A complex affine projection adaptive signal processing method based on kernel function

Abstract

This invention discloses a kernel-function-based complex affine projection adaptive signal processing method. This method applies a complex Gaussian kernel function to a generalized linear affine complex projection algorithm. This method leverages the superior performance of the Gaussian kernel function in non-Gaussian environments, particularly those with impulse noise, to improve the robustness of the adaptive filtering system. Furthermore, when the number of iterations, k, is small, the identity matrix in the update formula is of order k; when k is large, the identity matrix in the update formula is of order p, improving the accuracy of weight vector calculation. Furthermore, a variable step size method is used to increase convergence speed and reduce misalignment errors.

Description

Technical Field

[0001] The present invention relates to the field of adaptive signal processing, in particular to a complex affine projection adaptive signal processing method based on kernel function. Background Art

[0002] Adaptive filtering algorithms are widely used in many fields, such as communications. The APA (af fineprojection algorithms) algorithm orthogonally projects the current weight coefficient vector onto an affine subspace defined by the projection order, and uses the input vectors of the current and past moments to update the weight coefficient vector. Compared with the traditional LMS and NLMS algorithms, APA converges faster, especially for highly correlated input signals. The APA algorithm is essentially a gradient descent algorithm. In order to reduce computational complexity and increase convergence speed, researchers have proposed the augmented affine projection algorithm (AAPA) in recent years. As a general extension of APA, AAPA uses augmented complex statistics and generalized linear models in the algorithm derivation process, and is more suitable for processing second-order complex circular and non-circular signals.

[0003] In the field of complex adaptive filtering, when the system is disturbed by impulse noise such as impulse noise, the existing affine projection family of algorithms is not robust. Summary of the Invention

[0004] In response to the defects in the existing technology, the present invention provides a complex affine projection adaptive signal processing method based on Gaussian kernel function, which utilizes the robustness of Gaussian kernel function to impulse noise to improve the stability of the algorithm in non-Gaussian noise environment.

[0005] The present invention provides a complex affine projection adaptive signal processing method based on a kernel function, comprising the following steps:

[0006] S100: Initialize the number of iterations k = 1; initialize the generalized linear model y(k) = w H x(k)+v H x * The standard weight vector w∈c of (k) m×1 and conjugate weight vector v∈c m×1 are all zero vectors, where m is the order of the filter, x(k)∈c m×1 Input signal vector at the current moment;

[0007] S200: Determine whether the number of iterations k is less than or equal to the projection order p. If so, proceed to step S300; otherwise, proceed to step S400.

[0008] S300: Set the input signal matrix at time k; calculate the error signal and the complex Gaussian kernel function according to the generalized linear model and the input signal matrix; set the identity matrix in the update formula to order k and update the weight vector; set k←k+1, and return to step S200;

[0009] S400: Setting the input signal matrix at time k; calculating the error signal and the complex Gaussian kernel function according to the generalized linear model and the input signal matrix; setting the unit matrix in the update formula to p order and updating the weight vector.

[0010] Preferably, the step S400 further includes:

[0011] If k<L, set k←k+1 and return to step S200; otherwise, end the loop; where L is the total number of samples.

[0012] Preferably, the step S300 specifically includes:

[0013] Construct the input signal matrix at the current moment from the input signal vectors at the current moment and the past moment

[0014] Calculate the error signal vector e(k) = d(k)- X T (k) w * and the complex Gaussian kernel function vector k(e(k))=exp(-|e(k)| 2 / 2σ 2 );

[0015] where d(k) = [dk), ..., d(1)] T ∈c k×1 is the expected signal vector at discrete time k, X (k)∈c 2m×k ;

[0016] σ is the core width;

[0017] Update the weight vector w(k+1) of the k-th cycle according to the following formula:

[0018] w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(k)] -1 g*(k),

[0019] Where μ is the step size factor, the regularization parameter δ is set to 0.001, I(k) is the identity matrix, g(k) = k(e(k))Θe(k), Θ represents the multiplication of corresponding elements;

[0020] Calculate the square of the weight vector error || w opt w (k)|| 2 ,in w opt ∈C m×1 and v opt ∈C n×1 are the optimal standard weight vector and the optimal conjugate weight vector respectively;

[0021] Set k←k+1 and return to step S200.

