Fourier spectrum resolution enhancement algorithm and system based on linear prediction

Abstract

The present invention provides a Fourier spectrum resolution enhancement algorithm and system for linear prediction. The algorithm includes: inputting existing interferogram data; estimating parameters using forward and backward prediction total least squares and singular value decomposition (SVD); establishing an interferogram signal autoregressive model based on the existing interferogram data to reduce false peaks generated by singular value inversion and suppress noise interference; defining a minimum prediction error principle for the interferogram signal, and processing the resulting autoregressive model order; and performing linear prediction with the autoregressive model using a multi-step prediction method for the interferogram signal based on a sliding window, thereby obtaining a prediction result of suitable accuracy. The present invention solves the technical problems of spectral instrument instability and the limitations of resolution improvement.

Description

Technical Field

[0001] The present invention relates to the field of environmental gas monitoring, and in particular to a Fourier spectrum resolution enhancement algorithm and system based on linear prediction. Background Art

[0002] Fourier transform infrared spectroscopy (FTIR) has been widely used for real-time monitoring of environmental gases. However, for quantitative analysis of certain gases, higher spectral resolution is required for more accurate quantification. The highest resolution achievable by an FTIR spectrometer is determined by the maximum optical path difference between the two arms of the interferometer. The prior invention application, published with publication number CN1837784A, "A Fourier Spectrometer Based on Dynamic Stable Scanning Technology," comprises: a white light and laser light source; an optical system primarily consisting of a Michelson interferometer; a feedback module for detecting weak optical signal changes and converting them into corresponding electrical signals; a control module for converting analog signals processed by the feedback module into digital signals; an execution module for controlling the phase difference between the two arms of the Michelson interferometer based on the processing results of the control module; and a module for detecting uniform changes in the phase difference between the two arms of the Michelson interferometer controlled by the feedback module and the execution module when the phase difference is locked. The spectrometer provided in this prior art can perform continuous scanning over a large range. However, increasing the interferometer's range reduces the instrument's stability and reliability. Therefore, using signal processing methods to extrapolate interference information is very meaningful for improving the resolution of FTIR spectrometers.

[0003] FTIR spectrometers, a type of frequency-modulated interferometer spectrometer, measure interference signals in the time domain and perform Fourier transforms on them to acquire and analyze spectra. Their superior spectroscopic technology has applications in fields such as biochemistry, environmental protection, aerospace, astronomy, medicine, agriculture, and plasma diagnostics. Certain applications, such as environmental gas analysis, often require higher spectral resolution for accurate quantitative analysis. The highest achievable resolution of an FTIR spectrometer is determined by the maximum optical path difference between the two arms of the interferometer. The prior invention patent application, "Optical Device for Optical Imaging Fourier Spectrometer and Method of Operating the Same," with publication number CN1265738A, utilizes a birefringence device to systematically vary the optical path difference across all regions of an image simultaneously to generate a Fourier transform spectrum. This allows for the acquisition of independent interference patterns for each image by combining the intensity values within multiple images. However, increasing resolution by increasing the optical path difference can also have negative consequences. Not only does it reduce instrument stability and reliability, but the resolution improvement is also limited. Therefore, signal processing techniques are being used to improve the resolution of FTIR spectrometers.

[0004] Many researchers have proposed effective methods for enhancing infrared spectral resolution to obtain more spectral detail and improve the accuracy of quantitative analysis. Currently, there are two main approaches to enhancing infrared spectral resolution. One approach involves improving the interferometer and its hardware structure to increase the maximum optical path difference, thereby obtaining high-resolution spectra. For example, Ahmet's microelectromechanical system (MEMS)-based in-plane (x- and y-axis) lens scanner has been developed to improve the resolution of long-wave infrared optical systems (8-12μm wavelength). Another category primarily uses advanced signal processing techniques on interferograms or spectra to improve spectral resolution. Skiba et al. used different types of neural networks (convolutional neural networks and multilayer perceptrons) to improve the spectral resolution of initial absorption spectra. Tingting Liu developed a spectral resolution enhancement algorithm regularized by contourlet transform for FTIR spectral imaging. Samar Elaraby used a neural network to take low-resolution spectra as input and generate high-resolution spectra, using a deep convolutional neural network (CNN) to improve the resolution of FTIR gas spectra. Guangpu Shao developed a resolution enhancement algorithm with a total variation (TV-norm) constraint to address infrared spectral degradation caused by overlap and noise degradation. Ning Liu proposed a resolution improvement method for enhancing interference fringes based on interpolation theory. In the study of using signal processing techniques to enhance spectral resolution, most studies directly perform signal processing on spectra to enhance resolution, while fewer studies use signal processing on interferograms.

[0005] In summary, the existing technology has technical problems such as instability of spectral instruments and limitations on improving resolution. Summary of the Invention

[0006] The technical problem to be solved by the present invention is how to solve the technical problems in the prior art of instability of spectral instruments and the limitation of the effect of improving resolution.

[0007] The present invention solves the above technical problems by adopting the following technical solutions: a Fourier spectrum resolution enhancement algorithm based on linear prediction includes:

[0008] S1. Input existing interference pattern data;

[0009] S2. Using the forward and backward prediction total least squares method and the singular value decomposition (SVD) to estimate parameters, an interference signal autoregressive model is established based on the existing interference pattern data to reduce false peaks generated by singular value inversion and suppress noise interference. The minimum prediction error principle of the interference signal is defined and the order of the autoregressive model is obtained based on the principle. Step S2 includes:

[0010] S21. Using the forward and backward prediction total least squares method, construct a forward and backward expansion matrix of the interference signal to obtain a forward and backward linear prediction equation, solve the forward and backward linear prediction equation using the influence total least squares method, and use the noise perturbation of the processed observation vector and data matrix of the interference signal to estimate the parameters of the AR model and decompose the singular values;

[0011] S22, using prediction error logic, adaptively determining the autoregressive model order of the AR model;

