CN112583381B - Self-adaptive filtering method based on deviation compensation auxiliary variable - Google Patents

Self-adaptive filtering method based on deviation compensation auxiliary variable Download PDF

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CN112583381B
CN112583381B CN202011481813.9A CN202011481813A CN112583381B CN 112583381 B CN112583381 B CN 112583381B CN 202011481813 A CN202011481813 A CN 202011481813A CN 112583381 B CN112583381 B CN 112583381B
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贾丽娟
李妍
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Beijing Institute of Technology BIT
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Abstract

The invention relates to a self-adaptive filtering method based on deviation compensation auxiliary variables, and belongs to the technical field of signal estimation and digital filters. The method comprises the following steps: presetting iteration times M and constructing a variable error-containing model; constructing a class auxiliary variable strongly related to an input signal vector; calculating an estimated value and an estimated deviation of a similar auxiliary variable method under the condition of colored output noise; calculating an estimated value of unknown input noise variance under the colored output noise condition; and compensating the deviation caused by the noise by using a deviation compensation principle to obtain an unbiased estimation of unknown system parameters. The method can stably work when the input signal is a white Gaussian process or a colored Gaussian process and the output noise signal is colored noise; the influence of the output noise variance is eliminated by introducing the similar auxiliary variable, only the input noise variance is required to be estimated, and the complexity of the method is reduced; the method for estimating the input noise variance in real time is provided, and the unknown input noise variance can be accurately estimated.

Description

Self-adaptive filtering method based on deviation compensation auxiliary variables
Technical Field
The invention relates to a self-adaptive filtering method based on deviation compensation auxiliary variables, and belongs to the technical field of signal estimation and digital filters.
Background
The traditional RLS adaptive filter has wide application in the field of adaptive filtering, such as adaptive equalization, echo cancellation, antenna array beam forming, parameter estimation, noise cancellation, spectrum estimation, etc. in the field of communication.
Convergence speed, steady state detuning and robustness are three important performance indexes of the adaptive filter. The convergence rate determines the time taken for the adaptive filter to approach an unknown system, the steady state maladjustment height determines the estimation precision of the proposed method for the unknown system, and the robustness determines the application range and the effectiveness of the proposed method, and the three indexes simultaneously influence the quality of signal processing.
When the output end of the adaptive filter is interfered by colored noise, the traditional RLS method cannot realize a good filtering effect, and the problem that the least square estimation value is biased when the output noise is interfered by the colored noise under some conditions is solved by introducing auxiliary variables. However, in a common situation, when the input signal is a white gaussian process, the auxiliary variable method is unstable, and cannot accurately estimate unknown parameters of the system, and the application range needs to be expanded.
In some applications, the adaptive filter is disturbed by input noise in addition to output noise. If the parameters of an unknown system are estimated by adopting the traditional RLS adaptive filtering method, the estimation deviation introduced by noise exists, and a more accurate estimation result cannot be obtained.
Wherein RLS is Recursive Least square;
disclosure of Invention
The invention aims to overcome the defect that the conventional deviation compensation RLS method cannot normally work under the condition of colored output noise, provides a self-adaptive filtering method based on deviation compensation auxiliary variables, and realizes unbiased estimation of an unknown system under the condition that additive noise interference of unknown variance exists at the input end and the output end.