[0022] Preferably, the step S400 includes:

[0023] Construct the input signal matrix at the current moment from the input signal vectors at the current moment and the past moment

[0024] Calculate the error signal vector e(k) = d(k)- X T (k) w * and the complex Gaussian kernel function vector k(e(k))=exp(-|e(k)| 2 / 2σ 2 );

[0025] where d(k) = [d(k), ..., d(k-p+1)] T ∈C p×1 is the expected signal vector at discrete time k,

[0026] X (k)∈C 2m×p ; σ is the core width;

[0027] Update the weight vector w of the k-th cycle according to the following formula:

[0028] w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(p)] -1 g ＊ (k),

[0029] Where μ is the step size factor, the regularization parameter δ is set to 0.001, I(p) is the identity matrix, g(k) = k(e(k))Θe(k), Θ represents the multiplication of the corresponding elements; the square of the weight vector error is calculated as || w opt - w (k)|| 2 .

[0030] Preferably, the step S400 further includes:

[0031] If k<L, set k←k+1 and return to step S200, otherwise end the loop, where L is the total number of samples.

[0032] Preferably, the step S400 includes:

[0033] According to the principle that the step size decreases as the number of iterations increases, the step size factor function is taken The weight vector at the kth moment w Update according to the following formula:

[0034] w (k+1)= w (k)+μ(k) X (k)[ X H (k) X (k)+δI(p)] -1 g ＊ (k);

[0035] in is an estimate of E[k(e(k))], calculated as follows: Where λ is the forgetting factor;

[0036] When k is equal to L, the weight vector w is obtained and output; the square difference of the weight error vector is as follows: || w opt - w (k)|| 2 .

[0037] Preferably, the step S400 further includes:

[0038] If k<L, set k←k+1 and return to step S200, otherwise end the loop, where L is the total number of samples.

[0039] Preferably, the step size factor function μ(k) is obtained by the following steps:

[0040] The cost function of the complex affine projection adaptive filtering algorithm based on the kernel function is constructed as follows:

[0041]

[0042] Where η(k) is the Lagrange multiplier vector,

[0043] Evaluate the cost function w * The conjugate derivative of (k+1) gives,

[0044]

[0045] make have to w (k+1)= w (k)+ X (k)η(k), and then we get,

[0046] X H (k) w (k+1)= X H (k) w (k)+ X H (k) X (k)η(k);

[0047] Therefore,

[0048]

[0049] Further, we can get,

[0050] w (k+1)= w (k)+μ X (k)[ X H (k) X (k)] -1 k(e(k))Θe * (k);

[0051] Using the idea of ​​fixed points, we can get

[0052] X (k)[ X H (k) X (k)] -1 k(e(k))Θe * (k) = 0, that is,

[0053] E{ X (k)[ X H (k) X (k)] -1k(e(k))Θd * (k)}=E{ X (k)[ X H (k) X (k)] -1 k(e(k))Θ X H w (k+1)};

[0054] Then we can get in

[0055] R k+1 =R k + X (k)[ X H (k) X (k)] -1 diag(k(e(k))) X H (k), diag(.) represents a diagonal matrix consisting of corresponding elements;

[0056] q k+1 =q k + X (k)[ X H (k) X (k)] -1 k(e(k))Θd ＊ (k);

[0057] By the matrix inversion lemma, we can get,

[0058]

[0059] definition:

[0060]

[0061] Multiply both sides of the equation by {I+ X H (k)R k -1 X (k)[ X H (k) X (k)] -1 diag(k(e(k)))}, we can get,

[0062]

[0063] Therefore,

[0064]

[0065] In addition, by R k+1 The definition of can be further obtained,

[0066]

[0067] Therefore,

[0068]

[0069] Available,

[0070] The beneficial effects of the present invention are:

[0071] The Gaussian kernel function is introduced into the complex affine projection algorithm to propose a generalized linear maximum correlation entropy affine projection algorithm. Furthermore, a variable step-size method is used to propose a variable step-size generalized linear maximum correlation entropy affine projection algorithm. This method leverages the robustness of the Gaussian kernel function to impulse noise, resulting in excellent steady-state performance. Furthermore, the use of a variable step-size improves convergence speed and reduces misalignment errors. Experimental results demonstrate that compared to traditional algorithms such as APA, this method offers superior performance, effectively resolving impulse noise interference and improving signal quality in fields such as communications. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following briefly describes the drawings required for the specific embodiments or the description of the prior art. Similar elements or parts are generally identified by similar reference numerals throughout the drawings. Elements or parts in the drawings are not necessarily drawn to scale.

