CN113193855B - Robust adaptive filtering method for identifying low-rank acoustic system - Google Patents

Robust adaptive filtering method for identifying low-rank acoustic system Download PDF

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CN113193855B
CN113193855B CN202110446659.XA CN202110446659A CN113193855B CN 113193855 B CN113193855 B CN 113193855B CN 202110446659 A CN202110446659 A CN 202110446659A CN 113193855 B CN113193855 B CN 113193855B
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CN113193855A (en
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何宏森
陈景东
喻翌
周颖玥
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Northwestern Polytechnical University
Southwest University of Science and Technology
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Abstract

The invention discloses a robust adaptive filtering method for identifying a low-rank sound system, which is used for establishing a cost function by utilizing a Cauchy estimator or a Welsch estimator to obtain a robust adaptive filtering algorithm with stable numerical value in order to enhance the robustness of an adaptive filter to non-Gaussian noise. Compared with a method based on recursive least squares, the method provided by the invention is more robust to non-Gaussian noise, and the effectiveness of the method is verified through experiments. At the same time, adopt
Figure DDA0003037171440000011
And
Figure DDA0003037171440000012
are respectively paired
Figure DDA0003037171440000013
And
Figure DDA0003037171440000014
performing approximation to calculate the weighting factor gamma1(n) and γ2(n) weighting factor gamma due to the memory function of adaptive parameters xi and c1(n) and γ2(n) reflects the statistical properties of recent noise samples, so this approximation has a negligible effect on the robustness of the invention.

Description

Robust adaptive filtering method for identifying low-rank acoustic system
Technical Field
The invention belongs to the technical field of adaptive filters, and particularly relates to a robust adaptive filtering method for identifying a low-rank sound system.
Background
The adaptive filter is widely applied to the technical fields of signal processing, communication, automatic control and the like. Researchers have developed a number of adaptive filtering methods to improve the convergence speed and stability of adaptive filters, reduce computational complexity, reduce steady-state errors, and reduce susceptibility to noise. The robustness of the adaptive filter to noise in real environments remains a very challenging problem.
Recently, a recursive least squares method based on kronecker product is proposed and used to identify adaptive filtering of low rank acoustic systems. In this method, the impulse response of the acoustic system is decomposed using kronecker sum and low rank approximation, whereby a high-dimensional system identification problem is converted into a low-dimensional system identification problem. The computational efficiency of this method and good tracking performance in white gaussian noise environments have been demonstrated in identifying long-tail echo channel impulse responses. However, this approach is sensitive to common non-gaussian noise such as keyboard strikes, room footsteps, telephone rings, slamming door closes, heavy objects falling, firecracker sounds, and impulsive noise generated by double talk detection failures in acoustic echo cancellation systems.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a robust adaptive filtering method for identifying a low-rank acoustic system so as to enhance the robustness to non-Gaussian noise.
In order to achieve the above object, the robust adaptive filtering method for identifying a low rank acoustic system according to the present invention comprises the following steps:
(1) adaptive filter parameter setting
Setting the length of the adaptive filter to L, and setting the length of the coefficients of two sets of adaptive sub-filters to L1、L2Wherein L ═ L1×L2The number of the two groups of self-adaptive sub-filters is P, and two forgetting factors lambda are set1、λ2Wherein 0 is<λ1，λ2<1；
(2) Initializing two sets of adaptive sub-filter coefficients and two parameterized inverse correlation matrices
Figure BDA0003037171420000011
And
Figure BDA0003037171420000012
n time lengths are respectively L1And L2P is 1,2, … P, and the coefficient vector at time 0 is used as the adaptive sub-filter coefficient vector
Figure BDA0003037171420000021
L in (1)1A coefficient sum
Figure BDA0003037171420000022
