Method and apparatus for estimating ar model parameters based on forward-backward linear prediction

Abstract

The application discloses an AR model parameter estimation method and device based on forward and backward linear prediction, and the method comprises the following steps: obtaining an interference signal of an infrared spectrometer and converting the interference signal into a matrix form, so as to complete construction of an AR model; a forward and backward expansion matrix of the interference signal is constructed by using a forward and backward prediction total least square method, and a forward and backward linear prediction equation is formed based on the forward and backward expansion matrix; the forward and backward expansion matrix is converted into a homogeneous linear equation by considering the noise disturbance influence of an observation vector and a data matrix, and a total least square problem of the homogeneous linear equation is constructed; and a singular value decomposition method is used to solve the solution of the total least square problem, so as to obtain parameter coefficients of the AR model. The application has the advantages that the problem of spectrum line separation of a restored spectrum is solved, the influence of noise on parameter estimation is reduced, and the accuracy of the restored spectrum is improved.

Description

Technical Field

[0001] This invention relates to the field of Fourier transform infrared spectral resolution enhancement technology, and more specifically to an AR model parameter estimation method and apparatus based on forward and backward linear prediction. Background Technology

[0002] Fourier transform infrared spectroscopy has been widely applied in many research fields, including petroleum industry, biomedicine, aerospace, and environmental science. However, the spectral resolution of infrared spectrometers is limited by the finite displacement of the spectrometer scanning system. To address the problem of low spectral resolution, researchers have proposed many methods to enhance it. Among these, digital signal processing techniques, such as linear prediction techniques, are important means of enhancing spectral resolution.

[0003] Linear prediction techniques improve spectral resolution by modeling the interferometric signal and using an autoregressive (AR) model of the interferometric signal to extrapolate the interferogram to a longer OPD (optical path difference). In linear prediction-based spectral resolution enhancement techniques, solving for the AR model parameters of the interferometric signal is a crucial prerequisite for interferogram extrapolation and directly affects the quality of the reconstructed spectrum. Currently, solving for the AR model parameters is equivalent to solving a system of linear equations, which can be estimated using different methods, such as the Burg method, the modified covariance method, and the YW method. The literature "IEEE, 2018. Dodds DE, Fretz T. Parametric Analysis of Frequency Domain Reflectometry Measurements [C] / / Conference on Electrical & Computer Engineering. IEEE, 2007." indicates that while the Burg method can achieve good spectral resolution enhancement, for interferometric signals with a certain signal-to-noise ratio, improper order selection may lead to spectral line separation after spectral resolution enhancement. The literature “Gao Xiaofeng, Xiang Libin. Comparison of Three Methods to Solve AR Model Parameters of Line Shape Optimized Maximum Entropy Spectral Estimation[J]. Acta Photonica Sinica, 2007, 36(3): 481-486.” shows that for high signal-to-noise ratio interference signals, the modified covariance method is better than the Burg method for solving the autoregressive model coefficients. However, this method is greatly affected by the noise level, so the accuracy of the restored spectrum is low. Summary of the Invention

[0004] The technical problem to be solved by this invention is that the parameter solving method of the existing AR model may have the problem of spectral line separation after the spectral resolution is enhanced. In addition, it is greatly affected by the noise level and the accuracy of the restored spectrum is low.

[0005] This invention solves the above-mentioned technical problems through the following technical means: an AR model parameter estimation method based on forward and backward linear prediction, the method comprising:

[0006] Step 1: Acquire the interference signal from the infrared spectrometer and convert it into matrix form to complete the construction of the AR model;

[0007] Step 2: Construct the forward and backward extended order matrix of the interference signal using the forward and backward prediction total least squares method, and form the forward and backward linear prediction equation based on the forward and backward extended order matrix;

[0008] Step 3: Consider the noise disturbance effect on the observation vector and data matrix, transform the forward and backward extended order matrix into a homogeneous linear equation, and construct the overall least squares problem of the homogeneous linear equation.

[0009] Step 4: Solve the overall least squares problem using singular value decomposition to obtain the parameter coefficients of the AR model.

[0010] Beneficial effects: This invention utilizes the idea of ​​linear prediction between preceding and following terms to construct a forward-and-following extended matrix. Since the coefficients of the forward-and-following linear prediction method satisfy the conjugate constraint condition, the problem of spectral line separation in the restored spectrum will not occur in essence. Then, the parameters of the extended matrix are solved based on the overall least squares method. This method considers the noise disturbance of the observation vector and the data matrix in the process of solving the parameters, effectively reducing the impact of noise on parameter estimation and improving the accuracy of the restored spectrum.

[0011] Further, step one includes:

[0012] In an infrared spectrometer, equal optical path difference sampling interferometric data, also known as the AR model, is represented as:

[0013]

[0014] Where I(n) represents the interferometric data, f k and a k Here, represents the frequency and amplitude of the k-th sine wave, n represents the sampling point of the n-th interference signal, W(n) represents the noise signal, and p refers to the order of the prediction model;

[0015] When noise is ignored, the AR model of the interferometric signal forms a linear difference equation.