[0073] Figure 1 This is a flow chart of the first embodiment of the present invention;

[0074] Figure 2 This is a simulation diagram of the third embodiment of the present invention in an impulse noise environment;

[0075] Figure 3 This is a simulation diagram of the third embodiment of the present invention in an alpha noise environment. DETAILED DESCRIPTION

[0076] The following embodiments of the technical solution of the present invention will be described in detail with reference to the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention and are therefore only examples and are not intended to limit the scope of protection of the present invention.

[0077] It should be noted that, unless otherwise specified, the technical or scientific terms used in this application should have the common meanings understood by those skilled in the art to which the present invention belongs.

[0078] Example 1

[0079] like Figure 1 As shown, the first embodiment of the present invention provides a complex affine projection adaptive signal processing method based on a kernel function, comprising the following steps:

[0080] S100: Initialize the number of iterations k = 1; initialize the generalized linear model y(k) = w H x(k)+v H x * The standard weight vector w∈c of (k) m×1 and conjugate weight vector v∈c m×1 are all zero vectors, where m is the order of the filter, x(k)∈c m×1 Input signal vector at the current moment;

[0081] S200: Determine whether the number of iterations k is less than or equal to the projection order p. If so, proceed to step S300; otherwise, proceed to step S400.

[0082] S300: Set the input signal matrix at time k; calculate the error signal and the complex Gaussian kernel function according to the generalized linear model and the input signal matrix; set the identity matrix in the update formula to order k and update the weight vector; set k←k+1, and return to step S200;

[0083] S400: Setting the input signal matrix at time k; calculating the error signal and the complex Gaussian kernel function according to the generalized linear model and the input signal matrix; setting the identity matrix in the update formula to p order and updating the weight vector.

[0084] Step S400 further includes: if k<L, setting k←k+1, and returning to step S200, where L is the total number of samples.

[0085] In an embodiment of the present invention, a Gaussian kernel function is applied to the generalized linear affine complex projection algorithm WL-APA algorithm, and the superior performance of the Gaussian kernel function in non-Gaussian environments, especially impulse noise environments, is utilized to improve the robustness of the adaptive filtering system.

[0086] Example 2

[0087] A second embodiment of the present invention provides a complex affine projection adaptive signal processing method based on a kernel function, comprising the following steps:

[0088] S110: Initialize the number of iterations k to 1; initialize the generalized linear model y(k)=w H x(k)+v H x * The standard weight vector w∈c of (k) m×1and conjugate weight vector v∈c m×1 are all zero vectors, where m is the order of the filter, x(k)∈c m×1 Input signal vector at the current moment;

[0089] S210: Determine whether the number of iterations k is less than or equal to the projection order p. If so, proceed to step S310; otherwise, proceed to step S410.

[0090] S310:

[0091] Construct the input signal matrix at the current moment from the input signal vectors at the current moment and the past moment Calculate the error signal vector e(k) = d(k)- X T (k) w * and the complex Gaussian kernel function vector k(e(k))=exp(-|e(k)| 2 / 2σ 2 );

[0092] where d(k) = [dk), ..., d(1)] T ∈C k×1 is the expected signal vector at discrete time k, X (k)∈C 2m×k ;

[0093] σ is the core width;

[0094] The weight vector of the kth cycle w (k+1) is updated according to the following formula:

[0095] w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(k)] -1 g*(k),

[0096] Where μ is the step size factor, the regularization parameter δ is set to 0.001, I(k) is the identity matrix, g(k) = k(e(k))Θe(k), Θ represents the multiplication of corresponding elements;

[0097] Calculate the square of the weight vector error || w opt - w (k)|| 2 ;in w opt ∈C m×1 and vopt ∈C m×1 are the optimal standard weight vector and the optimal conjugate weight vector respectively;

[0098] Set k←k+1 and return to step S210.

