CN112583381A - Self-adaptive filtering method based on deviation compensation auxiliary variable - Google Patents

Abstract

The invention relates to an adaptive filtering method based on a bias compensation auxiliary variable, belonging to the technical field of signal estimation and digital filters. The method includes: presetting the number of iterations M and constructing a variable containing error model; constructing a class auxiliary variable strongly correlated with the input signal vector; calculating the estimated value and estimated deviation of the class auxiliary variable method under the condition of colored output noise; The estimated value of the unknown input noise variance under the condition of output noise; the deviation caused by the noise is compensated by the principle of deviation compensation, and the unbiased estimate of the unknown system parameters is obtained. The method can work stably when the input signal is a white Gaussian process or a colored Gaussian process, and the output noise signal is colored noise; and the influence of the output noise variance is eliminated by introducing a class auxiliary variable, only the input noise variance needs to be estimated, and the method is reduced. A real-time method for estimating the variance of input noise is proposed, which can more accurately estimate the variance of unknown input noise.

Description

An Adaptive Filtering Method Based on Bias Compensation Auxiliary Variables

technical field

The invention relates to an adaptive filtering method based on a bias compensation auxiliary variable, belonging to the technical field of signal estimation and digital filters.

Background technique

The traditional RLS adaptive filter has a wide range of applications in the field of adaptive filtering, such as adaptive equalization, echo cancellation, antenna array beamforming, parameter estimation, noise cancellation, and spectrum estimation in the communication field.

Convergence speed, steady state offset and robustness are three important performance indicators of adaptive filter. The convergence speed determines the time it takes for the adaptive filter to approach the unknown system, the level of steady-state imbalance determines the estimation accuracy of the proposed method for the unknown system, and the robustness determines the scope and effectiveness of the proposed method. These three indicators simultaneously affects the quality of signal processing.

When the output end of the adaptive filter is disturbed by colored noise, the traditional RLS method cannot achieve a better filtering effect. By introducing auxiliary variables, the problem of biased least squares estimation in some cases when the output noise is disturbed by colored noise is solved. . But for a general situation, when the input signal is a white Gaussian process, the auxiliary variable method is not stable and cannot accurately estimate the unknown parameters of the system, and its application range needs to be expanded.

In some applications, the adaptive filter is disturbed by input noise in addition to output noise. If the traditional RLS adaptive filtering method is used to estimate the parameters of the unknown system, there will be estimation deviations introduced by noise, so that more accurate estimation results cannot be obtained.

Among them, RLS is Recursive Least Squares, which means recursive least squares;

SUMMARY OF THE INVENTION

The purpose of the present invention is to overcome the defect that the existing bias compensation RLS method cannot work normally under the condition of colored output noise, and proposes an adaptive filtering method based on bias compensation auxiliary variables, which realizes the existence of both input and output terminals. Unbiased estimation of unknown systems in the presence of additive noise interference of unknown variance.

In order to realize the above-mentioned technical purpose, the present invention adopts the following technical scheme to realize:

Described a kind of self-adaptive filtering method based on bias compensation auxiliary variable, comprises the following steps:

Step A, presetting the number of iterations M, constructing a variable error model;

Among them, the variable containing error model, namely the EIV-IIR filter model, can be expressed as:

Among them, EIV is Errors in Variables, which means the variable contains errors; IIR is Infinite ImpulseResponse, which is the infinite impulse response; y(i) is the noisy output signal at the i-th moment, and p _i is the EIV-IIR filter model at the i-th moment. The input signal vector of , h is the L-order unknown system parameter vector to be estimated, v(i) is the composite noise at the ith moment, the superscript T represents the transposition operation, and L is the order of the EIV-IIR filter model;

The vectors p _i and v(i) are specifically:

p _i = [y(i-1)y(i-2)...y(iL)x(i-1)x(i-2)...x(iL)] ^T (2)

n _i =[e(i-1)e(i-2)...e(iL)n(i-1)n(i-2)...n(iL)] ^T (4)