[0016]

[0017] The matrix form of this linear difference equation is as follows:

[0018]

[0019] Among them, c k Let i(p) represent the AR model parameters, i(p) represent the p-th element in the observation vector i, and I(p) represent the p-th element in the data matrix I.

[0020] Furthermore, step two includes:

[0021] Using the forward and backward prediction total least squares method to construct the forward and backward extended order matrix of each matrix in the AR model of the interferometric signal, the corresponding forward and backward linear prediction equation is:

[0022]

[0023] Right now

[0024] Ac≈b (5)

[0025] Where * denotes complex conjugate, A is equivalent to data matrix I, c is the prediction coefficient vector, and b is equivalent to observation vector i.

[0026] Furthermore, step three includes:

[0027] Considering the noise disturbances in the data matrix I and the observation vector i, denoted as △A and △b respectively, equation (5) is rewritten as:

[0028] (A+ΔA)c=b+Δb (6)

[0029] Formula (6) can be transformed into:

[0030]

[0031] Let B = [A:b] be the data augmentation matrix, and let E = [ΔA:Δb] be the perturbation augmentation matrix. Then there is

[0032] (B+E)z=0 (8)

[0033] The total least squares problem of the above homogeneous linear equation is to find a perturbation augmented matrix E with the minimum norm square such that (B+E) is non-full rank.

[0034] Furthermore, step four includes:

[0035] If we use SVD(M) to denote the singular value decomposition of matrix M, and represent the result as a singular value diagonal matrix, then the singular value decompositions of matrix A and the data augmented matrix B are respectively

[0036] SVD(A)=diag(σ A1 ,...,σ An ),σ A1 ≥...≥σ An (9)

[0037] SVD(B)=diag(σ1,...,σ n+1 ),σ1≥...≥σ n+1 (10)

[0038] Where diag() represents a diagonal matrix, σ An Let σ represent the eigenvalues ​​of matrix A. n+1 Represents the eigenvalues ​​of the augmented matrix B;

[0039] According to the properties of singular values

[0040]

[0041] Among them, ||E|| F Let r(B+E) be the Fronenius norm of E, and r(B+E) be the rank of the matrix (B+E).

[0042] If let

[0043]

[0044] Where ν is related to the eigenvalue σ n+1 2 Any unit vector in the corresponding feature subspace Sc;

[0045] but

[0046] in, yes The Fronenius norm;

[0047] From formulas (8), (11), (12), and (13), we can obtain

[0048] (B-Bvv T z = B(I - vv) T z = 0 (14)

[0049] Solving

[0050] z∈null(B)=Sc (15)

[0051] Where null(B) is the general solution of equation (14);

[0052] If v is represented as y∈R n If α∈R and α≠0, then the general solution obtained by the total least squares method is:

[0053]

[0054] That is, to find the prediction coefficient vector c, which is also the parameter coefficient of the AR model.

[0055] This invention also provides an AR model parameter estimation device based on forward and backward linear prediction, the device comprising:

[0056] The model building module is used to acquire the interference signal from the infrared spectrometer and convert it into matrix form, thereby completing the construction of the AR model;

[0057] The extended order module is used to construct the extended order matrix of the interference signal using the forward and backward prediction total least squares method, and to form the forward and backward linear prediction equation based on the extended order matrix.

[0058] The least squares module is used to consider the noise perturbation effect on the observation vector and data matrix, transform the forward and backward extended matrices into homogeneous linear equations, and construct the overall least squares problem of the homogeneous linear equations.

[0059] The solver module is used to solve the overall least squares problem using the singular value decomposition method and obtain the parameter coefficients of the AR model.

[0060] Furthermore, the model building module is used for:

[0061] In an infrared spectrometer, equal optical path difference sampling interferometric data, also known as the AR model, is represented as:

[0062]

[0063] Where I(n) represents the interferometric data, f k and a k Here, represents the frequency and amplitude of the k-th sine wave, n represents the sampling point of the n-th interference signal, W(n) represents the noise signal, and p refers to the order of the prediction model;

[0064] When noise is ignored, the AR model of the interferometric signal forms a linear difference equation.

[0065]

[0066] The matrix form of this linear difference equation is as follows:

[0067]

[0068] Among them, ck Let i(p) represent the AR model parameters, i(p) represent the p-th element in the observation vector i, and I(p) represent the p-th element in the data matrix I.

[0069] Furthermore, the extended-order module is used for:

[0070] Using the forward and backward prediction total least squares method to construct the forward and backward extended order matrix of each matrix in the AR model of the interferometric signal, the corresponding forward and backward linear prediction equation is:

[0071]

[0072] Right now

[0073] Ac≈b (5)

[0074] Where * denotes complex conjugate, A is equivalent to data matrix I, c is the prediction coefficient vector, and b is equivalent to observation vector i.

[0075] Furthermore, the least squares module is used for:

[0076] Considering the noise disturbances in the data matrix I and the observation vector i, denoted as △A and △b respectively, equation (5) is rewritten as:

[0077] (A+ΔA)c=b+Δb (6)

[0078] Formula (6) can be transformed into:

[0079]

[0080] Let B = [A:b] be the data augmentation matrix, and let E = [ΔA:Δb] be the perturbation augmentation matrix. Then there is

[0081] (B+E)z=0 (8)

[0082] The total least squares problem of the above homogeneous linear equation is to find a perturbation augmented matrix E with the minimum norm square such that (B+E) is non-full rank.

