Underwater acoustic reception signal detection method based on joint channel estimation, data decoding and impulse noise estimation

Abstract

The present invention discloses a method for detecting underwater acoustic reception signals that combines channel estimation, data decoding, and impulse noise estimation. The method first performs impulse noise initialization, then estimates the channel of the 0th sub-block based on training symbols. It then iterates between the combined channel, symbol, and impulse noise estimation modules and the decoding module, ultimately achieving combined channel estimation, data decoding, and impulse noise estimation. This method achieves smaller channel estimation errors and lower coding bit error rates under impulse noise.

Description

Technical Field

[0001] The present invention relates to the field of underwater acoustic communication, and in particular to an underwater acoustic reception signal detection method combining channel estimation, data decoding and impulse noise estimation. Background Art

[0002] Traditional underwater acoustic communication systems generally assume Gaussian noise and design their receiving algorithms accordingly. Common receiving algorithms for underwater acoustic communication include zero-forcing equalization, linear minimum mean square error equalization, and adaptive decision feedback equalization. These algorithms all assume Gaussian noise, but impulse noise exists in the ocean environment, such as sonar noise, underwater explosions, ice breaking, and the sounds of marine life. Impulse noise does not conform to the Gaussian noise assumption, and therefore the performance of traditional underwater acoustic communication algorithms is significantly affected in impulse noise environments. Existing underwater acoustic communication receiving algorithms are unable to cope with the strong interference of impulse noise. In the presence of impulse noise, their output results suffer from severe symbol estimation errors. Existing receiving algorithms that consider impulse noise generally only perform channel estimation, such as using sparse Bayesian learning for joint channel and impulse noise estimation or the robust l1-norm adaptive algorithm for channel estimation. However, they do not consider symbol estimation, resulting in high bit error rates in the presence of impulse noise. Summary of the Invention

[0003] The present invention aims to solve the problem of underwater acoustic communication under impulse noise and proposes an underwater acoustic receiving signal detection method combining channel estimation, data decoding and impulse noise estimation, thereby improving the coding bit error rate performance under impulse noise.

[0004] The object of the present invention is achieved through the following technical solutions:

[0005] A method for detecting underwater acoustic reception signals by combining channel estimation, data decoding, and impulse noise estimation, the method comprising the following steps:

[0006] S1: Initialize the estimated channels of different sub-blocks in, Indicates L c ×1-dimensional zero vector, parameter L c Indicates the channel length, parameter N sb Indicates the number of sub-blocks; initializes the hard decision symbol vector Among them, the vector x p Represents the training symbol vector, the superscript T means transpose, and the parameter N d Represents the number of data symbols; assuming the modulation order is M, initialize the prior likelihood ratio Initialize the first-order autoregressive coefficient of the 0th sub-block α0=0;

[0007] S2: Based on the training symbol vector x p and training symbol received signal y p Estimate the channel h0 and noise variance of the 0th sub-block

[0008] S3: Estimate the channel based on the received signal y Hard decision symbol vector Noise variance The first-order autoregressive coefficient α0 and the prior likelihood ratio L of the 0th sub-block a Calculate the estimated symbol x and estimated channel Among them, h i The subscript i of represents the i-th sub-block;

[0009] S4: Based on the estimated symbol x and the prior likelihood ratio L a Calculate the external likelihood ratio L e ;

[0010] S5: According to the external likelihood ratio L e Calculate the estimated bit b and map the estimated bit b to a hard decision symbol

[0011] S6: Update the prior likelihood ratio L a =L e ;

[0012] S7: Repeat steps (3)-(6) until the number of iterations reaches the maximum number of iterations N iter , and obtain the final estimated bit b.

[0013] Furthermore, the step S2 includes the following sub-steps:

[0014] S2.1: Construct the Toeplitz matrix x of the training symbols p :

[0015]

[0016] Where x p，n Represents the training symbol vector x p The nth element of ;

[0017] S2.2: Calculate the estimated channel h0 and noise variance of the 0th sub-block

[0018]

[0019]

[0020] Where, the superscript H means conjugate transpose, the superscript -1 means matrix inversion, and the symbol |||| means the l2 norm.