Among them, y(i-1)y(i-2)...y(i-L) are the delays of the noisy output signal at the i-1th time and the interval from 1 to the i-Lth time; x(i-1)x( i-2)...x(i-L) are the delays of the noisy input signal at the i-1th time and the interval is 1 until the i-Lth time; e(i)e(i-1)e(i-2)...e (i-L) are the delay of the output noise at the i-th time and the interval is 1 until the i-Lth time; n(i-1)n(i-2)...n(i-L) are the i-1th time and the interval is 1, respectively the delay of the input noise until the i-Lth time;

Step B, constructing auxiliary variables of the class, specifically: for the situation that the input signal is a white Gaussian process or a colored Gaussian process, and the output noise signal is colored noise, based on the auxiliary variables, the parameters of the auxiliary variables are adjusted to form the auxiliary variables of the construction class, so that the jth The class auxiliary variables ξ _j at time are all strongly correlated with the input signal vector p _j at the jth time of the EIV-IIR filter model;

Among them, class auxiliary variables, namely Instrumental Variable-like, abbreviated as IV-like, class auxiliary variables must meet the following conditions to ensure the consistency of parameter estimation:

是非奇异矩阵

is a nonsingular matrix

E[ξ _j v(j)]=0

Among them, v(j) is the composite noise at the jth moment;

For the case where the input signal is a white Gaussian process or a colored Gaussian process, and the output noise signal is a colored noise, construct the auxiliary variable ξ _j at the jth time, as shown in (5):

ξ _j =[x(jL-1)x(jL-2)...x(j-2L)

x(j-1)x(j-2)…x(jL)] ^T (5)

Wherein, x(j-L-1)x(j-L-2)...x(j-2L) are the delays of the noisy input signal at the j-L-1 time and the interval from 1 to the j-2L time respectively;

Step C, calculate the parameter estimation value of IV-like under the EIV-IIR filter model through (6):

表示第i时刻IV-like的估计值，

表示估计值，y(j)表示第j时刻的含噪输出信号，y(j)通过(1)计算，用j替换公式(1)中的i；in,

represents the estimated value of IV-like at time i,

represents the estimated value, y(j) represents the noisy output signal at the jth time, y(j) is calculated by (1), and i is replaced by j in formula (1);

Step D, calculate the estimated deviation of parameter estimation using IV-like under the condition of colored output noise. The calculation process includes the following: take the limit of (6), and obtain (7):

为第i时刻IV-like方法估计值

与未知系统参数向量h之间的偏差，记为：Δh；

为未知输入噪声方差，I_L为L×L维的单位矩阵；从(7)可知，由于EIV-IIR滤波器模型中噪声的存在，

故Δh≠0；由于h未知，偏差Δh可通过用第i-1时刻的无偏估计值

代替Δh表达式中的未知系统参数向量h来获得，偏差Δh的估计值可记为

in,

Estimated value for the IV-like method at time i

The deviation from the unknown system parameter vector h, denoted as: Δh;

is the unknown input noise variance, _IL is the L×L-dimensional unit matrix; it can be seen from (7) that due to the existence of noise in the EIV-IIR filter model,

So Δh≠0; since h is unknown, the deviation Δh can be obtained by using the unbiased estimate at the i-1th moment

Obtained instead of the unknown system parameter vector h in the expression of Δh, the estimated value of the deviation Δh can be written as

的实时估计值，包括以下子步骤：Step E, calculate the unknown input noise variance

A real-time estimate of , including the following sub-steps:

Step E1, calculate the estimated value of the backward input estimated vector a _i at the i-th time

分别为第i、i-1时刻后向输入估计向量的估计值，

的初始值为0向量，

为第i时刻类辅助变量ξ_i的转置；in,

are the estimated values of the backward input estimated vector at the i-th and i-1 times, respectively,

The initial value of the 0 vector,

is the transpose of the auxiliary variable ξ _i of the class at the i-th time;

和后向输入估计误差

相乘得到的第i时刻互相关函数g(i)的估计值

Step E2, calculate the estimated error by the IV-like method

and backward input estimation error

The estimated value of the cross-correlation function g(i) at the i-th time obtained by multiplying

分别是第i、i-1时刻IV-like方法估计误差和后向输入估计误差的互相关函数的估计值，

初始值为0；in,

are the estimated value of the cross-correlation function of the IV-like method estimation error and the backward input estimation error at the i and i-1 times, respectively,

The initial value is 0;

的实时估计值：Step E3, calculate unknown input noise variance

A real-time estimate of :