[0083] Furthermore, the solution module is used for:

[0084] If we use SVD(M) to denote the singular value decomposition of matrix M, and represent the result as a singular value diagonal matrix, then the singular value decompositions of matrix A and the data augmented matrix B are respectively

[0085] SVD(A)=diag(σ A1 ,...,σ An ),σ A1 ≥...≥σ An (9)

[0086] SVD(B)=diag(σ1,...,σ n+1 ),σ1≥...≥σ n+1 (10)

[0087] Where diag() represents a diagonal matrix, σ An Let σ represent the eigenvalues ​​of matrix A. n+1 Represents the eigenvalues ​​of the augmented matrix B;

[0088] According to the properties of singular values

[0089]

[0090] Among them, ||E|| F Let r(B+E) be the Fronenius norm of E, and r(B+E) be the rank of the matrix (B+E).

[0091] If let

[0092]

[0093] Where ν is related to the eigenvalue σ n+1 2 Any unit vector in the corresponding feature subspace Sc;

[0094] but

[0095] in, yes The Fronenius norm;

[0096] From formulas (8), (11), (12), and (13), we can obtain

[0097] (B-Bvv T z = B(I - vv) T z = 0 (14)

[0098] Solving

[0099] z∈null(B)=Sc (15)

[0100] Where null(B) is the general solution of equation (14);

[0101] If v is represented as y∈R n If α∈R and α≠0, then the general solution obtained by the total least squares method is:

[0102]

[0103] That is, to find the prediction coefficient vector c, which is also the parameter coefficient of the AR model.

[0104] The advantages of this invention are as follows: This invention utilizes the idea of ​​linear prediction between preceding and following terms to construct a forward-and-backward extended matrix. Since the coefficients of the forward-and-backward linear prediction method satisfy the conjugate constraint condition, the problem of spectral line separation in the restored spectrum will not occur in essence. Then, the parameters of the extended matrix are solved based on the overall least squares method. This method considers the noise disturbance of the observation vector and the data matrix in the process of solving the parameters, effectively reducing the impact of noise on parameter estimation and improving the accuracy of the restored spectrum. Attached Figure Description

[0105] Figure 1 This is a flowchart of the AR model parameter estimation method based on forward and backward linear prediction disclosed in Embodiment 1 of the present invention;

[0106] Figure 2 This is a schematic diagram of an ideal spectrum with controllable noise conditions and no wireless characteristics in the AR model parameter estimation method based on forward and backward linear prediction disclosed in Embodiment 1 of the present invention.

[0107] Figure 3 This is a schematic diagram of interference signals with different signal-to-noise ratios (SNR = 100, 1000, 10000, 100000) in the AR model parameter estimation method based on forward and backward linear prediction disclosed in Embodiment 1 of the present invention.

[0108] Figure 4 This is a spectral restoration effect diagram at the optimal order in the AR model parameter estimation method based on forward and backward linear prediction disclosed in Embodiment 1 of the present invention. Figure 4 (a) shows the spectral resolution enhancement effect of FB-TLS, Burg, and LS parameter estimation methods at low signal-to-noise ratio. Figure 4 (b), (c), and (d) are the spectral resolution enhancement effects of the FB-TLS method (SNR=100, p=280), the Burg method (SNR=100, p=241), and the LS method (SNR=100, p=212), respectively.

[0109] Figure 5 This is a diagram showing the spectral resolution enhancement effect at the optimal order (p=6, p=13, p=24) in the AR model parameter estimation method based on forward and backward linear prediction disclosed in Embodiment 1 of the present invention.

[0110] Figure 6 This is a diagram illustrating the spectral enhancement effects of high-order (p=22, p=17, p=27) AR model parameter estimation methods based on forward and backward linear prediction as disclosed in Embodiment 1 of the present invention. Figure 6 (a) shows the spectral enhancement effect of the FB-TLS method (p=22); Figure 6(b) shows the spectral enhancement effect of the Burg method (p=17); Figure 6 (c) shows the spectral enhancement effect of the LS method (p=27);

[0111] Figure 7 This is a schematic diagram of the experimental equipment used in the AR model parameter estimation method based on forward and backward linear prediction disclosed in Embodiment 1 of the present invention.

[0112] Figure 8 In Embodiment 1 of the present invention, the AR model parameter estimation method based on forward and backward linear prediction is used to enhance the 1cm post-processing. -1 The spectrum and the measured 2cm -1 Spectrum, measured at 1cm -1 A schematic diagram comparing the spectra. Detailed Implementation

[0113] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of 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 some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0114] Example 1

[0115] This invention provides a parameter estimation method for AR models based on forward and backward linear prediction, such as... Figure 1 The diagram illustrates the AR parameter estimation process based on FB-TLS (Forward and Backward Linear Prediction Total Least Squares) of this invention. The method includes the following steps:

[0116] S1. Acquire the interference signal from the infrared spectrometer and convert it into matrix form; the specific process is as follows:

[0117] In an infrared spectrometer, equal optical path difference sampling interferometric data can be represented as:

[0118]

[0119] Where I(n) represents the interferometric data, f k and a k denoted by , where is the frequency and amplitude of the k-th sine wave, n is the sampling point of the n-th interference signal, W(n) represents the noise signal, and p refers to the order of the prediction model.

