CN113078884A - Adaptive algorithm with addition of non-linear fitting - Google Patents

Adaptive algorithm with addition of non-linear fitting Download PDF

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CN113078884A
CN113078884A CN202110277276.4A CN202110277276A CN113078884A CN 113078884 A CN113078884 A CN 113078884A CN 202110277276 A CN202110277276 A CN 202110277276A CN 113078884 A CN113078884 A CN 113078884A
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CN113078884B (en
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毛鑫
向阳
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Wuhan University of Technology WUT
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Abstract

The invention discloses a self-adaptive algorithm for adding nonlinear fitting. The algorithm comprises the following steps: step S1, adding a nonlinear filtering module in the linear filtering module of the adaptive filter, collecting the input signal received by the adaptive filter, and intercepting the input signal to obtain the input signal of the nonlinear filtering module; step S2, carrying out nonlinear transformation on the input signal of the nonlinear filtering module to obtain a nonlinear transformation signal; step S3, calculating output signals of the nonlinear filtering module and the linear filtering module; step S4, calculating an error signal; step S5, calculating a filter update value; step S6, calculating the convergence weight coefficients of the linear filtering module and the nonlinear filtering module; and step S7, obtaining output signals of a linear filtering module and a nonlinear filtering module according to filtering operation, and superposing the two output signals to highly fit the expected signal. The algorithm is applied to a scene with a nonlinear effect, and a better system modeling effect is obtained.

Description

Adaptive algorithm with addition of non-linear fitting
Technical Field
The invention relates to a method for processing signals through an adaptive filter, in particular to an adaptive algorithm for adding nonlinear fitting.
Background
An adaptive filter refers to a filter that changes parameters and structure of the filter using an adaptive algorithm according to a change in environment. In general, the structure of the adaptive filter is not changed, and the weight coefficients of the adaptive filter are time-varying coefficients updated by an adaptive algorithm. The adaptive algorithm is an algorithm which takes the estimation of the statistical characteristics of the input signal and the output signal as the basis and adopts a specific algorithm to automatically adjust the weight coefficient of the filter so as to achieve the optimal filter characteristic. The least mean square algorithm (LMS algorithm) is rapidly becoming a standard algorithm for adaptive filtering because of its easy implementation. The algorithm utilizes the steepest descent method, and the weight coefficient vector of the next moment is calculated iteratively from the weight coefficient vector of the filter at the current moment by the gradient estimation of the mean square error. The method is mainly applied to the fields of acoustic system modeling, active noise control, acoustic echo cancellation and the like.
For a linear system, the frequency components in the output signal are the same as the frequency components in the input signal. For non-linear systems, however, the frequency in the output signal is usually not equal to the frequency in the input signal, and if the input signal has more than one frequency component, the output signal will have intermodulation terms as well as harmonics of the input signal frequency. For example, loudspeaker drivers are prone to nonlinear distortion, especially in the low frequency range. The main cause of the nonlinear distortion is the generation of new frequency components by harmonics and intermodulation terms. In a control strategy of the LMS algorithm commonly used at present, the influence of nonlinear distortion is not considered, and the application of the LMS algorithm in a nonlinear distortion scene is restricted.
Disclosure of Invention
The invention aims to consider the nonlinear effect and provide a self-adaptive algorithm for optimizing the addition of nonlinear fitting of an LMS algorithm in a nonlinear distortion scene.
To achieve the above object, the present invention provides an adaptive algorithm for adding a non-linear fit, which is characterized by comprising:
step S1, input signal acquisition: adding a nonlinear filtering module in a linear filtering module of the adaptive filter, collecting an input signal received by the adaptive filter, and intercepting the input signal to obtain an input signal of the nonlinear filtering module;
step S2, nonlinear transformation: carrying out nonlinear transformation on the input signal of the nonlinear filtering module to obtain a nonlinear transformation signal;
step S3, signal filtering: filtering to obtain an output signal of a nonlinear filtering module according to the nonlinear transformation signal and the weight coefficient of the nonlinear filtering module, and filtering to obtain an output signal of a linear filtering module;
step S4, error signal calculation: and obtaining an error signal at the output moment according to the output signal of the linear filtering module, the output signal of the nonlinear filtering module and the expected signal at the output moment.