[0012] S3. Based on the sliding window, linear prediction is performed using an autoregressive model and a multi-step prediction method of the interference signal to obtain a prediction result of applicable accuracy. Step S2 includes:

[0013] S31, using the AR model of the interference signal to form a linear difference equation, thereby obtaining an interference output signal matrix, and extrapolating the interference signal;

[0014] S32, using the minimum prediction error criterion of the interference signal and the resolution requirement data, preselecting the applicable window length T and the number of sliding times of the sliding window;

[0015] S33, within the applicable window length T, using the AR model to approximate the time series of the interference signal, so as to form a multi-step prediction model of the interference signal based on the sliding window as the sliding window slides forward;

[0016] S34, using a sliding window-based interference signal multi-step prediction model and a preset sliding window prediction logic, respectively, to perform AR model order re-determination and parameter re-estimation operations, thereby obtaining an AR iterative model and a new interference pattern.

[0017] The present invention uses a signal processing method to extrapolate interference information, which is beneficial to improving the resolution of the FTIR spectrometer. In the process of establishing the autoregressive model of the present invention, the forward and backward prediction total least squares method and singular value decomposition (SVD) parameter estimation are used, which plays an important role in reducing false peaks caused by singular value inversion and suppressing noise interference. At the same time, the present invention proposes the principle of minimum prediction error of the interference signal to determine the order of the autoregressive model. The present invention uses the autoregressive model to perform linear prediction: a multi-step prediction method of the interference signal based on a sliding window is proposed to improve the prediction accuracy. The present invention has good applicability in improving spectral resolution and can improve the shortcomings of low-resolution spectra in the quantitative analysis of gas concentration to a certain extent.

[0018] In a more specific technical solution, step S1 uses the following logic to represent the equal optical path difference sampling interference data:

[0019]

[0020] Where {I(n)} is the interference data, W(n) means the noise signal, p refers to the order of the prediction model, fk and ak are the frequency and amplitude of the kth sine wave respectively.

[0021] In a more specific technical solution, step S21 includes:

[0022] S211, constructing a forward and backward expansion matrix using the forward and backward prediction total least squares method;

[0023] S212, obtaining the forward and backward linear prediction equation FBLP according to the forward and backward expansion matrix:

[0024]

[0025] That is: Ac k ≈b

[0026] Where * represents complex conjugate, A is the I data matrix, c k is the prediction coefficient vector, b is the observation vector;

[0027] S213. The noise perturbation of the data matrix and the noise perturbation of the observation vector are respectively denoted as ΔA and Δb, thereby obtaining the following equation:

[0028] (A+ΔA)c k =b+Δb

[0029] S214, using the following logic to influence the total least squares method to obtain a perturbation matrix E with the minimum norm square, so that the matrix (B+E) is not full rank, and the coefficient vector c is obtained by using the SVD method k :

[0030]

[0031] In order to solve the problem in the prior art that both the data matrix and the observation vector may be subject to noise disturbance, the present invention adopts the influence-total least squares method, which takes into account the noise disturbance of the observation vector and the data matrix at the same time, and realizes the suppression of noise disturbance more comprehensively.

[0032] In a more specific technical solution, in step S224, where B=[A:b] is the data augmentation matrix, E=[ΔA:Δb] is the perturbation augmentation matrix, To obtain the following formula:

[0033] (B+E)z=0.

[0034] In a more specific technical solution, step S22 includes:

[0035] S221, using the following judgment function to predict the prediction error between the interference pattern and the original interference pattern:

[0036]

[0037] Where N refers to the number of interference signal data points;

[0038] S222. Select the order that minimizes the prediction error E as the order of the autoregressive model.

[0039] The present invention addresses the defects in the prior art: given a data length N, selecting too low an order will result in a simple AR system with a large fitting error, while selecting too high an order may cause false peaks in the spectrum due to additional poles. The present invention uses a decision function to predict the prediction error between the interferogram and the original interferogram, and selects a suitable order for the AR model based on the prediction error, which is beneficial to achieving correct fitting and extrapolation.

[0040] In a more specific technical solution, step S31 includes:

[0041] S311. Use the AR model of the interference signal to form a linear difference equation:

[0042]

[0043] Where {ck} is the parameter vector of the AR model;

[0044] S312. The signal i(n) outputted from the current interference pattern is expressed as the sum of p past output interference signals I with different weights to obtain an interference output signal matrix:

[0045] i=c k I

[0046] in

[0047]

[0048] S313. Calculate the coefficient vector {ck} according to the interference output signal matrix, determine the order p of the linear prediction model, and implement the extrapolation of the interference signal accordingly.

[0049] The present invention calculates the coefficient vector {ck} and determines the linear prediction model order p to achieve extrapolation of the interference signal. This effectively solves the noise interference problem and can extrapolate the interference signal to a great distance without error, thereby enhancing spectral resolution.

[0050] In a more specific technical solution, step S32 includes:

[0051] S321, using the minimum prediction error criterion of the interference signal to determine the sliding window length T, respectively establish linear prediction models from the short window to the long window, and calculate the corresponding prediction error E;

[0052] S322. Obtain the maximum window length that makes the prediction error E satisfy E < Ep (Ep is the set maximum error value), and use it as the applicable window length T.

[0053] The present invention establishes a linear prediction model from a short window to a long window, calculates the corresponding value of E, and determines the maximum window length that makes E satisfy E < Ep (Ep is the set maximum error value) as the length of the sliding window, avoiding the problems of decreased accuracy, reduced estimation accuracy, and longer prediction time caused by excessive data in a single prediction window in the traditional technology.

[0054] In a more specific technical solution, step S34 includes:

[0055] S341. Use the predicted value of the preset small window as the new observed value, and use the information of the new observed value to predict the future sequence value;

[0056] S342. Assume that the length of the sliding window at time t is T, and use the following prediction logic to perform the order re-determination and parameter re-estimation operations of the AR model at times t + T, t + 2T,..., respectively, and calculate it + 2T, it + 3T,... to form an AR iterative model:

[0057]

[0058] where c k represents the parameter vector of the AR model.

