A bias compensation adaptive filtering method based on matrix eigenvalue decomposition

Abstract

The application discloses a bias compensation adaptive filtering method based on a matrix eigenvalue decomposition method and belongs to the field of digital filters.The application is implemented in the following manner: a corresponding bias compensation method is selected according to a filter type, and a corresponding bias compensation adaptive filtering step is executed; the filter type is divided into a finite impulse response filter and an infinite impulse response filter, wherein the infinite impulse response filter includes two cases of white output noise and colored output noise.The application adopts the eigenvalue decomposition of a matrix to obtain unknown parameters of a system, and realizes unbiased estimation of system parameters of a bias compensation adaptive filter; in the calculation process, the application does not need to estimate the variance of output noise, does not need to derive a cost function multiple times, and only needs to estimate the variance of input noise, thereby eliminating the influence of the variance of output noise on an unknown parameter estimated value and reducing the amount of operation parameters.The application has the advantages of high estimation precision, good filtering effect, good robustness and wide application range.

Description

Technical Field

[0001] This invention relates to a bias-compensated adaptive filtering method based on matrix eigenvalue decomposition, belonging to the field of digital filters. Background Technology

[0002] In recent years, adaptive filters have attracted widespread attention due to their excellent learning, tracking, and adaptive performance, and have been successfully applied in fields such as communication, control, radar, sonar, seismology, and biomedical engineering. However, how to reduce or even eliminate the impact of input and output noise on adaptive filters has always been a topic of great interest. When errors or noise interference exist at both the input and output of a linear system, it can usually be described by an "Error-in-Variables" (EIV) model. For linear dynamic systems where both the input and output are affected by noise, parameter estimation is a challenging problem, and the parameter estimation problem of the EIV model has become a research hotspot in system identification. Because the EIV model is closer to the models used in practical engineering applications, it is widely used in fields such as econometrics, finance and management, image processing, time series analysis, industrial modeling, biomedicine, and ecology.

[0003] For filters where the input and output noise are stationary additive colored noise, researchers have proposed traditional recursive least squares (RLS) algorithms, least mean squares (LMS) algorithms, and their derivatives. These algorithms show that, without considering the influence of interference noise, the least squares (LS) estimation of the unknown system parameters is biased. However, in practical applications, nodes inevitably suffer from noise interference at the input and output ends, and the variance of the interference noise is often unknown. To address this issue, researchers have proposed a series of algorithms. For finite impulse response (FIR) adaptive filters under the EIV model, Jia et al. proposed the bias-compensated RLS (BCRLS) and bias-compensated LMS (BCLMS) algorithms. Theoretical analysis proves that the estimation bias of the RLS and LMS algorithms is caused by input noise and is closely related to the variance of the input noise. These bias compensation algorithms can estimate the noise variance in real time to compensate for biased algorithm estimates and achieve unbiased estimation of unknown parameters.

[0004] In related literature, Jia et al. further proposed an adaptive filtering algorithm for bias compensation auxiliary variables, which can achieve unbiased estimation of unknown parameters. However, the above algorithms use gradient descent to continuously update parameters to achieve unbiased estimation, which has the problems of complex algorithm process and large amount of computation. Summary of the Invention

[0005] The main objective of this invention is to provide a bias-compensated adaptive filtering method based on matrix eigenvalue decomposition. The problem it addresses is how to perform unbiased estimation of unknown system parameters under an EIV adaptive filter model. Compared to traditional gradient descent methods, this invention eliminates the need for multiple derivatives of the cost function, significantly reducing computational costs. It directly obtains the eigenvectors (i.e., the unknown parameters) through matrix eigenvalue decomposition, improving estimation accuracy and convergence performance. Furthermore, this invention is applicable not only to finite impulse response (FIR) adaptive filters under EIV models but also to infinite impulse response (IIR) adaptive filters under EIV models, demonstrating good universality.

[0006] To achieve the above objectives, the present invention adopts the following technical solution.

[0007] This invention discloses a bias compensation adaptive filtering method based on matrix eigenvalue decomposition. The method selects the corresponding bias compensation method according to the filter type and executes the corresponding bias compensation adaptive filtering steps. The filter types are divided into finite impulse response filters and infinite impulse response filters. Since the system parameter estimation of infinite impulse response filters is related not only to the input noise but also to the output noise, the filtering of infinite impulse response filters includes two cases: colorless output noise and colored output noise.

[0008] For the EIV-FIR adaptive filter model, the least squares estimate of the unknown parameter h of the system at time i is first obtained. Then, the bias compensation principle is used to compensate for the bias, which requires knowing the variance of the input noise. To obtain an estimate of the input noise variance, a backward output variable γ is introduced. i Finally, an unbiased estimate of h is obtained using the bias-compensated recursive least squares algorithm (BCRLS). The expression is transformed and rearranged into a matrix form based on eigenvalue decomposition. The unbiased estimate of parameter h is obtained by solving the eigenvectors of the matrix.

[0009] For the EIV-IIR adaptive filter model, the estimated value of the unknown system parameter h at time i is first obtained using the instrumental variable-like (IV-like) algorithm. Then, bias compensation is applied using the bias compensation principle. Since the algorithm already includes the output noise information in the auxiliary variable, only the variance of the input noise needs to be known. There are two cases here: one is when the output noise is white noise. To obtain an estimate of the input noise variance, a backward output variable β is introduced. i and class auxiliary variable ξ j Finally, an unbiased estimate of h is obtained through a bias-compensated recursive auxiliary variable algorithm (BCRIV-like). The expression is transformed and rearranged into a matrix form based on eigenvalue decomposition. The unbiased estimate of parameter h is obtained by solving for the eigenvectors of the matrix. Another case involves colored noise as the output noise. To obtain an estimate of the input noise variance, a backward input variable α is introduced. i And adjust the class auxiliary variable ξ j Finally, an unbiased estimate of h is obtained through a bias-compensated recursive auxiliary variable algorithm (BCRIV-like). The expression is transformed and rearranged into a matrix form based on eigenvalue decomposition. The unbiased estimate of parameter h is obtained by solving the eigenvectors of the matrix.