[0099] S410:

[0100] Construct the input signal matrix at the current moment from the input signal vectors at the current moment and the past moment Calculate the error signal vector e(k) = d(k)- X T (k) w * and the complex Gaussian kernel function vector k(e(k))=exp(-|e(k)| 2 / 2σ 2 );

[0101] where d(k) = [d(k), ..., d(k-p+1)] T ∈C p×1 is the expected signal vector at discrete time k,

[0102] X (k)∈C 2m×p ; σ is the core width;

[0103] Update the weight vector w of the kth cycle according to the following formula

[0104] w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(p)] -1 g*(k),

[0105] Where μ is the step size factor, the regularization parameter δ is set to 0.001, I(p) is the identity matrix, g(k) = k(e(k))Θe(k), Θ represents the multiplication of corresponding elements;

[0106] Calculate the square of the weight vector error || w opt - w (k)|| 2 ;

[0107] If k<L, set k←k+1 and return to step S210, otherwise end the loop, where L is the total number of samples.

[0108] Example 3

[0109] An embodiment of the present invention provides a variable-step-size complex affine projection adaptive signal processing method based on a kernel function, comprising the following steps:

[0110] S120: Initialize the number of iterations k=1; initialize the generalized linear model y(k)=w H x(k)+v H x * The standard weight vector w∈C of (k) m×1 and conjugate weight vector v∈C m×1 are all zero vectors, where m is the order of the filter, x(k)∈C m×1 Input signal vector at the current moment;

[0111] S220: Determine whether the number of iterations k is less than or equal to the projection order p. If so, proceed to step S320; otherwise, proceed to step S420.

[0112] S320:

[0113] Construct the input signal matrix at the current moment from the input signal vectors at the current moment and the past moment Calculate the error signal vector e(k) = d(k)- X T (k) w * and the complex Gaussian kernel function vector k(e(k))=exp(-|e(k)| 2 / 2σ 2 );

[0114] where d(k) = [d(k), ..., d(1)] T ∈C k×1 is the expected signal vector at discrete time k, X (k)∈C 2m×k ;

[0115] σ is the core width;

[0116] The weight vector of the kth cycle w (k+1) is updated according to the following formula:

[0117] w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(k)] -1 g*(k),

[0118] Where μ is the step size factor, the regularization parameter δ is set to 0.001, I(k) is the identity matrix, g(k) = k(e(k))Θe(k), Θ represents the multiplication of corresponding elements;

[0119] The square of the weight vector error is calculated and set to k←k+1, and the process returns to step S220.

[0120] S420:

[0121] Construct the input signal matrix at the current moment from the input signal vectors at the current moment and the past moment Calculate the error signal vector e(k) = d(k)- X T (k)w * and the complex Gaussian kernel function vector k(e(k))=exp(-|e(k)| 2 / 2σ 2 );

[0122] where d(k) = [d(k), ..., d(k-p+1)] T ∈C p×1 is the expected signal vector at discrete time k,

[0123] X (k)∈C 2m×p ; σ is the core width;

[0124] Update the step size μ(k) of the kth cycle according to the following formula

[0125]

[0126] in is E[exp(-|e(k)| 2 / 2σ 2 )],

[0127] The calculation is performed as follows:

[0128] Where λ is the forgetting factor.

[0129] Update the weight vector w of the kth cycle according to the following formula w (k+1)= w (k)+μ(k) X (k)[ X H (k) X (k)+δI(k)] -1 g*(k); calculate the square of the weight vector error || w opt -w (k)|| 2 ;

[0130] If k<L, set k←k+1 and return to step S220, otherwise end the loop, where L is the total number of samples.

[0131] The step size factor μ(k) is obtained by the following steps:

[0132] The cost function of the complex affine projection adaptive filtering algorithm based on the kernel function is constructed as follows: Where η(k) is the Lagrange multiplier vector,

[0133] Evaluate the cost function w * The conjugate derivative of (k+1) gives,

[0134]

[0135] make have to w (k+1)= w (k)+ X (k)η(k), and then we get,

[0136] X H (k) w (k+1)= X H (k) w (k)+ X H (k) X (k)η(k); therefore,

[0137]

[0138] Further, we can get,

[0139] w (k+1)= w (k)+μ X (k)[ X H (k) X (k)] -1 K(e(k))⊙e * (k); Using the idea of ​​fixed points, we can get,

[0140] X (k)[ X H (k) X (k)] -1 K(e(k))⊙e* (k) = 0, that is,

[0141] E{ X (k)[ X H (k) X (k)] -1 k(e(k))Θd * (k)}=E{ X (k)[ X H (k) X (k)] -1 k(e(k))Θ X H w (k+1)};