[0059] Based on the original observed values, the present invention uses the predicted value of the small window as the new observed value, and these new observed values bring more information, so that the unknown information to be predicted at future times gradually decreases. Therefore, using the information of the new observed values, the present invention can better predict the future sequence values and improve the prediction accuracy.

[0060] In a more specific technical solution, in step S342, use the following prediction logic to perform the order re-determination and parameter re-estimation operations of the AR model at times t + T, t + 2T,..., respectively, and calculate it + 2T, it + 3T,... to form an AR iterative model:

[0061]

[0062] In a more specific technical solution, the Fourier transform spectral resolution enhancement system based on linear prediction includes:

[0063] An input data module for inputting existing interferogram data;

[0064] The interference signal autoregressive model establishment module is used to estimate parameters using the forward and backward prediction total least squares method and singular value decomposition (SVD). The interference signal autoregressive model is established based on the existing interference pattern data to reduce the false peaks generated by singular value inversion and suppress noise interference. The interference signal minimum prediction error principle is defined and the autoregressive model order is obtained based on the principle. The interference signal autoregressive model establishment module is connected to the input data module and includes:

[0065] The parameter estimation and singular value decomposition module is used to construct the forward and backward expansion matrix of the interference signal using the forward and backward prediction total least squares method to obtain the forward and backward linear prediction equation. The forward and backward linear prediction equation is solved using the influence total least squares method to process the noise disturbance of the observation vector and data matrix of the interference signal, thereby estimating the parameters of the AR model and decomposing the singular values;

[0066] An order determination module is used to adaptively determine the autoregressive model order of the AR model using prediction error logic;

[0067] The linear prediction module is used to perform linear prediction based on a sliding window and an autoregressive model using an interference signal multi-step prediction method to obtain a prediction result with applicable accuracy. The linear prediction module establishes a module connection with the interference signal autoregressive model. The linear prediction module includes:

[0068] An interference signal extrapolation module is used to form a linear differential equation using the AR model of the interference signal, thereby obtaining the interference output signal matrix and extrapolating the interference signal;

[0069] A sliding window parameter preselection module is used to preselect the applicable window length T and number of sliding times of the sliding window by using the minimum prediction error criterion of the interference signal and the resolution requirement. The sliding window parameter preselection module is connected to the interference signal extrapolation module;

[0070] a multi-step prediction model acquisition module, configured to approximate the time series of the interference signal using the AR model within the applicable window length T, so as to form a multi-step prediction model of the interference signal based on the sliding window as the sliding window slides forward, and the multi-step prediction model acquisition module is connected to the sliding window parameter preselection module;

[0071] The AR iterative model prediction module is used to use a sliding window-based interference signal multi-step prediction model and a preset sliding window prediction logic to respectively perform AR model order redetermination and parameter re-estimation operations, thereby obtaining an AR iterative model and a new interference graph. The AR iterative model prediction module is connected to the multi-step prediction model acquisition module.

[0072] The present invention has the following advantages compared with the prior art: The present invention uses a signal processing method for extrapolating interference information, which is beneficial to improving the resolution of the FTIR spectrometer. In the process of establishing the autoregressive model of the present invention, the forward and backward prediction total least squares method and singular value decomposition (SVD) are used to estimate parameters, which play an important role in reducing false peaks caused by singular value inversion and suppressing noise interference. At the same time, the present invention proposes the principle of minimum prediction error of interference signals to determine the order of the autoregressive model. The present invention uses the autoregressive model for linear prediction: a multi-step prediction method for interference signals based on a sliding window is proposed, which improves the prediction accuracy. The present invention has good applicability in improving spectral resolution and can improve the deficiencies of low-resolution spectra in quantitative analysis of gas concentration to a certain extent.

[0073] In view of the problem that both the data matrix and the observation vector may have noise perturbations in the prior art, the present invention adopts the influence-total least squares method, taking into account the noise perturbations of both the observation vector and the data matrix, and more comprehensively realizes the suppression of noise perturbations.

[0074] In view of the deficiencies in the prior art: for a given data length N, selecting too low an order will result in a simple AR system with a large fitting error, while selecting too high an order may cause false peaks in the spectrum due to additional poles. The present invention uses a decision function to predict the prediction error between the interferogram and the original interferogram, and accordingly selects an appropriate order for the AR model, which is beneficial to achieving correct fitting and extrapolation.

[0075] The present invention determines the order p of the linear prediction model by obtaining the coefficient vector {ck} to realize the extrapolation of interference signals. And it effectively solves the problem of noise interference, and can extrapolate the interference signals to a long distance without error, thereby realizing the enhancement of spectral resolution.

[0076] The present invention establishes a linear prediction model from a short window to a long window, calculates the corresponding value of E, and determines the maximum length of the window that satisfies E < Ep as the length of the sliding window, avoiding the problems of decreased accuracy, reduced estimation accuracy, and longer prediction time caused by too much data in the prediction window at one time in the traditional technology.

[0077] Based on the original observed values, the present invention uses the small-window prediction values as new observed values. These new observed values bring more information, so that the unknown information to be predicted at future times gradually decreases. Therefore, using the information of the new observed values, the present invention can better predict the sequence values in the future and improve the prediction accuracy.