[0010] This invention discloses a bias compensation adaptive filtering method based on matrix eigenvalue decomposition, comprising the following steps:

[0011] Step 0: Select the corresponding deviation compensation method according to the filter type, and execute the corresponding deviation compensation adaptive filtering step. The filter types are divided into finite impulse response filters and infinite impulse response filters. The infinite impulse response filters are further divided into infinite impulse response filters with white output noise and infinite impulse response filters with colored output noise.

[0012] For a finite impulse response filter, the bias-compensated adaptive filtering method based on matrix eigenvalue decomposition includes steps A to E:

[0013] Step A: Construct an FIR filter under the EIV model with constraints, and obtain the relationship between the input, output and filter system parameters.

[0014] Step A1: To construct an EIV-FIR filter, the filter must satisfy the following conditions:

[0015] Condition ①: The order of the adaptive FIR filter is known;

[0016] Condition ②: The input signal s(i) is a generalized stationary process;

[0017] Condition ③: n(i) and e(i) are Gaussian white noise with zero mean and no correlation, and their noise variances are unknown. and

[0018] Step A2: Represent the data as a vector form as follows

[0019] s i =[s(i)s(i-1)…s(i-L+1)] T

[0020]

[0021] h = [h0h1…h] L-1 ] T

[0022] Where s(i) is the noiseless input signal at time i, x(i) is the noisy input signal at time i, n(i) is the input noise at time i, i.e., x(i) = s(i) + n(i); e(i) is the output noise at time i, h is the weight vector characterizing the filter system, and is the unknown parameter to be estimated; L is the order of the filter, and the superscript T indicates the transpose operator.

[0023] Step A3: The EIV-FIR filter model can be expressed as follows:

[0024]

[0025] Where y(i) is the noisy output signal at time i, and v(i) is the composite noise, which can be expressed as:

[0026]

[0027] Step B: Based on the EIV-FIR filter model, and according to the least squares (LS) principle, obtain the least squares estimate of the unknown weight vector h at time i. Analysis shows that... It is biased.

[0028] Step B1: According to the least squares (LS) principle, the LS estimate of the unknown weight vector h is expressed as:

[0029]

[0030] Substituting formula (2) into formula (4), we obtain the following relationship.

[0031]

[0032] Step B2: In order to obtain the estimated value The deviation from the true value h can be obtained by taking the limit of the above equation using conditions ①~③ and the ergodicity of stationary random processes.

[0033]

[0034] Analysis of formula (6) shows that if the RLS algorithm is used to estimate the unknown parameters, the estimated value under the least squares criterion will be as i→∞ in the absence of noise interference. It converges to h, but in the EIV model, both the input and output are affected by noise, i.e. Therefore, the estimates from the traditional RLS algorithm are biased. The difference between h and h is At time i, replace h in equation (6) with the unbiased estimate from the previous time. The deviation is then expressed as The estimated value of h Further expressed as

[0035]

[0036] Wherein, the inverse correlation matrix P i It is expressed as follows

[0037]

[0038] Analysis of the results obtained in step B shows that the estimated values ​​of the unknown parameters obtained by the recursive least squares algorithm (RLS) are biased, and this bias is related to the variance of the input noise.

[0039] Step C: Based on the bias-compensated recursive least squares principle (BCRLS), introduce the backward output estimate variable γ. i The backward output estimation error ζ(j) is obtained. By defining the cross-correlation function g(i) between the least squares estimation error ε(j) and ζ(j), the estimated value of the input noise variance is obtained. This leads to the biased estimate obtained in step B. Perform bias compensation to obtain an unbiased estimate.

[0040] Step C1: Based on the bias-compensated recursive least squares principle (BCRLS), introduce the backward output estimation variable γ. i =[γ1γ2…γ L ] T Based on linear prediction theory, expressions for the backward output estimation error ζ(j) and the least squares estimation error ε(j) are obtained.

[0041] γ i The estimated value Represented as

[0042]

[0043] According to linear prediction theory, the backward output estimation error ζ(j) and the least squares estimation error ε(j) are expressed as follows:

[0044]

[0045]

[0046] Step C2: Obtain the cross-correlation function g(i) by using the backward output estimation error ζ(j) and the least squares estimation error ε(j), and take the limit to obtain the estimated value of the input noise variance.

[0047] Let g(i) denote the cross-correlation function between ε(j) and ζ(j), defined as follows:

[0048]

[0049] Under conditions ① and ②, as i→∞, taking the limit, we have: Then the estimated value of the input noise variance It can be represented as

[0050]

[0051] Step C3: Under the EIV-FIR filter model, obtain the unbiased estimate based on the steps above. It is expressed as follows

[0052]

[0053] Step D: Multiply both sides of the expression for the unbiased estimate of the unknown parameter h obtained in step C by the scalar in formula (13). Transform it into matrix eigenvalue decomposition form, and solve for the normalized eigenvectors. and its corresponding coefficient k l The eigenvectors of the matrix are obtained, and the eigenvectors are the unbiased estimates of the unknown parameter h.

[0054] Step D1: In the expression (13) for the unbiased estimate of the unknown parameter h, the denominator It is a scalar, so both sides of formula (13) are multiplied by the scalar. The expression for the unbiased estimate of the unknown parameter h is transformed into the eigenvalue decomposition form of the constructed matrix, as shown in formula (14):

[0055]

[0056] Step D2: Analyze the above equation based on the principles of matrix analysis, scalar with vector The product equals the matrix with vector The product of, i.e., scalar sum vector Representing matrices respectively The eigenvalues ​​and eigenvectors. Analyzing formula (14), we obtain the following characteristics of the unknown parameter h: unbiased estimation of the unknown parameter h. It is a matrix An eigenvector of .

[0057] Step D3: Matrix It is an L×L dimensional square matrix with L eigenvalues ​​λ1, λ2, ..., λ3. L The corresponding feature vector is in It is a normalized eigenvector, k l The eigenvalue is a real number to be estimated. The eigenvalue k can be obtained by performing eigenvalue decomposition on the matrix. l and The following will determine k l .

[0058] Depend on The following relationship can be obtained:

[0059]

[0060] Therefore, real number k l Calculated as

[0061]

[0062] Define function

[0063]

[0064] have

[0065]

[0066] There is only one stable point, which is the unique solution to formula (18), i.e., the unbiased estimate of the unknown parameter. This is achieved by testing the formula. In each normalized eigenvector The convergence of the surrounding area leads to a unique solution, because as the number of iterations increases, only the unique solution converges.