[0142] Then we can get in

[0143] R k+1 =R k + X (k)[ X H (k) X (k)] -1 diag(K(e(k))) X H (k), diag(.) represents a diagonal matrix consisting of corresponding elements;

[0144] q k+1 =q k + X (k)[ X H (k) X (k)] -1 K(e(k))⊙d*(k); By the matrix inversion lemma, we can get,

[0145]

[0146] definition:

[0147]

[0148] Multiply both sides of the equation by {I+ X H (k)R k -1 X (k)[ X H (k) X (k)] -1 diag(k(e(k)))}, we can get,

[0149]

[0150] Therefore,

[0151]

[0152] In addition, by R k+1 The definition of can be further obtained,

[0153]

[0154] Therefore,

[0155]

[0156] Available,

[0157] Further deduction shows that

[0158]

[0159] Then the weight update based on variable step size is obtained as follows

[0160]

[0161] The third embodiment of the present invention uses a variable step size to further improve the convergence speed of the aforementioned method and reduce the offset error. When the number of iterations is small, the step size μ is large and constant. When the number of iterations is large, the step size μ decreases with the number of iterations. In this process, the error is gradually reduced.

[0162] The algorithm in the second embodiment of the present invention is defined as a generalized linear maximum correlation entropy affine projection algorithm (WL-MCCC-APA algorithm), and the algorithm in the third embodiment of the present invention is defined as a variable step-size generalized linear maximum correlation entropy affine projection algorithm (WL-MCCC-APA-V algorithm). The WL-MCCC-APA algorithm, the WL-MCCC-APA-V algorithm, and the traditional CLMS algorithm, MCCC algorithm, and CLMS-APA algorithm are simulated under mixed Gaussian impulse noise and α-stable noise conditions, respectively. The simulation conditions are: the filter order is 5th order, the optimal standard weight vector w opt and the conjugate weight vector v optThe real and imaginary parts of each element in the mixed Gaussian impulse noise follow a Gaussian distribution with mean 0 and variance 1. The optimal standard weight vector and conjugate weight vector are normalized. The input signal is a classic first-order autoregressive process with a correlation coefficient of 0.95. The real and imaginary parts of each element follow a Gaussian distribution with mean 0 and variance 1. The number of samples is 20,000. The probability of occurrence of large Gaussian noise in the mixed Gaussian impulse noise is 0.06 and the variance is 200. The probability of occurrence of small Gaussian noise is 0.94 and the variance is 0.1. It is a first-order autoregressive process with a correlation coefficient of 0.9. The real and imaginary parts of each element in the α-stable noise follow an α-stable distribution with a variance of 1. It is a first-order autoregressive process with a correlation coefficient of 0.95. The number of Monte Carlo experiments is set to 100.

[0163] Update the weight vector of each algorithm:

[0164] Step 1: Set k←1, initialize w(1) and v(1) to be 5×1 order 0 matrices;

[0165] Step 2: Update the weight vector in the CLMS algorithm

[0166] a) w (k+1)= w (k)+μe ＊ (k) x (k), where μ is the step size factor, e(k)=d(k)- X H w (k);

[0167] b) Calculate the square of the weight vector error MSD(k) = 10log 10 (||w(k)- w opt || 2 );

[0168] c) Set k←k+1. If k<20000, return to step a) in step 2; otherwise, terminate the operation.

[0169] Step 3: Update the weight vector in the MCCC algorithm

[0170] a)w(k+1)=w(k)+μK(e(k))e*(k)x(k). Wherein, the Gaussian kernel function is

[0171] K(e(k))=exp(-|e(k)| 2 / 2σ 2 ), σ is the kernel width; the other parameters are the same as step 2;

[0172] b) Calculate the square of the weight vector error MSD(k) = 10log 10 (||w(k)- w opt || 2 );

[0173] c) Set k←k+1. If k<20000, return to step a) in step 3; otherwise, terminate the operation.

[0174] Step 4: Update the weight vector in the CLMS-APA algorithm

[0175] a) When 1≤k≤p, w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(k)] -1 e*(k), where μ is the step size factor, d(k)=[d(k), ..., d(l)] T , e(k)=d(k)- X T (k) w * ;

[0176] When p<k≤20000, w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(p)] -1 e*(k), where

[0177] d(k)=[d(k),...,d(k-p+1)] T , the rest of the parameters are the same as above.