Step S5, filter update value calculation: calculating the gradient value of each filter updating, and updating the filter coefficients of the linear module and the nonlinear module, wherein the derivation principle of the filter updating value is to square the error signal function, obtain an expected loss function, and respectively calculate the bias derivative operation of the loss function on the weight coefficient of the linear filter module and the weight coefficient of the nonlinear filter module to obtain the gradient updating value of the weight coefficient of the linear filter module and the gradient updating value of the weight coefficient of the nonlinear filter module;
step S6, filter update: obtaining an updating formula of the weight coefficient of the linear filtering module by adopting a gradient descent method according to the gradient updating value of the weight coefficient of the linear filtering module, and obtaining a convergence weight coefficient of the linear filtering module by repeatedly operating according to the updating formula of the weight coefficient of the linear filtering module; obtaining an updating formula of the weight coefficient of the nonlinear filter module by adopting a gradient descent method according to the gradient updating value of the weight coefficient of the nonlinear filter module, and obtaining a convergence weight coefficient of the nonlinear filter module by repeatedly operating according to the updating formula of the weight coefficient of the nonlinear filter module;
and step S7, respectively carrying out filtering operation on the linear filtering module convergence weight coefficient and the nonlinear filtering module convergence weight coefficient to obtain a linear filtering module output signal and a nonlinear filtering module output signal, wherein the two output signals can be highly fitted to the expected signal after being superposed.
Preferably, in step S2, the nonlinear transformation signal is:
u(n)＝f(xcut(n))
wherein ,
xcut(n)＝[x(n)，x(n-1)，…，x(n-M+1)]T，x(n)＝[x(n)，x(n-1)，…，x(n-N+1)]Tfor MRepresenting the order of the nonlinear filtering module, N is used for representing the order of the linear filtering module, M is less than N, the superscript T is used for representing the transposition operation, x (N) is used for representing the input signal of the linear filtering module, xcut(n) for representing an input signal of the non-linear filtering module;
n is used for representing time;
f is used to represent a non-linear transformation;
u (n) is used to represent the non-linearly transformed signal.
Preferably, in the nonlinear transformation, f may adopt a square nonlinear transformation which may generate a second harmonic frequency for a single frequency signal, and is used for modeling a nonlinear system, as follows:
Figure BDA0002977154830000031
wherein ,
ω is used to represent the frequency of the input signal of the non-linear filtering module;
2 ω is used to represent the second harmonic frequency generated after the nonlinear transformation;
()2for representing a squaring operation;
preferably, in the nonlinear transformation, f may adopt RELU nonlinear transformation which may generate even harmonic components for a single frequency signal for modeling a nonlinear system as follows:
Figure BDA0002977154830000032
wherein ,
ω t is used to represent a frequency of the nonlinear filtering module;
2 ω t is used to denote another frequency of the non-linear filtering module;
n is used for representing time;
RELU is used for representing signal negative value zero setting operation;
preferably, in step S3, the output signal of the linear filtering module and the output signal of the non-linear filtering module obtained by filtering are calculated according to the following formulas:
yl(n)＝x(n)Twl(n)
ynl(n)＝u(n)Twnl(n)
wherein ,
x (n) is used to represent the input signal of the linear filtering module;
wl(n) for representing linear filter module weight coefficients;
u (n) a non-linear transformed signal representing the non-linear filtering module;
wnl(n) for representing nonlinear filter module weight coefficients;
n is used for representing time;
superscript T is used to denote transpose operations;
yl(n) is used to represent the output signal of the linear filtering module.
ynl(n) is used to represent the output signal of the non-linear filtering module.
Preferably, in step S4, the error signal at the output time is calculated according to the following formula:
e(n)＝d(n)+yl(n)+ynl(n)
wherein ,
d (n) a desired signal representing an output time;
e (n) represents an error signal at the output timing.
Preferably, in step S5, the loss function is calculated according to the following formula:
J＝E(e2(n))
wherein ,
e is used to represent the desired operation;
preferably, in step S5, the gradient update values of the linear filtering module weight coefficients and the gradient update values of the nonlinear filtering module weight coefficients are respectively:
Figure BDA0002977154830000041
Figure BDA0002977154830000042
wherein ,
x (n) is used to represent the input signal of the linear filtering module;
u (n) a non-linear transformed signal representing the non-linear filtering module;
Figure BDA0002977154830000043
gradient update values representing linear filter module weight coefficients;
Figure BDA0002977154830000044
gradient update values representing the weight coefficients of the nonlinear filter module;
the superscript T is used to denote the transpose operation.