[0067] Step E: Based on the unbiased estimate h obtained in step D, construct an EIV-FIR adaptive filter. Using the bias-compensated recursive least squares (BCRLS) algorithm, for a given input signal sample x(i), the optimal estimate of the expected response is given by the real-time updated system weight vector parameter h, which minimizes the mean square value of the estimation error ε(i) and improves the estimation accuracy. At the same time, compared with the previous method of estimating h by gradient descent, the computational complexity is reduced and the signal processing cost is saved.

[0068] The above describes the process of solving the unknown parameter h using the matrix eigenvalue decomposition method in the BCRLS algorithm for the EIV-FIR adaptive filter. The following section uses the matrix eigenvalue decomposition method to estimate the unknown parameter h for the infinite impulse response (EIV-IIR) adaptive filter model with error variables, thus broadening the applicability of this invention from finite impulse response filters to infinite impulse response filters.

[0069] For an infinite impulse response filter, the bias-compensated adaptive filtering method based on matrix eigenvalue decomposition includes steps F to G:

[0070] Step F: For the EIV-IIR filter model, if the BCRLS algorithm is used for unbiased estimation of h, both input and output noise need to be estimated simultaneously, which is computationally complex and difficult. Therefore, for the EIV-IIR filter, we use the BCRIV-like algorithm to perform unbiased estimation of h using a matrix eigenvalue decomposition method. The biased estimate of the unknown system parameter h at time i under the (IV-like) algorithm is... Since the algorithm has included the output noise information in the auxiliary variable, it only needs to know the variance information of the input noise.

[0071] There are two cases here. One case is that the output noise is white noise. In order to obtain an estimate of the variance of the input noise, the backward output variable β is introduced. i and class auxiliary variable ξ j Finally, an unbiased estimate of h is obtained through a bias-compensated recursive auxiliary variable algorithm (BCRIV-like). The expression is transformed and rearranged into a matrix form based on eigenvalue decomposition. The unbiased estimate of parameter h is obtained by solving for the eigenvectors of the matrix. Another case involves colored noise as the output noise. To obtain an estimate of the input noise variance, a backward input variable α is introduced. i And adjust the class auxiliary variable ξ j Finally, an unbiased estimate of h is obtained through a bias-compensated recursive auxiliary variable algorithm (BCRIV-like). The expression is transformed and rearranged into a matrix form based on eigenvalue decomposition. The unbiased estimate of parameter h is obtained by solving the eigenvectors of the matrix.

[0072] Step F1: Under white output noise, for the EIV-IIR-BCRIV-Like model, define the class auxiliary variable ξ. j The inverse correlation matrix Q is obtained. i Then, by introducing the backward output estimate variable β i The backward output estimation error ζ(j) is obtained, and the input noise variance is estimated by defining the cross-correlation function g(i) between ε(j) and ζ(j). Further, an unbiased estimation expression for h is obtained. After converting the expression into a matrix form based on eigenvalue decomposition, the eigenvectors of the matrix are obtained by solving the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h.

[0073] The input data vector p at time i i Defined as follows

[0074] p i =[-y(i-1)-y(i-2)…-y(iL)x(i-1)x(i-2)…x(iL)] T

[0075] The auxiliary variable representation when the output noise is white noise is as follows:

[0076] ξ j =[-y(jL-1)-y(jL-2)…-y(j-2L)x(j-1)x(j-2)…x(jL)] T

[0077] iTimep i and ξ j The inverse correlation matrix Q i Represented as

[0078]

[0079] Backward output estimate variable β i =[β1β2…β 2L ] T The estimated value Defined as

[0080]

[0081] Corresponding backward output estimation for

[0082]

[0083] The estimation error ε(j) of the auxiliary variable is

[0084]

[0085] in It is a biased estimate of parameter h under class auxiliary variables (IV-like).

[0086] The backward output estimation error ζ(j) is the difference between the backward output y(j-1) and the backward output estimate. The difference is expressed as follows:

[0087]

[0088] Define the cross-correlation function g(i) between the estimation error of the auxiliary variable and the estimation error of the backward output as follows:

[0089]

[0090] As i→∞, taking the limit of formula (24) yields

[0091]

[0092] From formula (25), the estimated value of the input noise variance can be obtained.

[0093]

[0094] When the output noise is white noise, the unbiased estimate of h under the BCRIV-like condition of the EIV-IIR filter is expressed as follows:

[0095]

[0096] Multiply both sides of formula (27) by the denominator in the formula. Transform the unbiased estimate of the unknown parameter h into a matrix. The eigenvalue decomposition form is formula (28).

[0097]

[0098] Repeat step D to obtain the matrix by solving for the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h.

[0099] Step F2: In the case of colored output noise, adjust the auxiliary variable ξ. j The inverse correlation matrix Q is obtained. i Then, by introducing the backward input estimation variable α i Obtain the backward input estimation error By defining ε(j) and The cross-correlation function f(i) is used to obtain an estimate of the input noise variance. Further, an unbiased estimation expression for h is obtained. After converting the expression into a form based on matrix eigenvalue decomposition, the eigenvectors of the matrix are obtained by solving the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h.

[0100] When the output noise is colored noise, the auxiliary variable is represented as follows:

[0101] ξ j =[x(jL-1)x(jL-2)…x(j-2L)x(j-1)x(j-2)…x(jL)] T

[0102] Backward input estimated variable α i =[α1α2…α 2L ] T The estimated value Defined as

[0103]

[0104] Corresponding backward input estimation for

[0105]

[0106] Backward input estimation error For backward input x(j-1) and backward input estimation The difference is expressed as follows:

[0107]

[0108] Define the estimation error ε(j) of the auxiliary variable and the estimation error of the backward input. The cross-correlation function f(i) is

[0109]

[0110] As i→∞, taking the limit of formula (32) yields

[0111]

[0112] From formula (33), the estimated value of the input noise variance can be obtained.

[0113]

[0114] When the output noise is colored noise, the unbiased estimate of h under the BCRIV-like EIV-IIR filter is expressed as follows:

[0115]

[0116] Multiply both sides of formula (35) by the denominator in the formula. Transform the unbiased estimate of the unknown parameter h into a matrix. The eigenvalue decomposition form is formula (36).