为未知输入噪声方差

的实时估计值；in,

is the unknown input noise variance

real-time estimates of

In step F, the deviation caused by noise in the IV-like estimated value is compensated by using the principle of deviation compensation, and the unbiased estimate of the unknown system parameter vector h is calculated, and the specific formula is (11):

为第i时刻未知系统参数向量h的无偏估计；in,

is the unbiased estimate of the unknown system parameter vector h at the i-th time;

Step G, looping from step B to step F, performing iterative update until the preset number of iterations M is reached, and the method ends.

beneficial effect

Compared with the prior art, an adaptive filtering method based on a bias compensation auxiliary variable according to the present invention has the following beneficial effects:

1. By introducing class auxiliary variable in described method step B, when the input signal is a white Gaussian process, the problem that the method is unstable and cannot work is overcome, and the proposed class auxiliary variable is a white Gaussian process or a white Gaussian process in the input signal. The colored Gaussian process is strongly correlated with the input signal vector, and the proposed method of auxiliary variables is stable, which expands the application scope of auxiliary variables in practice;

2. in the described method step D, by calculating the estimated deviation of parameter estimation using the IV-like method in the EIV-IIR filter model, it is seen from (7) that this method has eliminated the impact of the output noise variance on the estimated value of the unknown parameter, Compared with the bias compensation RLS method, this method can work under the condition of colored output noise and only need to estimate the input noise variance to obtain an unbiased estimate of the unknown parameters, which reduces the complexity of the method and expands the scope of application of the method;

3. The method can estimate the variance of unknown input noise in real time, and can estimate the variance of input noise more accurately without prior knowledge of input noise, and then realize unbiased estimation of unknown parameters through bias compensation.

Description of drawings

Fig. 1 is a kind of EIV-IIR filter model adopted in the adaptive filtering method step A based on bias compensation auxiliary variable of the present invention;

Fig. 2 is a kind of adaptive filtering method based on the bias compensation auxiliary variable of the present invention. During specific implementation, under the conditions described, when the input signal is a white Gaussian process and the output noise is colored noise, the recursive bias compensation auxiliary variable is performed. The comparison between the method and the RLS method without bias compensation, the recursive auxiliary variable method, and the recursive auxiliary variable method;

Fig. 3 is a kind of adaptive filtering method based on the bias compensation auxiliary variable of the present invention. During specific implementation, under the conditions described, when the input signal is a colored Gaussian process and the output noise is colored noise, the recursive bias compensation auxiliary variable is performed. A comparison of the method and the RLS method without bias compensation, the recursive auxiliary variable method, and the recursive auxiliary variable method.

Detailed ways

The technical solutions of the present invention will be described in detail below with reference to the accompanying drawings and embodiments. This embodiment is implemented on the premise of the technical solution of the present invention, and provides a detailed implementation manner and a specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

Example

In this embodiment, the mean square error criterion is used as the performance index, which is used for system identification for noisy input in a colored output noise environment.

Figure 1 shows the EIV-IIR filter model adopted by the present invention, which is an adaptive filter model under the system identification framework. A specific implementation of the adaptive filtering method based on a bias compensation auxiliary variable proposed in the present invention will be described below in conjunction with FIG. 1, which is summarized as follows:

Step A, presetting the number of iterations M, constructing a variable error model;

Among them, the variable containing error model, namely the EIV-IIR filter model, can be expressed as:

Among them, EIV is Errors in Variables, which means the variable contains errors; IIR is Infinite ImpulseResponse, which is the infinite impulse response; y(i) is the noisy output signal at the i-th moment, and p _i is the EIV-IIR filter model at the i-th moment. The input signal vector of , h is the L-order unknown system parameter vector to be estimated, v(i) is the composite noise at the ith moment, the superscript T represents the transposition operation, and L is the order of the EIV-IIR filter model;

The vectors p _i and v(i) are specifically:

p _i = [y(i-1)y(i-2)...y(iL)x(i-1)x(i-2)...x(iL)] ^T (2)

n _i =[e(i-1)e(i-2)...e(iL)n(i-1)n(i-2)...n(iL)] ^T (4)