[0078] The present invention solves the technical problems existing in the prior art, such as instability of spectral instruments and restricted improvement of resolution effect. Brief Description of the Drawings

[0079] Figure 1 Schematic diagram of the basic steps of the Fourier spectrum resolution enhancement algorithm based on linear prediction in Example 1 of the present invention;

[0080] Figure 2 Schematic diagram of the specific steps of the Fourier spectrum resolution enhancement algorithm based on linear prediction in Example 1 of the present invention;

[0081] Figure 3 Schematic diagram of simulated interference signal according to embodiment 2 of the present invention;

[0082] Figure 4a This is a schematic diagram of interference error obtained using a linear prediction model in Example 2 of the present invention;

[0083] Figure 4b This is a schematic diagram of spectral errors obtained using a linear prediction model in Example 2 of the present invention;

[0084] Figure 5 Schematic diagram of the working principle of the portable Fourier transform infrared spectrometer according to Example 3 of the present invention;

[0085] Figure 6a The embodiment 3 of the present invention applies the Fourier spectrum resolution enhancement algorithm based on linear prediction to extrapolate the measured interferogram forward and backward;

[0086] Figure 6b A schematic diagram of applying FFT to an extrapolated interferogram to obtain a resolution-enhanced spectrum according to embodiment 3 of the present invention;

[0087] Figure 7a 1 is a comparison diagram of enhanced interference data and measured interference data according to Example 3 of the present invention;

[0088] Figure 7b This is a schematic diagram of applying FFT to compare two sets of interferometric spectra in Example 3 of the present invention. DETAILED DESCRIPTION

[0089] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0090] Example 1

[0091] like Figure 1 As shown, the Fourier spectrum resolution enhancement algorithm based on linear prediction provided by the present invention includes the following basic steps:

[0092] S1. Establishing an autoregressive model for the interference signal: The forward and backward prediction total least squares method and singular value decomposition (SVD) are used to estimate parameters in the autoregressive model. This plays an important role in reducing false peaks caused by singular value inversion and suppressing noise interference. At the same time, the principle of minimum prediction error of the interference signal is proposed to determine the order of the autoregressive model.

[0093] S2. Linear Prediction Using Autoregressive Models: A sliding window-based multi-step prediction method for interferometric signals was proposed to improve prediction accuracy. The effects of signal-to-noise ratio, enhancement factor, and resolution on the accuracy of a Fourier spectral resolution enhancement algorithm based on linear prediction were investigated theoretically and experimentally.

[0094] like Figure 2 As shown, in this embodiment, the Fourier spectrum resolution enhancement algorithm based on linear prediction provided by the present invention further includes the following specific steps:

[0095] S1', input the existing interference pattern data and assign m=1;

[0096] S2', adaptively determine the order of the AR model through the prediction error formula;

[0097] S3', AR model coefficient estimation;

[0098] S4', determine the length of the prediction window T and the number of slides M according to the resolution requirements;

[0099] S5', predicting the interference data of window T based on the existing interference data I;

[0100] S6', obtain the interference data of window T, and form a new interference graph I = (I T1 ,I,I T2 ); In this embodiment, T1 refers to forward prediction and T2 refers to backward prediction.

[0101] S7', determine whether the value of m is less than the sliding number M of the prediction window T. If so, assign m=m+1 and repeat the above steps S2' to S6';

[0102] S8', if not, output the target resolution interference pattern.

[0103] 1 Linear Prediction

[0104] 1.1 Relationship between interference signal and linear prediction

[0105] In this embodiment, the Prony method indicates that a linear combination of a set of exponential terms can be used to fit the equally spaced sampling data. In the FTIR spectrometer, the equal optical path difference sampling interference data can be expressed as:

[0106]

[0107] where {I(n)} is the interference data, fk and ak are the frequency and amplitude of the kth sine wave, respectively.

[0108] In this embodiment, ignoring the noise, the AR model of the interference signal forms a linear difference equation:

[0109]

[0110] {ck} is the parameter vector of the AR model.

[0111] According to equation (2), the signal i(n) output by the current interference pattern is expressed as the sum of p past output interference signals I with different weights.

[0112] In this embodiment, the above equation can be expressed in matrix form:

[0113] i=c k I (4)

[0114] in

[0115]

[0116] In this embodiment, when N = 2p, the equation has a unique, exact solution; when N > 2p, the equation has a least-squares solution. Extrapolation of the interference signal can be achieved by simply finding the coefficient vector {ck} and determining the order p of the linear prediction model. If noise interference can be resolved, the interference signal can be extrapolated to significant distances without error, thereby enhancing spectral resolution.

[0117] 1.2 Parameter Estimation and Singular Value Decomposition

[0118] In this embodiment, the linear prediction derivation process of the interference signal ignores the influence of noise. In order to suppress noise interference, it is proposed to use the forward and backward prediction total least squares method to construct the forward and backward expansion matrix (L>p), then the forward and backward linear prediction equation (FBLP) is:

[0119]

[0120] Right now

[0121] Ac k ≈b

[0122] Where * represents complex conjugate, A is the I data matrix, c k is the prediction coefficient vector, and b is the observation vector.

[0123] In this embodiment, there are two main methods for solving the above equations: the first method is the least squares (Ls) method, which assumes that there is a disturbance term Δb in the observation vector b; the second method assumes that there is a disturbance term ΔA in the data matrix A, which was proposed by Tufst and Kumaresan and is called the principal eigenvector (PE) method. These two compensation methods only consider the noise disturbance effects of the observation vector and the data matrix respectively, which is insufficient. Usually, both the data matrix and the observation vector may have noise disturbances, denoted as ΔA and Δb respectively, so Equation (3) can be written as:

[0124] (A+ΔA)c k =b+Δb

[0125] This method is a total least squares method because it considers the effects of noise disturbances on both the observation vector and the data matrix.

[0126]

[0127] Where B = [A:b] data augmentation matrix, E = [ΔA:Δb] perturbation augmentation matrix, Then there is

[0128] (B+E)z=0

[0129] In this embodiment, the total least squares problem of the above homogeneous linear equation is to find a perturbation matrix E with the minimum square norm so that (B+E) is non-full rank. The solution of the total least squares problem can be obtained by applying the SVD method, which is the coefficient vector c k .