[0117]

[0118] Repeat step D to obtain the matrix by solving for the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h.

[0119] Step G: Based on the unbiased estimate h obtained in step F, construct an EIV-IIR adaptive filter. Using the bias-compensated recursive auxiliary variable algorithm (BCRIV-like), for a given input signal sample x(i), the optimal estimate of the expected response is given by updating the system weight vector parameter h in real time, so that the mean square value of the estimation error ε(i) is minimized, thus improving the estimation accuracy. At the same time, compared with the previous method of estimating h by gradient descent, the computational complexity is reduced and the signal processing cost is saved.

[0120] Beneficial effects:

[0121] 1. The present invention discloses a bias compensation adaptive filtering method based on matrix eigenvalue decomposition. The method uses eigenvalue decomposition of the matrix to obtain the unknown parameter h of the system. Compared with the traditional gradient descent method to approximate the unknown parameter h, it does not require the use of a cost function and its derivative, which significantly saves computational costs. The advantage will be more obvious when the filter order is large.

[0122] 2. The deviation compensation adaptive filtering method disclosed in this invention is applicable not only to deviation compensation EIV-FIR adaptive filters, but also to deviation compensation EIV-IIR adaptive filters, and has good universality.

[0123] 3. This invention discloses a bias-compensated adaptive filtering method based on matrix eigenvalue decomposition. Under EIV-FIR, matrix eigenvalue decomposition is used to achieve unbiased estimation of unknown parameters. Simulation results show that the invention has high computational accuracy and good curve fitting. For input and output noise of different intensities, it highlights the difference between the BCRLS algorithm and the traditional RLS algorithm in terms of mean square deviation (MSD), and has good estimation accuracy and robustness.

[0124] 4. The present invention discloses a bias compensation adaptive filtering method based on matrix eigenvalue decomposition. For different filters, an appropriate bias compensation algorithm is selected. For EIV-FIR filters, the BCRLS algorithm is used, and for the case of output noise interference in EIV-IIR filters, the BCRIV-like algorithm is selected. In the calculation process, there is no need to estimate the output noise variance, only the input noise variance needs to be estimated, thus eliminating the influence of the output noise variance on the estimated value of unknown parameters, reducing the number of computational parameters, and reducing computational complexity. Attached Figure Description

[0125] Figure 1 This invention relates to the variable error-containing finite impulse response (EIV-FIR) adaptive filter model.

[0126] Figure 2 This invention relates to an adaptive filter model for infinite impulse response (EIV-IIR) with variable error.

[0127] Figure 3 This paper compares the performance of the bias-compensated recursive least squares algorithm based on matrix solving with that of the traditional recursive least squares algorithm under different input noise conditions in an EIV-FIR filter, where the input signal is a colored Gaussian signal and the output noise is constant.

[0128] Figure 4 This paper compares the performance of the bias-compensated recursive least squares algorithm based on matrix solving with that of the traditional recursive least squares algorithm under different output noise conditions in an EIV-FIR filter, where the input signal is a colored Gaussian signal and the input noise is constant.

[0129] Figure 5 This is a flowchart of a deviation compensation FIR adaptive filtering method based on matrix eigenvalue decomposition disclosed in this invention.

[0130] Figure 6 This is a flowchart of a bias compensation IIR adaptive filtering method based on matrix eigenvalue decomposition disclosed in this invention. Detailed Implementation

[0131] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments. This embodiment is implemented based on the technical solution of the present invention, providing detailed implementation methods and specific operating procedures; however, the scope of protection of the present invention is not limited to the following embodiments.

[0132] like Figure 5 , 6 As shown in the figure, the deviation compensation adaptive filtering method based on matrix eigenvalue decomposition disclosed in this embodiment includes the following steps:

[0133] Step 0: Select the corresponding deviation compensation method according to the filter type, and execute the corresponding deviation compensation adaptive filtering steps. The filter type is divided into finite impulse response filters (FIR). Figure 1 ) and infinite impulse response filter ( Figure 2 Among them, the infinite impulse response filter is further divided into an infinite impulse response filter with white output noise and an infinite impulse response filter with colored output noise.

[0134] Combination Figure 5 For finite impulse response filters, the bias-compensated adaptive filtering method based on matrix eigenvalue decomposition includes steps A to E:

[0135] Step A: As Figure 1 As shown, an FIR filter under the EIV model is constructed under constraints to obtain the relationship between the input, output and filter system parameters.

[0136] Step A1: To construct an EIV-FIR filter, the filter must satisfy the following conditions:

[0137] Condition ①: The order of the adaptive FIR filter is known;

[0138] Condition ②: The input signal s(i) is a generalized stationary process;

[0139] Condition ③: n(i) and e(i) are Gaussian white noise with zero mean and no correlation, and their noise variances are unknown. and

[0140] Step A2: Represent the data as a vector form as follows

[0141] s i =[s(i)s(i-1)…s(i-L+1)] T

[0142]

[0143] h = [h0h1…h] L-1 ] T

[0144] Where s(i) is the noiseless input signal at time i, x(i) is the noisy input signal at time i, n(i) is the input noise at time i, i.e., x(i) = s(i) + n(i); e(i) is the output noise at time i, h is the weight vector characterizing the filter system, and is the unknown parameter to be estimated; L is the order of the filter, and the superscript T indicates the transpose operator.

[0145] Step A3: The EIV-FIR filter model can be expressed as follows:

[0146]

[0147] Where y(i) is the noisy output signal at time i, and v(i) is the composite noise, which can be expressed as:

[0148]

[0149] Step B: Based on the EIV-FIR filter model, and according to the least squares (LS) principle, obtain the least squares estimate of the unknown weight vector h at time i. Analysis shows that... It is biased.

[0150] Step B1: According to the least squares (LS) principle, the LS estimate of the unknown weight vector h is expressed as:

[0151]

[0152] Substituting formula (2) into formula (4), we obtain the following relationship.

[0153]

[0154] Step B2: In order to obtain the estimated value The deviation from the true value h can be obtained by taking the limit of the above equation using conditions ①~③ and the ergodicity of stationary random processes.