[0120] When noise is ignored, the AR model of the interferometric signal forms a linear difference equation.

[0121]

[0122] The matrix form of this linear difference equation is as follows:

[0123]

[0124] Among them, c k To represent the AR model parameters, simply obtain the parameter vector {c} k Furthermore, by determining the order p of the AR model, the extrapolation of the interference signal can be achieved. If the noise interference problem can be solved, the interference signal can be extrapolated without error, thereby improving the spectral resolution.

[0125] S2. Construct the forward and backward extended order matrix of the interferometric signal using the forward and backward prediction total least squares method, and form the forward and backward linear prediction equation based on the forward and backward extended order matrix; the specific process is as follows:

[0126] Using the forward and backward prediction total least squares method to construct the forward and backward extended order matrix (L>p) of each matrix in the AR model of the interferometric signal, the corresponding forward and backward linear prediction (FBLP) equation is:

[0127]

[0128] Right now

[0129] Ac≈b (5)

[0130] Where * denotes complex conjugate, A is equivalent to data matrix I, c is the prediction coefficient vector, and b is equivalent to observation vector i.

[0131] S3. Considering the noise disturbance effect on the observation vector and data matrix, the forward and backward extended matrices are transformed into homogeneous linear equations, and the overall least squares problem of the homogeneous linear equations is constructed; the specific process is as follows:

[0132] There are two main methods to solve the equation (5) above: the first method is the least squares (LS) method, which assumes that there is a perturbation term △b in the observation vector b; the second method assumes that there is a perturbation term △A 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 perturbation effects of the observation vector and the data matrix respectively, which is insufficient. Usually, both the data matrix I and the observation vector i may have noise perturbations, denoted as △A and △b respectively, so equation (5) can be written as:

[0133] (A+ΔA)c=b+Δb (6)

[0134] This method is a total least squares method because it considers the effects of noise disturbances in both the observation vector and the data matrix.

[0135] Formula (6) can be transformed into:

[0136]

[0137] Let B = [A:b] be the data augmentation matrix, and let E = [ΔA:Δb] be the perturbation augmentation matrix. Then there is

[0138] (B+E)z=0 (8)

[0139] The total least squares problem for the homogeneous linear equations described above is to find a perturbation matrix E with the smallest norm square such that (B+E) is not full rank. The solution to the total least squares problem, namely the coefficient vector c, can be obtained by applying singular value decomposition (SVD).

[0140] S4. Solve the overall least squares problem using singular value decomposition, which yields the parameter coefficients of the AR model; the specific process is as follows:

[0141] If we use SVD(M) to denote the singular value decomposition of matrix M, and represent the result as a singular value diagonal matrix, then the singular value decompositions of matrix A and the data augmented matrix B are respectively

[0142] SVD(A)=diag(σ A1 ,...,σ An ),σ A1 ≥...≥σ An (9)

[0143] SVD(B)=diag(σ1,...,σ n+1 ),σ1≥...≥σ n+1 (10)

[0144] Where diag() represents a diagonal matrix, σ An Let σ represent the eigenvalues ​​of matrix A. n+1 Represents the eigenvalues ​​of the augmented matrix B;

[0145] According to the properties of singular values

[0146]

[0147] Among them, ||E|| F r is the Fronenius norm of E, and r(B+E) is the rank of the matrix (B+E).

[0148] If let

[0149]

[0150] Where ν is related to the eigenvalue σ n+1 2Any unit vector in the corresponding feature subspace Sc;

[0151] but

[0152]

[0153] in, yes The Fronenius norm;

[0154] From formulas (8), (11), (12), and (13), we can obtain

[0155] (B-Bvv T z = B(I - vv) T z = 0 (14)

[0156] Solving

[0157] z∈null(B)=Sc (15)

[0158] Among them, null(B) is the general solution of equation (14).

[0159] If v can be expressed as y∈R n If α∈R and α≠0, then the general solution obtained by the total least squares method is:

[0160]

[0161] In other words, find the parameter vector {c}.

[0162] The above methods and steps have explained how to solve for the parameters of the AR model. The following simulations and experiments will verify the effectiveness of the present invention.

[0163] 1. Simulation verification of the effectiveness of the method of the present invention

[0164] In reality, spectral lines have specific line shapes and linewidths, and adjacent lines may overlap. To compare the effectiveness of different parameter estimation methods in spectral resolution enhancement techniques, the simulation experiments used ideal spectra with controllable noise conditions and no line shape, such as... Figure 2 As shown.

[0165] When an ideal spectral line is interfered with by an interferometer (i.e., a discrete Fourier transform is performed), an interference signal is obtained, which can be expressed as:

[0166]

[0167] In the formula, I n Indicates the intensity of the interference signal, x n Indicates the sampling position, ω nThe noise signal is represented by ν1 = 400cm⁻¹, ν2 = 550cm⁻¹, ν3 = 400cm⁻¹, B1 = B3 = 2, and B2 = 1.