L in (1)2The individual coefficients are all set to 0; time n is set to 1;
Q1(n)、Q2(n) two parameterized inverse correlation matrices at n times, respectively, and an inverse correlation matrix Q at 0 time1(n)、Q2(n) initializing to an identity matrix;
(3) signal acquisition
Collecting acoustic signals, taking the latest L sampling values at the moment of n input ends of an acoustic system as input signal vectors, and recording as x (n); taking an output signal which is acquired at the time n and contains additive noise as an observation signal and recording the signal as d (n);
(4) calculating two equivalent prior error signals
First calculate the vector x2，p(n)、x1,p(n)：
Figure BDA0003037171420000023
Figure BDA0003037171420000024
Wherein the content of the first and second substances,
Figure BDA0003037171420000025
and
Figure BDA0003037171420000026
respectively is dimension L1×L1And L2×L2The unit matrix of (a) is,
Figure BDA0003037171420000027
is kronecker product;
then concatenated as vector x2(n)、x1(n)：
Figure BDA0003037171420000028
Figure BDA0003037171420000029
Finally calculating prior error signal epsilon1(n)、ε2(n)：
Figure BDA00030371714200000210
Figure BDA00030371714200000211
(5) Calculating two weighting factors
Calculating two weighting factors gamma according to the prior error1(n)、γ2(n)：
Figure BDA00030371714200000212
Figure BDA00030371714200000213
Where ρ (-) is an M estimator, ρ' () is the first derivative of ρ (-);
the M estimator is a Cauchy estimator, rho [ epsilon ]1(n)]Is rhoC[ε1(n)]，ρ[ε2(n)]Is rhoC[ε2(n)]The values are respectively:
Figure BDA00030371714200000214
Figure BDA0003037171420000031
or the M estimator is a Welsch estimator, p [ epsilon ]1(n)]Is rhoW[ε1(n)]，ρ[ε2(n)]Is rhoW[ε2(n)]The values are respectively:
Figure BDA0003037171420000032
Figure BDA0003037171420000033
using error signals epsilon1(n)、ε2(n) adaptively estimating parameters xi and c in the variable (x) by the variance;
(6) calculating two gain vectors
Weighting factor gamma according to n time1(n)、γ2(n), vector x2(n)、x1Two inverse correlation matrices Q at (n), n-1 time instants1(n-1)、Q2(n-1) and two forgetting factors lambda1、λ2Computing a gain vector k1(n)、k2(n)：
Figure BDA0003037171420000034
Figure BDA0003037171420000035
(7) Updating two parameterized inverse correlation matrices
According to the gain vector k1(n)、k2(n) and vector x2(n)、x1Two inverse correlation matrices Q at (n), n-1 time instants1(n-1)、Q2(n-1) and two forgetting factors lambda1、λ2Computing (updating) an n-time inverse correlation matrix Q1(n)、Q2(n)：
Figure BDA0003037171420000036
Figure BDA0003037171420000037
(8) Updating two adaptive sub-filter coefficient vectors
First, an inverse correlation matrix Q is obtained from the n time instants1(n)、Q2(n) weighting factor gamma1(n)、γ2(n), vector x2(n)、x1(n) a priori error signal ε1(n)、ε2(n) and the adaptive sub-filter coefficient vector at time n-1 (previous time)
Figure BDA0003037171420000038
Computing (updating) adaptive sub-filter coefficient vectors
Figure BDA0003037171420000039
Figure BDA00030371714200000310
Figure BDA00030371714200000311
Then, the adaptive sub-filter coefficient vector is applied
Figure BDA00030371714200000312
The representation (split into P vectors) is:
Figure BDA0003037171420000041
Figure BDA0003037171420000042
(9) calculating adaptive filter coefficient vector
Computing adaptive filter coefficient vectors from adaptive sub-filter coefficient vectors
Figure BDA0003037171420000043
Figure BDA0003037171420000044
And (4) returning to the step (3) when n is n + 1.
The invention aims to realize the following steps:
the robust adaptive filtering method for identifying the low-rank sound system is used for building a cost function by utilizing a Cauchy estimator or a Welsch estimator to obtain a class of adaptive algorithms with stable values in order to enhance the robustness of adaptive filtering to non-Gaussian noise. Compared with a method based on recursive least squares, the method provided by the invention is more robust to non-Gaussian noise, and the effectiveness of the method is verified through experiments. At the same time, adopt
Figure BDA0003037171420000045
And
Figure BDA0003037171420000046
are respectively paired
Figure BDA0003037171420000047
And
Figure BDA0003037171420000048
performing approximation to calculate the weighting factor gamma1(n) and γ2(n) weighting factor gamma due to the memory function of adaptive parameters xi and c1(n) and γ2(n) reflects the statistical properties of recent noise samples, so this approximation has a negligible effect on the robustness of the invention.