[0155]

[0156] Analysis of formula (6) shows that if the RLS algorithm is used to estimate the unknown parameters, the estimated value under the least squares criterion will be as i→∞ in the absence of noise interference. It converges to h, but in the EIV model, both the input and output are affected by noise, i.e. Therefore, the estimates from the traditional RLS algorithm are biased. The difference between h and h is At time i, replace h in equation (6) with the unbiased estimate from the previous time. The deviation is then expressed as The estimated value of h Further expressed as

[0157]

[0158] Wherein, the inverse correlation matrix P i It is expressed as follows

[0159]

[0160] Analysis of the results obtained in step B shows that the estimated values ​​of the unknown parameters obtained by the recursive least squares algorithm (RLS) are biased, and this bias is related to the variance of the input noise.

[0161] Step C: Based on the bias-compensated recursive least squares principle (BCRLS), introduce the backward output estimate variable γ. i The backward output estimation error ζ(j) is obtained. By defining the cross-correlation function g(i) between the least squares estimation error ε(j) and ζ(j), the estimated value of the input noise variance is obtained. This leads to the biased estimate obtained in step B. Perform bias compensation to obtain an unbiased estimate.

[0162] Step C1: Based on the bias-compensated recursive least squares principle (BCRLS), introduce the backward output estimation variable γ. i =[γ1γ2…γ L ] T Based on linear prediction theory, expressions for the backward output estimation error ζ(j) and the least squares estimation error ε(j) are obtained.

[0163] γ i The estimated value Represented as

[0164]

[0165] According to linear prediction theory, the backward output estimation error ζ(j) and the least squares estimation error ε(j) are expressed as follows:

[0166]

[0167]

[0168] Step C2: Obtain the cross-correlation function g(i) by using the backward output estimation error ζ(j) and the least squares estimation error ε(j), and take the limit to obtain the estimated value of the input noise variance.

[0169] Let g(i) denote the cross-correlation function between ε(j) and ζ(j), defined as follows:

[0170]

[0171] Under conditions ① and ②, as i→∞, taking the limit, we have: Then the estimated value of the input noise variance It can be represented as

[0172]

[0173] Step C3: Under the EIV-FIR filter model, obtain the unbiased estimate based on the steps above. It is expressed as follows

[0174]

[0175] Step D: Multiply both sides of the expression for the unbiased estimate of the unknown parameter h obtained in step C by the scalar in formula (13). Transform it into matrix eigenvalue decomposition form, and solve for the normalized eigenvectors. and its corresponding coefficient k l The eigenvectors of the matrix are obtained, and the eigenvectors are the unbiased estimates of the unknown parameter h.

[0176] Step D1: In the expression (13) for the unbiased estimate of the unknown parameter h, the denominator It is a scalar, so both sides of formula (13) are multiplied by the scalar. The expression for the unbiased estimate of the unknown parameter h is transformed into the eigenvalue decomposition form of the constructed matrix, as shown in formula (14):

[0177]

[0178] Step D2: Analyze the above equation based on the principles of matrix analysis, scalar with vector The product equals the matrix with vector The product of, i.e., scalar sum vector Representing matrices respectively The eigenvalues ​​and eigenvectors. Analyzing formula (14), we obtain the following characteristics of the unknown parameter h: unbiased estimation of the unknown parameter h. It is a matrix An eigenvector of .

[0179] Step D3: Matrix It is an L×L dimensional square matrix with L eigenvalues ​​λ1, λ2, ..., λ3. L The corresponding feature vector is in It is a normalized eigenvector, k l The eigenvalue is a real number to be estimated. The eigenvalue k can be obtained by performing eigenvalue decomposition on the matrix. l and The following will determine k l .

[0180] Depend on The following relationship can be obtained:

[0181]

[0182] Therefore, real number k l Calculated as

[0183]

[0184] Define function

[0185]

[0186] have

[0187]

[0188] There is only one stable point, which is the unique solution to formula (18), i.e., the unbiased estimate of the unknown parameter. This is achieved by testing the formula. In each normalized eigenvector The convergence of the surrounding area leads to a unique solution, because as the number of iterations increases, only the unique solution converges.

[0189] Step E: Based on the unbiased estimate h obtained in step D, construct an EIV-FIR adaptive filter. Using the bias-compensated recursive least squares (BCRLS) algorithm, for a given input signal sample x(i), the optimal estimate of the expected response is given by the real-time updated system weight vector parameter h, which minimizes the mean square value of the estimation error ε(i), thus improving the estimation accuracy. At the same time, compared with the previous method of estimating h by gradient descent, the computational complexity is reduced, and the signal processing cost is saved.

[0190] The above describes the process of solving the unknown parameter h using the matrix eigenvalue decomposition method in the BCRLS algorithm for the EIV-FIR adaptive filter. The following section uses the matrix eigenvalue decomposition method to estimate the unknown parameter h for the infinite impulse response (EIV-IIR) adaptive filter model with error variables, thus broadening the applicability of this invention from finite impulse response filters to infinite impulse response filters.

[0191] Combination Figure 6 For an infinite impulse response filter, the bias-compensated adaptive filtering method based on matrix eigenvalue decomposition includes steps F to G:

[0192] Step F: As Figure 2As shown, for the EIV-IIR filter model, if the BCRLS algorithm is used for unbiased estimation of h, both input and output noise need to be estimated simultaneously, which is computationally complex and difficult. Therefore, for the EIV-IIR filter, we use the BCRIV-like algorithm to perform unbiased estimation of h using a matrix eigenvalue decomposition method. The biased estimate of the unknown system parameter h at time i under the (IV-like) algorithm is... Since the algorithm has included the output noise information in the auxiliary variable, it only needs to know the variance information of the input noise.

[0193] There are two cases here. One case is that the output noise is white noise. In order to obtain an estimate of the variance of the input noise, the backward output variable β is introduced. i and class auxiliary variable ξ j Finally, an unbiased estimate of h is obtained through a bias-compensated recursive auxiliary variable algorithm (BCRIV-like). The expression is transformed and rearranged into a matrix form based on eigenvalue decomposition. The unbiased estimate of parameter h is obtained by solving for the eigenvectors of the matrix. Another case involves colored noise as the output noise. To obtain an estimate of the input noise variance, a backward input variable α is introduced. i And adjust the class auxiliary variable ξ j Finally, an unbiased estimate of h is obtained through a bias-compensated recursive auxiliary variable algorithm (BCRIV-like). The expression is transformed and rearranged into a matrix form based on eigenvalue decomposition. The unbiased estimate of parameter h is obtained by solving the eigenvectors of the matrix.