Among them, y(i-1)y(i-2)...y(i-L) are the delays of the noisy output signal at the i-1th time and the interval from 1 to the i-Lth time; x(i-1)x( i-2)...x(i-L) are the delays of the noisy input signal at the i-1th time and the interval is 1 until the i-Lth time; e(i)e(i-1)e(i-2)...e (i-L) are the delay of the output noise at the i-th time and the interval is 1 until the i-Lth time; n(i-1)n(i-2)...n(i-L) are the i-1th time and the interval is 1, respectively the delay of the input noise until the i-Lth time;

Step B, constructing auxiliary variables of the class, specifically: for the situation that the input signal is a white Gaussian process or a colored Gaussian process, and the output noise signal is colored noise, based on the auxiliary variables, the parameters of the auxiliary variables are adjusted to form the auxiliary variables of the construction class, so that the jth The class auxiliary variables ξ _j at time are all strongly correlated with the input signal vector p _j at the jth time of the EIV-IIR filter model;

Among them, class auxiliary variables, namely Instrumental Variable-like, abbreviated as IV-like, class auxiliary variables must meet the following conditions to ensure the consistency of parameter estimation:

是非奇异矩阵

is a nonsingular matrix

E[ξ _j v(j)]=0

Among them, v(j) is the composite noise at the jth moment;

For the case where the input signal is a white Gaussian process or a colored Gaussian process, and the output noise signal is a colored noise, construct the auxiliary variable ξ _j at the jth time, as shown in (5):

ξ _j =[x(jL-1)x(jL-2)...x(j-2L)

x(j-1)x(j-2)…x(jL)] ^T (5)

Wherein, x(j-L-1)x(j-L-2)...x(j-2L) are the delays of the noisy input signal at the j-L-1 time and the interval from 1 to the j-2L time respectively;

Step C, calculate the parameter estimation value of IV-like under the EIV-IIR filter model through (6):

表示第i时刻IV-like的估计值，

表示估计值，y(j)表示第j时刻的含噪输出信号，y(j)通过(1)计算，用j替换公式(1)中的i；in,

represents the estimated value of IV-like at time i,

represents the estimated value, y(j) represents the noisy output signal at the jth time, y(j) is calculated by (1), and i is replaced by j in formula (1);

Step D, calculate the estimated deviation of parameter estimation using IV-like under the condition of colored output noise. The calculation process includes the following: take the limit of (6), and obtain (7):

为第i时刻IV-like方法估计值

与未知系统参数向量h之间的偏差，记为：Δh；

为未知输入噪声方差，I_L为L×L维的单位矩阵；从(7)可知，由于EIV-IIR滤波器模型中噪声的存在，

故Δh≠0；由于h未知，偏差Δh可通过用第i-1时刻的无偏估计值

代替Δh表达式中的未知系统参数向量h来获得，偏差Δh的估计值可记为

in,

Estimated value for the IV-like method at time i

The deviation from the unknown system parameter vector h, denoted as: Δh;

is the unknown input noise variance, _IL is the L×L-dimensional unit matrix; it can be seen from (7) that due to the existence of noise in the EIV-IIR filter model,

So Δh≠0; since h is unknown, the deviation Δh can be obtained by using the unbiased estimate at the i-1th moment

Obtained instead of the unknown system parameter vector h in the expression of Δh, the estimated value of the deviation Δh can be written as

的实时估计值，包括以下子步骤：Step E, calculate the unknown input noise variance

A real-time estimate of , including the following sub-steps:

Step E1, calculate the estimated value of the backward input estimated vector a _i at the i-th time

分别为第i、i-1时刻后向输入估计向量的估计值，

的初始值为0向量，

为第i时刻类辅助变量ξ_i的转置；in,

are the estimated values of the backward input estimated vector at the i-th and i-1 times, respectively,

The initial value of the 0 vector,

is the transpose of the auxiliary variable ξ _i of the class at the i-th time;

和后向输入估计误差

相乘得到的第i时刻互相关函数g(i)的估计值

Step E2, calculate the estimated error by the IV-like method

and backward input estimation error

The estimated value of the cross-correlation function g(i) at the i-th time obtained by multiplying

分别是第i、i-1时刻IV-like方法估计误差和后向输入估计误差的互相关函数的估计值，

初始值为0；in,

are the estimated value of the cross-correlation function of the IV-like method estimation error and the backward input estimation error at the i and i-1 times, respectively,