Drawings
FIG. 1 is a flow diagram of one embodiment of a robust adaptive filtering method for identifying low rank acoustic systems in accordance with the present invention;
fig. 2 is a graph comparing the convergence of the present invention and the conventional RLS-NKP algorithm in identifying a sudden acoustic system when a white gaussian sequence is used as an input signal under the condition of a signal-to-noise ratio of 10dB, wherein a sub-graph (a) is a non-gaussian noise (α ═ 1) environment, and a sub-graph (b) is a gaussian noise environment;
fig. 3 is a graph comparing the convergence of the present invention and the conventional RLS-NKP algorithm in recognizing an abrupt acoustic system when a speech signal is used as an input signal under the condition of 15dB snr, wherein the sub-graph (a) is a non-gaussian noise (α ═ 1) environment, and the sub-graph (b) is a gaussian noise environment.
Detailed Description
The following description of the embodiments of the present invention is provided in order to better understand the present invention for those skilled in the art with reference to the accompanying drawings. It is to be expressly noted that in the following description, a detailed description of known functions and designs will be omitted when it may obscure the subject matter of the present invention.
1. Signal model and optimal optimization criterion
Without loss of generality, the adaptive filtering method is researched under the framework of acoustic system identification, and an error signal between an unknown linear system and the output of a filter at the moment n can be represented as
Figure BDA0003037171420000051
Where d (n) is an observed signal derived from the output of the unknown system, y (n) hTx (n) and additive noise v (n), i.e. d (n) ═ y (n) + v (n)) H is an impulse response vector of unknown system length L, and x (n) is an input signal vector containing the nearest L samples, which is x (n) ═ x (n) x (n-1) x (n-2) … x (n-L + 1)]The vector h (n-1) of length L is an estimate of the filter at time n-1. In the present invention, the vectors are column vectors, and the superscript T represents transposition.
In common acoustic applications, most acoustic systems are low-rank due to room wall reflections and redundancy caused by the sparsity of the acoustic system. The acoustic channel impulse response of such a system can be approximated with a low rank model, and the low rank model is related to the kronecker product between a set of short pulse responses. Therefore, the decomposition of the adaptive filter coefficient vector into a kronecker product may be considered
Figure BDA0003037171420000052
In the formula
Figure BDA0003037171420000053
And
Figure BDA0003037171420000054
respectively, is of length L1And L2The p-th coefficient vectors of the two sets of adaptive sub-filters,
Figure BDA0003037171420000055
representing the kronecker product, the number of both sets of adaptive sub-filters is P. The length of the adaptive filter is L, L ═ L1L2，P<min{L1,L2}. The following relationship is used:
Figure BDA0003037171420000056
in the formula
Figure BDA0003037171420000057
And
Figure BDA0003037171420000058
respectively is dimension L1×L1And L2×L2Then the error signal in (1) can be expressed as two equivalent a priori error signals:
Figure BDA0003037171420000059
Figure BDA00030371714200000510
in the formula:
Figure BDA0003037171420000061
Figure BDA0003037171420000062
Figure BDA0003037171420000063
Figure BDA0003037171420000064
Figure BDA0003037171420000065
Figure BDA0003037171420000066
in a recursive least squares algorithm based on the kronecker product, a squared error signal is used to define a cost function, thereby producing an adaptive filtering algorithm that is robust to gaussian noise. However, such algorithms degrade in non-gaussian noise environments. To solve this problem, the present invention uses an M estimator to define a set of robust cost functions as follows:
Figure BDA0003037171420000067
Figure BDA0003037171420000068
in the formula
Figure BDA0003037171420000069
Figure BDA00030371714200000610
e1(i)、e2(i) Is two a posteriori error signals, 0<λ1,λ2<1 is two forgetting factors and ρ (-) is an M estimator. The purpose of defining the cost function using the M-estimation function is to smooth transient fluctuations caused by large amplitude transients in non-gaussian noise, thereby limiting its adverse impact on the cost function.