[0194] Step F1: Under white output noise, for the EIV-IIR-BCRIV-Like model, define the class auxiliary variable ξ. j The inverse correlation matrix Q is obtained. i Then, by introducing the backward output estimate variable β i The backward output estimation error ζ(j) is obtained, and the input noise variance is estimated by defining the cross-correlation function g(i) between ε(j) and ζ(j). Further, an unbiased estimation expression for h is obtained. After converting the expression into a matrix form based on eigenvalue decomposition, the eigenvectors of the matrix are obtained by solving the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h.

[0195] The input data vector p at time i i Defined as follows

[0196] p i=[-y(i-1)-y(i-2)…-y(iL)x(i-1)x(i-2)…x(iL)] T

[0197] The auxiliary variable representation when the output noise is white noise is as follows:

[0198] ξ j =[-y(jL-1)-y(jL-2)…-y(j-2L)x(j-1)x(j-2)…x(jL)] T

[0199] iTimep i and ξ j The inverse correlation matrix Q i Represented as

[0200]

[0201] Backward output estimate variable β i =[β1β2…β 2L ] T The estimated value Defined as

[0202]

[0203] Corresponding backward output estimation for

[0204]

[0205] The estimation error ε(j) of the auxiliary variable is

[0206]

[0207] in It is a biased estimate of parameter h under class auxiliary variables (IV-like).

[0208] The backward output estimation error ζ(j) is the difference between the backward output y(j-1) and the backward output estimate. The difference is expressed as follows:

[0209]

[0210] Define the cross-correlation function g(i) between the estimation error of the auxiliary variable and the estimation error of the backward output as follows:

[0211]

[0212] As i→∞, taking the limit of formula (24) yields

[0213]

[0214] From formula (25), the estimated value of the input noise variance can be obtained.

[0215]

[0216] When the output noise is white noise, the unbiased estimate of h under the BCRIV-like condition of the EIV-IIR filter is expressed as follows:

[0217]

[0218] Multiply both sides of formula (27) by the denominator in the formula. Transform the unbiased estimate of the unknown parameter h into a matrix. The eigenvalue decomposition form is formula (28).

[0219]

[0220] Repeat step D to obtain the matrix by solving for the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h.

[0221] Step F2: In the case of colored output noise, adjust the auxiliary variable ξ. j The inverse correlation matrix Q is obtained. i Then, by introducing the backward input estimation variable α i Obtain the backward input estimation error By defining ε(j) and The cross-correlation function f(i) is used to obtain an estimate of the input noise variance. Further, an unbiased estimation expression for h is obtained. After converting the expression into a form based on matrix eigenvalue decomposition, the eigenvectors of the matrix are obtained by solving the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h.

[0222] When the output noise is colored noise, the auxiliary variable is represented as follows:

[0223] ξ j =[x(jL-1)x(jL-2)…x(j-2L)x(j-1)x(j-2)…x(jL)] T

[0224] Backward input estimated variable α i =[α1α2…α 2L ] T The estimated value Defined as

[0225]

[0226] Corresponding backward input estimation for

[0227]

[0228] Backward input estimation error For backward input x(j-1) and backward input estimation The difference is expressed as follows:

[0229]

[0230] Define the estimation error ε(j) of the auxiliary variable and the estimation error of the backward input. The cross-correlation function f(i) is

[0231]

[0232] As i→∞, taking the limit of formula (32) yields

[0233]

[0234] From formula (33), the estimated value of the input noise variance can be obtained.

[0235]

[0236] When the output noise is colored noise, the unbiased estimate of h under the BCRIV-like EIV-IIR filter is expressed as follows:

[0237]

[0238] Multiply both sides of formula (35) by the denominator in the formula. Transform the unbiased estimate of the unknown parameter h into a matrix. The eigenvalue decomposition form is formula (36).

[0239]

[0240] Repeat step D to obtain the matrix by solving for the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h.

[0241] Step G: Based on the unbiased estimate h obtained in step F, construct an EIV-IIR adaptive filter. Using the bias-compensated recursive auxiliary variable algorithm (BCRIV-like), for a given input signal sample x(i), the optimal estimate of the expected response is given by the real-time updated system weight vector parameter h, which minimizes the mean square value of the estimation error ε(i), thus improving the estimation accuracy. At the same time, compared with the previous method of estimating h by gradient descent, the computational complexity is reduced, and the signal processing cost is saved.

[0242] The effectiveness of this embodiment can be verified through the following experiments:

[0243] The input signal s(i) is a zero-mean Gaussian colored signal, generated as follows:

[0244] s(i)=randn(i)-0.3×randn(i-1)+0.5×randn(i-2)-0.7×randn(i-3)+0.9×randn(i-4)

[0245] The simulation experiment had 5000 iterations and 100 independent experiments. The FIR filter was specified as order 5, and the true values ​​of the unknown parameters were set to [-0.3, -0.9, 0.8, -0.7, 0.6]. The matrix eigenvalue decomposition method proposed in this paper was used to conduct experiments in two cases. (1) Output noise variance Input noise variance Analysis of the mean square deviation (MSD) of the EIV-FIR-BCRLS algorithm and the traditional RLS algorithm for the local adaptation filter node; (2) Input noise variance Output noise variance This paper analyzes the node-local EIV-FIR-BCRLS algorithm of the adaptive filter and the MSD under the traditional RLS algorithm.

[0246] Figure 3 The figure presents the node-local EIV-FIR-BCRLS algorithm and the MSD of the traditional RLS algorithm for adaptive filters with different variances of input Gaussian white noise, given a fixed output noise. As can be seen from the figure, the estimation accuracy and convergence performance of both algorithms steadily improve with the increase of the number of iterations. At the same time, the mean square deviation of both algorithms increases and the estimation accuracy decreases with the increase of the input noise intensity.

