CN115800957A - Deviation compensation adaptive filtering method based on matrix eigenvalue solution - Google Patents

Deviation compensation adaptive filtering method based on matrix eigenvalue solution Download PDF

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CN115800957A
CN115800957A CN202211294211.1A CN202211294211A CN115800957A CN 115800957 A CN115800957 A CN 115800957A CN 202211294211 A CN202211294211 A CN 202211294211A CN 115800957 A CN115800957 A CN 115800957A
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贾丽娟
王丽茹
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a deviation compensation self-adaptive filtering method based on a matrix eigenvalue decomposition method, and belongs to the field of digital filters. The implementation method of the invention comprises the following steps: selecting a corresponding deviation compensation method according to filter categories, and executing a corresponding deviation compensation self-adaptive filtering step, wherein the filter categories are divided into a finite impulse response filter and an infinite impulse response filter, and the infinite impulse response filter comprises two cases of white output noise and colored output noise. The method adopts the characteristic value decomposition of the matrix to obtain system unknown parameters, and realizes unbiased estimation of the system parameters of the deviation compensation adaptive filter; in the calculation process, the output noise variance is not required to be estimated, the cost function is not required to be subjected to derivation for multiple times, the input noise variance is only required to be estimated, the influence of the output noise variance on the estimation value of the unknown parameter is eliminated, and the number of operation parameters is reduced. The method has the advantages of high estimation precision, good filtering effect, good robustness and wide application range.

Description

Deviation compensation adaptive filtering method based on matrix eigenvalue decomposition method
Technical Field
The invention relates to a deviation compensation self-adaptive filtering method based on a matrix eigenvalue decomposition method, and belongs to the field of digital filters.
Background
In recent years, due to the excellent learning performance, tracking performance and adaptive performance of the adaptive filter, the adaptive filter has been widely used in the fields of communication, control, radar, sonar, earthquake, biomedical engineering and the like. However, how to reduce or even eliminate the influence of the input and output noise of the filter on the adaptive filter is always the topic of people's jinjin music. When errors or noise interference exists at the input and output ends of a linear system, the system can be generally described by an Error-in-Errors (EIV) model. For a linear dynamic system with an input end and an output end affected by noise, parameter estimation is a relatively difficult problem, and the problem of parameter estimation of an EIV model becomes a research hotspot of system identification. The EIV model is more similar to a model in practical engineering applications, and thus is widely used in the fields of economic metering, finance and management, image processing, time series analysis, industrial modeling, biomedicine, ecology, and the like.
For the condition that the input noise and the output noise of the filter are stable additive colored noise, the scholars propose a traditional Recursive Least Square (RLS) algorithm, a Least Mean Square (LMS) algorithm and a derivative algorithm thereof, and the like, wherein the algorithms show that the Least Square (LS) estimation of unknown parameters of the system is biased if the influence caused by interference noise is not considered. However, in practical applications, each node is inevitably interfered by noise at the input end and the output end, and the variance of the interference noise is often unknown, so a series of algorithms are proposed by the scholars to solve the problem, and theoretical analysis proves that the estimated deviations of the RLS algorithm and the LMS algorithm are caused by the input noise and are closely related to the variance of the input noise by aiming at a Finite Impulse Response (FIR) adaptive filter under an EIV model, a Bias-Compensated RLS (BCRLS) algorithm and a Bias-Compensated LMS (bics-Compensated LMS) algorithm proposed by Jia et al. The deviation compensation algorithms can estimate the noise variance in real time, so as to compensate the biased algorithm estimation value and realize unbiased estimation of unknown parameters.
In related documents, a scholars such as Jia further provides an adaptive filtering algorithm of a deviation compensation auxiliary variable, and unbiased estimation of an unknown parameter can be achieved.
Disclosure of Invention
The invention mainly aims to provide a deviation compensation adaptive filtering method based on a matrix eigenvalue solution, and aims to solve the problem of how to carry out unbiased estimation on unknown parameters of a system under an EIV adaptive filter model. Meanwhile, the invention is not only suitable for the Finite Impulse Response (FIR) adaptive filter under the variable Error (EIV) model, but also suitable for the Infinite Impulse Response (IIR) adaptive filter under the variable Error (EIV) model, and has good universality.
In order to achieve the above purpose, the invention adopts the following technical scheme.
The invention discloses a deviation compensation self-adaptive filtering method based on a matrix eigenvalue decomposition method, which selects a corresponding deviation compensation method according to filter categories and executes a corresponding deviation compensation self-adaptive filtering step, wherein the filter categories are divided into a finite impulse response filter and an infinite impulse response filter, and system parameter estimation of the infinite impulse response filter is not only related to input noise but also related to output noise, so that filtering of the infinite impulse response filter comprises two conditions of colorless output noise and colored output noise.
Aiming at an EIV-FIR self-adaptive filter model, firstly, a least square estimation value of an unknown parameter h of a transmission system at the moment i of the system is obtained
Figure BDA0003902514790000021
Then, the variance compensation is carried out by utilizing the principle of the variance compensation, wherein the variance information of the input noise is required to be known, and in order to obtain the estimated value of the variance of the input noise, a backward output variable gamma is introduced i Finally obtaining an unbiased estimate of h by a bias compensation recursive least squares algorithm BCRLS
Figure BDA0003902514790000022
The expression (b) is arranged into a matrix form based on eigenvalue decomposition through transformation, and an unbiased estimated value of the parameter h is obtained by solving an eigenvector of the matrix.
Aiming at an EIV-IIR adaptive filter model, firstly, obtaining an estimated value of an unknown parameter h of a system at the moment i of the system under an auxiliary Variable-like (IV-like) algorithm
Figure BDA0003902514790000023
And then the deviation compensation is carried out by utilizing the deviation compensation principle, and because the algorithm contains the output noise information in the class auxiliary variable, only the variance information of the input noise needs to be known. There are two cases, one in which the output noise is white noise, and a backward output variable β is introduced in order to obtain an estimate of the variance of the input noise i Auxiliary variable xi of sum class j Finally, obtaining unbiased estimation of h through a bias compensation hierarchical auxiliary variable algorithm (BCRIV-like)
Figure BDA0003902514790000024
The expression (b) is arranged into a matrix form based on eigenvalue decomposition through transformation, and an unbiased estimated value of the parameter h is obtained by solving an eigenvector of the matrix. Alternatively, the output noise is colored noise, and in order to obtain an estimate of the variance of the input noise, a backward input variable α is introduced i And adjusts the class assist variable ξ j Finally, obtaining unbiased estimation of h through a bias compensation hierarchical auxiliary variable algorithm (BCRIV-like)
Figure BDA0003902514790000025
The expression (b) is arranged into a matrix form based on eigenvalue decomposition through transformation, and an unbiased estimated value of the parameter h is obtained by solving an eigenvector of the matrix.
The invention discloses a deviation compensation self-adaptive filtering method based on a matrix eigenvalue solution method, which comprises the following steps:
step 0: selecting a corresponding deviation compensation method according to filter categories, and executing a corresponding deviation compensation self-adaptive filtering step, wherein the filter categories are divided into a finite impulse response filter and an infinite impulse response filter, and the infinite impulse response filter is further divided into an infinite impulse response filter for white output noise and an infinite impulse response filter for colored output noise.
For finite impulse response filter, the deviation compensation adaptive filtering method based on matrix eigenvalue decomposition comprises the following steps A to E:
and A, constructing an FIR filter under an EIV model under constraint conditions to obtain the relation between input and output and system parameters of the filter.