[0021] Furthermore, the step S3 includes the following sub-steps:

[0022] S3.1: Estimate the channel based on the received signal y and hard decision symbol vector Calculate and estimate the impulse noise o;

[0023] S3.2. Initialize sub-block number i=1;

[0024] S3.3: Define the received signal of the i-th sub-block Among them, the variable y n Represents the nth element of the received signal y, parameter L i Indicates the interval between adjacent sub-blocks;

[0025] Define the estimated impulse noise of the i-th sub-block

[0026]

[0027] In the formula, the variable o n Represents the nth element of the estimated impulse noise o

[0028] Define the estimated symbol of the i-th sub-block

[0029]

[0030] In the formula, the variable x n Represents the nth element of the estimated symbol x;

[0031] According to the received signal y of the i-th sub-block i , the estimated impulse noise o of the i-th sub-block i and the estimated channel h of the i-1th sub-block i-1 Calculate the estimated symbol x of the i-th sub-block i ;

[0032] S3.4: Define hard decision function This function outputs the constellation point closest to x and updates the elements of the hard decision symbol vector

[0033]

[0034] S3.5: Based on the hard decision symbol vector The first-order autoregressive coefficient α of the i-1th sub-block i-1 and the noise variance of the received signal Calculate the estimated channel h of the i-th sub-block i and the first-order autoregressive coefficient α of the i-th sub-blocki , and assign i+1 to i, and update the sub-block index;

[0035] S3.6: Repeat (3.3)-(3.5) until i=N sb +1.

[0036] Furthermore, the S3.1 includes the following sub-steps:

[0037] S3.1.1: Let L y =N d +L c -1, parameter L y The meaning is to consider the length of the data block of the channel tail and calculate the smoothing parameter μ:

[0038]

[0039] β=λ2(L y -1) / 2

[0040]

[0041]

[0042] Where, parameter λ2 represents the regularization parameter of the sparse term of the impulse noise group, variable α represents the first parameter of the smooth approximation, variable β represents the second parameter of the smooth approximation, parameter ε represents the allowable error, and variable L f Lipschitz constant representing the smooth function of the received signal portion;

[0043] S3.1.2: Compute the Lipschitz Constant for Smooth Functions

[0044] S3.1.3: Calculate the real-valued residual received signal after removing the transmitted symbol components

[0045]

[0046]

[0047]

[0048] Where, variable H represents the channel convolution matrix, variable y in represents the residual received signal in complex form after the transmitted symbol components are eliminated, represents the residual received signal y in The vector consisting of the real and imaginary parts of the nth element of y in，n represents the residual received signal y inThe nth element of Indicates taking the real part, the symbol Indicates taking the imaginary part;

[0049] S3.1.4: Initialize the impulse noise estimate after Nesterov momentum acceleration for the 0th iteration of the fast iterative soft threshold algorithm: z 0 =0, initialize the number of iterations of the fast iterative soft threshold algorithm: k=0, initialize the intermediate variable t of the Nesterov momentum acceleration of the 0th fast iterative soft threshold algorithm iteration 0 :t 0 =1;

[0050] S3.1.5: Definitions

[0051]

[0052]

[0053] A n =BC n , n=2,3,…,L y

[0054] Where, 0 a×b represents an a×b dimensional zero matrix, I a represents the a-dimensional unit matrix, B, C n and A n are all constant matrices;

[0055] S3.1.6: Compute the gradient of a smooth function

[0056]

[0057]

[0058] Where, is the gradient of the smooth function of the sparse term of the impulse noise group, symbol The meaning of is gradient;

[0059] S3.1.7: Calculate the impulse noise for the kth iteration

[0060]

[0061]

[0062]

[0063]

[0064]

[0065]

[0066] Where, For temporarily defined intermediate variables, variables Represents an intermediate variable The nth element of , the symbol j means The parameter λ1 represents the regularization parameter of the sparse term of the impulse noise, and the symbol ∠ means the phase angle of the complex number;

[0067] S3.1.8: Calculate the estimated impulse noise z after Nesterov momentum acceleration for the kth iteration of the fast iterative soft thresholding algorithm k+1 :

[0068]

[0069]

[0070] Update the number of iterations of the fast iterative soft threshold algorithm: k←k+1.