In the present invention, two M estimators are used to define the robust cost function, one is the Cauchy estimator, which is defined as
Figure BDA00030371714200000611
Another is the Welsch estimator, which is defined as
Figure BDA00030371714200000612
The invention uses a priori error signal epsilon1(n)、ε2The variance of (n) adaptively estimates the parameters xi and c in the equation.Compared with a quadratic cost function, the adaptive Cauchy function and the Welsch function can track the statistical characteristics of the short-time error block, and the robustness of the method to different types of noise is improved.
Although some robust adaptive filters can cope with the adverse effects of non-gaussian noise, their computational complexity is too high to be applied to real-time acoustic systems. The invention develops an effective robust self-adaptive filtering algorithm from the pulse response decomposition angle, and converts the high-dimensional system identification problem into the low-dimensional problem, so the calculated amount is reduced.
2. Adaptive algorithm
Based on the formula (12a), J1(n) pairs
Figure BDA0003037171420000071
The partial derivative of (c) is calculated as follows:
Figure BDA0003037171420000072
where ρ' (. cndot.) is the first derivative of ρ (-). Let the weighting factor
Figure BDA0003037171420000073
And (17) equation equals zero, a generalized regular equation can be obtained as follows:
Figure BDA0003037171420000074
in the formula:
Figure BDA0003037171420000075
Figure BDA0003037171420000076
similarly, according to (12b), another generalized regular equation can be obtained as follows:
Figure BDA0003037171420000077
in the formula:
Figure BDA0003037171420000078
Figure BDA0003037171420000079
Figure BDA00030371714200000710
note that when e1(i) And e2(i) Approaching zero may result in unstable values because they are each located at a weighting factor y1(i) And gamma2(i) The denominator of (a). However, the M estimator selected by the present invention can avoid this problem. It can be checked that: for the Cauchy estimator,
Figure BDA00030371714200000711
for the purpose of the Welsch estimator,
Figure BDA0003037171420000081
using the matrix inversion theorem, the update equations for the adaptive filter coefficient vectors can be derived from the recursion equations (20) and (23). Order to
Figure BDA0003037171420000082
And
Figure BDA0003037171420000083
are two parameterized inverse correlation matrices, which can be obtained after calculation:
Figure BDA0003037171420000084
Figure BDA0003037171420000085
in the formula:
Figure BDA0003037171420000086
Figure BDA0003037171420000087
are two gain vectors. Using (21), (28), and (30), a sub-filter coefficient vector can be obtained
Figure BDA0003037171420000088
Update equation of (1):
Figure BDA0003037171420000089
similarly, a sub-filter coefficient vector may be obtained
Figure BDA00030371714200000810
Update equation of (1):
Figure BDA00030371714200000811
note that (19) and (22) are related to
Figure BDA00030371714200000812
And
Figure BDA00030371714200000813
however, in the derivation process described above, the coefficient γ1(n) and γ2(n) is considered time invariant. In the algorithm, use is made of
Figure BDA00030371714200000814
And
Figure BDA00030371714200000815
are respectively paired
Figure BDA00030371714200000816
And
Figure BDA00030371714200000817
approximation is made to calculate the weighting factor gamma1(n) and γ2(n) of (a). Due to the memory function of the adaptive parameters xi and c, the weighting factor gamma1(n) and γ2(n) reflects the statistical properties of recent noise samples. This approximation has a negligible effect on the robustness of the invention.
FIG. 1 is a flow chart of an embodiment of a robust adaptive filtering method for identifying low rank acoustic systems according to the present invention.