[0130] 1.3 Model Applicability Test

[0131] In order to achieve correct fitting and extrapolation, the AR model should first select a suitable order. Relevant literature shows that for a given data length N, selecting an order that is too low will result in a simple AR system with a large fitting error, while selecting an order that is too high may cause false peaks in the spectrum due to additional poles. The determination of the appropriate order depends on the statistical characteristics of the data. Some AR model order judgment criteria based on information theory and statistics have been developed to select the optimal order, such as the final prediction error (FPE), the criterion autoregressive transfer (CAT) function, the minimum information theory criterion (AIC), and the Bayesian information criterion (BIC). Based on the characteristics of the interference signal extrapolation, the judgment function shown in formula (8) is proposed to select the order that minimizes the function value. Formula (8) is the prediction error formula between the predicted interferogram and the original interferogram.

[0132]

[0133] 1.4 Multi-step Prediction Method of Interference Signal Based on Sliding Window

[0134] In this embodiment, for a linear model, it cannot fully describe the global characteristics of the time series. As a prediction model, if there is too much prediction window data at one time, it will inevitably lead to a decrease in accuracy, and the prediction data needs to be divided into multiple prediction windows. The longer the prediction step length, the more unknown information there is, and thus the worse the estimation accuracy. However, the shorter the prediction step length, the longer the prediction time required. Suppose a suitable window length T is selected. Within this window time, the interference signal time series is approximated by an AR model. As time goes by, the window also slides forward accordingly, forming a multi-step prediction model of the interference signal based on the sliding window.

[0135] As the prediction progresses, based on the original observed values, the present invention uses the small window prediction values as new observed values. These new observed values bring more information, so that the unknown information at future times gradually decreases. Therefore, by using the information of the new observed values, the present invention can better predict the future sequence values and improve the prediction accuracy.

[0136] Let the sliding window length at time t be T, and the prediction is as shown in Equation (9)

[0137]

[0138] In the formula, ck is the parameter vector of the above AR model.

[0139] Then at times t+T, t+2T,..., the order determination and parameter estimation are re-performed respectively, and i is calculated t+2T , i t+3T ,..., thus forming an AR iterative model.

[0140] In this embodiment, the interference signal minimum prediction error criterion determines the sliding window length T. In specific applications, usually linear prediction models from short windows to long windows are established respectively, and the corresponding E values are calculated. Then, the maximum window length that makes E satisfy E<Ep is determined as the sliding window length.

[0141] Embodiment 2

[0142] 2 Simulation Study

[0143] 2.2 Signal-to-Noise Ratio of Interference Signal

[0144] In the Fourier transform spectroscopy resolution enhancement algorithm based on linear prediction, although the influence factors of noise are considered and suppressed, in fact, the signal-to-noise ratio of the interference signal will still have a certain impact on the effect of linear prediction.

[0145] Such as Figure 3As shown, in this embodiment, in order to study the influence of the signal-to-noise ratio of the interference signal on the Fourier spectral resolution enhancement algorithm based on linear prediction, MATLAB is used to simulate an interference pattern with a maximum optical path difference of 2 cm (i.e., a spectral resolution of 0.5 cm-1), and it is truncated into an interference pattern with a maximum optical path difference of 1 cm (i.e., a resolution of 1 cm-1).

[0146] Then, white Gaussian noise (AWGN) is added to the generated interferogram to form an interferogram with a certain signal-to-noise ratio. The signal-to-noise ratio is defined as:

[0147]

[0148] Where Ps is the interference signal power and Pn is the noise power, I represents the interference signal, noise represents the noise signal, and N represents the number of collected interference data points.

[0149] Changing the signal-to-noise ratio of the interference pattern, using the sliding window-based multi-step prediction method of the interference signal under different signal-to-noise ratios to extrapolate the interference pattern to the maximum optical path difference of 2 cm, and calculate the corresponding interference signal prediction error and spectral signal error The relationship between SNR and prediction error is shown in Table 1.

[0150] Table 1 Effect of different interference signal noise ratios on linear prediction accuracy

[0151] SNR 35 40 45 50 55 60 Interference error 1.4494 1.3647 0.9682 0.6926 0.6106 0.4285 Spectral error 1.96％ 1.11％ 0.75％ 0.42％ 0.29％ 0.24％

[0152] like Figure 4a and Figure 4b As shown in the figure, in this embodiment, as the signal-to-noise ratio decreases, the errors of the interferogram and spectrogram obtained using the linear prediction model increase significantly. This is because the algorithm predicts future samples based on past samples, and lowering the SNR means that past samples are more susceptible to noise, which leads to error accumulation, manifested in the interferogram and spectrogram as increased prediction errors.

[0153] 2.3 Enhancement Factor

[0154] For a given interferogram, the length of the empirical regression data is fixed. However, different resolution enhancement factors have different effects on the prediction errors of the enhanced interferogram and spectral graph. To study the effect of the enhancement factor on the Fourier spectral resolution enhancement algorithm based on linear prediction, we simulated interferometric data with a resolution of 4 cm⁻¹. Interferometric data with resolutions of 2 cm⁻¹, 1 cm⁻¹, and 0.5 cm⁻¹ were predicted in turn. The corresponding enhancement factors were 2, 4, and 8, respectively. The errors compared with the directly simulated interferometric data of 2 cm⁻¹, 1 cm⁻¹, and 0.5 cm⁻¹, as well as the inverted spectral data, were calculated. The relationship between the enhancement factor and the prediction error is shown in Table 2.

[0155] Table 2 Effects of different enhancement factors on linear prediction accuracy

[0156] Enhancement Factor 2 4 8 Interference error 0.4863 1.3784 2.6501 Spectral error 0.74％ 1.45％ 2.11％

[0157] In this embodiment, simulation results show that, for the same interferometric data, the larger the enhancement factor, the greater the prediction error of the Fourier spectral resolution enhancement algorithm based on linear prediction. Because the same interferometric data provides the same real historical data to the autoregressive model, lower spectral resolution requires the interferogram to be extrapolated to a larger OPD, which will lead to increased cumulative error. The larger the enhancement factor, the greater the cumulative error, and the lower the accuracy of the enhanced interferogram. If FFT is applied to the extrapolated interferogram with a large prediction error, the error in the reconstructed spectrum will naturally increase.