In order to achieve the technical purpose, the invention adopts the following technical scheme:
the self-adaptive filtering method based on the deviation compensation auxiliary variables comprises the following steps:
step A, presetting iteration times M, and constructing a variable error-containing model;
wherein the variable error-containing model, i.e. the EIV-IIR filter model, can be expressed as:
Figure BDA0002837805710000021
wherein, EIV is Errors in Variables, meaning that the Variables contain Errors; IIR is Infine Impulse Response which is Infinite Impulse Response; y (i) is the noisy output signal at time i, piAn input signal vector at the ith moment of the EIV-IIR filter model is represented as h, an L-order unknown system parameter vector to be estimated is represented as v (i), composite noise at the ith moment is represented as v (i), superscript T represents transposition operation, and L is the order of the EIV-IIR filter model;
vector piAnd v (i) in particular:
pi＝[y(i-1)y(i-2)…y(i-L)x(i-1)x(i-2)…x(i-L)]T (2)
Figure BDA0002837805710000022
ni＝[e(i-1)e(i-2)…e(i-L)n(i-1)n(i-2)…n(i-L)]T (4)
wherein y (i-1) y (i-2) … y (i-L) is the delay of the noisy output signal at time instant i-1 and at an interval of 1 up to time instant i-L, respectively; x (i-1) x (i-2) … x (i-L) are the delay of the noisy input signal at time i-1 and at an interval of 1 up to time i-L, respectively; e (i) e (i-1) e (i-2) … e (i-L) are the delay of the output noise at the ith time and the interval from 1 to the ith-L time, respectively; n (i-1) n (i-2) … n (i-L) are the delays of the input noise at time i-1 and at intervals of 1 through time i-L, respectively;
step B, constructing a class auxiliary variable, specifically: aiming at the condition that the input signal is a white Gaussian process or a colored Gaussian process and the output noise signal is colored noise, adjusting the parameters of the auxiliary variables to form a structural auxiliary variable based on the auxiliary variables, so that the class auxiliary variable xi at the j momentjInput signal vector p at the j time of EIV-IIR filter modeljStrong correlation;
wherein, the class auxiliary Variable, i.e. the instrument Variable-like, abbreviated as IV-like, must satisfy the following conditions to ensure the consistency of parameter estimation:
Figure BDA0002837805710000023
is a non-singular matrix
E[ξjv(j)]＝0
Wherein v (j) is the composite noise at the j time;
aiming at the condition that the input signal is a white Gaussian process or a colored Gaussian process and the output noise signal is colored noise, constructing a j-th time auxiliary variable xijAs shown in (5):
ξj＝[x(j-L-1)x(j-L-2)…x(j-2L)
x(j-1)x(j-2)…x(j-L)]T (5)
wherein x (j-L-1) x (j-L-2) … x (j-2L) is the delay of the noisy input signal at the j-L-1 th time and the interval from 1 to the j-2L th time respectively;
and C, calculating the parameter estimation value of IV-like under the EIV-IIR filter model through the step (6):
Figure BDA0002837805710000031
wherein the content of the first and second substances,
Figure BDA0002837805710000032
representing the estimated value of IV-like at time i,
Figure BDA0002837805710000033
representing an estimated value, y (j) representing a noisy output signal at the j-th time, y (j) calculating by (1), and replacing i in formula (1) by j;
step D, calculating the estimation deviation of parameter estimation by using IV-like under the condition of colored output noise, wherein the calculation process comprises the following steps: taking the limit for (6), we get (7):
Figure BDA0002837805710000034
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002837805710000035
for the estimation of the IV-like method at the ith time
Figure BDA0002837805710000036
The deviation from the unknown system parameter vector h is recorded as: Δ h;
Figure BDA0002837805710000037
for unknown input noise variance, ILAn identity matrix of dimension L x L; from (7), it can be seen that, due to the presence of noise in the EIV-IIR filter model,
Figure BDA0002837805710000038
therefore,. DELTA.h.not equal to 0; since h is unknown, the deviation Δ h can be estimated by using the unbiased estimate of the time at i-1
Figure BDA0002837805710000039
Instead of obtaining the unknown system parameter vector h in the Δ h expression, the estimated value of the deviation Δ h can be recorded as
Figure BDA00028378057100000310
Step E, calculating the variance of the unknown input noise
Figure BDA00028378057100000311
Comprises the following substeps:
step E1, calculating the i-th time backward input estimation vector aiIs estimated by
Figure BDA00028378057100000312
Figure BDA00028378057100000313
Wherein the content of the first and second substances,
Figure BDA00028378057100000314
the estimated values of the estimation vector are input backward at the ith and the i-1 st time respectively,
Figure BDA00028378057100000315
is a vector of 0 s as an initial value,
Figure BDA00028378057100000316
as an i-th time class auxiliary variable xiiTransposing;
step E2, calculating the error estimated by the IV-like method
Figure BDA0002837805710000041
And backward input estimation error
Figure BDA0002837805710000042
The estimated value of the cross-correlation function g (i) at the ith time obtained by multiplication
Figure BDA0002837805710000043
Figure BDA0002837805710000044
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002837805710000045
the estimated values of the cross-correlation function of the IV-like method estimated error and the backward input estimated error at the ith and the ith-1 time respectively,
Figure BDA0002837805710000046
the initial value is 0;
step E3, calculating the unknown input noise variance
Figure BDA0002837805710000047
Real-time estimation of (c):
Figure BDA0002837805710000048
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002837805710000049
for unknown input noise variance
Figure BDA00028378057100000410
Real-time estimation of (a);
step F, compensating the deviation caused by noise in the IV-like estimated value by using a deviation compensation principle, and calculating the unbiased estimation of the unknown system parameter vector h, wherein the unbiased estimation specifically comprises the following formula (11):
Figure BDA00028378057100000411
wherein the content of the first and second substances,
Figure BDA00028378057100000412
unbiased estimation is carried out on the unknown system parameter vector h at the ith moment;
and G, circulating the step B to the step F, executing iterative updating until a preset iteration number M is reached, and ending the method.
