which can effectively save the calculation time. Perform an inverse Fourier transform on Equation (5) to convert the measurement data into the time domain.

[0032]

[0033] In the above formula, I m(t) represents the mth group of measured free induction decay signals. The redundant data of the measured free induction decay signals of the M groups are removed, and only the N most effective data points are retained. The measured free induction decay signals are arranged into a single vector, which is expressed as:

[0034] I(t)=[I1(t)I2(t)…I M (t)]=[i1(t)i2(t)…i MN (t)] (12)

[0035] Obviously, I(t) is a vector consisting of M groups of measured free induction decay signals of length N, and its length is M×N (assuming it is an even number). Subsequently, I(t) is constructed into a Hankel matrix:

[0036]

[0037] The role of the Hankel matrix is ​​to preserve all the details of the original data while expanding it to a high dimension. The Hankel matrix in equation (13) is subjected to singular value decomposition to effectively extract the main component features of the signal:

[0038]

[0039] Where U s ,∑ s and V s They represent the left singular matrix, singular matrix, and right singular matrix corresponding to the first s largest eigenvalues. p ,∑ p and V p They represent the left singular matrix, singular matrix and right singular matrix corresponding to the last p eigenvalues ​​respectively. When there is no noise, multiple groups of free induction decay signals are perfectly overlapped, that is, the free induction decay signal sequence in equation (12) is periodically repeated and has a very high autocorrelation. This means that the singular value of the principal component of the free induction decay signal is large. As for the noise, it has no autocorrelation and the singular value is close to 0. Therefore, for the free induction decay signal sequence with noise, the singular values ​​of the principal components of the signal are mainly concentrated in ∑ s The singular values ​​of the noise are mainly concentrated in ∑ p In other words, through SVD decomposition, the separation of signal and noise subspaces is achieved.

[0040] The main component of the signal is extracted from equation (14), which is expressed as:

[0041]

[0042] Finally, the reconstructed free induction decay signal is expressed as:

[0043]

[0044] In the formula, h ij Denotes the matrix H s The element in row i and column j. k (t) is re-segmented into M periodic sequences of length N to obtain M groups of denoised free induction decay signals, denoted as I dm (t), where m = 1, 2…M. The data of each cycle is padded with zeros to restore the original data length and the original spectral resolution. The de-noised free induction decay signal is Fourier transformed to obtain the logarithmic absorption spectrum, which is expressed as:

[0045]

[0046] Where real{f} represents the real part of the extracted function f.

[0047] Beneficial effects of the present invention: The present invention performs noise reduction processing on the measured absorption spectrum data, which can effectively separate the signal and noise subspaces, suppress the influence of noise, reduce the requirements of the dual-comb spectroscopy measurement technology on the light source system and the signal acquisition system, and significantly improve the measurement accuracy of gas parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 This is a flow chart of a high-precision extraction method of dual-comb absorption spectra for gas parameter measurement.

[0049] Figure 2 Schematic diagram of the acetylene gas chamber temperature measuring device in the embodiment

[0050] Figure 3 This is the acetylene absorption spectrum before noise reduction in the embodiment

[0051] Figure 4 This is the acetylene absorption spectrum after noise reduction in the embodiment

[0052] Figure 5 is the standard deviation of the acetylene chamber temperature measurement in the embodiment DETAILED DESCRIPTION

[0053] The technical solutions in the embodiments of the present invention are described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0054] The acetylene chamber temperature measuring device used in this embodiment is as follows Figure 2 The device consists of a dual-wavelength mode-locked dual-comb laser, a polarization controller, a collimator, an acetylene gas chamber (heatable), a photodetector, a low-pass filter, a 16-bit data acquisition card, and a temperature controller.

[0055] In this example, the dual-wavelength mode-locked dual-comb laser can simultaneously generate two pairs of asynchronous pulse sequences with slightly different repetition rates, with a repetition rate of about 34.3339 MHz and a repetition rate difference of about 940 Hz. One of the optical pulse sequences is used as the local oscillator light comb, and the other is regarded as the detection light comb. The polarization controller is used to adjust the polarization direction of the pulse to ensure the coherence between the local oscillator light comb and the detection light comb. After the local oscillator light comb and the detection light comb pass through a heatable acetylene gas chamber at the same time, the dual-comb interference signal is detected by a photodetector, where the temperature of the acetylene gas chamber can be monitored and controlled by a temperature controller. Then, the dual-comb interference signal is filtered by a low-pass filter and then collected by a 16-bit data acquisition card. Next, the collected dual-comb interference signal is phase corrected and Fourier transformed to extract it. Figure 3 Finally, the principal component analysis denoising method proposed in the present invention is used to denoise the measured absorption spectrum data, and then combined with the multi-spectrum fitting algorithm, the gas parameter information can be calculated.