[0071] S3.1.9: Repeat (3.1.6)-(3.1.8) until k = K + 1; calculate the estimated impulse noise

[0072]

[0073] Furthermore, step S3.3 includes the following sub-steps:

[0074] S3.3.1: Initialize the estimated symbols of the i-th sub-block of the 0th main optimal minimization iteration Initialize the number of iterations of the main optimization minimization: p = 0; define the channel convolution matrix H of the i-1th sub-block i-1

[0075]

[0076] Among them, h i，l Denotes the estimated channel h of the i-th sub-block i The lth element of ;

[0077] Calculate the matrix The maximum eigenvalue λ max ;

[0078] The matrix H that defines the tail part of the channel convolution matrix of the i-1th sub-block i-1，tail :

[0079]

[0080]

[0081] Where H i-1，tail，tmp The minimum non-zero submatrix of the matrix representing the tail part of the channel convolution matrix of the i-1th sub-block; Calculate the residual received signal y after eliminating impulse noise and inter-block inter-code interference r :

[0082] y r =y i -H i-1，tail x i-1，tail -o i ;

[0083] S3.3.2: Compute the optimal value of the estimated sign for the pth principal minimization iteration

[0084]

[0085] Note vector The nth element of vector The nth element of if In the constraints , then take otherwise Take constraints Within the range of The nearest point. Calculate the estimated symbol of the i-th sub-block at the p+1th main optimal minimization iteration

[0086]

[0087] Update the number of iterations of the main optimization minimization: p←p+1;

[0088] S3.3.3: Repeat (3.3.2) until p = K MM +1, where parameter K MM The meaning is the maximum number of main optimal minimization iterations; calculate the estimated symbol x of the i-th sub-block i

[0089]

[0090] Furthermore, in step S3.5, the estimated channel h of the i-th sub-block i and the first-order autoregressive coefficient α of the i-th sub-block i The calculation process includes the following steps:

[0091] S3.5.1: Based on the hard decision symbol vector Construct the Toeplitz matrix X of the hard decision symbols of the i-th sub-block i :

[0092]

[0093] Define the received signal of the i-th non-overlapping sub-block The estimated channel h is calculated based on the noise variance of the received signal. i-1 and the estimated impulse noise o of the i-th sub-block i , calculate the estimated channel h of the i-th sub-block i :

[0094]

[0095] S3.5.2: Calculate the first-order autoregressive coefficient of the i-th sub-block:

[0096] Furthermore, step S4 includes the following sub-steps:

[0097] S4.1: Calculate the gain factor g of the estimated symbol:

[0098] S4.2: Calculate the variance of the estimated symbol

[0099] S4.3: Assume that the set of constellation points where the mth bit is 0 is The set of constellation points where the mth bit is 1 is For n=1, 2, ..., N d ,m=1,2,…,M,calculate the outer likelihood ratio L of the estimated symbol e :

[0100]

[0101] L e =[L e (1), L e (2),…,L e (N d M)] T in, function It represents the m-th bit in the bits mapped to the constellation point α, the function ln means the natural logarithm function, the symbol ∑ means summation, and the symbol π means cumulative multiplication.

[0102] The beneficial effects of the present invention are as follows:

[0103] The underwater acoustic reception signal detection method of the present invention combines channel estimation, data decoding and impulse noise estimation, because it takes channel estimation, data decoding and impulse noise estimation into consideration at the same time, thus, the channel estimation error under impulse noise is smaller and the coding bit error rate is lower. BRIEF DESCRIPTION OF THE DRAWINGS

[0104] Figure 1 It is a schematic diagram of the overall flow of the underwater acoustic receiving signal detection method of the present invention.

[0105] Figure 2 Schematic diagram of the process of step S3.

[0106] Figure 3 Schematic diagram of the calculation flow of estimated impact noise in step S3.1.