Preferably, the gradient update value of the linear filtering module weight coefficient and the gradient update value of the nonlinear filtering module weight coefficient are obtained according to the following formulas:
Figure BDA0002977154830000045
Figure BDA0002977154830000046
wherein ,
Figure BDA0002977154830000047
the device is used for representing the partial derivative operation of the weight coefficient of the linear filtering module;
Figure BDA0002977154830000051
the device is used for representing the partial derivative operation of the weight coefficient of the nonlinear filtering module;
n is used for representing the order of the linear filtering module;
m is used for representing the order of the nonlinear filtering module, and M is less than N;
the superscript T is used to denote the transpose operation.
Preferably, in step S6, the update formula of the linear module filter weight coefficients and the update formula of the nonlinear module filter weight coefficients are respectively as follows:
wl(n+1)＝wl(n)-2μle(n)x(n)T
wnl(n+1)＝wnl(n)-2μnle(n)u(n)T
wherein ,
wl(n) for representing linear filter module weight coefficients;
wnl(n) for representing nonlinear filter module weight coefficients;
μlthe iteration step length is used for representing the control convergence speed in the linear module gradient descent method;
μnlthe iteration step length is used for representing the control convergence speed in the nonlinear module gradient descent method;
e (n) an error signal indicating an output timing;
x (n)) is used to represent the input signal of the linear filtering module;
u (n) is used to represent the nonlinear transformed signal of the nonlinear filtering module.
The invention has the advantages that:
1. the adaptive algorithm with the addition of the nonlinear fitting provided by the invention is characterized in that the nonlinear filtering module is added in the linear filtering module of the adaptive filter, the nonlinear transformation is carried out on the input signal of the nonlinear filtering module, and the nonlinear fitting term is added on the basis of the filtering algorithm, so that the algorithm obtains a better system modeling effect in a scene with the nonlinear effect.
2. The non-linear transformation strategy provided by the invention has the advantages of taking the square sum and setting the negative numerical value to zero.
3. The invention carries out strict mathematical proof on the self-adaptive algorithm added with the nonlinear fitting and provides a derivation process of the algorithm.
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FIG. 1 is a block diagram of an adaptive algorithm for linear fitting;
FIG. 2 is a block diagram of an adaptive algorithm for adding non-linear fits proposed by the present invention;
FIG. 3 is a power spectrum estimate of a white noise signal as an input signal;
FIG. 4 is an estimate of the power spectrum of the desired signal at the time of white noise signal output in FIG. 3;
FIG. 5 is a diagram showing a MSE convergence curve comparison in an adaptive algorithm and an adaptive algorithm with non-linear fitting added according to the present invention;
in the figure: the system comprises a loudspeaker 1, a microphone 2, an MSE convergence curve 3 (an adaptive algorithm with an N value of 3072), an MSE convergence curve 4 (an adaptive algorithm with an N value of 2048), an MSE convergence curve 5 (an adaptive algorithm with square operation and non-linear fitting addition adopted by nonlinear transformation), and an MSE convergence curve 6 (an adaptive algorithm with negative value zero setting operation and non-linear fitting addition adopted by nonlinear transformation).
Detailed Description
The invention is described in further detail below with reference to the following figures and specific examples:
as shown in fig. 1, the speaker-enclosure-microphone system is modeled using an adaptive algorithm. The input signal x (n) is played through the loudspeaker 1, sound is propagated, the expected signal at the microphone 2 is d (n), and the system through which x (n) passes mainly comprises the loudspeaker 1, air and the microphone 2. The process is modeled by adopting an adaptive algorithm, and the updating formula of the weight coefficient of the adaptive filter is as follows:
w(n+1)＝w(n)-2μe(n)x(n)
wherein ,
w (n) is used to represent the adaptive filter weight coefficients;
w (n +1) is used for representing the weight coefficient of the next time instant of the adaptive filter;
mu is used for representing the iteration step length for controlling the convergence rate in the adaptive filtering gradient descent method;
e (n) an error signal indicating an output time, e (n) ═ d (n) + x (n)Tw(n)；
x (N) is an input signal for the adaptive filter, x (N) ═ x (N), x (N-1), …, x (N-N +1)]TThe superscript T is used to denote the transpose operation and N is used to denote the adaptive filter order.