In this embodiment, the robust adaptive filtering method for identifying a low rank acoustic system of the present invention includes the following steps:
step S1: adaptive filter parameter setting
Setting the length of the adaptive filter to L, and setting the length of the coefficients of two sets of adaptive sub-filters to L1、L2Wherein L ═ L1×L2The number of the two groups of self-adaptive sub-filters is P, and two forgetting factors lambda are set1、λ2Wherein 0 is<λ1,λ2<1；
Step S2: initializing two sets of adaptive sub-filter coefficient vectors and two parameterized inverse correlation matrices
Figure BDA0003037171420000091
And
Figure BDA0003037171420000092
n time lengths are respectively L1And L2P is 1,2, … P, and the coefficient vector at time 0 is used as the adaptive sub-filter coefficient vector
Figure BDA0003037171420000093
L in (1)1A coefficient sum
Figure BDA0003037171420000094
L in (1)2The individual coefficients are all set to 0; time n is set to 1;
Q1(n)、Q2(n) two parameterized inverse correlation matrices at n times, respectively, and an inverse correlation matrix Q at 0 time1(n)、Q2(n) initializing to an identity matrix;
step S3: signal acquisition
Collecting acoustic signals, taking the latest L sampling values at the moment of n input ends of an acoustic system as input signal vectors, and recording as x (n); taking an output signal which is acquired at the time n and contains additive noise as an observation signal and recording the signal as d (n);
step S4: calculating two equivalent a priori error signals
First calculate the vector x2，p(n)、x1，p(n)：
Figure BDA0003037171420000095
Figure BDA0003037171420000096
Wherein the content of the first and second substances,
Figure BDA0003037171420000097
and
Figure BDA0003037171420000098
respectively is dimension L1×L1And L2×L2The unit matrix of (a) is,
Figure BDA0003037171420000099
is kronecker product;
then concatenated as vector x2(n)、x1(n)：
Figure BDA00030371714200000910
Figure BDA00030371714200000911
Finally calculating prior error signal epsilon1(n)、ε2(n)：
Figure BDA00030371714200000912
Figure BDA00030371714200000913
Step S5: calculating two weighting factors
Calculating two weighting factors gamma according to the prior error1(n)、γ2(n)：
Figure BDA0003037171420000101
Figure BDA0003037171420000102
Where ρ (-) is an M estimator, ρ' () is the first derivative of ρ (-);
the M estimator is a Cauchy estimator, rho [ epsilon ]1(n)]Is rhoC[ε1(n)]，ρ[ε2(n)]Is rhoC[ε2(n)]The values are respectively:
Figure BDA0003037171420000103
Figure BDA0003037171420000104
or the M estimator is a Welsch estimator, p [ epsilon ]1(n)]Is rhoW[ε1(n)]，ρ[ε2(n)]Is rhoW[ε2(n)]The values are respectively:
Figure BDA0003037171420000105
Figure BDA0003037171420000106
using error signals epsilon1(n)、ε2(n) adaptively estimating parameters xi and c in the variable (x) by the variance;
step S6: calculating two gain vectors
Weighting factor gamma according to n time1(n)、γ2(n), vector x2(n)、x1Two inverse correlation matrices Q at (n), n-1 time instants1(n-1)、Q2(n-1) and two forgetting factors lambda1、λ2Computing a gain vector k1(n)、k2(n)：
Figure BDA0003037171420000107
Figure BDA0003037171420000108
Step S7: updating two parameterized inverse correlation matrices
According to the gain vector k1(n)、k2(n) and vector x2(n)、x1Two inverse correlation matrices Q at (n), n-1 time instants1(n-1)、Q2(n-1) and two forgetting factors lambda1、λ2Computing (updating) an n-time inverse correlation matrix Q1(n)、Q2(n)：
Figure BDA0003037171420000109
Figure BDA00030371714200001010
Step S8: updating two adaptive sub-filter coefficient vectors
First, an inverse correlation matrix Q is obtained from the n time instants1(n)、Q2(n) weighting factor gamma1(n)、γ2(n), vector x2(n)、x1(n) a priori error signal ε1(n)、ε2(n) and the adaptive sub-filter coefficient vector at time n-1 (previous time)
Figure BDA0003037171420000111
Computing (updating) adaptive sub-filter coefficient vectors
Figure BDA0003037171420000112
Figure BDA0003037171420000113
Figure BDA0003037171420000114
Then, the adaptive sub-filter coefficient vector is applied
Figure BDA0003037171420000115
To represent(divided into P vectors) is:
Figure BDA0003037171420000116
Figure BDA0003037171420000117
step S9: computing adaptive filter coefficient vectors
Computing adaptive filter coefficient vectors from adaptive sub-filter coefficient vectors
Figure BDA0003037171420000118
Figure BDA0003037171420000119
And (4) returning to the step (3) when n is n + 1.