Step A1, in order to construct an EIV-FIR filter, defining the filter to meet 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 noises having a mean value of 0 and being uncorrelated with each other, and the noise variances are respectively unknown
Figure BDA0003902514790000031
And
Figure BDA0003902514790000032
step A2, expressing the data into the following vector form
s i ＝[s(i)s(i-1)…s(i-L+1)] T
Figure BDA0003902514790000033
h＝[h 0 h 1 …h L-1 ] T
Wherein s (i) is a noiseless input signal at time i, x (i) is a noisy input signal at time i, and n (i) is an input noise at time i, that is, x (i) = s (i) + n (i) is satisfied; e (i) is the output noise at the moment i, and h is a weight vector representing the filter system and is an unknown parameter to be estimated; l is the order of the filter and the superscript T represents the transpose operator.
Step A3 the EIV-FIR filter model can be expressed as
Figure BDA0003902514790000034
Where y (i) is the noisy output signal at time i, and v (i) is the composite noise, which can be expressed as
Figure BDA0003902514790000035
B, on the basis of the EIV-FIR filter model, obtaining the least square estimation value of the unknown weight vector h at the moment i according to the Least Square (LS) principle
Figure BDA0003902514790000036
As can be seen by the analysis, the method,
Figure BDA0003902514790000037
is biased.
Step B1, according to the least square LS principle, the LS estimation value of the unknown weight vector h is expressed as
Figure BDA0003902514790000041
Substituting the formula (2) into the formula (4) results in the following relationship
Figure BDA0003902514790000042
Step B2, in order to obtain an estimated value
Figure BDA0003902514790000043
The deviation from the true value h is obtained by limiting the above formula by using the conditions (1) to (3) and the ergodicity of the stable random process
Figure BDA0003902514790000044
As can be seen from the analysis of equation (6), if the unknown parameters are estimated using the RLS algorithm, the estimated value under the least-squares criterion follows i → ∞ without noise interference
Figure BDA0003902514790000045
Converge to h, but because in the EIV model both the input and the output are disturbed by noise, i.e.
Figure BDA0003902514790000046
The estimation of the conventional RLS algorithm is biased,
Figure BDA0003902514790000047
the difference between h and h is
Figure BDA0003902514790000048
At time i, h in equation (6) is replaced with the unbiased estimate of the previous time
Figure BDA0003902514790000049
The deviation is expressed as
Figure BDA00039025147900000410
Estimated value of h
Figure BDA00039025147900000411
Is further shown as
Figure BDA00039025147900000412
Wherein the inverse correlation matrix P i Is shown below
Figure BDA00039025147900000413
Analyzing the result obtained in step B shows that the estimation value of the unknown parameter obtained by the Recursive Least Squares (RLS) has deviation, and the deviation is related to the variance of the input noise.
Step C, according to the deviation compensation recursive least square principle (BCRLS), introducing a backward output estimation variable gamma i Obtaining backward output estimation error zeta (j), and obtaining estimation value of input noise variance by defining cross-correlation function g (i) of least square estimation error epsilon (j) and zeta (j)
Figure BDA00039025147900000414
Further on the biased estimation obtained in step B
Figure BDA00039025147900000415
Performing deviation compensation to obtain unbiased estimation
Figure BDA0003902514790000051
Step C1, introducing backward output estimation variable gamma according to deviation compensation recursive least squares (BCRLS) i ＝[γ 1 γ 2 …γ L ] T (ii) a According to the linear prediction theory, expressions of backward output estimation errors zeta (j) and least square estimation errors epsilon (j) are obtained.
γ i Is estimated value of
Figure BDA0003902514790000052
Is shown as
Figure BDA0003902514790000053
According to the linear prediction theory, the backward output estimation error ζ (j) and the least square estimation error ε (j) are expressed as follows
Figure BDA0003902514790000054
Figure BDA0003902514790000055
And C2, obtaining a cross-correlation function g (i) of the backward output estimation error zeta (j) and the least square estimation error epsilon (j), and obtaining an estimated value of the input noise variance by taking a limit.
The cross-correlation function of ε (j) and ζ (j) is represented by g (i) and is defined as follows
Figure BDA0003902514790000056
Under the conditions (1) and (2), the limit is taken when i → ∞ is
Figure BDA0003902514790000057
Then an estimate of the variance of the input noise
Figure BDA0003902514790000058
Can be expressed as
Figure BDA0003902514790000059
Step C3, under the EIV-FIR filter model, the unbiased estimation obtained according to the steps
Figure BDA00039025147900000510
Is shown below
Figure BDA0003902514790000061
Step D, multiplying the two sides of the equation of the expression of the unbiased estimated value of the unknown parameter h obtained in the step C by the scalar in the formula (13) respectively
Figure BDA0003902514790000062
Transforming into matrix eigenvalue decomposition form, and solving normalized eigenvector
Figure BDA0003902514790000063
And corresponding theretoCoefficient k l And obtaining the eigenvector of the matrix, wherein the eigenvector is the unbiased estimation value of the unknown parameter h.
Step D1: in the obtained unbiased estimated value expression (13) of the unknown parameter h, the denominator
Figure BDA0003902514790000064
Is a scalar quantity, and both sides of the formula (13) are simultaneously multiplied by the scalar quantity
Figure BDA0003902514790000065
The expression of the unbiased estimated value of the unknown parameter h is transformed into a constructed matrix eigenvalue decomposition form as shown in formula (14):
Figure BDA0003902514790000066
step D2: analysis of the above equation, scalar, according to the principle of matrix analysis
Figure BDA0003902514790000067
And vector
Figure BDA0003902514790000068
Is equal to the matrix
Figure BDA0003902514790000069
And vector
Figure BDA00039025147900000610
Product of, i.e. scalar
Figure BDA00039025147900000611
Sum vector
Figure BDA00039025147900000612
Respectively represent a matrix
Figure BDA00039025147900000613
The eigenvalues and eigenvectors. The analysis formula (14) obtains the unknown parameter h with the following characteristics: unbiased estimation of unknown parameter h
Figure BDA00039025147900000614
Is a matrix
Figure BDA00039025147900000615
A feature vector of (2).
And D3: matrix of
Figure BDA00039025147900000616
Is a L x L dimensional square matrix whose L eigenvalues are lambda 1 ,λ 2 ,…,λ L The corresponding feature vector is
Figure BDA00039025147900000617
Wherein
Figure BDA00039025147900000618
Is a normalized feature vector, k l Is a real number to be estimated, and an eigenvalue k can be obtained by performing eigenvalue decomposition on the matrix l And
Figure BDA00039025147900000619
k will be determined below l 。
By
Figure BDA00039025147900000620
The following relationship can be obtained:
Figure BDA00039025147900000621
hence, the real number k l Is calculated as
Figure BDA00039025147900000622
Defining functions
Figure BDA00039025147900000623
Figure BDA0003902514790000071
Is provided with
Figure BDA0003902514790000072
Figure BDA0003902514790000073
There is only a single stable point, i.e. the only solution of equation (18), i.e. the unbiased estimated value of the unknown parameter. By test formulas
Figure BDA0003902514790000074
At each normalized feature vector
Figure BDA0003902514790000075
Near convergence, a unique solution can be obtained because as the number of iterations increases, only the unique solution is converged.
And E, step E: and D, constructing an EIV-FIR (equivalent irregular value-finite Impulse response) adaptive filter according to the unbiased estimated value h obtained in the step D, giving an optimal estimate of an expected response through a system weight vector parameter h updated in real time under the condition of giving an input signal sample x (i) by using the Bias Compensation Recursive Least Squares (BCRLS), so that the mean square value of the estimation error epsilon (i) is minimum, improving the estimation precision, and meanwhile, estimating h through a gradient descent method in the past, reducing the operation complexity and saving the signal processing cost.