As can be seen from the calculation formula of the adaptive algorithm, the algorithm only considers a linear fit to the signal. As shown in fig. 2, the system of speaker-enclosure-microphone is modeled again, and a non-linear fitting term is added in the LMS algorithm to obtain a new adaptive algorithm, which comprises the following steps:
step S1, input signal acquisition: adding a nonlinear filtering module in a linear filtering module of the adaptive filter, collecting an input signal received by the adaptive filter, and intercepting the input signal to obtain an input signal of the nonlinear filtering module;
step S2, nonlinear transformation: carrying out nonlinear transformation on an input signal of a nonlinear filtering module to obtain a nonlinear transformation signal;
step S3, signal filtering: filtering according to the nonlinear transformation signal and the weight coefficient of the nonlinear filtering module to obtain an output signal of the nonlinear filtering module; filtering to obtain an output signal of the linear filtering module;
step S4, error signal calculation: and obtaining an error signal at the output moment according to the output signal of the linear filtering module, the output signal of the nonlinear filtering module and the expected signal at the output moment.
Step S5, filter update value calculation: calculating the gradient value of each filter updating, and updating the filter coefficients of the linear module and the nonlinear module, wherein the derivation principle of the filter updating value is to square the error signal function, obtain an expected loss function, and respectively obtain a partial derivative operation on the linear filter module weight coefficient and the nonlinear filter module weight coefficient by the loss function to obtain a gradient updating value of the linear filter module weight coefficient and a gradient updating value of the nonlinear filter module weight coefficient;
step S6, filter update: obtaining an updating formula of the weight coefficient of the linear filtering module by adopting a gradient descent method according to the gradient updating value of the weight coefficient of the linear filtering module, and obtaining a convergence weight coefficient of the linear filtering module by repeatedly operating according to the updating formula of the weight coefficient of the linear filtering module; obtaining an updating formula of the weight coefficient of the nonlinear filter module by adopting a gradient descent method according to the gradient updating value of the weight coefficient of the nonlinear filter module, and obtaining a convergence weight coefficient of the nonlinear filter module by repeatedly operating according to the updating formula of the weight coefficient of the nonlinear filter module;
and step S7, respectively carrying out filtering operation on the linear filtering module convergence weight coefficient and the nonlinear filtering module convergence weight coefficient to obtain a linear filtering module output signal and a nonlinear filtering module output signal, wherein the two output signals can be highly fitted with an expected signal after being superposed.
In the preferred embodiment of the present invention, in step S2, the nonlinear transformation signal is:
u(n)＝f(xcut(n))
wherein ,
xcut(n)＝[x(n)，x(n-1)，…，x(n-M+1)]T，x(n)＝[x(n)，x(n-1)，…，x(n-N+1)]Tm for the order of the non-linear filtering module, N for the order of the linear filtering module, and M < N, the superscript T for the transposition operation, x (N) for the input signal of the linear filtering module, xcut(n) for representing an input signal of the non-linear filtering module;
n is used for representing time;
f is used to represent a non-linear transformation;
u (n) is used to represent the non-linearly transformed signal.
In the preferred embodiment of the present invention, the nonlinear transformation may be a squaring nonlinear transformation that generates a second harmonic frequency for a single frequency signal, which is used to model a nonlinear system, as follows:
Figure BDA0002977154830000081
wherein ,
ω is used to represent the frequency of the input signal of the non-linear filtering module;
2 ω is used to represent the second harmonic frequency generated after the nonlinear transformation;
()2for representing a squaring operation;
in the preferred embodiment of the present invention, the nonlinear transformation, f, can be a RELU nonlinear transformation that can generate even harmonic components for a single frequency signal for modeling a nonlinear system as follows:
Figure BDA0002977154830000082
wherein ,
ω t is used to represent a frequency of the nonlinear filtering module;
2 ω t is used to denote another frequency of the non-linear filtering module;
n is used for representing time;
RELU is used for representing signal negative value zero setting operation;
in the preferred embodiment of the present invention, in step S3, the filtered output signal of the linear filtering module and the filtered output signal of the non-linear filtering module are calculated according to the following formula:
yl(n)＝x(n)Twl(n)
ynl(n)＝u(n)Twnl(n)
wherein ,
x (n) is used to represent the input signal of the linear filtering module;
wl(n) for representing linear filter module weight coefficients;
u (n) a non-linear transformed signal representing the non-linear filtering module;
wnl(n) for representing nonlinear filter module weight coefficients;
n is used for representing time;
superscript T is used to denote transpose operations;
yl(n) for representing the output signal of the linear filtering module;
ynl(n) is used to represent the output signal of the non-linear filtering module.