[0168] In the simulation experiment, the sampling interval of the interference signal was set to Δx = 0.00003125 cm, and the number of sampling data points was N = 800. Therefore, the spectral resolution of the sampled spectral signal was 40 cm⁻¹. -1 By controlling the noise signal magnitude, interference signals with different signal-to-noise ratios (SNR = 100, 1000, 10000, 100000) are generated, such as... Figure 3 As shown.

[0169] 1.1 Low signal-to-noise ratio (SNR=100)

[0170] The orders p=280, p=241, and p=212 are the optimal orders for the FB-TLS method, Burg method, and LS method when SNR=100, respectively (the order determination reference is "YANG Wu. Study on Modal Parameter Estimation and On-Line Identification Technology for Linear Time-Varying Structures in Time-domain[D]. Beijing Institute of Technology, 2015."). The spectral restoration effect under the optimal order is as follows: Figure 4 As shown in the figure, Orig represents the original spectrum. Figure 4 (a) shows the spectral resolution enhancement effect of FB-TLS, Burg, and LS parameter estimation methods at low signal-to-noise ratio. Figure 4 (b), (c), and (d) show the spectral resolution enhancement effects of the FB-TLS method (SNR=100, p=280), the Burg method (SNR=100, p=241), and the LS method (SNR=100, p=212), respectively. Figure 4 As can be seen, the FB-TLS method and the Burg method have good noise resistance and the restoration results are basically similar. The LS method restored the spectrum with relative amplitude distortion and wavenumber position shift, and was greatly affected by noise.

[0171] 1.2 High signal-to-noise ratio (SNR = 100000)

[0172] The orders p=6, p=13, and p=24 are the optimal orders for the FB-TLS method, Burg method, and LS method at SNR=100000, respectively (the order determination reference is "YANG Wu. Study on Modal Parameter Estimation and On-Line Identification Technology for Linear Time-Varying Structures in Time-domain[D]. Beijing Institute of Technology, 2015."). The spectral resolution enhancement effect under the optimal order is as follows: Figure 5 As shown. From Figure 5 As can be seen, under high signal-to-noise ratio background, the FB-TLS method, Burg method, and LS method have basically the same spectral resolution enhancement effect.

[0173] 1.3 Spectral line separation and spectral pseudo-peak problems

[0174] When the signal-to-noise ratio (SNR) is fixed, the resolution of spectral estimation using an autoregressive model can be improved by increasing the model order. However, this can easily lead to spectral line splitting and the generation of spurious peaks. In spectral resolution enhancement techniques, spectral line splitting and spurious peak problems typically occur when the SNR is high (SNR = 100,000) and the order is determined to be too high.

[0175] Orders p=22, p=17, and p=27 were used as the criteria for excessively high orders in the FB-TLS method, Burg method, and LS method at SNR=100000, respectively. The spectral resolution enhancement effect is as follows: Figure 6 As shown, Figure 6 (a) is a spectral enhancement effect diagram of FB-TLS (p=22); Figure 6 (b) shows the spectral enhancement effect of the Burg method (p=17); Figure 6 (c) shows the spectral enhancement effect of the LS method (p=27). From Figure 6 As can be seen, under high signal-to-noise ratio (SNR) conditions, the Burg method and the LS method may produce spurious peaks and cause spectral line separation when the order determination is too high. However, the FB-TLS method applies a conjugate constraint condition to the forward linear prediction (FLP) coefficients and the backward linear prediction (BLP) coefficients. Since the coefficients satisfy the conjugate constraint condition, the problem of spectral line separation and spurious peaks under high SNR and high order determination conditions is solved.

[0176] 2. The method of this invention is applied to actual spectral resolution enhancement and compared with the measured spectrum.

[0177] 2.1 Experimental Equipment

[0178] The simulations above demonstrate that FB-TLS performs well in low signal-to-noise ratio (SNR), high SNR, and in mitigating spectral pseudo-peak effects. The proposed FB-TLS-based linear prediction parameter estimation method is then applied to practical spectral resolution enhancement, and the results are compared with measured spectra.

[0179] To demonstrate the practicality of FB-TLS parameter estimation in linear prediction-based spectral resolution enhancement techniques, a portable Fourier transform infrared spectrometer was used for verification. This portable instrument, designed around a Michelson interferometer, can perform quantitative analysis of atmospheric pollutants. Its core equipment (interferometer system) consists of a first beam splitter 2, a moving mirror 3, and a fixed mirror 5. Its working principle is as follows: Figure 7 As shown in the figure, 1 is the infrared light source, i.e., the infrared radiation signal; 2 is the first reflector; 3 is the moving mirror; 4 is the beam splitter; 5 is the fixed mirror; 6 is the second reflector; 7 is the gas cell; and 8 is the detector.