The method aims at solving the unknown parameter h of the BCRLS algorithm of the EIV-FIR adaptive filter based on the matrix eigenvalue solution, and estimates the unknown parameter h of the variable infinite impulse response adaptive filter (EIV-IIR) model with errors by using the matrix eigenvalue solution, so that the application range of the method is widened from the finite impulse response filter to the infinite impulse response filter.
For an infinite impulse response filter, the deviation compensation adaptive filtering method based on the matrix eigenvalue solution comprises the following steps of F to G:
and F, for the EIV-IIR filter model, if the BCRLS algorithm is used for carrying out unbiased estimation on h, input noise and output noise need to be estimated simultaneously, and the calculation complexity is high, so that for the EIV-IIR filter, a BCRIV-like algorithm is used for carrying out unbiased estimation on h by using a matrix eigenvalue decomposition-based method. Biased estimation value of system unknown parameter h under (IV-like) algorithm at moment i
Figure BDA0003902514790000076
Since the algorithm already includes the output noise information in the class auxiliary variable, it is only necessary to know the variance information of the input noise.
There are two cases, one in which the output noise is white noise, and a backward output variable β is introduced in order to obtain an estimate of the variance of the input noise i Auxiliary variable xi of sum class j Finally, obtaining unbiased estimation of h through a bias compensation hierarchical auxiliary variable algorithm (BCRIV-like)
Figure BDA0003902514790000077
The expression (b) is transformed and arranged into a matrix form based on eigenvalue decomposition, and an unbiased estimated value of the parameter h is obtained by solving an eigenvector of the matrix. Alternatively, the output noise is colored noise, and in order to obtain an estimate of the variance of the input noise, a backward input variable α is introduced i And adjusts class assist variable ξ j Finally, obtaining unbiased estimation of h through deviation compensation hierarchical auxiliary variable algorithm (BCRIV-like)
Figure BDA0003902514790000078
The expression (b) is arranged into a matrix form based on eigenvalue decomposition through transformation, and an unbiased estimated value of the parameter h is obtained by solving an eigenvector of the matrix.
Step F1: defining a class auxiliary variable xi for an EIV-IIR-BCRIV-Like model in the case of white output noise j To find an inverse correlation matrix Q i Then, by introducing backward output estimationMeasuring variable beta i Obtaining backward output estimation error zeta (j), and obtaining estimation value of input noise variance by defining cross-correlation function g (i) of epsilon (j) and zeta (j)
Figure BDA0003902514790000081
And further obtaining an unbiased estimation expression of h, converting the expression into a matrix form based on eigenvalue decomposition, and then solving the normalized eigenvector and the corresponding coefficient thereof to further obtain the eigenvector of the matrix, wherein the eigenvector is the unbiased estimation value of the unknown parameter h.
i moment input data vector p i Is 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 quasi-auxiliary variable when the output noise is white noise is expressed as follows
ξ j ＝[-y(j-L-1)-y(j-L-2)…-y(j-2L)x(j-1)x(j-2)…x(j-L)] T
i time p i And xi j Of the inverse correlation matrix Q i Is shown as
Figure BDA0003902514790000082
Backward output estimation variable beta i ＝[β 1 β 2 …β 2L ] T Is estimated by
Figure BDA0003902514790000083
Is defined as
Figure BDA0003902514790000084
Corresponding backward output estimation
Figure BDA0003902514790000085
Is composed of
Figure BDA0003902514790000086
The auxiliary-like variable estimation error epsilon (j) is
Figure BDA0003902514790000087
Wherein
Figure BDA0003902514790000088
Is a biased estimate of the parameter h under the class assist variable (IV-like).
The backward output estimation error ζ (j) is the backward output y (j-1) and the backward output estimation
Figure BDA0003902514790000089
Is expressed as follows
Figure BDA0003902514790000091
Defining a cross-correlation function g (i) of the class auxiliary variable estimation error and the backward output estimation error as
Figure BDA0003902514790000092
When i → ∞ is satisfied, the limit is set to formula (24)
Figure BDA0003902514790000093
The estimate of the variance of the input noise from equation (25) is
Figure BDA0003902514790000094
When the output noise is white noise, the unbiased estimation of h under BCRIV-like of EIV-IIR filter is expressed as follows
Figure BDA0003902514790000095
Multiplying both sides of the formula (27) by the denominator in the formula
Figure BDA0003902514790000096
Transforming an unbiased estimate expression of an unknown parameter h into a matrix
Figure BDA0003902514790000097
Eigenvalue decomposition form, equation (28).
Figure BDA0003902514790000098
And D, repeating the step D, solving the normalized eigenvector and the corresponding coefficient thereof to further obtain a matrix
Figure BDA0003902514790000101
The eigenvector is an unbiased estimation value of the unknown parameter h.
Step F2: in case of colored output noise, the class assist variable ξ is adjusted j To find an inverse correlation matrix Q i Then, the variable α is estimated by introducing backward input i Obtaining a backward input estimation error
Figure BDA0003902514790000102
By defining ε (j) and
Figure BDA0003902514790000103
to obtain an estimate of the variance of the input noise
Figure BDA0003902514790000104
Further obtaining an unbiased estimation expression of h, converting the expression into a form based on matrix eigenvalue decomposition, and then obtaining the eigenvector of the matrix by solving the normalized eigenvector and the corresponding coefficient thereof, wherein the eigenvector is the direction of the characteristicThe quantity is an unbiased estimated value of the unknown parameter h.
The auxiliary variables for the case where the output noise is colored noise are expressed as follows
ξ j ＝[x(j-L-1)x(j-L-2)…x(j-2L)x(j-1)x(j-2)…x(j-L)] T
Backward input estimation variable alpha i ＝[α 1 α 2 …α 2L ] T Is estimated by
Figure BDA0003902514790000105
Is defined as
Figure BDA0003902514790000106
Corresponding backward input estimation
Figure BDA0003902514790000107
Is composed of
Figure BDA0003902514790000108
Backward input estimation error
Figure BDA0003902514790000109
For backward input x (j-1) and backward input estimation
Figure BDA00039025147900001010
Is expressed as follows
Figure BDA00039025147900001011
Defining class auxiliary variable estimation error epsilon (j) and backward input estimation error
Figure BDA00039025147900001012
The cross-correlation function f (i) of
Figure BDA00039025147900001013
When i → ∞ is satisfied, the limit is set to formula (32)
Figure BDA0003902514790000111
The estimate of the variance of the input noise, which can be obtained from equation (33), is
Figure BDA0003902514790000112
When the output noise is colored noise, the unbiased estimation of h under BCRIV-like of the EIV-IIR filter is expressed as follows
Figure BDA0003902514790000113
Multiplying both sides of the formula (35) by the denominator in the formula
Figure BDA0003902514790000114
Transforming an unbiased estimate expression of an unknown parameter h into a matrix
Figure BDA0003902514790000115
The eigenvalue decomposition form, equation (36).
Figure BDA0003902514790000116
And D, repeating the step D, solving the normalized eigenvector and the corresponding coefficient thereof to further obtain a matrix
Figure BDA0003902514790000117
The feature vector of (2), which is an unbiased estimation value of the unknown parameter h.