In the preferred embodiment of the present invention, in step S4, the error signal at the output time is calculated according to the following formula:
e(n)＝d(n)+yl(n)+ynl(n)
wherein ,
d (n) a desired signal representing an output time;
e (n) represents an error signal at the output timing.
In the preferred embodiment of the present invention, in step S5, the loss function is calculated according to the following formula:
J＝E(e2(n))
wherein ,
e is used to represent the desired operation;
in the preferred embodiment of the present invention, in step S5, the gradient update values of the linear filter module weight coefficients and the gradient update values of the non-linear filter module weight coefficients are respectively:
Figure BDA0002977154830000091
Figure BDA0002977154830000092
wherein ,
x (n) is used to represent the input signal of the linear filtering module;
u (n) a non-linear transformed signal representing the non-linear filtering module;
Figure BDA0002977154830000093
the gradient of the weight coefficient of the linear filter module is shownA new value;
Figure BDA0002977154830000094
gradient update values representing the weight coefficients of the nonlinear filter module;
the superscript T is used to denote the transpose operation.
In the preferred embodiment of the present invention, the gradient update value of the weight coefficient of the linear filter module and the gradient update value of the weight coefficient of the non-linear filter module are obtained according to the following formulas:
Figure BDA0002977154830000095
Figure BDA0002977154830000096
wherein ,
Figure BDA0002977154830000097
the device is used for representing the partial derivative operation of the weight coefficient of the linear filtering module;
Figure BDA0002977154830000098
the device is used for representing the partial derivative operation of the weight coefficient of the nonlinear filtering module;
n is used for representing the order of the linear filtering module;
m is used for representing the order of the nonlinear filtering module, and M is less than N;
the superscript T is used to denote the transpose operation.
In the preferred embodiment of the present invention, in step S6, the update formula of the linear module filter weight coefficients and the update formula of the nonlinear module filter weight coefficients are respectively as follows:
wl(n+1)＝wl(n)-2μle(n)x(n)T
wnl(n+1)＝wnl(n)-2μnle(n)u(n)T
wherein ,
wl(n) for representing linear filter module weight coefficients;
wnl(n) for representing nonlinear filter module weight coefficients;
μlthe iteration step length is used for representing the control convergence speed in the linear module gradient descent method;
μnlthe iteration step length is used for representing the control convergence speed in the nonlinear module gradient descent method;
e (n) an error signal indicating an output timing;
x (n) is used to represent the input signal of the linear filtering module;
u (n) is used to represent the nonlinear transformed signal of the nonlinear filtering module.
The above technical solution is specifically described by a specific embodiment as follows:
as shown in fig. 3 to 4, in the speaker-enclosure-microphone system, a white noise signal is used as an input signal x (n) passing through the speaker 1, and a power spectrum thereof is shown in fig. 3. The expected signal of the played white noise propagated to the microphone is d (n), and the power spectrum thereof is shown in fig. 4. To increase the non-linear components in the system, less performing loudspeakers and microphones are used. Comparing fig. 3 and fig. 4, it can be seen that the desired signal d (n) has a severe high frequency loss with respect to the input signal x (n).
Respectively substituting x (n) and d (n) into an adaptive algorithm and the adaptive algorithm for adding the nonlinear fitting, and obtaining y (n), yl(n)、ynl(n) determining the Mean Square Error (MSE), i.e.