Advantageous effects
Compared with the prior art, the self-adaptive filtering method based on the deviation compensation auxiliary variable has the following beneficial effects that:
1. in the step B of the method, a class auxiliary variable is introduced, so that the problem that the method cannot work normally due to instability of the auxiliary variable when an input signal is a white Gaussian process is solved, the proposed class auxiliary variable is strongly correlated with an input signal vector when the input signal is the white Gaussian process or a colored Gaussian process, the proposed class auxiliary variable method is stable, and the application range of the auxiliary variable in practice is expanded;
2. in the method step D, the estimation deviation of parameter estimation is carried out by using an IV-like method in an EIV-IIR filter model through calculation, and the influence of the output noise variance on the estimation value of the unknown parameter is eliminated by the method shown in the step (7). compared with a deviation compensation RLS method, the method can work under the condition of colored output noise, only the input noise variance needs to be estimated, the unbiased estimation of the unknown parameter can be obtained, the complexity of the method is reduced, and the application range of the method is expanded;
3. the method can estimate the unknown input noise variance in real time, can accurately estimate the input noise variance under the condition of no need of input noise prior knowledge, and further realizes unbiased estimation of unknown parameters through deviation compensation.
Drawings
FIG. 1 is an EIV-IIR filter model used in step A of an adaptive filtering method based on bias compensation type auxiliary variables according to the present invention;
fig. 2 is a diagram of an adaptive filtering method based on offset compensation-like auxiliary variables, which is implemented specifically by comparing a recursive offset compensation-like auxiliary variable method with an RLS method, a recursive auxiliary variable method, and a recursive classification-like auxiliary variable method without offset compensation under the condition that an input signal is a white gaussian process and output noise is colored noise;
fig. 3 is a diagram of an adaptive filtering method based on a bias compensation-like auxiliary variable, in which, in implementation, when an input signal is a colored gaussian process and an output noise is colored noise under the conditions, a recursive bias compensation-like auxiliary variable method is performed and a RLS method, a recursive auxiliary variable method, and a recursive classification auxiliary variable method that are not performed with bias compensation are compared.
Detailed Description
The technical scheme of the invention is explained in detail by combining the drawings and the embodiment. 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 protection scope of the present invention is not limited to the following embodiments.
Examples
The present embodiment uses the mean square error criterion as a performance indicator, and is used for system identification for noisy inputs in a colored output noise environment.
Fig. 1 shows an EIV-IIR filter model adopted by the present invention, which is an adaptive filter model under a system identification framework. The following describes a specific implementation of the adaptive filtering method based on offset compensation auxiliary variables proposed in the present invention with reference to fig. 1, which is summarized as follows:
step A, presetting iteration times M, and constructing a variable error-containing model;
wherein the variable error-containing model, i.e. the EIV-IIR filter model, can be expressed as:
Figure BDA0002837805710000051
wherein, EIV is Errors in Variables, meaning that the Variables contain Errors; IIR is Infine Impulse Response which is Infinite Impulse Response; y (i) is the noisy output signal at time i, piIs an input signal vector of EIV-IIR filter model at the ith moment, h is an L-order unknown system parameter vector to be estimated, and v (i) is the ith momentIn the composite noise of (1), the superscript T represents transposition operation, and L is the order of the EIV-IIR filter model;
vector piAnd v (i) in particular:
pi＝[y(i-1)y(i-2)…y(i-L)x(i-1)x(i-2)…x(i-L)]T (2)
Figure BDA0002837805710000061
ni＝[e(i-1)e(i-2)…e(i-L)n(i-1)n(i-2)…n(i-L)]T (4)
wherein y (i-1) y (i-2) … y (i-L) is the delay of the noisy output signal at time i-1 and at an interval of 1 up to time i-L, respectively; x (i-1) x (i-2) … x (i-L) are the delays of the noisy input signal at time instant i-1 and at an interval of 1 up to time instant i-L, respectively; e (i) e (i-1) e (i-2) … e (i-L) are the delay of the output noise at the ith time and the interval from 1 to the ith-L time, respectively; n (i-1) n (i-2) … n (i-L) are the delay of the input noise at time i-1 and at an interval of 1 up to time i-L, respectively;