And G, constructing an EIV-IIR adaptive filter according to the unbiased estimated value h obtained in the step F, and giving an optimal estimation of expected response through a system weight vector parameter h updated in real time under the condition of a given input signal sample x (i) by using the bias compensation recursive auxiliary variable algorithm (BCRIV-like), so that the mean square value of an estimation error epsilon (i) is minimum, the estimation precision is improved, and meanwhile, compared with the prior art, the h is estimated through a gradient descent method, the operation complexity is reduced, and the signal processing cost is saved.
Has the advantages that:
1. compared with the traditional method for carrying out approximation estimation on the unknown parameter h based on the gradient descent method, the method does not need to use a cost function and derive the cost function, obviously saves the calculation cost, and has more obvious advantages when the order of a filter is large.
2. The deviation compensation type adaptive filtering method based on the matrix eigenvalue decomposition is not only suitable for a deviation compensation EIV-FIR adaptive filter, but also suitable for the deviation compensation EIV-IIR adaptive filter, and has good universality.
3. The invention discloses a bias compensation type self-adaptive filtering method based on a matrix eigenvalue decomposition, which realizes unbiased estimation of unknown parameters by using the matrix eigenvalue decomposition under an EIV-FIR (extreme empirical mode decomposition-finite impulse response) condition, and simulation results show that the method has high calculation precision and good curve fitting degree, and for input noise and output noise with different intensities, the difference between a BCRLS (binary redundancy standard) algorithm and a traditional RLS (recursive redundancy standard) algorithm on MSD (Mean Square Deviation) is prominently realized, and the method has good estimation precision and robustness.
4. The invention discloses a bias compensation adaptive filtering method based on a matrix eigenvalue solution, which selects an adaptive bias compensation algorithm for different filters, uses a BCRLS algorithm for an EIV-FIR filter, selects a BCRIV-like algorithm for the condition of output noise interference in the EIV-IIR filter, does not need to estimate and calculate the output noise variance in the calculation process, only needs to estimate the input noise variance, eliminates the influence of the output noise variance on the estimation value of unknown parameters, reduces the number of operation parameters and reduces the calculation complexity.
Drawings
FIG. 1 is a variable error-containing finite impulse response (EIV-FIR) adaptive filter model to which the present invention relates.
FIG. 2 is a variable infinite impulse response (EIV-IIR) adaptive filter model with error according to the present invention.
Fig. 3 is a performance comparison of a deviation compensation recursive least square algorithm based on a matrix solution method and a conventional recursive least square algorithm under the condition that an input signal is a colored gaussian signal and different input noises exist in an EIV-FIR filter at a certain output noise.
Fig. 4 is a performance comparison of a bias compensation recursive least square algorithm based on a matrix solution method with a conventional recursive least square algorithm in the case that an input signal is a colored gaussian signal and different output noises exist in an EIV-FIR filter at a certain input noise.
FIG. 5 is a flow chart of a bias compensation FIR adaptive filtering method based on matrix eigenvalue decomposition disclosed in the present invention.
FIG. 6 is a flow chart of a bias compensation IIR adaptive filtering method based on a matrix eigenvalue decomposition method disclosed by the invention.
Detailed Description
The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and examples. The present embodiment is implemented on the premise of the technical solution of the present invention, and a detailed implementation manner and a specific operation process are given, but the scope of the present invention is not limited to the following embodiments.
As shown in fig. 5 and 6, the deviation compensation adaptive filtering method based on the matrix eigenvalue decomposition disclosed in this embodiment includes the following steps:
step 0: a corresponding offset compensation method is selected according to a filter class, which is classified into a finite impulse response filter (fig. 1) and an infinite impulse response filter (fig. 2), in which an infinite impulse response filter for white output noise and an infinite impulse response filter for colored output noise are further classified, and a corresponding offset compensation adaptive filtering step is performed.
Referring to fig. 5, for the fir filter, the bias compensation adaptive filtering method based on the matrix eigenvalue decomposition includes steps a to E:
step A, as shown in figure 1, constructing an FIR filter under an EIV model under constraint conditions to obtain the relationship among input, output and filter system parameters.
Step A1, in order to construct an EIV-FIR filter, defining the filter to meet 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 noises having a mean value of 0 and being uncorrelated with each other, and the noise variances are respectively unknown
Figure BDA0003902514790000131
And
Figure BDA0003902514790000132
step A2, expressing the data into the following vector form
s i ＝[s(i)s(i-1)…s(i-L+1)] T
Figure BDA0003902514790000133
h＝[h 0 h 1 …h L-1 ] T
Wherein s (i) is a noiseless input signal at time i, x (i) is a noiseless input signal at time i, and n (i) is input noise at time i, that is, x (i) = s (i) + n (i) is satisfied; e (i) is the output noise at the moment i, and h is a weight vector representing the filter system and is an unknown parameter to be estimated; l is the order of the filter and the superscript T represents the transpose operator.
Step A3 the EIV-FIR filter model can be expressed as
Figure BDA0003902514790000141
Where y (i) is the noisy output signal at time i, and v (i) is the complex noise, which can be expressed as
Figure BDA0003902514790000142
Step B, obtaining a least square estimation value of the unknown weight vector h at the moment i according to a Least Square (LS) principle on the basis of an EIV-FIR filter model
Figure BDA0003902514790000143
As can be seen by the analysis, the method,
Figure BDA0003902514790000144
is biased.
Step B1, according to the least square LS principle, the LS estimation value of the unknown weight vector h is expressed as
Figure BDA0003902514790000145
Substituting the formula (2) into the formula (4) results in the following relationship
Figure BDA0003902514790000146
Step B2, in order to obtain an estimated value
Figure BDA0003902514790000147
The deviation from the true value h, using the conditions (1) to (3) and the ergodic performance of the stable random process, the limit of the above equation can be obtained
Figure BDA0003902514790000148
As can be seen from the analysis of equation (6), if the unknown parameters are estimated using the RLS algorithm, the estimated value under the least-squares criterion follows i → ∞ without noise interference
Figure BDA0003902514790000149
Converge to h, but since in the EIV model both the input and the output are disturbed by noise, i.e.
Figure BDA00039025147900001410
The estimation of the conventional RLS algorithm is biased,
Figure BDA00039025147900001411
the difference between h and h is
Figure BDA00039025147900001412
At time i, h in equation (6) is replaced with the unbiased estimate of the previous time
Figure BDA00039025147900001413
The deviation is expressed as
Figure BDA00039025147900001414
Estimated value of h
Figure BDA00039025147900001415
Is further shown as
Figure BDA00039025147900001416
Wherein the inverse correlation matrix P i Is shown as follows
Figure BDA0003902514790000151
Analyzing the result obtained in step B shows that the estimation value of the unknown parameter obtained by the Recursive Least Squares (RLS) has deviation, and the deviation is related to the variance of the input noise.
Step C, according to the deviation compensation recursive least square principle (BCRLS), introducing a backward output estimation variable gamma i Obtaining backward output estimation error zeta (j), obtaining estimation value of input noise variance by defining cross-correlation function g (i) of least square estimation error epsilon (j) and zeta (j)
Figure BDA0003902514790000152
Further to the biased estimation obtained in step B
Figure BDA0003902514790000153
Performing deviation compensation to obtain unbiased estimation
Figure BDA0003902514790000154
Step C1, introducing backward output estimation variable gamma according to deviation compensation recursive least squares (BCRLS) i ＝[γ 1 γ 2 …γ L ] T (ii) a According to the linear prediction theory, expressions of backward output estimation errors zeta (j) and least square estimation errors epsilon (j) are obtained.