Figure BDA0002977154830000101
A plot of the MSE convergence curve as shown in figure 5 is obtained.
wherein ,
curve 3 represents: in the self-adaptive algorithm, N takes the value of 3072, mu takes the value of 0.001 MSE convergence curve, and the converged noise reduction amount is 7.59 dB;
curve 4 represents: in the self-adaptive algorithm, N takes 2048, mu takes an MSE convergence curve of 0.001, and the converged noise reduction amount is 9.28 dB;
curve 5 represents: in the adaptive algorithm with nonlinear fitting, N takes 2048 and M takes 1024 and mulThe value is 0.001, munlThe value is 0.001, the nonlinear transformation adopts an MSE convergence curve of square operation, and the noise reduction amount of convergence is 11.48 dB.
Curve 6 represents: in the adaptive algorithm with nonlinear fitting, N takes 2048 and M takes 1024 and mulThe value is 0.001, munlThe value is 0.001, the nonlinear transformation adopts an MSE convergence curve of negative value zero setting operation, and the noise reduction amount of convergence is 11.87 dB.
Therefore, the noise reduction amount of the MSE convergence curve obtained by adding the nonlinear fitting adaptive algorithm is obviously improved, and the modeling capability of the adaptive filter on a nonlinear system can be effectively improved.
While the invention has been described with reference to a preferred embodiment, it will be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope of the invention.

Claims (10)

1. An adaptive algorithm for adding a non-linear fit, comprising:
step S1, input signal acquisition: adding a nonlinear filtering module in a linear filtering module of the adaptive filter, collecting an input signal received by the adaptive filter, and intercepting the input signal to obtain an input signal of the nonlinear filtering module;
step S2, nonlinear transformation: carrying out nonlinear transformation on the input signal of the nonlinear filtering module to obtain a nonlinear transformation signal;
step S3, signal filtering: filtering to obtain an output signal of a nonlinear filtering module according to the nonlinear transformation signal and the weight coefficient of the nonlinear filtering module, and filtering to obtain an output signal of a linear filtering module;
step S4, error signal calculation: and obtaining an error signal at the output moment according to the output signal of the linear filtering module, the output signal of the nonlinear filtering module and the expected signal at the output moment.
Step S5, filter update value calculation: calculating the gradient value of each filter updating, and updating the filter coefficients of the linear module and the nonlinear module, wherein the derivation principle of the filter updating value is to square the error signal function, obtain an expected loss function, and respectively calculate the bias derivative operation of the loss function on the weight coefficient of the linear filter module and the weight coefficient of the nonlinear filter module to obtain the gradient updating value of the weight coefficient of the linear filter module and the gradient updating value of the weight coefficient of the nonlinear filter module;
step S6, filter update: obtaining an updating formula of the weight coefficient of the linear filtering module by adopting a gradient descent method according to the gradient updating value of the weight coefficient of the linear filtering module, and obtaining a convergence weight coefficient of the linear filtering module by repeatedly operating according to the updating formula of the weight coefficient of the linear filtering module; obtaining an updating formula of the weight coefficient of the nonlinear filter module by adopting a gradient descent method according to the gradient updating value of the weight coefficient of the nonlinear filter module, and obtaining a convergence weight coefficient of the nonlinear filter module by repeatedly operating according to the updating formula of the weight coefficient of the nonlinear filter module;
and step S7, respectively carrying out filtering operation on the linear filtering module convergence weight coefficient and the nonlinear filtering module convergence weight coefficient to obtain a linear filtering module output signal and a nonlinear filtering module output signal, wherein the two output signals can be highly fitted to the expected signal after being superposed.
2. The adaptive algorithm with non-linear fit added according to claim 1, characterized in that: in step S2, the nonlinear conversion signal is:
u(n)＝f(xcut(n))
wherein ,
xcut(n)＝[x(n)，x(n-1)，…，x(n-M+1)]T，x(n)＝[x(n)，x(n-1)，…，x(n-N+1)]Tm for the order of the non-linear filtering module, N for the order of the linear filtering module, and M < N, the superscript T for the transposition operation, x (N) for the input signal of the linear filtering module, xcut(n) for representing an input signal of the non-linear filtering module;
n is used for representing time;
f is used to represent a non-linear transformation;
u (n) is used to represent the non-linearly transformed signal.