step B, constructing a class auxiliary variable, specifically: aiming at the condition that the input signal is a white Gaussian process or a colored Gaussian process and the output noise signal is colored noise, adjusting the parameters of the auxiliary variables to form a structural auxiliary variable based on the auxiliary variables, so that the class auxiliary variable xi at the j momentjInput signal vector p at the j time of EIV-IIR filter modeljStrong correlation;
wherein, a class auxiliary Variable, i.e. an instrument Variable-like, abbreviated as IV-like, must satisfy the following conditions to ensure consistency of parameter estimation:
Figure BDA0002837805710000062
is a non-singular matrix
E[ξjv(j)]＝0
Wherein, v (j) is the composite noise at the j time;
white gaussian process or colored high for input signalIn the case of a Gaussian process and colored noise as an output noise signal, constructing an auxiliary variable xi of the j-th time classjAs shown in (5):
ξj＝[x(j-L-1)x(j-L-2)…x(j-2L)
x(j-1)x(j-2)…x(j-L)]T (5)
wherein x (j-L-1) x (j-L-2) … x (j-2L) is the delay of the noisy input signal at the j-L-1 th time and the interval of 1 to the j-2L th time respectively;
and C, calculating the parameter estimation value of IV-like under the EIV-IIR filter model through the step (6):
Figure BDA0002837805710000071
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002837805710000072
representing an estimate of IV-like at time i,
Figure BDA0002837805710000073
representing an estimated value, y (j) represents a noisy output signal at the j-th moment, and y (j) replaces i in the formula (1) by j through (1) calculation;
step D, calculating the estimation deviation of parameter estimation by using IV-like under the condition of colored output noise, wherein the calculation process comprises the following steps: taking the limit for (6), we get (7):
Figure BDA0002837805710000074
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002837805710000075
IV-like method estimation for the ith time
Figure BDA0002837805710000076
The deviation from the unknown system parameter vector h is recorded as: Δ h;
Figure BDA0002837805710000077
for unknown input noise variance, ILAn identity matrix of dimension L x L; from (7), it can be seen that, due to the presence of noise in the EIV-IIR filter model,
Figure BDA0002837805710000078
therefore,. DELTA.h.not equal to 0; since h is unknown, the deviation Δ h can be estimated by using the unbiased estimate at time i-1
Figure BDA0002837805710000079
Instead of obtaining the unknown system parameter vector h in the Δ h expression, the estimated value of the deviation Δ h can be recorded as
Figure BDA00028378057100000710
Step E, calculating the variance of the unknown input noise
Figure BDA00028378057100000711
Comprises the following substeps:
step E1, calculating the i-th backward input estimation vector aiIs estimated by
Figure BDA00028378057100000712
Figure BDA00028378057100000713
Wherein the content of the first and second substances,
Figure BDA00028378057100000714
the estimated values of the estimated vectors are input backwards at the ith and the i-1 time respectively,
Figure BDA00028378057100000715
is a vector of 0 (0) as an initial value,
Figure BDA00028378057100000716
is an auxiliary variable xi of the ith time classiIs transferred to；
Step E2, calculating the error estimated by the IV-like method
Figure BDA00028378057100000717
And backward input estimation error
Figure BDA00028378057100000718
The estimated value of the cross-correlation function g (i) at the ith moment obtained by multiplication
Figure BDA00028378057100000719
Figure BDA00028378057100000720
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00028378057100000721
the estimated values of the cross-correlation function of the IV-like method estimated error and the backward input estimated error at the ith and the ith-1 time respectively,
Figure BDA0002837805710000081
the initial value is 0;
step E3, calculating the variance of the unknown input noise
Figure BDA0002837805710000082
Real-time estimation of (c):
Figure BDA0002837805710000083
wherein the content of the first and second substances,
Figure BDA0002837805710000084
for unknown input noise variance
Figure BDA0002837805710000085
Real-time estimate of (a);
step F, compensating the deviation caused by noise in the IV-like estimated value by using a deviation compensation principle, and calculating the unbiased estimation of the unknown system parameter vector h, wherein the unbiased estimation specifically comprises the following formula (11):
Figure BDA0002837805710000086
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002837805710000087
unbiased estimation is carried out on the unknown system parameter vector h at the ith moment;
and G, circulating the step B to the step F, executing iterative updating until a preset iteration number M is reached, and ending the method.