γ i Is estimated value of
Figure BDA0003902514790000155
Is shown as
Figure BDA0003902514790000156
According to the linear prediction theory, the backward output estimation error ζ (j) and the least square estimation error ε (j) are expressed as follows
Figure BDA0003902514790000157
Figure BDA0003902514790000158
And C2, obtaining a cross-correlation function g (i) of the backward output estimation error zeta (j) and the least square estimation error epsilon (j), and obtaining an estimation value of the variance of the input noise by taking a limit.
The cross-correlation function of ε (j) and ζ (j) is represented by g (i), which is defined as follows
Figure BDA0003902514790000159
Under the conditions (1) and (2), the limit is taken when i → ∞ is
Figure BDA00039025147900001510
Then an estimate of the variance of the input noise is made
Figure BDA0003902514790000161
Can be expressed as
Figure BDA0003902514790000162
Step C3, under the EIV-FIR filter model, the unbiased estimation obtained according to the above steps
Figure BDA0003902514790000163
Is shown below
Figure BDA0003902514790000164
Step D, multiplying the two sides of the equation of the expression of the unbiased estimated value of the unknown parameter h obtained in the step C by the scalar in the formula (13) respectively
Figure BDA0003902514790000165
Transforming into matrix eigenvalue decomposition form, and solving normalized eigenvector
Figure BDA0003902514790000166
And its corresponding coefficient k l And obtaining a characteristic vector of the matrix, wherein the characteristic vector is an unbiased estimation value of the unknown parameter h.
Step D1: in the obtained unbiased estimated value expression (13) of the unknown parameter h, the denominator
Figure BDA0003902514790000167
Is a scalar quantity, and both sides of the formula (13) are simultaneously multiplied by the scalar quantity
Figure BDA0003902514790000168
The unbiased estimated value expression of the unknown parameter h is transformed into a constructed matrix eigenvalue decomposition form as shown in formula (14):
Figure BDA0003902514790000169
step D2: analyzing the above equation, scalar, according to the principle of matrix analysis
Figure BDA00039025147900001610
And vector
Figure BDA00039025147900001611
Is equal to the matrix
Figure BDA00039025147900001612
And vector
Figure BDA00039025147900001613
Product of (i.e. scalar)
Figure BDA00039025147900001614
Sum vector
Figure BDA00039025147900001615
Respectively represent a matrix
Figure BDA00039025147900001616
The eigenvalues and eigenvectors. The analysis formula (14) obtains the unknown parameter h with the following characteristics: unbiased estimation of unknown parameter h
Figure BDA00039025147900001617
Is a matrix
Figure BDA00039025147900001618
A feature vector of (2).
And D3: matrix array
Figure BDA00039025147900001619
Is a L x L dimensional square matrix whose L eigenvalues are lambda 1 ,λ 2 ,…,λ L The corresponding feature vector is
Figure BDA00039025147900001620
Wherein
Figure BDA00039025147900001621
Is a normalized feature vector, k l Is a real number to be estimated, and an eigenvalue k can be obtained by performing eigenvalue decomposition on the matrix l And
Figure BDA00039025147900001622
k will be determined below l 。
By
Figure BDA00039025147900001623
The following relationship can be obtained:
Figure BDA0003902514790000171
hence, a real number k l Is calculated as
Figure BDA0003902514790000172
Defining functions
Figure BDA0003902514790000173
Figure BDA0003902514790000174
Is provided with
Figure BDA0003902514790000175
Figure BDA0003902514790000176
There is only a single stable point, i.e. a single solution of equation (18), i.e. an unbiased estimate of the unknown parameter. By means of test formulas
Figure BDA0003902514790000177
At each normalized feature vector
Figure BDA0003902514790000178
Near convergence, a unique solution can be obtained because only the unique solution converges as the number of iterations increases.
And E, step E: and D, constructing an EIV-FIR (equivalent irregular value-finite Impulse response) adaptive filter according to the unbiased estimated value h obtained in the step D, and giving an optimal estimate of an expected response through a system weight vector parameter h updated in real time under the condition of giving an input signal sample x (i) by using the Bias Compensation Recursive Least Squares (BCRLS), so that the mean square value of the estimation error epsilon (i) is minimum, the estimation precision is improved, and meanwhile, compared with the prior method of estimating h through a gradient descent method, the method reduces the operation complexity and saves the signal processing cost.
The method aims at solving the unknown parameter h of the BCRLS algorithm of the EIV-FIR adaptive filter based on the matrix eigenvalue decomposition, and estimates the unknown parameter h of the variable error-containing infinite impulse response adaptive filter (EIV-IIR) model by using the matrix eigenvalue decomposition, so that the application range of the method is widened from a finite impulse response filter to an infinite impulse response filter.
Referring to fig. 6, for the infinite impulse response filter, the bias compensation adaptive filtering method based on the matrix eigenvalue solution includes steps F to G:
step (ii) ofAs shown in fig. 2, for the EIV-IIR filter model, if the BCRLS algorithm is used to estimate h unbiased, it is necessary to estimate input and output noise simultaneously, and the calculation is complex and difficult, so for the EIV-IIR filter, we use the BCRIV-like algorithm to estimate h unbiased by using a matrix eigenvalue decomposition-based method. Biased estimation value of unknown parameter h of system under (IV-like) algorithm at moment i
Figure BDA0003902514790000179
Since the algorithm already includes the output noise information in the class auxiliary variable, only the variance information of the input noise needs to be known.
There are two cases, one in which the output noise is white noise, and a backward output variable β is introduced in order to obtain an estimate of the variance of the input noise i Auxiliary variable xi of sum class j Finally, obtaining unbiased estimation of h through deviation compensation hierarchical auxiliary variable algorithm (BCRIV-like)
Figure BDA0003902514790000181
The expression (b) is transformed and arranged into a matrix form based on eigenvalue decomposition, and an unbiased estimated value of the parameter h is obtained by solving an eigenvector of the matrix. Alternatively, the output noise is colored noise, and in order to obtain an estimate of the variance of the input noise, a backward input variable α is introduced i And adjusts class assist variable ξ j Finally, obtaining unbiased estimation of h through deviation compensation hierarchical auxiliary variable algorithm (BCRIV-like)
Figure BDA0003902514790000182
The expression (b) is arranged into a matrix form based on eigenvalue decomposition through transformation, and an unbiased estimated value of the parameter h is obtained by solving an eigenvector of the matrix.
Step F1: in the case of white output noise, a class auxiliary variable ξ is defined for the EIV-IIR-BCRIV-Like model j To find an inverse correlation matrix Q i Thereafter, the variable β is estimated by introducing a backward output i Obtaining a backward output estimation error ζ (j) by defining ∈ (j) andzeta (j) cross-correlation function g (i) to obtain an estimate of the input noise variance
Figure BDA0003902514790000183
And further obtaining an unbiased estimation expression of h, converting the expression into a matrix form based on eigenvalue decomposition, and then obtaining an eigenvector of the matrix by solving the normalized eigenvector and a coefficient corresponding to the normalized eigenvector, wherein the eigenvector is the unbiased estimation value of the unknown parameter h.
i data vector p input at time i Is 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 class auxiliary variable when the output noise is white noise is expressed as follows
ξ j ＝[-y(j-L-1)-y(j-L-2)…-y(j-2L)x(j-1)x(j-2)…x(j-L)] T
i moment p i And xi j Of (2) an inverse correlation matrix Q i Is shown as
Figure BDA0003902514790000184
Backward output estimation variable beta i ＝[β 1 β 2 …β 2L ] T Is estimated value of
Figure BDA0003902514790000185
Is defined as
Figure BDA0003902514790000186
Corresponding backward output estimation
Figure BDA0003902514790000187
Is composed of
Figure BDA0003902514790000188
The class auxiliary variable estimation error epsilon (j) is
Figure BDA0003902514790000191
Wherein
Figure BDA0003902514790000193
Is a biased estimate of the parameter h under the class auxiliary variable (IV-like).