[0056] In this embodiment, the principal component analysis denoising method used includes the following three steps:

[0057] Step 1: Calculate the free induction decay signal sequence. Perform logarithmic operation and baseline shift operation on the measured absorption spectrum data to suppress the correlation between the signal and the noise, so as to facilitate the subsequent separation of the noise and signal subspaces. Perform inverse Fourier transform on the absorption spectrum data after logarithmic operation to the time domain to obtain the free induction decay signal;

[0058] Step 2: Separate the signal and noise subspaces. Arrange the free induction decay signal sequence into a Hankel matrix to achieve high-dimensional expansion of the signal; perform singular value decomposition on the obtained Hankel matrix to obtain the corresponding singular value matrix. Among them, the larger singular values ​​mainly come from the signal, while the smaller singular values ​​mainly come from the noise. By distinguishing the size of the singular values, the noise and signal subspaces can be effectively separated;

[0059] Step 3: Extract the de-noised absorption spectrum data. Keep the signal subspace obtained in step 2 and arrange the de-noised free induction decay signal. Perform Fourier transform and inverse logarithmic transform on the de-noised free induction decay signal to obtain the de-noised absorption spectrum data.

[0060] The absorption spectrum data after noise reduction is as follows Figure 4 Compared with the data before denoising, the standard deviation of the absorption spectrum data after denoising is significantly reduced, and the data overlap is increased. Using the multi-spectrum fitting algorithm, we can obtain the following Figure 5The calculation results of the acetylene gas chamber temperature are shown in Figure 2. Compared with the temperature before noise reduction, the standard deviation of the acetylene gas chamber temperature after noise reduction is reduced from 6K to 18.9K to 0.9K to 9.4K, and the temperature uncertainty is significantly reduced.

[0061] The above description of the present invention and its embodiments is not limited thereto, and the accompanying drawings are only one embodiment of the present invention. Without departing from the purpose of the present invention, any structure or embodiment similar to the technical solution designed without creativity shall fall within the protection scope of the present invention.

Claims

1. A high-precision extraction method of dual-comb absorption spectrum for gas parameter measurement mainly includes the following three steps: Step 1, calculating the free induction decay signal sequence, which requires baseline shift and logarithmic operation of the measured absorption spectrum data to suppress the correlation between the signal and the noise: f m (v)=(1+σ m (v))a m (v)+β(v) (1) in, f m (v) represents the measured absorption spectrum data, β(v) represents the baseline of the absorption spectrum, α m (v) and σ m (v) represent the real absorption spectrum signal and multiplicative noise coefficient of the mth group respectively, β0 represents the baseline shift factor, and the value of β0 satisfies the relationship: min{f m (v)+β0}≥10; According to the second term on the right side of formula (2), when the measured absorption spectrum contains a baseline, the logarithmic operation will m (v) and β(v) / α m (v) are coupled together. In other words, the multiplicative noise factor σ m (v) and absorption signal α m There is still a correlation between (v), which will lead to the failure of noise reduction. In order to suppress the multiplicative noise coefficient σ m (v) and absorption signal α m (v) and a baseline shift factor, denoted as β0, is applied to the measured absorption spectrum to move the baseline position. Then, the free induction decay signal sequence is obtained by inverse Fourier transforming equation (2): I(t)=[I1(t) I2(t)…I M (t)]=[i1(t) i2(t)…i MN (t)] (4) In the above formula, I m (t) represents the mth group of measured free induction decay signal signals; redundant data of the measured free induction decay signals of the M groups are removed, and only N valid data near the main lobe are retained and arranged into a single vector to obtain a free induction decay signal sequence; Step 2: Separate the signal and noise subspaces, arrange the free induction decay signal sequence into a Hankel matrix, and thus realize the high-dimensional expansion of the signal; perform singular value decomposition on the Hankel matrix to obtain the corresponding singular value matrix; by distinguishing the size of the singular values, the noise and signal subspaces can be effectively separated: Perform singular value decomposition on the Hankel matrix in equation (5): Where U s ,∑ s and V s They represent the left singular matrix, singular matrix and right singular matrix corresponding to the first s largest eigenvalues ​​respectively; U p ,Σ p and V p They represent the left singular matrix, singular matrix and right singular matrix corresponding to the last p eigenvalues ​​respectively; for formula (6), the singular values ​​of the principal components of the signal are mainly concentrated in Σ s The singular values ​​of the noise are mainly concentrated in Σ p ;Therefore, through SVD decomposition, the separation of signal and noise subspaces is achieved; Step 3, extracting the denoised absorption spectrum data, removing the noise subspace in step 2, calculating the denoised Hankel matrix, and obtaining the denoised free induction decay signal sequence; performing Fourier transform and inverse logarithmic transform on the denoised free induction decay signal sequence to obtain the denoised absorption spectrum data; The denoised Hankel matrix is ​​expressed as: The free induction decay signal sequence after noise reduction is expressed as: In the formula, h ij Denotes the matrix H s The i-th row and j-th column element; R k (t) is re-segmented into M periodic sequences of length N to obtain M groups of denoised free induction decay signals, denoted as I dm (t), where m = 1, 2…M; and use zero padding operation to restore the original data length and original spectral resolution for the data of each cycle; perform Fourier transform and inverse logarithmic transform on the denoised free induction decay signal to extract the denoised absorption spectrum data, which is expressed as:

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