[0178] b) Calculate the square of the weight vector error MSD(k) = 10log 10 (||w(k)- w opt || 2 );

[0179] c) Set k←k+1. If k<20000, return to step a) in step 4; otherwise, end the operation.

[0180] Step 5: Update the weight vector in the WL-MCCC-APA algorithm

[0181] a) When 1≤k≤p, w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(k)] -1 g(k), where μ is the step size factor, d(k)=[d(k), ..., d(l)] T , e(k)=d(k)- X T (k) w * , K(e(k))=exp(-|e(k)| 2 / 2σ 2 ), g(k)=K(e(k))Θe(k);

[0182] When p<k≤20000, w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(p)] -1 g(k), where

[0183] d(k)=[d(k),...,d(k-p+1)] T , the rest of the parameters are the same as above.

[0184] b) Calculate the square of the weight vector error MSD(k) = 10log 10 (||w(k)- w opt || 2 );

[0185] c) Set k←k+1. If k<20000, return to step a) in step 5; otherwise, end the operation.

[0186] Step 6: Update the weight vector in the MCCC-APA-V algorithm

[0187] a) When 1≤k≤p, w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(k)] -1 g(k), where μ is the step size factor, d(k)=[d(k), ..., d(l)] T , e(k)=d(k)- X T (k) w * , K(e(k))=exp(-|e(k)| 2 / 2σ 2 ), g(k)=K(e(k))Θe(k);

[0188] When p<k≤20000, w (k+1)= w (k)+μ(k) X (k)[ X H (k) X (k)+δI(k)] -1 g(k), where d(k)=[d(k),...,d(k-p+1)] T , m is the order of the filter, which is 5 in this case, and the other parameters are the same as above.

[0189] b) Calculate the square of the weight vector error MSD(k) = 10log 10 (||w(k)- w opt || 2 );

[0190] c) Set k←k+1. If k<20000, return to step a) in step 6; otherwise, end the operation.

[0191] like Figure 2 and Figure 3 As shown in the figure, the two algorithms with the smallest mean square deviation are the WL-MCCC-APA algorithm and the WL-MCCC-APA-V algorithm. Compared with traditional algorithms, the WL-MCCC-APA and WL-MCCC-APA-V algorithms provided by the embodiments of the present invention can achieve excellent steady-state results in the presence of different noise conditions. In addition, in both noise environments, the WL-MCCC-APA-V algorithm has lower offset error than the WL-MCCC-APA algorithm.

[0192] The present invention provides a kernel-function-based complex affine projection adaptive signal processing method. This method introduces a Gaussian kernel function into the complex affine projection algorithm, proposing a generalized linear maximum correlation entropy affine projection algorithm. Furthermore, a variable step-size method is used to propose a variable step-size generalized linear maximum correlation entropy affine projection algorithm. By utilizing the robustness of the Gaussian kernel function to impulse noise, the present invention achieves excellent steady-state performance. Furthermore, a variable step-size is used to increase convergence speed and reduce misalignment errors. Experimental results demonstrate that compared with traditional algorithms such as APA, the present invention offers superior performance and can effectively address the issue of impulse noise interference.

[0193] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or make equivalent replacements for some or all of the technical features therein. These modifications or replacements do not deviate the essence of the corresponding technical solutions from the scope of the technical solutions of the embodiments of the present invention, and they should all be included in the scope of the claims and description of the present invention.