The backward output estimation error zeta (j) is the backward output y (j-1) and the backward output estimation
Figure BDA0003902514790000194
Is expressed as follows
Figure BDA0003902514790000195
Defining a cross-correlation function g (i) of the like auxiliary variable estimation error and the backward output estimation error as
Figure BDA0003902514790000196
When i → ∞ is satisfied, the limit is set to formula (24)
Figure BDA0003902514790000197
The estimate of the variance of the input noise from equation (25) is
Figure BDA0003902514790000198
When the output noise is white noise, the unbiased estimation of h under BCRIV-like of the EIV-IIR filter is expressed as follows
Figure BDA0003902514790000199
Multiplying both sides of the formula (27) by the denominator in the formula
Figure BDA00039025147900001910
Transforming an unbiased estimate expression of an unknown parameter h into a matrix
Figure BDA0003902514790000201
Eigenvalue decomposition form, equation (28).
Figure BDA0003902514790000202
And D, repeating the step D, solving the normalized characteristic vector and the corresponding coefficient thereof to further obtain a matrix
Figure BDA0003902514790000203
The feature vector of (2), which is an unbiased estimation value of the unknown parameter h.
Step F2: in case of colored output noise, the class assist variable ξ is adjusted j To find an inverse correlation matrix Q i Thereafter, the variable α is estimated by introducing a backward input i Obtaining a backward input estimation error
Figure BDA0003902514790000204
By defining ε (j) and
Figure BDA0003902514790000205
obtaining an estimate of the variance of the input noise
Figure BDA0003902514790000206
And further obtaining an unbiased estimation expression of h, converting the expression into a form based on matrix eigenvalue decomposition, and then obtaining an eigenvector of the matrix by solving the normalized eigenvector and a coefficient corresponding to the normalized eigenvector, wherein the eigenvector is the unbiased estimation value of the unknown parameter h.
The auxiliary variables for the case where the output noise is colored noise are expressed as follows
ξ j ＝[x(j-L-1)x(j-L-2)…x(j-2L)x(j-1)x(j-2)…x(j-L)] T
Backward input estimation variable alpha i ＝[α 1 α 2 …α 2L ] T Is estimated by
Figure BDA0003902514790000207
Is defined as
Figure BDA0003902514790000208
Corresponding backward input estimation
Figure BDA0003902514790000209
Is composed of
Figure BDA00039025147900002010
Backward input estimation error
Figure BDA00039025147900002011
For backward input x (j-1) and backward input estimation
Figure BDA00039025147900002012
Is expressed as follows
Figure BDA00039025147900002013
Defining class auxiliary variable estimation error epsilon (j) and backward input estimation error
Figure BDA00039025147900002014
The cross-correlation function f (i) of
Figure BDA0003902514790000211
When i → ∞ is reached, the limit is taken on the formula (32)
Figure BDA0003902514790000212
The estimate of the variance of the input noise from equation (33) is
Figure BDA0003902514790000213
When the output noise is colored noise, the unbiased estimation of h under BCRIV-like of the EIV-IIR filter is expressed as follows
Figure BDA0003902514790000214
Multiplying both sides of the formula (35) by the denominator in the formula
Figure BDA0003902514790000215
Transforming an unbiased estimate expression of an unknown parameter h into a matrix
Figure BDA0003902514790000216
The eigenvalue decomposition form, equation (36).
Figure BDA0003902514790000217
And D, repeating the step D, solving the normalized eigenvector and the corresponding coefficient thereof to further obtain a matrix
Figure BDA0003902514790000218
The feature vector of (2), which is an unbiased estimation value of the unknown parameter h.
And G, constructing an EIV-IIR adaptive filter according to the unbiased estimated value h obtained in the step F, and giving an optimal estimation of expected response through a system weight vector parameter h updated in real time under the condition of giving an input signal sample x (i) by using the bias compensation recursive auxiliary variable algorithm (BCRIV-like), so that the mean square value of an estimation error epsilon (i) is minimum, the estimation precision is improved, and meanwhile, compared with the conventional method of estimating h through a gradient descent method, the operation complexity is reduced, and the signal processing cost is saved.
The effect of this embodiment can be verified by the following experiment:
the input signal s (i) is a zero-mean gaussian colored signal, generated as follows:
s(i)＝randn(i)-0.3×randn(i-1)+0.5×randn(i-2)-0.7×randn(i-3)+0.9×randn(i-4)
the iteration number of the simulation experiment is 5000, the independent experiment number is 100, the order of the FIR filter is specified to be 5, and the true value of the unknown parameter is set to be [ -0.3, -0.9,0.8, -0.7 and 0.6]The experiments were performed in two cases using the matrix eigenvalue based solution presented herein. (1) Variance of output noise
Figure BDA0003902514790000221
Variance of input noise
Figure BDA0003902514790000222
Analyzing the local EIV-FIR-BCRLS algorithm of the adaptive filter node and the Mean Square Deviation (MSD) of the traditional RLS algorithm; (2) Variance of input noise
Figure BDA0003902514790000223
Variance of output noise
Figure BDA0003902514790000224
And analyzing the local EIV-FIR-BCRLS algorithm of the adaptive filter node and the MSD under the traditional RLS algorithm.
Fig. 3 shows the local EIV-FIR-BCRLS algorithm of the adaptive filter node and the MSD under the traditional RLS algorithm under the condition of white input gaussian noise with different variances at a certain output noise, and it can be seen from the figure that the estimation accuracy and the convergence performance of the two algorithms are steadily improved along with the increase of the number of iterations, and meanwhile, the mean square variance of the two algorithms is increased along with the increase of the input noise intensity, and the estimation accuracy is reduced.
Fig. 4 shows that when input noise is constant, the local EIV-FIR-BCRLS algorithm of the adaptive filter node under the condition of output white gaussian noise with different variances and the MSD under the conventional RLS algorithm, it can be seen from the figure that as the number of iterations increases, the estimation accuracy and the convergence performance of the two algorithms are steadily improved, and meanwhile, as the output noise strength increases, the mean square variance of the two algorithms increases, and the estimation accuracy decreases. All algorithms of the simulation use unknown parameters which are provided by the method and are solved based on matrix eigenvalue decomposition, and the method has good effect as can be seen from experimental results.
The above-mentioned embodiments are merely preferred embodiments of the present invention, and the present invention is not limited thereto, and the technical means disclosed in the present invention is not limited to the technical means disclosed in the above-mentioned embodiments, but also includes technical means formed by any combination of the above technical features. It will be understood by those skilled in the art that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the invention.