[0180] Infrared light emitted from infrared source 1 passes through first reflector 2 and reaches beam splitter 4. Ideally, 50% of the light is reflected back to beam splitter 4 after passing through moving mirror 3, while the other 50% passes through beam splitter 4, reaches fixed mirror 5, and is reflected back to beam splitter 4, thus creating an optical path difference and causing interference. When moving mirror 3 moves at a constant speed, the interference signal intensity exhibits a cosine variation, forming an interference light signal. This interference light signal passes through second reflector 6 and reaches gas cell 7, where it reacts with the sample to be tested. An interference light signal carrying sample information is emitted from gas cell 7, and the interference light signal is acquired by detector 8 and displayed as an interferogram in the computer.

[0181] The main parameters of the experimental instrument are as follows: the spectral range is 500-5000 cm-1; the scanning speed is 0.2875 cm / s; detector 8 is mercury cadmium telluride (MCT), and the MCT uses Stirling cooling.

[0182] Figure 7 The instrument operation process is as follows: the gas to be tested passes through the sampling head and filter to the multiple reflection absorption sample cell. At the same time, the infrared light source 1 is collimated by the optical system and introduced into the Fourier transform infrared spectrometer. After being interfered and modulated by the spectrometer, it is introduced into the multiple reflection absorption sample cell. The infrared radiation signal extracted from the sample cell after interference modulation carries the absorption information of the sample to be tested. By focusing the signal onto the detector 8, the interferogram can be obtained.

[0183] Set the maximum optical path difference of the instrument to 0.5 cm (i.e., set the spectral resolution to 2 cm). -1 The NH3 gas to be measured was introduced into the gas cell, and interference data was acquired. Then, the maximum optical path difference of the instrument was set to 1 cm (i.e., the spectral resolution was set to 1 cm). -1The experiment was repeated. To reduce random noise interference, the average value of eight consecutive sets of interference data was used as an interferogram.

[0184] 2.2 Experimental Analysis

[0185] First, for 2cm -1 The interference signal was used to perform linear prediction modeling, and the FB-TLS method was applied for parameter estimation. Then, after obtaining the linear model, the interference signal with the forward and backward outward inferences was used to estimate the maximum optical path difference to be 1 cm. -1 (i.e., spectral resolution set to 1cm) -1 Finally, the FFT was applied to extrapolate the interferometric signal to obtain a resolution-enhanced spectrum with a resolution of 1 cm⁻¹. -1 1cm after enhancement -1 Spectrum, measured at 2cm -1 Spectrum, measured at 1cm -1 Compare the spectra, such as Figure 8 As shown.

[0186] By calculation, the linear prediction spectral resolution enhanced to 1 cm⁻¹ using the FB-TLS method for parameter estimation was obtained. -1 The NH3 spectrum and the measured 1cm -1 The spectral error of the NH3 spectrum in the characteristic absorption band is 0.21%. Experimental results show that the FB-TLS parameter estimation method has good applicability in linear prediction spectral resolution enhancement algorithms.

[0187] In the above technical solutions, parameter estimation is a crucial step in linear prediction-based spectral resolution enhancement techniques. This invention proposes a parameter estimation method based on FB-TLS. Simulation results show that, compared with the Burg method and LS method, the FB-TLS-based parameter estimation method can effectively solve the noise interference and potential spectral pseudo-peak problems under low signal-to-noise ratio conditions. Experimental results demonstrate that the FB-TLS parameter estimation method has good applicability in linear prediction spectral resolution enhancement algorithms, achieving enhanced 1cm... -1 The NH3 spectrum and the measured 1cm -1 The spectral error of the NH3 spectrum in the characteristic absorption band is only 0.21%.

[0188] Example 2

[0189] Based on Embodiment 1, Embodiment 2 of the present invention also provides an AR model parameter estimation device based on forward and backward linear prediction, the device comprising:

[0190] The model building module is used to acquire the interference signal from the infrared spectrometer and convert it into matrix form, thereby completing the construction of the AR model;

[0191] The extended order module is used to construct the extended order matrix of the interference signal using the forward and backward prediction total least squares method, and to form the forward and backward linear prediction equation based on the extended order matrix.

[0192] The least squares module is used to consider the noise perturbation effect on the observation vector and data matrix, transform the forward and backward extended matrices into homogeneous linear equations, and construct the overall least squares problem of the homogeneous linear equations.

[0193] The solver module is used to solve the overall least squares problem using the singular value decomposition method and obtain the parameter coefficients of the AR model.

[0194] Specifically, the model building module is used for:

[0195] In an infrared spectrometer, equal optical path difference sampling interferometric data, also known as the AR model, is represented as:

[0196]

[0197] Where I(n) represents the interferometric data, f k and a k Here, represents the frequency and amplitude of the k-th sine wave, n represents the sampling point of the n-th interference signal, W(n) represents the noise signal, and p refers to the order of the prediction model;

[0198] When noise is ignored, the AR model of the interferometric signal forms a linear difference equation.

[0199]

[0200] The matrix form of this linear difference equation is as follows:

[0201]

[0202] Among them, c k Let i(p) represent the AR model parameters, i(p) represent the p-th element in the observation vector i, and I(p) represent the p-th element in the data matrix I.