The initial value is 0;

的实时估计值：Step E3, calculate unknown input noise variance

A real-time estimate of :

为未知输入噪声方差

的实时估计值；in,

is the unknown input noise variance

real-time estimates of

In step F, the deviation caused by noise in the IV-like estimated value is compensated by using the principle of deviation compensation, and the unbiased estimate of the unknown system parameter vector h is calculated, and the specific formula is (11):

为第i时刻未知系统参数向量h的无偏估计；in,

is the unbiased estimate of the unknown system parameter vector h at the i-th time;

Step G, looping from step B to step F, performing iterative update until the preset number of iterations M is reached, and the method ends.

Simulation

The effect of the present invention is verified by the following simulation experiments:

的高斯白噪声。输入信号分为两种情况：(1)输入信号

为零均值的白高斯过程，其方差为1；(2)输入信号

为有色高斯过程，

自适应IIR滤波器可用如下模型描述：The output noise is colored noise, e(i)=u(i)-0.3u(i-1), where u(i) is white noise with mean 0 and variance 1. The input noise n(i) has zero mean,

Gaussian white noise. The input signal is divided into two cases: (1) input signal

A white Gaussian process with zero mean whose variance is 1; (2) the input signal

is a colored Gaussian process,

The adaptive IIR filter can be described by the following model:

The number of iterations M of the simulation experiment is 20000, the number of independent experiments is 100, and the mean square error criterion is used as the performance index.

FIG. 2 shows the case where the input signal is a white Gaussian process and the output noise is colored noise in the embodiment of the present invention, the auxiliary variable method based on recursive deviation compensation and the RLS method without deviation compensation, the recursive auxiliary variable method, the recursive class The auxiliary variable method is used to compare the estimation accuracy of unknown system parameters. It can be seen from the figure that after compensating for the deviation caused by noise, the estimation accuracy of the bias compensation auxiliary variable adaptive filtering method proposed by the present invention is obviously improved, and unbiased estimation of unknown parameters can be realized.

3 shows the case where the input signal is a colored Gaussian process and the output noise is colored noise in the embodiment of the present invention, the auxiliary variable method based on recursive bias compensation and the RLS method without bias compensation, the recursive auxiliary variable method, the recursive bias compensation method The auxiliary variable method is used to compare the estimation accuracy of unknown system parameters. It can be seen from the figure that the bias compensation auxiliary variable adaptive filtering method proposed by the present invention has the highest estimation accuracy, and can realize unbiased estimation of unknown system parameters under the condition of colored output noise.

The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. The technical means disclosed in the solution of the present invention are not limited to the technical means disclosed in the above embodiments, but also include any combination of the above technical features. technical solution. It should be pointed out that for those skilled in the art, without departing from the principles of the present invention, any corrections, equivalent replacements and improvements made within the spirit and principles of the present invention shall be included in the scope of the present invention. within the scope of protection.

Claims (6)

1. an adaptive filtering method based on a bias compensation class auxiliary variable, is characterized in that: comprise the following steps: Step A, presetting the number of iterations M, constructing a variable error model; Step B, constructing auxiliary variables of the class, specifically: for the situation that the input signal is a white Gaussian process or a colored Gaussian process, and the output noise signal is colored noise, based on the auxiliary variables, the parameters of the auxiliary variables are adjusted to form the auxiliary variables of the construction class, so that the jth The class auxiliary variables ξ _j at time are all strongly correlated with the input signal vector p _j at the jth time of the EIV-IIR filter model; Among them, the class auxiliary variable, namely Instrumental Variable-like, abbreviated as IV-like; For the case where the input signal is a white Gaussian process or a colored Gaussian process, and the output noise signal is a colored noise, construct the auxiliary variable ξ _j at the jth time, as shown in (5): ξ _j =[x(jL-1)x(jL-2)...x(j-2L) x(j-1)x(j-2)…x(jL)] ^T (5) Wherein, x(j-L-1)x(j-L-2)...x(j-2L) are the delays of the noisy input signal at the j-L-1 time and the interval from 1 to the j-2L time respectively; Step C, calculate the parameter estimates of IV-like under the EIV-IIR filter model

