CN112929006B - Variable step size selection update kernel least mean square adaptive filter - Google Patents
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Abstract
The invention discloses a variable step size selection updating kernel least mean square self-adaptive filter, and belongs to the field of digital filter design. The filter mainly adopts time-varying step parameters to enable the core self-adaptive filter to better approach a nonlinear system, and a variable threshold selection updating strategy enables the filter to reduce the calculation cost and ensure that the performance of the filter is not affected. The variable step size selection updating kernel least mean square self-adaptive filter disclosed by the invention can be applied to electronic and communication systems which are interfered by white noise.
Description
Technical Field
The invention discloses a self-adaptive filter, in particular a variable step size selection updating kernel least mean square self-adaptive filter, and belongs to the field of digital filter design.
Background
The adaptive filter plays an important role in signal processing application, and is widely applied to the aspects of echo cancellation, communication channel equalization, system identification and the like of hands-free telephones and video conference systems at present. Adaptive filters can be classified into linear and nonlinear categories, with the criteria for classification being whether the input-output mapping follows the principle of superposition. Conventional adaptive filter filters focus mainly on linear filters, but linear filter filters are not suitable for the large number of non-linearity problems encountered in practice. On the other hand, the traditional nonlinear adaptive filtering models, such as Hammerstein, wiener and Volterra, have limited modeling capability, and problems of local minima, large computational complexity and the like can occur, and the defects limit the wide application of the models.
The kernel method has been successfully applied to nonlinear adaptive filter, and a kernel adaptive filter is proposed. The filter has attracted extensive research interest in the fields of machine learning and signal processing as a powerful tool for solving the nonlinear problem. The kernel adaptive filter maps input data to a high-dimensional feature space, in which the kernel adaptive filter based on a conventional linear framework is widely studied to solve various nonlinear applications including pattern classification, system identification, time series prediction, etc.
To date, several kinds of kernel adaptive filter filters have been proposed. Examples: a kernel least mean square (noted as KLMS) filter, a kernel least squares filter, a kernel affine projection filter, and the like. Conventional kernel least mean square filters use a fixed step size that is set in various situations, which typically results in non-ideal filter performance. To improve the performance of the kernel least mean square filter, K.Chen et al have proposed a new kernel-integrated normalized least mean square (denoted as K-SM-NLMS) filter [ Nonlinear Adaptive Filtering With Kernel Set-Membership Approach,2020, (68): 1515-1528]. There is still room for further improvement in the performance of this filter.
Disclosure of Invention
In order to further improve the performance of the kernel least mean square filter, the invention discloses a variable step size selection updating kernel least mean square adaptive filter (abbreviated as VSS-SU-KLMS). The filter adopts a variable step length method and a variable threshold value selection updating strategy to update the weight vector, thereby improving the identification performance of the nonlinear system. The filter also effectively reduces the computational cost of the filter.
In order to achieve the above purpose, the present application adopts the following technical scheme:
the VSS-SU-KLMS adaptive filter uses a combination of variable threshold selection updates and variable step size parameters to update its weight vector.
Preferably, the updating weight vector of the VSS-SU-KLMS filter comprises the following steps:
1) The a priori error e (n) is calculated from the input vector u (n) and the desired signal d (n), i.e. e (n) =d (n) -w T (n)k w (n), wherein T represents a transpose operation; w (n) represents a weight vector of the adaptive filter at the moment n; k (k) w (n)＝[k(u(n),u w (1)),…,k(u(n),u w (m))] T A nucleated input vector with a length m at time n; u (u) w (1),…,u w (m) is a vector satisfying the correlation criterion selected from all input vectors from time 0 to time n; k () represents a gaussian kernel function, i.e. the calculation formula for any two vectors u and u' is
ζ represents the core width and satisfies ζ > 0;
2) According to
Calculating a time-varying step size parameter eta * (n) wherein σ v Represents the standard deviation of the noise signal v (n), E (E) 2 (n)) represents a mean square error, which is estimated using the following calculation formula: e (E) 2 (n))＝λE(e 2 (n-1))+(1-λ)e 2 (n) wherein λ represents a smoothing factor and satisfies 0.ltoreq.λ < 1;
3) According to the formula η (n) =min { βη (n-1) + (1- β) max { η (n) * ,0},η max Calculating smoothed time-varying step size parameter eta (n), wherein min { ·, · } represents taking the minimum value, max { ·, · } represents taking the maximum value, beta is a smoothing factor and satisfies 0.ltoreq.beta < 1, eta max Is the maximum value of the allowed step size parameter;
4) According to
Calculating a time-varying threshold parameter gamma (n), wherein alpha represents a smoothing factor and satisfies 0.ltoreq.alpha < 1;
5) According to
The select update parameter s (n) is calculated.