[0107] Figure 4 The graph is a graph showing the change of the channel estimation error under impulse noise with the signal-to-noise ratio for the method of the present invention and the least squares channel estimation method without considering impulse noise.

[0108] Figure 5 The graph is a graph showing the variation of the coding bit error rate with the signal-to-noise ratio under impulse noise for the method of the present invention and four existing methods. DETAILED DESCRIPTION

[0109] The present invention will be described in detail below with reference to the accompanying drawings and preferred embodiments, and the purpose and effects of the present invention will become more apparent. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0110] like Figures 1 to 3 As shown, the underwater acoustic reception signal detection method of this embodiment for joint channel estimation, data decoding, and impulse noise estimation includes the following steps:

[0111] Step S1: Initialize the estimated channels of different sub-blocks in, Indicates I c ×1-dimensional zero vector, parameter L c Indicates the channel length, parameter N sb Indicates the number of sub-blocks; initializes the hard decision symbol vector Among them, the vector x p Represents the training symbol vector, the superscript T means transpose, and the parameter N d Represents the number of data symbols; assuming the modulation order is M, initialize the prior likelihood ratio Initialize the first-order autoregressive coefficient of the 0th sub-block α0=0;

[0112] S2: Based on the training symbol vector x p and training symbol received signal y pEstimate the channel h0 and noise variance of the 0th sub-block Step S2 includes the following sub-steps:

[0113] S2.1: Construct the Toeplitz matrix X of the training symbols p :

[0114]

[0115] Where x p，n Represents the training symbol vector x p The nth element of ;

[0116] S2.2: Calculate the estimated channel h0 and noise variance of the 0th sub-block

[0117]

[0118]

[0119] Where, the superscript H means conjugate transpose, the superscript -1 means matrix inversion, and the symbol |||| means the l2 norm.

[0120] S3: Estimate the channel based on the received signal y Hard decision symbol vector Noise variance First-order autoregressive coefficient α0 and prior likelihood ratio L a Calculate the estimated symbol x and estimated channel Among them, h i The subscript i represents the i-th sub-block; step S3 includes the following sub-steps:

[0121] S3.1: Estimate the channel based on the received signal y and hard decision symbol vector Calculate and estimate the impulse noise o; S3.1 includes the following sub-steps:

[0122] S3.1.1: Let L y =N d +L c -1, parameter L y The meaning is to consider the length of the data block of the channel tail and calculate the smoothing parameter μ:

[0123]

[0124] β=λ2(L y -1) / 2

[0125]

[0126]

[0127] Where, parameter λ2 represents the regularization parameter of the sparse term of the impulse noise group, variable α represents the first parameter of the smooth approximation, variable β represents the second parameter of the smooth approximation, parameter ε represents the allowable error, and variable L f Lipschitz constant representing the smooth function of the received signal portion;

[0128] S3.1.2: Compute the Lipschitz Constant for Smooth Functions

[0129] S3.1.3: Calculate the real-valued residual received signal after removing the transmitted symbol components

[0130]

[0131]

[0132]

[0133] Where, variable H represents the channel convolution matrix, variable y in represents the residual received signal in complex form after the transmitted symbol components are eliminated, represents the residual received signal y in The vector consisting of the real and imaginary parts of the nth element of y in，n represents the residual received signal y in The nth element of Indicates taking the real part, the symbol Indicates taking the imaginary part;

[0134] S3.1.4: Initialize the impulse noise estimate after Nesterov momentum acceleration for the 0th iteration of the fast iterative soft threshold algorithm: z 0 =0, initialize the number of iterations of the fast iterative soft threshold algorithm: k=0, initialize the intermediate variable t of the Nesterov momentum acceleration of the 0th fast iterative soft threshold algorithm iteration 0 :t 0 =1;

[0135] S3.1.5: Definitions

[0136]

[0137]

[0138] A n =BC n , n=2,3,…,L y

[0139] Where, 0 a×b represents an a×b dimensional zero matrix, I a represents the a-dimensional unit matrix, B, C n and A n are all constant matrices;