Claims (6)

1. A deviation compensation adaptive filtering method based on a matrix eigenvalue solution is characterized in that: comprises the following steps of (a) preparing a solution,
step 0: selecting a corresponding deviation compensation method according to the filter category, and executing a corresponding deviation compensation self-adaptive filtering step, wherein the filter category is divided into a finite impulse response filter and an infinite impulse response filter, and the infinite impulse response filter is further divided into an infinite impulse response filter for white output noise and an infinite impulse response filter for colored output noise;
aiming at an EIV-FIR adaptive filter, the deviation compensation adaptive filtering method based on the matrix eigenvalue solution comprises the following steps of A to E:
a, providing constraint conditions, constructing an FIR filter under an EIV model, and obtaining the relation between input and output and system parameters of the filter;
b, obtaining a least square estimation value of the unknown weight vector h at the moment i according to a least square LS principle on the basis of an EIV-FIR filter model
Figure FDA0003902514780000011
As can be seen by the analysis, the method,
Figure FDA0003902514780000012
is biased, the bias being related to the input noise variance;
step C, introducing backward output estimation variables according to the deviation compensation recursive least square principle BCRLS
Figure FDA0003902514780000013
Obtaining backward output estimation error zeta (j), and obtaining estimation value of input noise variance by defining cross-correlation function g (i) of least square estimation error epsilon (j) and zeta (j)
Figure FDA0003902514780000014
Further on the biased estimation obtained in step B
Figure FDA0003902514780000015
Performing deviation compensation to obtain unbiased estimation
Figure FDA0003902514780000016
D, multiplying the two sides of the equation of the unbiased estimated value expression of the unknown parameter h obtained in the step C by scalars respectively
Figure FDA00039025147800000112
Transforming the unbiased estimated value expression of the unknown parameter h into a matrix eigenvalue decomposition form, and solving the normalized eigenvector
Figure FDA0003902514780000018
And its corresponding coefficient k l Obtaining a characteristic vector of the matrix, wherein the characteristic vector is an unbiased estimation value of an unknown parameter h;
and E, step E: d, constructing an EIV-FIR (EIV-FIR) self-adaptive filter according to the unbiased estimation value h obtained in the step D, and giving an optimal estimation of an expected response through a system weight vector parameter h updated in real time under the condition of giving an input signal sample x (i) by using the bias compensation recursive least squares algorithm BCRLS so as to minimize the mean square value of an estimation error epsilon (i);
for an infinite impulse response filter, the deviation compensation adaptive filtering method based on the matrix eigenvalue solution comprises the following steps of F to G:
f, carrying out biased estimation value on unknown parameters h of the system under the IV-like algorithm at the moment i
Figure FDA0003902514780000019
Because the algorithm already contains the output noise information in the class auxiliary variable, only the variance information of the input noise needs to be known; there are two cases, one in which the output noise is white noise, and in order to obtain an estimate of the variance of the input noise, a backward output variable β is introduced i And class auxiliary variable xi j Obtaining an unbiased estimate of h by a bias compensation hierarchical auxiliary variable algorithm BCRIV-like
Figure FDA00039025147800000110
The expression (b) is arranged into a matrix form based on eigenvalue decomposition through transformation, and an unbiased estimation value of the parameter h is obtained by solving an eigenvector of the matrix; another case is that the output noise is colored noise, and in order to obtain an estimate of the variance of the input noise, a backward input variable α is introduced i And adjusts class assist variable ξ j Obtaining an unbiased estimate of h by a bias compensation hierarchical auxiliary variable algorithm BCRIV-like
Figure FDA00039025147800000111
Is transformed byArranging the parameters into a matrix form based on eigenvalue decomposition, and solving eigenvectors of the matrix to obtain an unbiased estimation value of the parameter h;
and G, constructing an EIV-IIR adaptive filter according to the unbiased estimated value h obtained in the step F, and giving an optimal estimate of an expected response through a system weight vector parameter h updated in real time under the condition that the bias compensation recursive auxiliary variable algorithm BCRIV-like is used for a given input signal sample x (i), so that the mean square value of an estimated error epsilon (i) is minimum.
2. The method of claim 1, wherein the adaptive filter method for bias compensation based on matrix eigenvalue decomposition comprises: the step A is realized by the following method that,
step A1, in order to construct an EIV-FIR filter, defining the filter to meet 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 white Gaussian noises having a mean value of 0 and being uncorrelated with each other, and the noise variances are respectively unknown
Figure FDA0003902514780000021
And
Figure FDA0003902514780000022
step A2, expressing the data into the following vector form
Figure FDA0003902514780000023
Wherein s (i) is a noiseless input signal at time i, x (i) is a noisy input signal at time i, and n (i) is an input noise at time i, that is, x (i) = s (i) + n (i) is satisfied; e (i) is the output noise at the moment i, and h is a weight vector representing the filter system and is an unknown parameter to be estimated; l is the order of the filter, and the superscript T represents the transpose operator;
step A3 the EIV-FIR filter model can be expressed as
Figure FDA0003902514780000024
Where y (i) is the noisy output signal at time i, and v (i) is the complex noise, denoted as
Figure FDA0003902514780000025
3. A method of bias-compensated adaptive filtering based on matrix eigenvalue decomposition according to claim 2 wherein: the step B is realized by the method that,
step B1, according to the least square LS principle, the LS estimation value of the unknown weight vector h is expressed as
Figure FDA0003902514780000026
Substituting the formula (2) into the formula (4) results in the following relationship
Figure FDA0003902514780000027
Step B2, in order to obtain an estimated value
Figure FDA0003902514780000028
The deviation from the true value h is limited by the conditions (1) to (3) and the ergodic property of the stable random process
Figure FDA0003902514780000031
Analyzing equation (6), if the estimation of unknowns is performed by using RLS algorithmParameter, estimate under least squares criterion with i → ∞, without noise interference
Figure FDA0003902514780000032
Converge to h, but since in the EIV model both the input and the output are disturbed by noise, i.e.
Figure FDA0003902514780000033
The estimation of the conventional RLS algorithm is biased,
Figure FDA0003902514780000034
the difference between h and h is
Figure FDA0003902514780000035
At time i, h in equation (6) is replaced with the unbiased estimate of the previous time
Figure FDA0003902514780000036
The deviation is expressed as
Figure FDA0003902514780000037
Estimated value of h
Figure FDA0003902514780000038
Is further shown as
Figure FDA0003902514780000039
Wherein the inverse correlation matrix P i Is shown below
Figure FDA00039025147800000310
Analyzing the result obtained in step B, the estimation value of the unknown parameter obtained by the recursive least square algorithm (RLS) has deviation, and the deviation is related to the variance of the input noise.