[0247] Figure 4The paper presents the mean squared error (MSD) of the EIV-FIR-BCRLS algorithm and the traditional RLS algorithm under different variances of output Gaussian white noise with a fixed input noise. As shown in the figures, the estimation accuracy and convergence performance of both algorithms steadily improve with increasing iteration count. However, the mean squared error of both algorithms increases and the estimation accuracy decreases with increasing output noise intensity. All algorithms used in this simulation employ the unknown parameters derived from matrix eigenvalue decomposition proposed in this paper. The experimental results also demonstrate the effectiveness of this method.

[0248] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. The technical means disclosed in the present invention are not limited to the technical means disclosed in the above embodiments, but also include technical solutions composed of any combination of the above technical features. It should be noted that for those skilled in the art, any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention without departing from the principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A bias-compensated adaptive filtering method based on matrix eigenvalue decomposition, characterized in that: Includes the following steps, Step 0: Select the corresponding deviation compensation method according to the filter type and execute the corresponding deviation compensation adaptive filtering step. The filter types are divided into finite impulse response filters and infinite impulse response filters. The infinite impulse response filters are further divided into infinite impulse response filters with white output noise and infinite impulse response filters with colored output noise. For the EIV-FIR adaptive filter, the bias compensation adaptive filtering method based on matrix eigenvalue decomposition includes steps A to E: Step A: Given the constraints, construct the FIR filter under the EIV model and obtain the relationship between the input, output and filter system parameters; Step B: Based on the EIV-FIR filter model, and according to the least squares (LS) principle, obtain the least squares estimate of the unknown weight vector h at time i. Analysis shows that... It is biased, and the bias is related to the variance of the input noise; Step C: Based on the bias-compensated recursive least squares principle (BCRLS), introduce the backward output estimate variable. The backward output estimation error ζ(j) is obtained. By defining the cross-correlation function g(i) between the least squares estimation error ε(j) and ζ(j), the estimated value of the input noise variance is obtained. This leads to the biased estimate obtained in step B. Perform bias compensation to obtain an unbiased estimate. Step D: Multiply both sides of the equation for the unbiased estimate of the unknown parameter h obtained in step C by a scalar. The expression for the unbiased estimate of the unknown parameter h is transformed into a matrix eigenvalue decomposition form, and the normalized eigenvectors are solved. and its corresponding coefficient k l The eigenvectors of the matrix are obtained, and the eigenvectors are the unbiased estimates of the unknown parameter h. Step E: Based on the unbiased estimate h obtained in step D, construct an EIV-FIR adaptive filter. Using the bias-compensated recursive least squares algorithm BCRLS, for a given input signal sample x(i), the optimal estimate of the expected response is given by the real-time updated system weight vector parameter h, so that the mean square value of the estimation error ε(i) is minimized. For an infinite impulse response filter, the bias-compensated adaptive filtering method based on matrix eigenvalue decomposition includes steps F to G: Step F: The biased estimate of the unknown system parameter h at time i using the IV-like algorithm. Since the algorithm already includes the output noise information in the auxiliary variable, only the variance of the input noise needs to be known. There are two cases here: one is when the output noise is white noise, in which case a backward output variable β is introduced to obtain an estimate of the input noise variance. i and class auxiliary variable ξ j An unbiased estimate of h is obtained through the bias-compensated recursive auxiliary variable algorithm BCRIV-like. The expression is transformed and rearranged into a matrix form based on eigenvalue decomposition. The unbiased estimate of parameter h is obtained by solving for the eigenvectors of the matrix. Another case involves colored noise as the output noise. To obtain an estimate of the input noise variance, a backward input variable α is introduced. i And adjust the class auxiliary variable ξ j An unbiased estimate of h is obtained through the bias-compensated recursive auxiliary variable algorithm BCRIV-like. The expression is transformed and rearranged into a matrix form based on eigenvalue decomposition. The unbiased estimate of parameter h is obtained by solving the eigenvectors of the matrix. Step G: Based on the unbiased estimate h obtained in step F, construct an EIV-IIR adaptive filter. Using the bias compensation recursive auxiliary variable algorithm BCRIV-like, for a given input signal sample x(i), the optimal estimate of the expected response is given by updating the system weight vector parameter h in real time, so that the mean square value of the estimation error ε(i) is minimized.

2. The deviation compensation adaptive filtering method based on matrix eigenvalue decomposition as described in claim 1, characterized in that: Step A is implemented as follows: Step A1: To construct an EIV-FIR filter, the filter must satisfy the following conditions: Condition ①: The order of the adaptive FIR filter is known; Condition ②: The input signal s(i) is a generalized stationary process; Condition ③: n(i) and e(i) are Gaussian white noise with zero mean and no correlation, and their noise variances are unknown. and Step A2: Represent the data as a vector form as follows Where s(i) is the noiseless input signal at time i, x(i) is the noisy input signal at time i, n(i) is the input noise at time i, i.e., x(i) = s(i) + n(i); e(i) is the output noise at time i, h is the weight vector characterizing the filter system, and is the unknown parameter to be estimated; L is the order of the filter, and the superscript T indicates the transpose operator; Step A3: The EIV-FIR filter model can be expressed as follows: Where y(i) is the noisy output signal at time i, and v(i) is the composite noise, expressed as: 。 3. The deviation compensation adaptive filtering method based on matrix eigenvalue decomposition as described in claim 2, characterized in that: Step B is implemented as follows: Step B1: According to the least squares (LS) principle, the LS estimate of the unknown weight vector h is expressed as: Substituting formula (2) into formula (4), we obtain the following relationship. Step B2: In order to obtain the estimated value The deviation from the true value h can be obtained by taking the limit of the above equation using conditions ①~③ and the ergodicity of stationary random processes. Analyzing formula (6), we know that if the RLS algorithm is used to estimate the unknown parameters, in the absence of noise interference, as i→∞, the estimated value under the least squares criterion... It converges to h, but in the EIV model, both the input and output are affected by noise, i.e. Therefore, the estimates from the traditional RLS algorithm are biased. The difference between h and h is At time i, replace h in equation (6) with the unbiased estimate from the previous time. The deviation is then expressed as The estimated value of h Further expressed as Wherein, the inverse correlation matrix P i It is expressed as follows Analysis of the results obtained in step B shows that the estimated values ​​of the unknown parameters obtained by the recursive least squares algorithm (RLS) are biased, and this bias is related to the variance of the input noise.