[0140] S3.1.6: Compute the gradient of a smooth function

[0141]

[0142]

[0143] Where, is the gradient of the smooth function of the sparse term of the impulse noise group, symbol The meaning of is to find the gradient;

[0144] S3.1.7: Calculate the impulse noise for the kth iteration

[0145]

[0146]

[0147]

[0148]

[0149]

[0150]

[0151] Where, For temporarily defined intermediate variables, variables Represents an intermediate variable The nth element of , the symbol j means The parameter λ1 represents the regularization parameter of the sparse term of the impulse noise, and the symbol ∠ means the phase angle of the complex number;

[0152] S3.1.8: Calculate the estimated impulse noise z after Nesterov momentum acceleration for the kth iteration of the fast iterative soft thresholding algorithm k+1 :

[0153]

[0154]

[0155] Update the number of iterations of the fast iterative soft threshold algorithm: k←k+1.

[0156] S3.1.9: Repeat (3.1.6)-(3.1.8) until k = K + 1; calculate the estimated impulse noise

[0157]

[0158] S3.2. Initialize sub-block number i=1;

[0159] S3.3: Define the received signal of the i-th sub-block Among them, the variable y n Represents the nth element of the received signal y, parameter L i Indicates the interval between adjacent sub-blocks;

[0160] Define the estimated impulse noise of the i-th sub-block

[0161]

[0162] In the formula, the variable o n Represents the nth element of the estimated impulse noise o

[0163] Define the estimated symbol of the i-th sub-block

[0164]

[0165] In the formula, the variable x n Represents the nth element of the estimated symbol x;

[0166] According to the received signal y of the i-th sub-block i , the estimated impulse noise o of the i-th sub-block i and the estimated channel h of the i-1th sub-block i-1 Calculate the estimated symbol x of the i-th sub-block i ;

[0167] Step S3.3 includes the following sub-steps:

[0168] S3.3.1: Initialize the estimated symbols of the i-th sub-block of the 0th main optimal minimization iteration Initialize the number of iterations of the main optimization minimization: p = 0; define the channel convolution matrix H of the i-1th sub-block i-1

[0169]

[0170] Among them, h i，l Denotes the estimated channel h of the i-th sub-block i The lth element of ;

[0171] Calculate the matrix The maximum eigenvalue λ max ;

[0172] The matrix H that defines the tail part of the channel convolution matrix of the i-1th sub-block i-1，tail :

[0173]

[0174]

[0175] Where H i-1，tail，tmp The minimum non-zero submatrix of the matrix representing the tail part of the channel convolution matrix of the i-1th sub-block; Calculate the residual received signal y after eliminating impulse noise and inter-block interference r :

[0176] y r =y i -H i-1，tail x i-1，tail -o i ;

[0177] S3.3.2: Compute the optimal value of the estimated sign for the pth principal minimization iteration Note vector The nth element of vector The nth element of if In the constraints , then take otherwise Take constraints Within the range of The nearest point. Calculate the estimated symbol of the i-th sub-block at the p+1th main optimal minimization iteration

[0178]

[0179] Update the number of iterations of the main optimization minimization: p←p+1;

[0180] S3.3.3: Repeat (3.3.2) until p = K MM +1, where parameter K MM The meaning is the maximum number of main optimal minimization iterations; calculate the estimated symbol x of the i-th sub-block i

[0181]

[0182] S3.4: Define hard decision function This function outputs the constellation point closest to x and updates the elements of the hard decision symbol vector

[0183]

[0184] S3.5: Based on the hard decision symbol vector The first-order autoregressive coefficient α of the i-1th sub-block i-1 and the noise variance of the received signal Calculate the estimated channel h of the i-th sub-block i and the first-order autoregressive coefficient α of the i-th sub-block i , and assign i+1 to i, update the sub-block index; in step S3.5, the estimated channel h of the i-th sub-block i and the first-order autoregressive coefficient α of the i-th sub-block i The calculation process includes the following steps:

[0185] S3.5.1: Based on the hard decision symbol vector Construct the Toeplitz matrix Xi of the hard decision symbols of the i-th sub-block:

[0186]

[0187] Define the received signal of the i-th non-overlapping sub-block The estimated channel h is calculated based on the noise variance of the received signal. i-1 and the estimated impulse noise o of the i-th sub-block i , calculate the estimated channel h of the i-th sub-block i :

[0188]

[0189] S3.5.2: Calculate the first-order autoregressive coefficient of the i-th sub-block:

[0190] S3.6: Repeat (3.3)-(3.5) until i=N sb +1.