Experimental verification
In order to evaluate the performance of the proposed adaptive filtering method, an ideal linear system was represented by an acoustic impulse response measured in a real room, which was measured at a sampling rate of 16 kHz. The reverberation time of the room is about 280 ms. Under the noisy environment, using recursive least square (RLS-NKP) algorithm based on kronecker product and robust recursive minimum M estimation (R) based on kronecker product provided by the invention2LM-NKP) algorithm identifies the acoustic system. Modeling additive noise with symmetric alpha stationary distribution, 0<α<2 corresponds to non-gaussian noise and α -2 corresponds to gaussian noise. The parameters of these two types of adaptive filters are set as follows: l1024, L1＝L2＝32，P＝15，λ1＝λ20.9998. The experiment verifies that the normalized mean square error is used as a performance criterion for evaluating the adaptive filtering method.
The first experiment was used to verify the convergence of the proposed adaptive filtering method when a white gaussian sequence was used as input signal. The acoustic impulse response of a linear system changes abruptly, i.e., from h to-h, at the intermediate moments of the excitation sequence.
FIG. 2 shows the proposed R of the present invention when a white Gaussian sequence is used as an input signal under the condition of a signal-to-noise ratio of 10dB2LM-NKP algorithm (R)2LM-NKP-Cauchy,R2LM-NKP-Welsch) and RLS-NKP algorithm, wherein, subgraph (a) is a non-gaussian noise (α ═ 1) environment, and subgraph (b) is a gaussian noise environment.
As can be seen from FIG. 2(a), the R proposed by the present invention is due to the adaptive identification of outliers in non-Gaussian noise by the M estimator2The LM-NKP algorithm exhibits better convergence than the RLS-NKP algorithm in a non-Gaussian noise environment. At the same time, R using two different M estimators2The LM-NKP algorithm achieves similar performance, indicating that both M estimators are effective for application in non-gaussian noise environments. As can be seen from FIG. 2(b), R is in a Gaussian noise environment2The LM-NKP algorithm and the RLS-NKP algorithm have similar convergence, which indicates that the approximation of the non-linearity in the proposed algorithm does not affect the robustness of the algorithm.
The second experiment was used to verify the convergence of the proposed adaptive filtering method when speech was used as the input signal.
FIG. 3 shows R as the input signal of a speech signal with a signal-to-noise ratio of 15dB2The convergence of the LM-NKP algorithm and the RLS-NKP algorithm in identifying a mutated acoustic system is compared, where subgraph (a) is a non-gaussian noise (α ═ 1) environment, and subgraph (b) is a gaussian noise environment.
As can be seen from fig. 3, the non-stationarity of the speech reduces the convergence of all adaptive filters. Similarly, R is mentioned due to the use of an adaptive M estimator2The LM-NKP algorithm shows better robustness to non-Gaussian noise than the RLS-NKP algorithm and is effective in Gaussian noise environment.
Although illustrative embodiments of the present invention have been described above to facilitate the understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the embodiments, and various changes may be made apparent to those skilled in the art as long as they are within the spirit and scope of the present invention as defined and defined by the appended claims, and all matters of the invention which utilize the inventive concepts are protected.