[0203] More specifically, the extended-order module is used for:

[0204] Using the forward and backward prediction total least squares method to construct the forward and backward extended order matrix of each matrix in the AR model of the interferometric signal, the corresponding forward and backward linear prediction equation is:

[0205]

[0206] Right now

[0207] Ac≈b (5)

[0208] Where * denotes complex conjugate, A is equivalent to data matrix I, c is the prediction coefficient vector, and b is equivalent to observation vector i.

[0209] More specifically, the least squares module is used for:

[0210] Considering the noise disturbances in the data matrix I and the observation vector i, denoted as △A and △b respectively, equation (5) is rewritten as:

[0211] (A+ΔA)c=b+Δb (6)

[0212] Formula (6) can be transformed into:

[0213]

[0214] Let B = [A:b] be the data augmentation matrix, and let E = [ΔA:Δb] be the perturbation augmentation matrix. Then there is

[0215] (B+E)z=0 (8)

[0216] The total least squares problem of the above homogeneous linear equation is to find a perturbation augmented matrix E with the minimum norm square such that (B+E) is non-full rank.

[0217] More specifically, the solver module is used for:

[0218] If we use SVD(M) to denote the singular value decomposition of matrix M, and represent the result as a singular value diagonal matrix, then the singular value decompositions of matrix A and the data augmented matrix B are respectively

[0219] SVD(A)=diag(σ A1 ,…,σ An ),σ A1 ≥…≥σ An (9)

[0220] SVD(B)=diag(σ1,…,σ n+1 ), σ1≥…≥σ n+1 (10)

[0221] Where diag() represents a diagonal matrix, σ An Let σ represent the eigenvalues ​​of matrix A. n+1 Represents the eigenvalues ​​of the augmented matrix B;

[0222] According to the properties of singular values

[0223]

[0224] Among them, ||E|| FLet r(B+E) be the Fronenius norm of E, and r(B+E) be the rank of the matrix (B+E).

[0225] If let

[0226]

[0227] Where ν is related to the eigenvalue σ n+1 2 Any unit vector in the corresponding feature subspace Sc;

[0228] but

[0229] in, yes The Fronenius norm;

[0230] From formulas (8), (11), (12), and (13), we can obtain

[0231] (B-Bvv T z = B(I - vv) T z = 0 (14)

[0232] Solving

[0233] z∈null(B)=Sc (15)

[0234] Where null(B) is the general solution of equation (14);

[0235] If v is represented as y∈R n If α∈R and α≠0, then the general solution obtained by the total least squares method is:

[0236]

[0237] That is, to find the prediction coefficient vector c, which is also the parameter coefficient of the AR model.

[0238] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A parameter estimation method for AR models based on forward and backward linear prediction, characterized in that, The method includes: Step 1: Acquire the interference signal from the infrared spectrometer and convert it into matrix form to complete the construction of the AR model; Step 2: Construct the forward and backward extended order matrix of the interference signal using the forward and backward prediction total least squares method, and form the forward and backward linear prediction equation based on the forward and backward extended order matrix; Step 3: Consider the noise disturbance effect on the observation vector and data matrix, transform the forward and backward extended order matrix into a homogeneous linear equation, and construct the overall least squares problem of the homogeneous linear equation. Step 4: Solve the overall least squares problem using singular value decomposition to obtain the parameter coefficients of the AR model.

2. The AR model parameter estimation method based on forward and backward linear prediction according to claim 1, characterized in that, Step one includes: In an infrared spectrometer, equal optical path difference sampling interferometric data, also known as the AR model, is represented as: Where I(n) represents the interferometric data, f k and a k Here, represents the frequency and amplitude of the k-th sine wave, n represents the sampling point of the n-th interference signal, W(n) represents the noise signal, and p refers to the order of the prediction model; When noise is ignored, the AR model of the interferometric signal forms a linear difference equation. The matrix form of this linear difference equation is as follows: Among them, c k Let i(p) represent the AR model parameters, i(p) represent the p-th element in the observation vector i, and I(p) represent the p-th element in the data matrix I.

3. The AR model parameter estimation method based on forward and backward linear prediction according to claim 2, characterized in that, Step two includes: Using the forward and backward prediction total least squares method to construct the forward and backward extended order matrix of each matrix in the AR model of the interferometric signal, the corresponding forward and backward linear prediction equation is: Right now Ac≈b (5) Where * denotes complex conjugate, A is equivalent to data matrix I, c is the prediction coefficient vector, and b is equivalent to observation vector i.

4. The AR model parameter estimation method based on forward and backward linear prediction according to claim 3, characterized in that, Step three includes: Considering the noise disturbances in the data matrix I and the observation vector i, denoted as △A and △b respectively, equation (5) is rewritten as: (A+ΔA)c=b+Δb (6) Formula (6) can be transformed into: Let B = [A:b] be the data augmentation matrix, and let E = [ΔA:Δb] be the perturbation augmentation matrix. Then there is (B+E)z=0 (8) The total least squares problem of the above homogeneous linear equation is to find a perturbation augmented matrix E with the minimum norm square such that (B+E) is non-full rank.