[0191] S4: Based on the estimated symbol x and the prior likelihood ratio L a Calculate the external likelihood ratio L e ; Step S4 includes the following sub-steps:

[0192] S4.1: Calculate the gain factor g of the estimated symbol:

[0193] S4.2: Calculate the variance of the estimated symbol

[0194] S4.3: Assume that the set of constellation points where the mth bit is 0 is The set of constellation points where the mth bit is 1 is For n=1, 2, ..., N d , m = 1, 2, ..., M, calculate the outer likelihood ratio L of the estimated symbol e :

[0195]

[0196] L e =[L e (1), L e (2),…,L e (N d M)] T

[0197] in, function It represents the m-th bit in the bits mapped to the constellation point α, the function ln means the natural logarithm function, the symbol ∑ means summation, and the symbol π means cumulative multiplication.

[0198] S5: According to the external likelihood ratio L e Calculate the estimated bit b and map the estimated bit b to a hard decision symbol

[0199] S6: Update the prior likelihood ratio L a =L e ;

[0200] S7: Repeat steps (3)-(6) until the number of iterations reaches the maximum number of iterations N iter , and obtain the final estimated bit b.

[0201] In order to further verify the beneficial effect of the method of the present invention, the method of the present invention is compared with the least squares channel estimation method without considering impulse noise, and the channel estimation error under impulse noise is obtained as follows: Figure 4 As shown in the figure, the horizontal axis is the signal-to-noise ratio (SNR), the vertical axis is the normalized mean square error (NMSE) of channel estimation, the JCDI curve represents the method of the present invention, and the JCD curve represents the least square channel estimation method without considering impulse noise. At a signal-to-noise ratio of 24dB, the normalized mean square error of the method of the present invention is about 13dB lower than that of the method without considering impulse noise. The excellent channel estimation performance of the receiver is a prerequisite for excellent coding bit error rate performance. The channel estimation error of the method of the present invention is better than that of the method without considering impulse noise, which lays the foundation for the superiority of the coding bit error rate performance of the method of the present invention. At the same time, the bit error rate of the method of the present invention is compared with that of the four existing methods, and the results are as follows. Figure 5 As shown in the figure, the horizontal axis is the signal-to-noise ratio, the vertical axis is the coding bit error ratio (BER), the JCDI curve represents the method of the present invention, the JCD curve represents the joint channel estimation and symbol decoding algorithm, which is a degenerate version of the method of the present invention, that is, it does not consider impulse noise, the LMMSE curve represents the linear minimum mean square error estimation algorithm, the RLS-DFE curve represents the recursive least squares decision feedback equalization algorithm, and the ZF curve represents the zero-forcing equalization algorithm. It can be seen that the coding bit error rate of the method of the present invention is the lowest, at 10 -3 At the coding error rate, the coding error rate performance of the method of the present invention is about 3dB higher than that of the joint channel estimation and symbol decoding algorithm, and about 6dB higher than that of the linear minimum mean square error estimation algorithm. Figure 4 and Figure 5 It can be shown that the method of the present invention has a smaller channel estimation error and a lower coding bit error rate under impulse noise.

[0202] Those skilled in the art will understand that the foregoing descriptions are merely preferred embodiments of the invention and are not intended to limit the invention. Although the invention has been described in detail with reference to the foregoing examples, those skilled in the art will still be able to modify the technical solutions described in the foregoing examples or substitute equivalents for some of the technical features therein. Any modifications, equivalent substitutions, etc. made within the spirit and principles of the invention shall be included within the scope of protection of the invention.