Simulation experiment
The effect of the invention is verified by the following simulation experiment:
the output noise is colored noise, and e (i) ═ u (i) -0.3u (i-1), where u (i) is white noise with a mean value of 0 and a variance of 1. The input noise n (i) is zero mean,
Figure BDA0002837805710000088
white gaussian noise. The input signal is divided into two cases: (1) input signal
Figure BDA0002837805710000089
A white gaussian process with zero mean with a variance of 1; (2) input signal
Figure BDA00028378057100000810
In order to be a colored gaussian process, the method comprises the following steps of,
Figure BDA00028378057100000811
Figure BDA00028378057100000812
the adaptive IIR filter can be described by the following model:
Figure BDA00028378057100000813
the iteration number M of the simulation experiment is 20000, the independent experiment number is 100, and the mean square error criterion is adopted as the performance index.
Fig. 2 shows a comparison of unknown system parameter estimation accuracy performed by an auxiliary variable method based on recursive bias compensation and an RLS method, a recursive auxiliary variable method, and a recursive auxiliary variable method that do not perform bias compensation in the case where an input signal is a white gaussian process and an output noise is colored noise according to an embodiment of the present invention. It is seen from the figure that after the deviation caused by noise is compensated, the estimation accuracy of the deviation compensation type auxiliary variable adaptive filtering method provided by the invention is obviously improved, and unbiased estimation of unknown parameters can be realized.
Fig. 3 shows a comparison of the estimation accuracy of unknown system parameters based on the recursive bias compensation type auxiliary variable method, the RLS method without bias compensation, the recursive auxiliary variable method, and the recursive classification type auxiliary variable method in the embodiment of the present invention, when the input signal is a colored gaussian process and the output noise is colored noise. As seen from the figure, the deviation compensation type auxiliary variable adaptive filtering method provided by the invention has the highest estimation precision, and can realize unbiased estimation on unknown system parameters under the condition of colored output noise.
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 should be understood that any modification, equivalent replacement, and improvement made by those skilled in the art without departing from the principle of the present invention shall fall within the protection scope of the present invention.

Claims (5)

1. An adaptive filtering method based on deviation compensation auxiliary variables is characterized in that: the method comprises the following steps:
step A, presetting iteration times M, and constructing a variable error-containing model;
step B, constructing a class auxiliary variable, specifically: aiming at the condition that the input signal is a white Gaussian process or a colored Gaussian process and the output noise signal is colored noise, adjusting the parameters of the auxiliary variables to form a structural auxiliary variable based on the auxiliary variables, so that the class auxiliary variable xi at the j momentjInput signal vector p at the j time of EIV-IIR filter modeljStrong correlation;
wherein, the auxiliary Variable of class, namely the instrument Variable-like, is abbreviated as IV-like;
aiming at the condition that the input signal is a white Gaussian process or a colored Gaussian process and the output noise signal is colored noise, constructing a j-th time auxiliary variable xijAs shown in (5):
Figure FDA0003605484940000011
wherein x (j-L-1) x (j-L-2) … x (j-2L) is the delay of the noisy input signal at the j-L-1 th time and the interval of 1 to the j-2L th time respectively;
step C, calculating the IV-like parameter estimation value under the EIV-IIR filter model through the step (6)
Figure FDA0003605484940000012
Figure FDA0003605484940000013
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003605484940000014
representing the estimated value of IV-like at time i,
Figure FDA0003605484940000019
representing an estimated value, y (j) representing a noisy output signal at the j-th time, y (j) calculating by (1), and replacing i in formula (1) by j;
step D, calculating the estimation deviation of parameter estimation by using IV-like under the condition of colored output noise, wherein the calculation process comprises the following steps: taking the limit for (6), we get (7):
Figure FDA0003605484940000015
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003605484940000016
IV-like method estimation for the ith time
Figure FDA0003605484940000017
The deviation from the unknown system parameter vector h is recorded as: Δ h;
Figure FDA0003605484940000018