4. The deviation compensation adaptive filtering method based on matrix eigenvalue decomposition as described in claim 3, characterized in that: Step C is implemented as follows: Step C1: Based on the bias-compensated recursive least squares principle (BCRLS), introduce the backward output estimate variable γ. i =[γ1γ2…γ L ] T Based on linear prediction theory, the expressions for the backward output estimation error ζ(j) and the least squares estimation error ε(j) are obtained. γ i The estimated value Represented as According to linear prediction theory, the backward output estimation error ζ(j) and the least squares estimation error ε(j) are expressed as follows: Step C2: Obtain the cross-correlation function g(i) by using the backward output estimation error ζ(j) and the least squares estimation error ε(j), and take the limit to obtain the estimated value of the input noise variance; Let g(i) denote the cross-correlation function between ε(j) and ζ(j), defined as follows: Under conditions ① and ②, as i→∞, taking the limit, we have: Then the estimated value of the input noise variance Represented as Step C3: Under the EIV-FIR filter model, obtain the unbiased estimate based on the steps above. It is expressed as follows 。 5. The deviation compensation adaptive filtering method based on matrix eigenvalue decomposition as described in claim 4, characterized in that: Step D is implemented as follows: Step D1: In the expression (13) for the unbiased estimate of the unknown parameter h, the denominator It is a scalar, so both sides of formula (13) are multiplied by the scalar. The expression for the unbiased estimate of the unknown parameter h is transformed into the eigenvalue decomposition form of the constructed matrix, as shown in formula (14). Step D2: Analyze the above equation based on the principles of matrix analysis, scalar with vector The product of is equal to the matrix [ with vector The product of, i.e., scalar sum vector Representing matrices respectively The eigenvalues ​​and eigenvectors; Analyzing formula (14), we obtain the following characteristics of the unknown parameter h: Unbiased estimation of the unknown parameter h It is a matrix A feature vector; Step D3: Matrix It is an L×L dimensional square matrix with L eigenvalues ​​λ1, λ2, ..., λ3. L The corresponding feature vector is in It is a normalized eigenvector, k l The eigenvalue is a real number to be estimated. The eigenvalue k can be obtained by performing eigenvalue decomposition on the matrix. l and The following will determine k l ; Depend on The following relationship was obtained: Therefore, real number k l Calculated as Define function have There is only one stable point, that is, the unique solution of formula (18), which is the unbiased estimate of the unknown parameter; by testing the formula In each normalized eigenvector The convergence of the surrounding area allows us to obtain a unique solution, because as the number of iterations increases, only a unique solution converges.

6. The deviation compensation adaptive filtering method based on matrix eigenvalue decomposition as described in claim 5, characterized in that: Step F is implemented as follows: Step F1: Under white output noise, for the EIV-IIR-BCRIV-Like model, define the class auxiliary variable ξ. j The inverse correlation matrix Q is obtained. i Then, by introducing the backward output estimate variable β i The backward output estimation error ζ(j) is obtained, and the input noise variance is estimated by defining the cross-correlation function g(i) between ε(j) and ζ(j). Further, an unbiased estimation expression for h is obtained. After converting the expression into a matrix form based on eigenvalue decomposition, the eigenvectors of the matrix are obtained by solving the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h. The input data vector p at time i i Defined as follows p i =[-y(i-1)-y(i-2)…-y(i-L)x(i-1)x(i-2)…x(i-L)] T The auxiliary variable representation when the output noise is white noise is as follows: ξ j =[-y(jL-1)-y(jL-2)…-y(j-2L)x(j-1)x(j-2)…x(jL)] T iTimep i and ξ j The inverse correlation matrix Q i Represented as Backward output estimate variable β i =[β1β2…β 2L ] T The estimated value Defined as Corresponding backward output estimation for The estimation error ε(j) of the auxiliary variable is in It is a biased estimate of parameter h under the class of auxiliary variables (IV-like); The backward output estimation error ζ(j) is the difference between the backward output y(j-1) and the backward output estimate. The difference is expressed as follows: Define the cross-correlation function g(i) between the estimation error of the auxiliary variable and the estimation error of the backward output as follows: As i→∞, taking the limit of formula (24) yields The estimated value of the input noise variance is obtained from formula (25). When the output noise is white noise, the unbiased estimate of h under the BCRIV-like condition of the EIV-IIR filter is expressed as follows: Multiply both sides of formula (27) by the denominator in the formula. Transform the unbiased estimate of the unknown parameter h into a matrix. The eigenvalue decomposition form, i.e., formula (28). Repeat step D to obtain the matrix by solving for the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h. Step F2: In the case of colored output noise, adjust the auxiliary variable ξ. j The inverse correlation matrix Q is obtained. i Then, by introducing the backward input estimation variable α i Obtain the backward input estimation error By defining ε(j) and The cross-correlation function f(i) is used to obtain an estimate of the input noise variance. Further, an unbiased estimation expression for h is obtained. After converting the expression into a form based on matrix eigenvalue decomposition, the normalized eigenvectors and their corresponding coefficients are solved to obtain the eigenvectors of the matrix. The eigenvectors are the unbiased estimates of the unknown parameter h. When the output noise is colored noise, the auxiliary variable is represented as follows: ξ j =[x(jL-1)x(jL-2)…x(j-2L)x(j-1)x(j-2)…x(jL)] T Backward input estimated variable α i =[α1α2…α 2L ] T The estimated value Defined as Corresponding backward input estimation for Backward input estimation error For backward input x(j-1) and backward input estimation The difference is expressed as follows: Define the estimation error ε(j) of the auxiliary variable and the estimation error of the backward input. The cross-correlation function f(i) is As i→∞, taking the limit of formula (32) yields The estimated value of the input noise variance is obtained from formula (33). When the output noise is colored noise, the unbiased estimate of h under the BCRIV-like condition of the EIV-IIR filter is expressed as follows: Multiply both sides of formula (35) by the denominator in the formula. Transform the unbiased estimate of the unknown parameter h into a matrix. The eigenvalue decomposition form, i.e., formula (36). Repeat step D to obtain the matrix by solving for the normalized eigenvectors and their corresponding coefficients. The eigenvectors are the unbiased estimates of the unknown parameter h.

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