Claims

1. A method for underwater acoustic reception signal detection combining channel estimation, data decoding and impulse noise estimation, characterized in that: The method comprises the following steps: S1: Initialize the estimated channels of different sub-blocks in, Indicates L c ×1-dimensional zero vector, parameter L c Indicates the channel length, parameter N sb Indicates the number of sub-blocks; initializes the hard decision symbol vector Among them, the vector x p Represents the training symbol vector, the superscript T means transpose, and the parameter N d Represents the number of data symbols; assuming the modulation order is M, initialize the prior likelihood ratio Initialize the first-order autoregressive coefficient of the 0th sub-block α0=0; S2: Based on the training symbol vector x p and training symbol received signal y p Estimate the channel h0 and noise variance of the 0th sub-block S3: Estimate the channel based on the received signal y Hard decision symbol vector Noise variance The first-order autoregressive coefficient α0 and the prior likelihood ratio L of the 0th sub-block a Calculate the estimated symbol x and estimated channel Among them, h i The subscript i of represents the i-th sub-block; S4: Based on the estimated symbol x and the prior likelihood ratio L a Calculate the external likelihood ratio L e ; S5: According to the external likelihood ratio L e Calculate the estimated bit b and map the estimated bit b to a hard decision symbol S6: Update the prior likelihood ratio L a =L e ; S7: Repeat steps (3)-(6) until the number of iterations reaches the maximum number of iterations N iter , get the final estimated bit b; The S2 includes the following sub-steps: S2.1: Construct the Toeplitz matrix X of the training symbols p : Where x p，n Represents the training symbol vector x p The nth element of ; S2.2: Calculate the estimated channel h0 and noise variance of the 0th sub-block In the formula, the superscript H means conjugate transpose, the superscript -1 means matrix inversion, and the symbol || || means the l2 norm; The S3 includes the following sub-steps: S3.1: Estimate the channel based on the received signal y and hard decision symbol vector Calculate and estimate the impulse noise o; S3.