[0158] 2.4 Initial Resolution

[0159] In this embodiment, for interferograms of different resolutions, even if the resolution enhancement factor is the same, since the empirical regression data length and the predicted data length are different, the prediction error of the enhanced interferogram and spectrogram will also have different effects. In order to study the impact of the initial resolution on the Fourier spectrum resolution enhancement algorithm based on linear prediction, interference data with resolutions of 4cm-1, 2cm-1, and 1cm-1 are simulated respectively, and interference data with resolutions of 2cm-1, 1cm-1, and 0.5cm-1 are predicted respectively. Under the same enhancement factor, the impact of the initial resolution on the prediction error of the interference data and spectral data is calculated, wherein the relationship between the initial resolution and the prediction error is shown in Table 3.

[0160] Table 3 Effect of different starting resolutions on linear prediction accuracy under the same enhancement factor

[0161] Enhancement method 4cm-1→2cm-1 2cm-1→1cm-1 1cm-1→0.5cm-1 Interference error 0.4863 0.4429 0.4285 Spectral error 0.74％ 0.39％ 0.24％

[0162] In this embodiment, simulation results show that for interferometric data with different starting resolutions, when the enhancement factor is the same, the higher the resolution of the interferogram, the greater the prediction error of the Fourier spectral resolution enhancement algorithm based on linear prediction. Interferometric data with higher resolution provides less real historical data to the autoregressive model, reducing the accuracy of the autoregressive model. This results in lower accuracy of the enhanced interferometric and spectral data, and greater error.

[0163] Example 3

[0164] 3 Experimental comparison

[0165] 3.1 Spectral enhancement experiment

[0166] In order to demonstrate the practicality of Fourier spectrum resolution enhancement algorithm based on linear prediction in improving spectral resolution, a portable Fourier transform infrared spectrometer was selected for verification. Its working principle is as follows Figure 5 As shown in Figure 1, this portable instrument, designed around a Michelson interferometer, can quantitatively analyze atmospheric pollutants. The interferometer's main components are a beam splitter, a moving mirror, and a fixed mirror. Infrared light emitted by an infrared light source passes through the beam splitter. Ideally, 50% of the light is reflected by the moving mirror and then back to the beam splitter, while the remaining 50% passes through the beam splitter, reaches the fixed mirror, and then reflects back to the beam splitter. This creates an optical path difference, leading to interference. When the optical path difference is zero, the two beams reflected from the fixed and moving mirrors back to the beam splitter have the same phase. When superimposed, no interference occurs, and the light intensity is the sum of the intensities of the two beams. When the moving mirror moves by 1 / 4 wavelength, the optical path difference becomes half a wavelength. At this point, the two beams have opposite phases, and when superimposed, they cancel each other out, resulting in a light intensity of zero. When the moving mirror moves another 1 / 4 wavelength, the optical path difference becomes one wavelength, and the two beams have the same phase difference, similar to the zero optical path difference. When the moving mirror moves at a constant speed, the signal intensity detected by the detector varies cosine-like, forming an interference pattern.

[0167] like Figure 5 The operation process of the instrument shown is as follows: the gas to be measured passes through the sampling head and filter to reach the multiple reflection absorption sample cell. At the same time, the infrared radiation signal is collimated by the optical system and then introduced into the Fourier transform infrared spectrometer. After being interferometrically modulated by the spectrometer, it is then introduced into the multiple reflection absorption sample cell. The interferometrically modulated infrared radiation signal derived from the sample cell carries the absorption information of the sample to be measured. The interference pattern can be obtained by converging the signal onto the detector.

[0168] The detector of the portable Fourier transform infrared spectrometer is MCT, and the maximum optical path difference is set to 0.5 cm (i.e., the spectral resolution is set to 2 cm-1). The NH3 gas to be measured is passed into the gas cell to obtain interference data. In order to reduce random noise interference, the average value of 8 consecutive sets of interference data is taken as an interference pattern. The Fourier spectrum resolution enhancement algorithm based on linear prediction is used to forward and backward extrapolate the measured interference pattern until the OPD is 1 cm, such as Figure 6a The FFT is then applied to the extrapolated interferogram to obtain a resolution-enhanced spectrum, as Figure 6b As shown, the obtained spectral resolution is 1 cm-1.

[0169] like Figure 6a and Figure 6b As shown in , under the same conditions, the maximum optical path difference of the spectrometer is set to 1 cm (that is, the spectral resolution is set to 1 cm-1), the interference data is obtained, and the enhanced interference data is compared with the measured interference data, as shown in Figure 7a Then FFT is applied to the two sets of interferometric data, and the spectra are compared as Figure 7b As shown, the interference error is 0.6419, and the spectrum error of the NH3 characteristic absorption band is 0.18%.

[0170] 4.2 Gas concentration inversion

[0171] In this example, to investigate the effect of resolution-enhanced spectra on gas concentration inversion, ammonia (which has a narrow spectral absorption peak and requires higher resolution) was used as an example. The ammonia sample used in the experiment was obtained from a standard gas diluted with a gas distribution device, with an ammonia concentration of 45 μmol·mol⁻¹.

[0172] In this example, the measured 1 cm⁻¹ spectrum, the measured 2 cm⁻¹ spectrum, and the enhanced 1 cm⁻¹ spectrum were imported into FTIR spectroscopy automated quantitative analysis software. This quantitative analysis software, independently developed by the laboratory, utilizes a nonlinear least squares quantitative analysis method based on a synthetic background spectrum. The NH⁃ inversion concentration was obtained using the FTIR spectroscopy automated quantitative analysis software.