6) Using w (n+1) =w (n) +s (n) η (n) e (n) k w (n) updating the weight vector w (n).
Advantageous effects
Compared with the scheme in the prior art, the VSS-SU-KLMS filter provided by the invention can obviously improve the filter performance, reduce the steady-state error of the estimated nonlinear system and ensure that the calculation cost is reduced on the premise of not influencing the filter performance.
Drawings
The invention is further described below with reference to the accompanying drawings and examples:
FIG. 1 is a schematic diagram of a variable step size selection update kernel least mean square adaptive filter according to an embodiment of the present invention;
fig. 2 is a comparison of the additional mean square error curve of the adaptive filter of the present invention with the KLMS filter of different step sizes under the colored signal input conditions of the embodiment.
Fig. 3 is a comparison of additional mean square error curves of an adaptive filter according to an embodiment of the present invention with a K-SM-NLMS filter of different thresholds under the input conditions of the colored signal according to the embodiment.
Detailed Description
Examples
The performance of the VSS-SU-KLMS filter is verified by adopting a computer experiment method. In the experiment, the VSS-SU-KLMS filter disclosed by the invention is used for identifying an unknown nonlinear system under the environment of white noise interference, and the performance of the unknown nonlinear system is compared with the performance of the KLMS and K-SM-NLMS adaptive filters.
Next, a detailed description of the VSS-SU-KLMS adaptive filter disclosed in embodiments of the present application identifies that the unknown nonlinear system includes the steps of:
1) The a priori error e (n) is calculated from the input vector u (n) and the desired signal d (n), i.e. e (n) =d (n) -w T (n)k w (n), wherein T represents a transpose operation; w (n) represents a weight vector of the adaptive filter at the moment n; k (k) w (n)＝[k(u(n),u w (1)),…,k(u(n),u w (m))] T A nucleated input vector with a length m at time n; u (u) w (1),…,u w (m) is a vector satisfying the correlation criterion selected from all input vectors from time 0 to time n; k () represents a gaussian kernel function, i.e. the calculation formula for any two vectors u and u' is
ζ represents the core width and satisfies ζ > 0;
2) According to
Calculating a time-varying step size parameter eta * (n) wherein σ v Represents the standard deviation of the noise signal v (n), E (E) 2 (n)) represents a mean square error, which is estimated using the following calculation formula: e (E) 2 (n))＝λE(e 2 (n-1))+(1-λ)e 2 (n) wherein λ represents a smoothing factor and satisfies 0.ltoreq.λ < 1;
3) According to the formula η (n) =min { βη (n-1) + (1- β) max { η (n) * ,0},η max Calculating smoothed time-varying step size parameter eta (n), wherein min { ·, · } represents taking the minimum value, max { ·, · } represents taking the maximum value, beta is a smoothing factor and satisfies 0.ltoreq.beta < 1, eta max Is the maximum value of the allowed step size parameter;
4) According to
Calculating a time-varying threshold parameter gamma (n), wherein alpha represents a smoothing factor and satisfies 0.ltoreq.alpha < 1;
5) According to
The select update parameter s (n) is calculated.
6) Using w (n+1) =w (n) +s (n) η (n) e (n) k w (n) updating the weight vector w (n).
Consider in experiments the nonlinear system identification problem that the input sequence of the system is generated by the following equation:
u(n)＝0.9u(n-1)+0.13z(n) (1)
where u (n) represents the input sequence and z (n) is a gaussian white signal with a mean of 0 and a variance of 1. The output signal of the nonlinear system (i.e. the desired signal of the adaptive filter) is
d(n)＝0.5d(n-1)+0.75x 2 (n)+0.8x 3 (n)+v(n) (2)
Where x (n) =0.5 u (n) -0.3u (n-1), v (n) represents a gaussian white signal with a mean of 0 and a variance of 0.001.
The additional mean square error is used as a measure of filter performance, defined as emse=10log 10 E[(e(n)-v(n)) 2 ]In dB, wherein E [ (E (n) -v (n)) 2 ]Is formed by (e (n) -v (n)) 2 The average value is obtained through 500 independent experiments. The weight vector update ratio is defined as r=c/L, where C represents the number of times the weight vector w (n) is updated in the experiment, and L represents the number of times the weight vector w (n) is updated without selection in the experiment. R is the result of averaging 500 independent experiments.