2. Initialize sub-block number i=1; S3.3: Define the received signal of the i-th sub-block Among them, the variable y n Represents the nth element of the received signal y, parameter L i Indicates the interval between adjacent sub-blocks; Define the estimated impulse noise of the i-th sub-block In the formula, the variable o n Represents the nth element of the estimated impulse noise o Define the estimated symbol of the i-th sub-block In the formula, the variable x n Represents the nth element of the estimated symbol x; According to the received signal y of the i-th sub-block i , the estimated impulse noise o of the i-th sub-block i and the estimated channel h of the i-1th sub-block i-1 Calculate the estimated symbol x of the i-th sub-block i ; S3.4: Define hard decision function This function outputs the constellation point closest to x and updates the elements of the hard decision symbol vector S3.5: Based on the hard decision symbol vector The first-order autoregressive coefficient α of the i-1th sub-block i-1 and the noise variance of the received signal Calculate the estimated channel h of the i-th sub-block i and the first-order autoregressive coefficient α of the i-th sub-block i , and assign i+1 to i, and update the sub-block index; S3.6: Repeat (3.3)-(3.5) until i=N sb +1; The S3.1 includes the following sub-steps: S3.1.1: Let L y =N d +L c -1, parameter L y The meaning is to consider the length of the data block of the channel tail and calculate the smoothing parameter μ: β=λ2(L y -1) / 2 Where, parameter λ2 represents the regularization parameter of the sparse term of the impulse noise group, variable α represents the first parameter of the smooth approximation, variable β represents the second parameter of the smooth approximation, parameter ε represents the allowable error, and variable L f Lipschitz constant representing the smooth function of the received signal portion; S3.1.2: Compute the Lipschitz Constant for Smooth Functions S3.1.3: Calculate the real-valued residual received signal after removing the transmitted symbol components Where, variable H represents the channel convolution matrix, variable y in represents the residual received signal in complex form after the transmitted symbol components are eliminated, represents the residual received signal y in The vector consisting of the real and imaginary parts of the nth element of y in，n represents the residual received signal y in The nth element of Indicates taking the real part, the symbol Indicates taking the imaginary part; S3.1.4: Initialize the impulse noise estimate after Nesterov momentum acceleration for the 0th iteration of the fast iterative soft threshold algorithm: z 0 =0, initialize the number of iterations of the fast iterative soft threshold algorithm: k=0, initialize the intermediate variable t of the Nesterov momentum acceleration of the 0th fast iterative soft threshold algorithm iteration 0 :t 0 =1; S3.1.5: Definitions HAS n =BC n ,n =2,3,…,L y Where, 0 a×b represents an a×b dimensional zero matrix, I a represents the a-dimensional unit matrix, B, C n and A n are all constant matrices; S3.1.6: Compute the gradient of a smooth function Where, is the gradient of the smooth function of the sparse term of the impulse noise group, symbol The meaning of is to find the gradient; S3.1.7: Calculate the impulse noise for the kth iteration Where, For temporarily defined intermediate variables, variables Represents an intermediate variable The nth element of , the symbol j means The parameter λ1 represents the regularization parameter of the sparse term of the impulse noise, and the symbol ∠ means the phase angle of the complex number; S3.1.8: Calculate the estimated impulse noise z after Nesterov momentum acceleration for the kth iteration of the fast iterative soft thresholding algorithm k+1 : Update the number of iterations of the fast iterative soft threshold algorithm: k←k+1; S3.1.9: Repeat (3.1.6)-(3.1.8) until k = K + 1; calculate the estimated impulse noise Step S3.3 includes the following sub-steps: S3.3.1: Initialize the estimated symbols of the i-th sub-block of the 0th main optimal minimization iteration Initialize the number of iterations of the main optimization minimization: p = 0; define the channel convolution matrix H of the i-1th sub-block i-1 Among them, h i，l Denotes the estimated channel h of the i-th sub-block i The lth element of ; Calculate the matrix The maximum eigenvalue λ max ; The matrix H that defines the tail part of the channel convolution matrix of the i-1th sub-block i-1,tail : Where H i-1,tail,tmp The minimum non-zero submatrix of the matrix representing the tail part of the channel convolution matrix of the i-1th sub-block; Calculate the residual received signal y after eliminating impulse noise and inter-block interference r : and r =and i -H i-1,tail x i-1,tail -either i ; S3.3.2: Compute the optimal value of the estimated sign for the pth principal minimization iteration Note vector The nth element of vector The nth element of if In the constraints , then take otherwise Take constraints Within the range of The nearest point; calculate the estimated symbol of the i-th sub-block of the p+1th main optimal minimization iteration Update the number of iterations of the main optimization minimization: p←p+1; S3.3.3: Repeat (3.3.2) until p = K MM +1, where parameter K MM The meaning is the maximum number of main optimal minimization iterations; calculate the estimated symbol x of the i-th sub-block i In step S3.5, the estimated channel h of the i-th sub-block i and the first-order autoregressive coefficient α of the i-th sub-block i The calculation process includes the following steps: S3.5.1: Based on the hard decision symbol vector Construct the Toeplitz matrix X of the hard decision symbols of the i-th sub-block i : Define the received signal of the i-th non-overlapping sub-block The estimated channel h is calculated based on the noise variance of the received signal. i-1 and the estimated impulse noise o of the i-th sub-block i , calculate the estimated channel h of the i-th sub-block i : S3.5.2: Calculate the first-order autoregressive coefficient of the i-th sub-block: Step S4 includes the following sub-steps: S4.1: Calculate the gain factor g of the estimated symbol: S4.2: Calculate the variance of the estimated symbol S4.3: Assume that the set of constellation points where the mth bit is 0 is The set of constellation points where the mth bit is 1 is For n=1, 2, ..., N d ,m=1,2,…,M,calculate the outer likelihood ratio L of the estimated symbol e : L e =[L e (1),L e (2),…,L e (N d M)] T in, function It represents the m-th bit in the bits mapped to the constellation point α, the function ln means the natural logarithm function, the symbol ∑ means summation, and the symbol ∏ means cumulative multiplication.

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