Step D, calculate the estimated deviation of parameter estimation using IV-like under the condition of colored output noise. The calculation process includes the following: take the limit of (6), and obtain (7): in,

Estimated value for the IV-like method at time i

The deviation from the unknown system parameter vector h, denoted as: Δh;

is the unknown input noise variance, _IL is the L×L-dimensional unit matrix; it can be seen from (7) that due to the existence of noise in the EIV-IIR filter model,

So Δh≠0; since h is unknown, the deviation Δh can be obtained by using the unbiased estimate at the i-1th moment

Obtained instead of the unknown system parameter vector h in the expression of Δh, the estimated value of the deviation Δh can be written as

Step E, calculate the unknown input noise variance

A real-time estimate of , including the following sub-steps: Step E1, calculate the estimated value of the backward input estimated vector a _i at the i-th time

in,

are the estimated values of the backward input estimated vector at the i-th and i-1 times, respectively,

The initial value of the 0 vector,

is the transpose of the auxiliary variable ξ _i of the class at the i-th time; Step E2, calculate the estimated error by the IV-like method

and backward input estimation error

The estimated value of the cross-correlation function g(i) at the i-th time obtained by multiplying

in,

are the estimated value of the cross-correlation function of the IV-like method estimation error and the backward input estimation error at the i and i-1 times, respectively,

The initial value is 0; Step E3, calculate unknown input noise variance

A real-time estimate of :

in,

is the unknown input noise variance

real-time estimates of Step F, using the principle of deviation compensation to compensate the deviation caused by noise in the IV-like estimated value, that is, to calculate the unbiased estimate of the unknown system parameter vector h; Step G, looping from step B to step F, performing iterative update until the preset number of iterations M is reached, and the method ends. 2. a kind of adaptive filtering method based on bias compensation class auxiliary variable according to claim 1, is characterized in that: in step A, variable contains error model namely EIV-IIR filter model can be expressed as:

Among them, EIV is Errors in Variables, which means the variable contains errors; IIR is Infinite ImpulseResponse, which is an infinite impulse response; y(i,) is the noisy output signal at the i-th moment, and p _i is the EIV-IIR filter model. The input signal vector at time, h is the L-order unknown system parameter vector to be estimated, v(i) is the composite noise at the i-th time, the superscript T represents the transpose operation, and L is the order of the EIV-IIR filter model. 3. a kind of adaptive filtering method based on bias compensation class auxiliary variable according to claim 1, is characterized in that: in step A, vector p _i and v (i) are specially: p _i = [y(i-1)y(i-2)...y(iL)x(i-1)x(i-2)...x(iL)] ^T (2)

in, n _i =[e(i-1)e(i-2)...e(iL)n(i-1)n(i-2)...n(iL)] ^T (4) y(i-1)y(i-2)...y(i-L) are the delays of the noisy output signal at the i-1th time and the interval of 1 until the i-Lth time; x(i-1)x(i- 2)...x(i-L) are the delays of the noisy input signal at the i-1th time and the interval of 1 until the i-Lth time; e(i)e(i-1)e(i-2)...e(i-L ) are the delay of the output noise at the i-th time and the interval is 1 until the i-Lth time; n(i-1)n(i-2)...n(i-L) are the i-1th time and the interval is 1 until the The delay of the input noise at time i-L. 4. a kind of adaptive filtering method based on bias compensation class auxiliary variable according to claim 3, is characterized in that: in step B, class auxiliary variable must satisfy following condition to ensure the consistency of parameter estimation:

is a nonsingular matrix E[ξ _j v(j)]=0 Among them, v(j) is the composite noise at the jth time. 5. a kind of adaptive filtering method based on bias compensation class auxiliary variable according to claim 4, is characterized in that: step C, by (6) calculate the parameter estimated value of IV-like under EIV-IIR filter model

in,

represents the estimated value of IV-like at time i,

represents the estimated value, y(j) represents the noisy output signal at the jth time, y(j) is calculated by (1), and i is replaced by j in formula (1); 6. a kind of adaptive filtering method based on bias compensation class auxiliary variable according to claim 5, is characterized in that: in step F, calculate the unbiased estimation of unknown system parameter vector h by (11):

in,

is an unbiased estimate of the unknown system parameter vector h at the i-th time.

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