The kernel width ζ of the gaussian kernel is taken to be 0.46, the dictionary dimension maximum of all filters is taken to be 16, and the threshold δ used in the generated dictionary is taken to be 0.3. The smoothing factors lambda, beta and alpha in the VSS-SU-KLMS filter are respectively taken as 0.8, 0.8 and 0.99, and the maximum value eta of the step length max Taking 1; the selective update threshold parameters gamma of the K-SM-NLMS filter are respectively set to sigma v And 2σ v The method comprises the steps of carrying out a first treatment on the surface of the The fixed step size η in the KLMS filter is taken as 0.12 and 0.8, respectively.
As can be seen from fig. 2, the VSS-SU-KLMS filter disclosed in the present invention has a lower steady state offset compared to the KLMS filter, while guaranteeing a similar convergence speed; the convergence speed can be increased when the steady state is similar to the steady state. At this point the VSS-SU-KLMS filter R is 40%, and the optional update also reduces the computational cost.
As can be seen from fig. 3, the threshold parameter γ affects the performance of the K-SM-NLMS filter. Compared with K-SM-NLMS filters with different gamma threshold parameters, the VSS-SU-KLMS filter disclosed by the invention adopts the self-adaptive threshold, does not need to be manually adjusted to obtain the optimal threshold by probing, and can obviously improve the system identification performance. When the threshold parameter gamma of the K-SM-NLMS filter is respectively set as sigma v And 2σ v When the R is 41% and 9%, respectively, the VSS-SU-KLMS filter R is 40%. Compared with the K-SM-NLMS filter, the VSS-SU-KLMS filter disclosed by the invention can obviously reduce steady state offset under the condition that the convergence speed and R are close.
From the experimental results, it can be seen that: the VSS-SU-KLMS filter disclosed by the invention has lower steady state mismatch and lower calculation cost.
The above embodiments are provided to illustrate the technical concept and features of the present invention and are intended to enable those skilled in the art to understand the content of the present invention and implement the same, and are not intended to limit the scope of the present invention. All equivalent changes or modifications made according to the spirit of the present invention should be included in the scope of the present invention.
Claims (2)
1. A variable step size selection update kernel least mean square adaptive filter, characterized by: the adaptive filter adopts the combination of variable threshold selection update and variable step size parameter to update the weight vector, and the update weight vector comprises the following steps:
1) The a priori error e (n) is calculated from the input vector u (n) and the desired signal d (n), i.e. e (n) =d (n) -w T (n)k w (n), wherein T represents a transpose operation; w (n) represents a weight vector of the adaptive filter at the moment n; k (k) w (n)＝[k(u(n),u w (1)),…,k(u(n),u w (m))] T A nucleated input vector with a length m at time n; u (u) w (1),…,u w (m) is a vector satisfying the correlation criterion selected from all input vectors from time 0 to time n; k () represents a gaussian kernel function, i.e. the calculation formula for any two vectors u and u' is
Xi represents the nuclear width and satisfies xi>0；
2) According to
Calculating a time-varying step size parameter eta * (n) wherein σ v Represents the standard deviation of the noise signal v (n), E (E) 2 (n)) represents a mean square error, and is estimated by using the following calculation formula: e (E) 2 (n))＝λE(e 2 (n-1))+(1-λ)e 2 (n) wherein λ represents a smoothing factor and satisfies 0.ltoreq.λ<1；
3) According to the formula η (n) =min { βη (n-1) + (1- β) max { η (n) * ,0},η max Calculating smoothed time-varying step size parameter eta (n), wherein min { ·, · } represents taking the minimum value, max { ·, · } represents taking the maximum value, beta is a smoothing factor and satisfies 0.ltoreq.beta<1，η max Is the maximum value of the allowed step size parameter;
4) According to
Calculating a time-varying threshold parameter gamma (n), wherein alpha represents a smoothing factor and satisfies 0.ltoreq.alpha<1；
5) According to
The select update parameter s (n) is calculated.
2. The adaptive filter of claim 1, wherein: the adaptive filter adopts a calculation formula w (n+1) =w (n) +s (n) eta (n) e (n) k w (n) updating the weight vector.
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| CN105391879A (en) * | 2015-12-09 | 2016-03-09 | 天津大学 | Echo residue-free double-end communication robust acoustic echo elimination method |
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| CN105391879A (en) * | 2015-12-09 | 2016-03-09 | 天津大学 | Echo residue-free double-end communication robust acoustic echo elimination method |
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