[0173] Table 2 NH3 concentration inversion results at different resolutions

[0174]

[0175]

[0176] Table 2 shows that the concentration inversion error of NH3 is small in high-resolution spectroscopy. The accuracy of NH3 gas quantitative analysis using 1cm-1 spectrum is higher than that using 2cm-1 spectrum. The error between the inverted concentration using the enhanced 1cm-1 spectrum and the measured 1cm-1 spectrum is only 2.07%, indicating that the spectrum with enhanced resolution using the AR iterative algorithm can improve the accuracy of gas concentration quantitative analysis to a certain extent.

[0177] In summary, based on the current state of research in spectral resolution enhancement, a linear prediction-based Fourier spectral resolution enhancement algorithm was proposed, and its performance and application were investigated. First, the effects of the signal-to-noise ratio, enhancement factor, and starting resolution on the algorithm's prediction accuracy were investigated. The results show that the prediction error increases significantly with decreasing signal-to-noise ratio and starting resolution, and also increases with increasing the required enhancement factor. The linear prediction-based Fourier spectral resolution enhancement algorithm was then applied to measured spectra from an FTIR spectrometer to improve the spectral resolution of NH₃. When the spectral resolution of ammonia was enhanced from 2 cm⁻¹ to 1 cm⁻¹, the spectral error in the characteristic absorption band was reduced to 0.18% compared to the measured 1 cm⁻¹ NH₃ spectrum. The enhanced ammonia spectrum was applied to quantitative gas concentration analysis. Compared to the concentration inversion results from the measured 2 cm⁻¹ spectrum, the inversion accuracy improved by 1.35%, and compared to the measured 1 cm⁻¹ spectrum, the relative error was only 2.07%. Experimental results show that the Fourier spectrum resolution enhancement algorithm based on linear prediction has good applicability in improving spectral resolution and can improve the shortcomings of low-resolution spectrum in quantitative analysis of gas concentration to a certain extent.

[0178] The present invention uses a signal processing method to extrapolate interference information, which is beneficial to improving the resolution of the FTIR spectrometer. In the process of establishing the autoregressive model of the present invention, the forward and backward prediction total least squares method and singular value decomposition (SVD) parameter estimation are used, which plays an important role in reducing false peaks caused by singular value inversion and suppressing noise interference. At the same time, the present invention proposes the principle of minimum prediction error of the interference signal to determine the order of the autoregressive model. The present invention uses the autoregressive model to perform linear prediction: a multi-step prediction method of the interference signal based on a sliding window is proposed to improve the prediction accuracy. The present invention has good applicability in improving spectral resolution and can improve the shortcomings of low-resolution spectra in the quantitative analysis of gas concentration to a certain extent.

[0179] In order to solve the problem in the prior art that both the data matrix and the observation vector may be subject to noise disturbance, the present invention adopts the influence-total least squares method, which takes into account the noise disturbance of the observation vector and the data matrix at the same time, and realizes the suppression of noise disturbance more comprehensively.

[0180] In view of the deficiencies in the prior art that for a given data length N, selecting too low an order will result in a simple AR system with a large fitting error, while selecting too high an order may lead to spurious peaks in the spectrum due to additional poles, the present invention uses a decision function to predict the prediction error between the interferogram and the original interferogram, and accordingly selects an appropriate order for the AR model, which is beneficial to achieving correct fitting and extrapolation.

[0181] The present invention determines the coefficient vector {ck} and the order p of the linear prediction model to achieve the extrapolation of the interference signal, and effectively solves the problem of noise interference. The interference signal can be extrapolated far away without error, thereby enhancing the spectral resolution.

[0182] The present invention establishes a linear prediction model from a short window to a long window, calculates the corresponding value of E, and determines the maximum length of the window that satisfies E < Ep (Ep is the set maximum error value) as the length of the sliding window, avoiding the problems in the prior art such as the decrease in accuracy, the reduction in estimation accuracy, and the longer prediction time caused by too much data in a single prediction window.

[0183] Based on the original observed values, the present invention uses the small-window prediction values as new observed values. These new observed values bring more information, so the unknown information to be predicted at future times gradually decreases. Therefore, by using the information of the new observed values, the present invention can better predict the future sequence values and improve the prediction accuracy.

[0184] The present invention solves the technical problems existing in the prior art, such as the instability of spectral instruments and the limitation of the effect of improving resolution.

[0185] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements for some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. Fourier spectrum resolution enhancement algorithm based on linear prediction, characterized by: The algorithm includes: S1. Input existing interference pattern data; S2, using the forward and backward prediction total least squares method and singular value decomposition (SVD) to estimate parameters, establishing an interference signal autoregressive model based on the existing interferogram data to reduce false peaks generated by singular value inversion and suppress noise interference, defining the interference signal minimum prediction error principle, and processing the autoregressive model order based on the principle, said step S2 includes: S21. Using a forward and backward prediction total least squares method, construct a forward and backward expansion matrix of the interference signal to obtain a forward and backward linear prediction equation, and using an influence total least squares method to solve the forward and backward linear prediction equation to process the observation vector and data matrix noise disturbance of the interference signal, thereby estimating the parameters of the AR model and decomposing the singular values; S22. Adaptively determining the autoregressive model order of the AR model using prediction error logic; S3. Based on a sliding window, linear prediction is performed using the autoregressive model and the interference signal multi-step prediction method to obtain a prediction result of applicable accuracy. Step S2 includes: S31, forming a linear differential equation using the AR model of the interference signal, thereby obtaining an interference output signal matrix, and extrapolating the interference signal based on the matrix; S32, using the minimum prediction error criterion of the interference signal and the resolution requirement data, preselecting an applicable window length T and number of sliding times of the sliding window; S33, within the applicable window length T, using the AR model to approximate the time series of the interference signal, so as to form a sliding window-based multi-step prediction model of the interference signal during the sliding window sliding forward; S34. Using the sliding window-based interference signal multi-step prediction model and a preset sliding window prediction logic, respectively, the AR model order is re-determined and the parameters are re-estimated to obtain an AR iterative model and a new interference pattern.