3. The adaptive algorithm with non-linear fit added according to claim 2, characterized in that: in the nonlinear transformation, f may be square nonlinear transformation as follows, which may generate a second harmonic frequency for a single frequency signal, and is used to model a nonlinear system, as follows:
Figure FDA0002977154820000021
wherein ,
ω is used to represent the frequency of the input signal of the non-linear filtering module;
2 ω is used to represent the second harmonic frequency generated after the nonlinear transformation;
()2for representing a squaring operation.
4. The adaptive algorithm with nonlinear fitting added according to claim 2, wherein in the nonlinear transformation, f can be adopted as RELU nonlinear transformation which can generate even harmonic components for a single frequency signal for modeling a nonlinear system as follows:
Figure FDA0002977154820000022
wherein ,
ω t is used to represent a frequency of the nonlinear filtering module;
2 ω t is used to denote another frequency of the non-linear filtering module;
n is used for representing time;
RELU is used to indicate the signal negative value zeroing operation.
5. The adaptive algorithm for adding nonlinear fitting according to claim 3 or 4, wherein in step S3, the filtered output signal of the linear filtering module and the filtered output signal of the nonlinear filtering module are calculated according to the following formula:
yl(n)＝x(n)Twl(n)
ynl(n)＝u(n)Twnl(n)
wherein ,
x (n) is used to represent the input signal of the linear filtering module;
wl(n) for representing linear filter module weight coefficients;
u (n) a non-linear transformed signal representing the non-linear filtering module;
wnl(n) for representing nonlinear filter module weight coefficients;
n is used for representing time;
superscript T is used to denote transpose operations;
yl(n) is used to represent the output signal of the linear filtering module.
ynl(n) is used to represent the output signal of the non-linear filtering module.
6. The adaptive algorithm with non-linear fit added as claimed in claim 5, wherein: in step S4, the error signal at the output time is calculated according to the following equation:
e(n)＝d(n)+yl(n)+ynl(n)
wherein ,
d (n) a desired signal representing an output time;
e (n) represents an error signal at the output timing.
7. The adaptive algorithm for adding nonlinear fitting according to claim 6, wherein in the step S5, the loss function is calculated according to the following formula:
J＝E(e2(n))
wherein ,
e is used to indicate the desired operation.
8. The adaptive algorithm for adding nonlinear fitting according to claim 7, wherein in step S5, the gradient update values of the linear filter module weight coefficients and the gradient update values of the nonlinear filter module weight coefficients are respectively:
Figure FDA0002977154820000031
Figure FDA0002977154820000032
wherein ,
x (n) is used to represent the input signal of the linear filtering module;
u (n) a non-linear transformed signal representing the non-linear filtering module;
Figure FDA0002977154820000033
gradient update values representing linear filter module weight coefficients;
Figure FDA0002977154820000034
gradient update values for representing the nonlinear filter module weight coefficients;
the superscript T is used to denote the transpose operation.
9. The adaptive algorithm for adding nonlinear fitting according to claim 8, wherein the gradient update values of the linear filter module weight coefficients and the gradient update values of the nonlinear filter module weight coefficients are obtained according to the following formulas:
Figure FDA0002977154820000041
Figure FDA0002977154820000042
wherein ,
Figure FDA0002977154820000043
the device is used for representing the partial derivative operation of the weight coefficient of the linear filtering module;
Figure FDA0002977154820000044
the device is used for representing the partial derivative operation of the weight coefficient of the nonlinear filtering module;
n is used for representing the order of the linear filtering module;
m is used for representing the order of the nonlinear filtering module, and M is less than N;
the superscript T is used to denote the transpose operation.
10. The adaptive algorithm for adding nonlinear fitting according to claim 9, wherein in step S6, the update formula of the linear module filter weight coefficients and the update formula of the nonlinear module filter weight coefficients are respectively as follows:
wl(n+1)＝wl(n)-2μle(n)x(n)T
wnl(n+1)＝wnl(n)-2μnle(n)u(n)T
wherein ,
wl(n) for representing linear filter module weight coefficients;
wnl(n) is used forRepresenting nonlinear filter module weight coefficients;
μlthe iteration step length is used for representing the control convergence speed in the linear module gradient descent method;
μnlthe iteration step length is used for representing the control convergence speed in the nonlinear module gradient descent method;
e (n) an error signal indicating an output timing;
x (n) is used to represent the input signal of the linear filtering module;
u (n) is used to represent the nonlinear transformed signal of the nonlinear filtering module.
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