for unknown input noise variance, ILAn identity matrix of dimension L × L; from (7), it can be seen that, due to the presence of noise in the EIV-IIR filter model,
Figure FDA0003605484940000021
therefore,. DELTA.h.not equal to 0; since h is unknown, the deviation Δ h can be estimated by using the unbiased estimate of the time at i-1
Figure FDA0003605484940000022
Instead of obtaining the unknown system parameter vector h in the Δ h expression, the estimated value of the deviation Δ h can be recorded as
Figure FDA0003605484940000023
Step E, calculating the variance of the unknown input noise
Figure FDA0003605484940000024
Comprises the following substeps:
step E1, calculating the i-th backward input estimation vector aiIs estimated by
Figure FDA0003605484940000025
Figure FDA0003605484940000026
Wherein the content of the first and second substances,
Figure FDA0003605484940000027
the estimated values of the estimated vectors are input backwards at the ith and the i-1 time respectively,
Figure FDA0003605484940000028
is a vector of 0 s as an initial value,
Figure FDA0003605484940000029
as an i-th time class auxiliary variable xiiTransposing;
step E2, calculating the error estimated by the IV-like method
Figure FDA00036054849400000210
And backward input estimation error
Figure FDA00036054849400000211
The estimated value of the cross-correlation function g (i) at the ith moment obtained by multiplication
Figure FDA00036054849400000212
Figure FDA00036054849400000213
Wherein the content of the first and second substances,
Figure FDA00036054849400000214
respectively estimating errors and backward input estimation by an IV-like method at the ith and the i-1 th timeAn estimate of the cross-correlation function of the error,
Figure FDA00036054849400000215
the initial value is 0;
step E3, calculating the variance of the unknown input noise
Figure FDA00036054849400000216
Real-time estimation of (c):
Figure FDA00036054849400000217
wherein the content of the first and second substances,
Figure FDA00036054849400000218
for unknown input noise variance
Figure FDA00036054849400000219
Real-time estimation of (a);
step F, compensating the deviation caused by noise in the IV-like estimated value by using a deviation compensation principle, namely calculating the unbiased estimation of the unknown system parameter vector h;
and G, circulating the step B to the step F, executing iterative updating until a preset iteration number M is reached, and ending the method.
2. The adaptive filtering method based on the bias compensation auxiliary-like variable according to claim 1, characterized in that: in step a, the variable error-containing model, i.e., the EIV-IIR filter model, can be expressed as:
Figure FDA00036054849400000220
wherein, EIV is Errors in Variables, meaning that the Variables contain Errors; IIR is Infine Impulse Response which is Infinite Impulse Response; y (i) is the noisy output signal at time i, piFor EIV-IIR filteringThe input signal vector of the wave filter model at the moment i, h is an L-order unknown system parameter vector to be estimated, v (i) is composite noise at the moment i, the superscript T represents transposition operation, and L is the order of the EIV-IIR filter model.
3. The adaptive filtering method based on the bias compensation auxiliary-like variable according to claim 1, characterized in that: in step A, the vector piAnd v (i) in particular:
pi＝[y(i-1) y(i-2) … y(i-L) x(i-1) x(i-2) … x(i-L)]T (2)
Figure FDA0003605484940000031
wherein, the first and the second end of the pipe are connected with each other,
ni＝[e(i-1) e(i-2) … e(i-L) n(i-1) n(i-2) … n(i-L)]T (4)
y (i-1) y (i-2) … y (i-L) are the delays of the noisy output signal at time instant i-1 and at an interval of 1 up to time instant i-L, respectively; x (i-1) x (i-2) … x (i-L) are the delays of the noisy input signal at time instant i-1 and at an interval of 1 up to time instant i-L, respectively; e (i) e (i-1) e (i-2) … e (i-L) are the delay of the output noise at time i and at intervals of 1 up to time i-L, respectively; n (i-1) n (i-2) … n (i-L) are the delays of the input noise at time instant i-1 and at intervals 1 through time instant i-L, respectively.
4. The adaptive filtering method based on the bias compensation auxiliary-like variable according to claim 3, characterized in that: in step B, the class auxiliary variables must satisfy the following conditions to ensure consistency of parameter estimation:
Figure FDA0003605484940000032
is a non-singular matrix
E[ξjv(j)]＝0
Where v (j) is the composite noise at time j.
5. The adaptive filtering method based on the bias compensation auxiliary-like variable according to claim 4, characterized in that: in step F, calculating an unbiased estimate of the unknown system parameter vector h by (11):
Figure FDA0003605484940000033
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003605484940000034
and (4) carrying out unbiased estimation on the unknown system parameter vector h at the ith moment.
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