2. The Fourier spectrum resolution enhancement algorithm based on linear prediction according to claim 1, characterized in that: In step S1, the following logic is used to represent the equal optical path difference sampling interference data: Where {I(n)} is the interference data, W(n) means the noise signal, p refers to the order of the prediction model, fk and ak are the frequency and amplitude of the kth sine wave, respectively.

3. The Fourier spectrum resolution enhancement algorithm based on linear prediction according to claim 1, characterized in that: The step S21 includes: S211, constructing the forward and backward expansion matrix using the forward and backward prediction total least squares method; S212, obtaining a forward and backward linear prediction equation FBLP according to the forward and backward expansion matrix: That is: Ac k ≈b Where * represents complex conjugate, A is the I data matrix, c k is the prediction coefficient vector, b is the observation vector; S213. The noise disturbance of the data matrix and the noise disturbance of the observation vector are respectively recorded as: ΔA, Δb, and the following equation is obtained: (A+ΔA)c k =b+Δb S214. Using the following logic, use the total least squares method to obtain a perturbation matrix E with the minimum norm square, so that the matrix (B+E) is not full rank, where B is the data augmentation matrix and E is the perturbation matrix, and the coefficient vector ck is obtained using the SVD method:

4. The Fourier spectrum resolution enhancement algorithm based on linear prediction according to claim 3, characterized in that: In step S214, where B=[A:b] is the data augmentation matrix, E=[ΔA:Δb] is the perturbation matrix, To obtain the following formula: (B+E)z=0.

5. The Fourier spectrum resolution enhancement algorithm based on linear prediction according to claim 3, characterized in that: The step S22 includes: S221, using the following judgment function to predict the prediction error between the interference pattern and the original interference pattern: Where N refers to the number of interference signal data points; S222. Select the order that minimizes the prediction error E as the order of the autoregressive model.

6. The Fourier spectrum resolution enhancement algorithm based on linear prediction according to claim 1, characterized in that: The step S31 includes: S311. Form a linear difference equation by using the AR model of the interference signal: where {ck} is the parameter vector of the AR model; S312. Represent the signal i(n) output by the current interference pattern as the sum of p past output interference signals I with different weights to obtain the interference output signal matrix: i=c k I where S313. Calculate the coefficient vector {ck} according to the interference output signal matrix, determine the order p of the linear prediction model, and extrapolate the interference signal based on this.

7. The Fourier spectrum resolution enhancement algorithm based on linear prediction according to claim 1, characterized in that: The step S32 includes: S321. Determine the sliding window length T by using the minimum prediction error criterion of the interference signal, establish linear prediction models from short windows to long windows respectively, and calculate the corresponding prediction error E; S322. Obtain the maximum window length that makes the prediction error E satisfy E < Ep, and use it as the applicable window length T.

8. The Fourier spectrum resolution enhancement algorithm based on linear prediction according to claim 6, characterized in that: The step S34 includes: S341. Use the prediction value of the preset small window as the new observation value, and use the information of the new observation value to predict the future sequence value; S342, assuming that the sliding window length at time t is T, use the following prediction logic to re-determine the order of the AR model and re-estimate the parameters at time t+T, t+2T, ..., respectively, and calculate i t+2T ,i t+3T ,…, thus forming the AR iterative model: Among them, c k Represents the parameter vector of the AR model.

9. A Fourier spectrum resolution enhancement system based on linear prediction, characterized in that: The system includes: An input data module for inputting existing interference pattern data; An autoregressive model establishment module for the interference signal, which is used to estimate parameters by using the total least squares method of forward and backward prediction and singular value decomposition SVD, establish an autoregressive model for the interference signal according to the existing interference pattern data, reduce the false peaks generated by the inversion of singular values and suppress noise interference, define the minimum prediction error principle of the interference signal, and process to obtain the order of the autoregressive model. The autoregressive model establishment module for the interference signal is connected to the input data module. The autoregressive model establishment module for the interference signal includes: A parameter estimation and singular value decomposition module for using the total least squares method of forward and backward prediction to construct the forward and backward extended order matrix of the interference signal to obtain the forward and backward linear prediction equations, solve the forward and backward linear prediction equations by using the total least squares method, process the observation vector and data matrix noise perturbation of the interference signal, and estimate the parameters of the AR model and decompose the singular values; An order determination module for adaptively determining the order of the autoregressive model of the AR model by using the prediction error logic; A linear prediction module for performing linear prediction based on a sliding window by using the interference signal multi-step prediction method with the autoregressive model to obtain a prediction result with applicable accuracy. The linear prediction module is connected to the autoregressive model establishment module for the interference signal. The linear prediction module includes: An interference signal extrapolation module for forming a linear difference equation by using the AR model of the interference signal to obtain an interference output signal matrix and extrapolate the interference signal based on this; A sliding window parameter preselection module for preselecting the applicable window length T and the number of sliding times of the sliding window by using the minimum prediction error criterion of the interference signal and the resolution requirement. The sliding window parameter preselection module is connected to the interference signal extrapolation module; a multi-step prediction model acquisition module, configured to use the AR model to approximate the time series of the interference signal within the applicable window length T, so as to form a sliding window-based multi-step prediction model of the interference signal as the sliding window slides forward, the multi-step prediction model acquisition module being connected to the sliding window parameter preselection module; An AR iterative model prediction module is used to use the sliding window-based interference signal multi-step prediction model and the preset sliding window prediction logic to respectively perform order redetermination and parameter re-estimation operations of the AR model, thereby obtaining an AR iterative model and a new interference graph. The AR iterative model prediction module is connected to the multi-step prediction model acquisition module.

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