Claims (1)

1. A robust adaptive filtering method for identifying a low rank acoustic system, comprising the steps of:
(1) adaptive filter parameter setting
Setting the length of the adaptive filter to L, and setting the length of the coefficients of two sets of adaptive sub-filters to L1、L2Wherein L ═ L1×L2The number of the two groups of self-adaptive sub-filters is P, and two forgetting factors lambda are set1、λ2Wherein 0 is<λ1,λ2<1；
(2) Initializing two sets of adaptive sub-filter coefficients and two parameterized inverse correlation matrices
Figure FDA0003391290310000011
And
Figure FDA0003391290310000012
n time lengths are respectively L1And L2P is 1,2, … P, and the coefficient vector at time 0 is used as the adaptive sub-filter coefficient vector
Figure FDA0003391290310000013
L in (1)1A coefficient sum
Figure FDA0003391290310000014
L in (1)2The individual coefficients are all set to 0; time n is set to 1;
Q1(n)、Q2(n) two parameterized inverse correlation matrices at n times, respectively, and an inverse correlation matrix Q at 0 time1(n)、Q2(n) initializing to an identity matrix;
(3) signal acquisition
Collecting acoustic signals, taking the latest L sampling values at the moment of n input ends of an acoustic system as input signal vectors, and recording as x (n); taking an output signal which is acquired at the time n and contains additive noise as an observation signal and recording the signal as d (n);
(4) calculating two equivalent prior error signals
First calculate the vector x2,p(n)、x1,p(n)：
Figure FDA0003391290310000015
Figure FDA0003391290310000016
Wherein the content of the first and second substances,
Figure FDA0003391290310000017
and
Figure FDA0003391290310000018
respectively is dimension L1×L1And L2×L2The unit matrix of (a) is,
Figure FDA0003391290310000019
is kronecker product;
then concatenated as vector x2(n)、x1(n)：
Figure FDA00033912903100000110
Figure FDA00033912903100000111
Finally calculating prior error signal epsilon1(n)、ε2(n)：
Figure FDA00033912903100000112
Figure FDA00033912903100000113
(5) Calculating two weighting factors
Calculating two weighting factors gamma according to the prior error1(n)、γ2(n)：
Figure FDA0003391290310000021
Figure FDA0003391290310000022
Where ρ (-) is an M estimator, ρ' () is the first derivative of ρ (-);
the M estimator is a Cauchy estimator, rho [ epsilon ]1(n)]Is rhoC[ε1(n)]，ρ[ε2(n)]Is rhoC[ε2(n)]The values are respectively:
Figure FDA0003391290310000023
Figure FDA0003391290310000024
or the M estimator is a Welsch estimator, p [ epsilon ]1(n)]Is rhoW[ε1(n)]，ρ[ε2(n)]Is v isW[ε2(n)]The values are respectively:
Figure FDA0003391290310000025
Figure FDA0003391290310000026
using error signals epsilon1(n)、ε2(n) adaptively estimating parameters xi and c in the variable (x) by the variance;
(6) calculating two gain vectors
Weighting factor gamma according to n time1(n)、γ2(n), vector x2(n)、x1Two inverse correlation matrices Q at (n), n-1 time instants1(n-1)、Q2(n-1) and two forgetting factors lambda1、λ2Computing a gain vector k1(n)、k2(n)：
Figure FDA0003391290310000027
Figure FDA0003391290310000028
(7) Updating two parameterized inverse correlation matrices
According to the gain vector k1(n)、k2(n) and vector x2(n)、x1Two inverse correlation matrices Q at (n), n-1 time instants1(n-1)、Q2(n-1) and two forgetting factors lambda1、λ2Computing an n-time inverse correlation matrix Q1(n)、Q2(n)：
Figure FDA0003391290310000029
Figure FDA00033912903100000210
(8) Updating two adaptive sub-filter coefficient vectors
First, an inverse correlation matrix Q is obtained from the n time instants1(n)、Q2(n) weighting factor gamma1(n)、γ2(n), vector x2(n)、x1(n) a priori error signal ε1(n)、ε2Adaptive sub-filter coefficient vector at time (n) and n-1
Figure FDA0003391290310000031
Computing adaptive sub-filter coefficient vectors
Figure FDA0003391290310000032
Figure FDA0003391290310000033
Figure FDA0003391290310000034
Then, the adaptive sub-filter coefficient vector is applied
Figure FDA0003391290310000035
Expressed as:
Figure FDA0003391290310000036
Figure FDA0003391290310000037
(9) calculating adaptive filter coefficient vector
Computing adaptive filter coefficient vectors from adaptive sub-filter coefficient vectors
Figure FDA0003391290310000038
Figure FDA0003391290310000039
And (4) returning to the step (3) when n is n + 1.
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