5. The AR model parameter estimation method based on forward and backward linear prediction according to claim 4, characterized in that, Step four includes: If we use SVD(M) to denote the singular value decomposition of matrix M, and represent the result as a singular value diagonal matrix, then the singular value decompositions of matrix A and the data augmented matrix B are respectively SVD(A)=diag(σ A1 ,...,s An ),s A1 ≥...≥σ An (9) SVD(B)=diag(σ1,...,σ n+1 ),σ1≥...≥σ n+1 (10) Where diag() represents a diagonal matrix, σ An Let σ represent the eigenvalues ​​of matrix A. n+1 Represents the eigenvalues ​​of the augmented matrix B; According to the properties of singular values Among them, ||E|| F Let r(B+E) be the Fronenius norm of E, and r(B+E) be the rank of the matrix (B+E). If let Where ν is related to the eigenvalue σ n+1 2 Any unit vector in the corresponding feature subspace Sc; but in, yes The Fronenius norm; From formulas (8), (11), (12), and (13), we can obtain (B-Bvv T )z=B(I-vv T )z=0 (14) Solving z∈null(B)=Sc (15) Where null(B) is the general solution of equation (14); If v is represented as y∈R n If α∈R and α≠0, then the general solution obtained by the total least squares method is: That is, to find the prediction coefficient vector c, which is also the parameter coefficient of the AR model.

6. An AR model parameter estimation device based on forward and backward linear prediction, characterized in that, The device includes: The model building module is used to acquire the interference signal from the infrared spectrometer and convert it into matrix form, thereby completing the construction of the AR model; The extended order module is used to construct the extended order matrix of the interference signal using the forward and backward prediction total least squares method, and to form the forward and backward linear prediction equation based on the extended order matrix. The least squares module is used to consider the noise perturbation effect on the observation vector and data matrix, transform the forward and backward extended matrices into homogeneous linear equations, and construct the overall least squares problem of the homogeneous linear equations. The solver module is used to solve the overall least squares problem using the singular value decomposition method and obtain the parameter coefficients of the AR model.

7. The AR model parameter estimation device based on forward and backward linear prediction according to claim 6, characterized in that, The model building module is used for: In an infrared spectrometer, equal optical path difference sampling interferometric data, also known as the AR model, is represented as: Where I(n) represents the interferometric data, f k and a k Here, represents the frequency and amplitude of the k-th sine wave, n represents the sampling point of the n-th interference signal, W(n) represents the noise signal, and p refers to the order of the prediction model; When noise is ignored, the AR model of the interferometric signal forms a linear difference equation. The matrix form of this linear difference equation is as follows: Among them, c k Let i(p) represent the AR model parameters, i(p) represent the p-th element in the observation vector i, and I(p) represent the p-th element in the data matrix I.

8. The AR model parameter estimation device based on forward and backward linear prediction according to claim 7, characterized in that, The extended order module is used for: Using the forward and backward prediction total least squares method to construct the forward and backward extended order matrix of each matrix in the AR model of the interferometric signal, the corresponding forward and backward linear prediction equation is: Right now Ac≈b (5) Where * denotes complex conjugate, A is equivalent to data matrix I, c is the prediction coefficient vector, and b is equivalent to observation vector i.

9. The AR model parameter estimation device based on forward and backward linear prediction according to claim 8, characterized in that, The least squares module is used for: Considering the noise disturbances in the data matrix I and the observation vector i, denoted as △A and △b respectively, equation (5) is rewritten as: (A+ΔA)c=b+Δb (6) Formula (6) can be transformed into: Let B = [A:b] be the data augmentation matrix, and let E = [ΔA:Δb] be the perturbation augmentation matrix. Then there is (B+E)z=0 (8) The total least squares problem of the above homogeneous linear equation is to find a perturbation augmented matrix E with the minimum norm square such that (B+E) is non-full rank.

10. The AR model parameter estimation device based on forward and backward linear prediction according to claim 9, characterized in that, The solution module is used for: If we use SVD(M) to denote the singular value decomposition of matrix M, and represent the result as a singular value diagonal matrix, then the singular value decompositions of matrix A and the data augmented matrix B are respectively SVD(A)=diag(σ A1 ,...,s An ),s A1 ≥...≥σ An (9) SVD(B)=diag(σ1,...,σ n+1 ),σ1≥...≥σ n+1 (10) Where diag() represents a diagonal matrix, σ An Let σ represent the eigenvalues ​​of matrix A. n+1 Represents the eigenvalues ​​of the augmented matrix B; According to the properties of singular values Among them, ||E|| F Let r(B+E) be the Fronenius norm of E, and r(B+E) be the rank of the matrix (B+E). If let Where ν is related to the eigenvalue σ n+1 2 Any unit vector in the corresponding feature subspace Sc; but in, yes The Fronenius norm; From formulas (8), (11), (12), and (13), we can obtain (B-Bvv T )z=B(I-vv T )z=0 (14) Solving z∈null(B)=Sc (15) Where null(B) is the general solution of equation (14); If v is represented as y∈R n If α∈R and α≠0, then the general solution obtained by the total least squares method is: That is, to find the prediction coefficient vector c, which is also the parameter coefficient of the AR model.

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