Claims

1. A complex affine projection adaptive signal processing method based on kernel function, characterized in that: The following steps are involved: S100: Initialize the number of iterations k = 1; initialize the generalized linear model y(k) = w H x(k)+v H x * The standard weight vector w∈c of (k) m×1 and conjugate weight vector v∈c m×1 are all zero vectors, where m is the order of the filter, x(k)∈c m×1 Input signal vector at the current moment; S200: Determine whether the number of iterations k is less than or equal to the projection order p. If so, proceed to step S300; otherwise, proceed to step S400. S300: Setting the input signal matrix at time k; calculating the error signal and the complex Gaussian kernel function according to the generalized linear model and the input signal matrix; Set the identity matrix in the update formula to order k and update the weight vector; Set k←k+1 and return to step S200; S400: Setting the input signal matrix at time k; calculating the error signal and the complex Gaussian kernel function according to the generalized linear model and the input signal matrix; Set the identity matrix in the update formula to p order and update the weight vector; The step S300 specifically includes: Construct the input signal matrix at the current moment from the input signal vectors at the current moment and the past moment Calculate the error signal vector e(k) = d(k)- X T (k) w * and the complex Gaussian kernel function vector k(e(k))=exp(-|e(k)| 2 / 2σ 2 ); where d(k) = [d(k), ..., d(1)] T ∈c k×1 is the expected signal vector at discrete time k, X (k)∈c 2m×k ; σ is the core width; Update the weight vector w(k+1) of the k-th cycle according to the following formula: w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(k)] -1 g*(k), Where μ is the step size factor, the regularization parameter δ is set to 0.001, I(k) is the identity matrix, g(k) = k(e(k))Θe(k), Θ represents the multiplication of corresponding elements; Calculate the square of the weight vector error || w opt - w (k)|| 2 ,in w opt ∈C m×1 and v opt ∈C m ×1 are the optimal standard weight vector and the optimal conjugate weight vector respectively; Set k←k+1, and return to step S200; The step S400 includes: Construct the input signal matrix at the current moment from the input signal vectors at the current moment and the past moment Calculate the error signal vector e(k) = d(k)- X T (k) w * and the complex Gaussian kernel function vector k(e(k))=exp(-|e(k)| 2 / 2σ 2 ); where d(k) = [d(k), ..., d(k-p+1)] T ∈C p×1 is the expected signal vector at discrete time k, X (k)∈C 2m×p ; σ is the core width; Update the weight vector w of the k-th cycle according to the following formula: w (k+1)= w (k)+μ X (k)[ X H (k) X (k)+δI(p)] -1 g * (k), Where μ is the step size factor, the regularization parameter δ is set to 0.001, I(p) is the identity matrix, g(k) = k(e(k))Θe(k), Θ represents the multiplication of the corresponding elements; the square of the weight vector error is calculated as || w opt - w (k)|| 2 ; The step S400 includes: According to the principle that the step size decreases as the number of iterations increases, the step size factor function is taken The weight vector at the kth moment w Update according to the following formula: w (k+1)= w (k)+μ(k) X (k)[ X H (k) X (k)+δI(p)] -1 g ＊ (k); in is an estimate of E[k(e(k))], calculated as follows: Where λ is the forgetting factor; When k is equal to L, the weight vector w is obtained and output; the square difference of the weight error vector is as follows: || w opt - w (k)|| 2 .

2. The complex affine projection adaptive signal processing method based on kernel function according to claim 1, characterized in that: The step S400 further includes: If k<L, set k←k+1 and return to step S200, otherwise end the loop, where L is the total number of samples.

3. The complex affine projection adaptive signal processing method based on kernel function according to claim 1, characterized in that: The step size factor function μ(k) is obtained by the following steps: The cost function of the complex affine projection adaptive filtering algorithm based on the kernel function is constructed as follows: Where η(k) is the Lagrange multiplier vector, Find the variable w for the cost function * The conjugate derivative of (k+1) gives, make have to w (k+1)= w (k)+ X (k)η(k), and then we get, X H (k)w(k+1)= X H (k) w (k)+ X H (k) X (k)η(k); Therefore, Further, we can get, w (k+1)= w (k)+μ X (k)[ X H (k) X (k)] -1 k(e(k))Θe * (k); Using the idea of ​​fixed points, we can get X (k) X H (k) X (k)] -1 k(e(k)) Θ e * (k) = 0, that is E{ x (k)[ X H (k) X (k)] -1 k(e(k))Θd * (k)}=E{ X (k)[ x H (k) X (k)] -1 k(e(k))Θ X H w (k+1)}; Then we can get in R k+1 =R k +X(k)[ X H (k)X(k)] -1 diag(k(e(k))) X H (k), diag(.) represents a diagonal matrix consisting of corresponding elements; q k+1 =q k + X (k)[ X H (k) X (k)] -1 k(e(k))Θd ＊ (k); By the matrix inversion lemma, we can get, definition: Multiply both sides of the equation by the same number on the right Available, Therefore, In addition, by R k+1 The definition of can be further obtained, Therefore, Available,

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