4. A method of bias-compensated adaptive filtering based on matrix eigenvalue decomposition according to claim 3 wherein: the method for realizing the step C comprises the following steps of,
step C1, introducing backward output estimation variable gamma according to deviation compensation recursive least squares (BCRLS) i ＝[γ 1 γ 2 …γ L ] T (ii) a Obtaining expressions of backward output estimation errors zeta (j) and least square estimation errors epsilon (j) according to a linear prediction theory;
γ i is estimated by
Figure FDA00039025147800000311
Is shown as
Figure FDA00039025147800000312
According to the linear prediction theory, the backward output estimation error ζ (j) and the least square estimation error ε (j) are expressed as follows
Figure FDA00039025147800000313
C2, obtaining a cross-correlation function g (i) of the backward output estimation error zeta (j) and the least square estimation error epsilon (j), and obtaining an estimation value of the input noise variance by taking a limit;
the cross-correlation function of ε (j) and ζ (j) is represented by g (i) and is defined as follows
Figure FDA0003902514780000041
Under the conditions (1) and (2), the limit is taken when i → ∞ is
Figure FDA0003902514780000042
Then an estimate of the variance of the input noise
Figure FDA0003902514780000043
Is shown as
Figure FDA0003902514780000044
Step C3, under the EIV-FIR filter model, the unbiased estimation obtained according to the steps
Figure FDA0003902514780000045
Is shown below
Figure FDA0003902514780000046
5. The method of claim 4, wherein the adaptive filtering method for bias compensation based on matrix eigenvalue decomposition comprises: the method for realizing the step D comprises the following steps of,
step D1: in the obtained unbiased estimated value expression (13) of the unknown parameter h, the denominator
Figure FDA0003902514780000047
Is a scalar quantity, and both sides of the formula (13) are simultaneously multiplied by the scalar quantity
Figure FDA0003902514780000048
The unbiased estimated value expression of the unknown parameter h is transformed into the constructed matrix eigenvalue decomposition form as shown in formula (14)
Figure FDA0003902514780000049
Step D2: analysis of the above equation, scalar, according to the principle of matrix analysis
Figure FDA00039025147800000410
And vector
Figure FDA00039025147800000411
The product of (A) is equal to matrix [ 2 ]
Figure FDA00039025147800000412
And vector
Figure FDA00039025147800000413
Product of (i.e. scalar)
Figure FDA00039025147800000414
Sum vector
Figure FDA00039025147800000415
Respectively represent a matrix
Figure FDA00039025147800000416
The eigenvalues and eigenvectors of (a); the analysis formula (14) obtains the unknown parameter h with the following characteristics: unbiased estimation of unknown parameter h
Figure FDA00039025147800000417
Is a matrix
Figure FDA00039025147800000418
A feature vector of (a);
and D3: matrix of
Figure FDA00039025147800000419
Is a L x L dimensional square matrix whose L eigenvalues are lambda 1 ,λ 2 ,…,λ L The corresponding feature vector is
Figure FDA00039025147800000420
Wherein
Figure FDA00039025147800000421
Is a normalized feature vector, k l Is a real number to be estimated, and an eigenvalue k can be obtained by performing eigenvalue decomposition on the matrix l And
Figure FDA00039025147800000422
k will be determined below l ；
By
Figure FDA00039025147800000423
The following relationship is obtained:
Figure FDA0003902514780000051
hence, the real number k l Calculated as,
Figure FDA0003902514780000052
Defining functions
Figure FDA0003902514780000053
Figure FDA0003902514780000054
Is provided with
Figure FDA0003902514780000055
Figure FDA0003902514780000056
Only one stable point, i.e. the only solution of equation (18), i.e. the unbiased estimated value of the unknown parameter; by test formulas
Figure FDA0003902514780000057
At each normalized feature vector
Figure FDA0003902514780000058
Nearby convergence, a unique solution can be obtained because only the unique solution converges as the number of iterations increases.
6. The bias compensation adaptive filtering method based on matrix eigenvalue decomposition of claim 5 wherein: the method for realizing the step F comprises the following steps of,
step F1: in the case of white output noise, a class auxiliary variable ξ is defined for the EIV-IIR-BCRIV-Like model j To find an inverse correlation matrix Q i Thereafter, the variable β is estimated by introducing a backward output i Obtaining a backward output estimation error zeta (j), and obtaining an estimation value of the variance of the input noise by defining a cross-correlation function g (i) of epsilon (j) and zeta (j)
Figure FDA0003902514780000059
Further obtaining an unbiased estimation expression of h, converting the expression into a matrix form based on eigenvalue decomposition, and then obtaining an eigenvector of the matrix by solving the normalized eigenvector and a coefficient corresponding to the normalized eigenvector, wherein the eigenvector is an unbiased estimation value of the unknown parameter h;
i moment input data vector p i Is 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 class auxiliary variable when the output noise is white noise is expressed as follows
ξ j ＝[-y(j-L-1)-y(j-L-2)…-y(j-2L)x(j-1)x(j-2)…x(j-L)] T
i time p i And xi j Of the inverse correlation matrix Q i Is shown as
Figure FDA00039025147800000510
Backward output estimation variable beta i ＝[β 1 β 2 …β 2L ] T Is estimated by
Figure FDA00039025147800000511
Is defined as
Figure FDA00039025147800000512
Corresponding backward output estimation
Figure FDA0003902514780000061
Is composed of
Figure FDA0003902514780000062
The class auxiliary variable estimation error epsilon (j) is
Figure FDA0003902514780000063
Wherein
Figure FDA0003902514780000064
Is the biased estimation of the parameter h under the class auxiliary variable (IV-like);
the backward output estimation error ζ (j) is the backward output y (j-1) and the backward output estimation
Figure FDA00039025147800000611
Is expressed as follows
Figure FDA0003902514780000065
Defining a cross-correlation function g (i) of the class auxiliary variable estimation error and the backward output estimation error as
Figure FDA0003902514780000066
When i → ∞ is satisfied, the limit is set to formula (24)
Figure FDA0003902514780000067
The estimate of the variance of the input noise is given by equation (25)
Figure FDA0003902514780000068
When the output noise is white noise, the unbiased estimation of h under BCRIV-like of the EIV-IIR filter is expressed as follows
Figure FDA0003902514780000069
Multiplying both sides of the formula (27) by the denominator in the formula
Figure FDA00039025147800000610
Transforming an unbiased estimate expression of an unknown parameter h into a matrix
Figure FDA0003902514780000071
Eigenvalue decomposition form, equation (28)
Figure FDA0003902514780000072
And D, repeating the step D, solving the normalized eigenvector and the corresponding coefficient thereof to further obtain a matrix
Figure FDA0003902514780000073
The feature vector of (a), which is an unbiased estimation value of the unknown parameter h;
step F2: adjusting a class assist variable ξ in the case of colored output noise j To find an inverse correlation matrix Q i Thereafter, the variable α is estimated by introducing a backward input i Obtaining backward input estimation error
Figure FDA00039025147800000714
By defining ε (j) and
Figure FDA00039025147800000715
to obtain an estimate of the variance of the input noise
Figure FDA0003902514780000074
Further obtaining an unbiased estimation expression of h, converting the expression into a form based on matrix eigenvalue decomposition, and then obtaining an eigenvector of the matrix by solving the normalized eigenvector and a coefficient corresponding to the normalized eigenvector, wherein the eigenvector is the unbiased estimation value of the unknown parameter h;
the auxiliary variables for the case where the output noise is colored noise are expressed as follows
ξ j ＝[x(j-L-1)x(j-L-2)…x(j-2L)x(j-1)x(j-2)…x(j-L)] T
Backward input estimation variable alpha i ＝[α 1 α 2 …α 2L ] T Is estimated by
Figure FDA0003902514780000075
Is defined as
Figure FDA0003902514780000076
Corresponding backward input estimation
Figure FDA0003902514780000077
Is composed of
Figure FDA0003902514780000078
Backward input estimation error
Figure FDA0003902514780000079
Estimate for the backward input x (j-1) and the backward input
Figure FDA00039025147800000710
Is expressed as follows
Figure FDA00039025147800000711
Defining class auxiliary variable estimation error epsilon (j) and backward input estimation error
Figure FDA00039025147800000712
The cross-correlation function f (i) of
Figure FDA00039025147800000713
When i → ∞ is reached, the limit is taken on the formula (32)
Figure FDA0003902514780000081
The estimate of the variance of the input noise is given by equation (33)
Figure FDA0003902514780000082
When the output noise is colored noise, the unbiased estimation of h under BCRIV-like of the EIV-IIR filter is expressed as follows
Figure FDA0003902514780000083
Multiplying both sides of the formula (35) by the denominator in the formula
Figure FDA0003902514780000084
Transforming an unbiased estimate expression of an unknown parameter h into a matrix
Figure FDA0003902514780000085
Eigenvalue decomposition form, equation (36)
Figure FDA0003902514780000086
And D, repeating the step D, solving the normalized eigenvector and the corresponding coefficient thereof to further obtain a matrix
Figure FDA0003902514780000087
The eigenvector is an unbiased estimation value of the unknown parameter h.
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CN116883469B (en) * 2023-07-20 2024-01-19 中国矿业大学 Point cloud registration method based on EIV model description under plane feature constraint

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