BCH code rapid blind identification method based on matrix transformation

Abstract

The invention discloses a BCH code rapid identification method based on matrix transformation. The method comprises the following steps: arranging BCH code bit streams into a matrix; simplifying the bit stream matrix into an upper triangular matrix; estimating a code length and a check matrix of the BCH code according to the matrix deficit rank characteristic; estimating the generated matrix by using the orthogonality of the check matrix and the generated matrix; and simplifying the generator matrix, and estimating to obtain a generator polynomial of the BCH code. According to the method, double traversal detection does not need to be carried out on coding parameters, the timeliness of BCH code recognition can be effectively improved, and compared with an existing method, the calculation complexity is low, and BCH code online recognition can be completed in a non-cooperative confrontation environment.

Description

Technical Field

[0001] The invention belongs to the field of communication countermeasure reconnaissance identification, and in particular relates to a BCH code fast identification method based on matrix transformation. Background Art

[0002] Blind recognition of BCH codes refers to estimating the code length, number of information bits, and generating polynomial without prior information. It is a key technology in the field of intelligent communications and information interception.

[0003] In the prior art, there are mainly the following methods for identifying BCH codes:

[0004] G.Wu, B.N.Zhang, X.Wen, et al. Blind Recognition of BCH Code Based on Galois Field Fourier Transform[C] / / International Conference on Wireless Communications & Signal Processing. Nanjing, China: IEEE, 2015. In this paper, the code length and primitive polynomial of the BCH code are first assumed, and then the received codeword sequence is subjected to Galois Field Fourier Transform (GFFT). The correctness of the code length and primitive polynomial is detected based on the number and position of consecutive zero coefficients in the GFFT result.

[0005] D.Jo, S.Kwon, DJ Shin. Blind Reconstruction of BCH Codes Based on Consecutive Roots of Generator Polynomials[J]. IEEE Communication Letters. 2018, 22(5): 894-897. In this paper, in the process of GFFT of codewords, codewords containing erroneous bits are continuously deleted, thereby improving the detection accuracy of the code length and primitive polynomial.

[0006] YYChang, W.Zhang, H.Wang, et al.Fast Blind Recognition of BCH CodeBased on Spectral Analysis and Probability Statistics[J].IEEE Communication Letters.2021,25(10):3170-3174. This paper proposes a fast GFFT. Compared with the original GFFT method, the amount of calculation required by the fast method is greatly reduced.

[0007] ADYardi, S. Vijayakumaran, A. Kumar. Blind Reconstruction of Binary Cyclic Codes from Unsynchronized Bitstream[J]. IEEE Transactions on Communications. 2016, 64(7): 2693-2706. This paper uses the property that the codeword polynomial and the generator polynomial contain common roots to estimate the BCH code generator polynomial based on hypothesis testing theory.

[0008] Existing methods mainly identify BCH codes by traversing and detecting possible code lengths and primitive polynomials of BCH codes. As the code length increases, the traversal search space increases rapidly, causing the computational complexity of the identification method to increase rapidly. Summary of the invention

[0009] In order to overcome the limitations of existing recognition methods that require traversal detection of BCH code encoding parameters and have high computational complexity, the present invention provides a BCH code fast recognition method based on matrix transformation, which does not require double traversal detection of encoding parameters and can effectively improve the timeliness of BCH code recognition. Compared with existing methods, the computational complexity is lower and BCH code online recognition can be completed in a non-cooperative adversarial environment.

[0010] The objective of the present invention is achieved through the following technical solutions:

[0011] A method for quickly identifying a BCH code based on matrix transformation comprises the following steps:

[0012] S1. Arrange the BCH code bit stream into a matrix C b ;

[0013] S2. Simplify the bitstream matrix into an upper triangular matrix C s ;

[0014] S3. Estimate the code length n and check matrix H of the BCH code according to the matrix rank deficiency characteristic;

[0015] S4. Using the orthogonality of the check matrix H and the generator matrix G, estimate the generator matrix G;

[0016] S5. Simplify the generator matrix G and estimate the generator polynomial of the BCH code.

[0017] Furthermore, the step S1 comprises:

[0018] The coded bit stream is grouped, and the number of bits N contained in each group is the number of columns of the matrix; each bit group is arranged in rows in sequence, and the number of groups M is the number of rows of the matrix, and the number of rows M of the matrix is ​​greater than the number of columns N, and the bit stream is arranged into a matrix C of size M×N b .

[0019] Furthermore, the step S2 comprises:

[0020] The BCH code bit stream matrix C b Perform elementary row transformations row by row to simplify to the upper triangular matrix C s , if the bitstream matrix C b Full rank, then the upper triangular matrix C s It is expressed as:

[0021]

[0022] In the formula, Q N×N represents an N×N upper triangular matrix, 0 (M-N)×N Represents a (MN)×N zero matrix.

[0023] Furthermore, the step S3 comprises:

[0024] If the number of columns N of the matrix is ​​equal to the code length n, the matrix C b There is rank deficiency, and the simplified upper triangular matrix C s It is expressed as:

[0025]

[0026] In the formula, the subscript k is the number of information bits of the BCH code, which is also the rank of the matrix at this time, and n is the code length of the BCH code;

[0027] The upper triangular matrix C s Simplified to the simplest form, the simplest matrix C s ′ is expressed as:

[0028]

[0029] In the formula, I k represents the unit matrix of size k×k, P k×(n-k)Represents a matrix of size k×(nk) consisting of 0 and 1 bits, and the matrix P k×(n-k) The check matrix H of the BCH code is constructed and expressed as:

[0030]

[0031] Alternatively, in step S3, if the number of matrix columns is not equal to the code length, then the bitstream matrix is ​​full rank. When the matrix is ​​full rank due to bit errors, a test statistic ρ is constructed:

[0032]

[0033] Among them, N c Represents the upper triangular matrix C s The number of 1 elements in each column, M represents the matrix C s The number of rows, i represents the index value of each column;

[0034] Set the threshold th, when ρ(i) is less than th, that is:

[0035] ρ(i)<th

[0036] Determine whether the column is an independent column;

[0037] When ρ(i) is greater than th, that is:

[0038] ρ(i)>th

[0039] Determine whether the column is a non-independent column. When the matrix is ​​rank deficient, there are non-independent columns, and the rank is equal to the number of independent columns.

[0040] When the matrix affected by the bit error is full rank, the BCH code check matrix is ​​estimated using the augmented matrix. N The same simplification process as the coded bit stream matrix in step S2 is performed to obtain:

[0041] I′ N =D1I N

[0042] Where D1 represents the elementary row transformation matrix;

[0043] In the matrix I′ N In the example, the columns corresponding to the non-independent columns in the simplest matrix constitute the check matrix H of the BCH code.

[0044] Furthermore, the step S4 comprises:

[0045] The check matrix H and the generator matrix G of the BCH code satisfy orthogonality, that is:

[0046] H.G. T =0

[0047] Combined with the expression of the check matrix H, the generator matrix G is:

[0048] G=[I k P k×(n-k) ]

[0049] Furthermore, the step S5 comprises:

[0050] The generating polynomial of BCH code is g(x)=g n-k x n-k +g n-k-1 x n-k-1 +…+g0 and the generator matrix G satisfy the following relationship:

[0051]

[0052] Simplifying the generator matrix G into the form of the above formula gives the generator polynomial, thus completing the BCH code recognition.

[0053] Preferably, in step S1, the coded bit stream is grouped, high reliability bits are selected using log-likelihood ratio, and then bit stream groups whose row reliability is greater than a threshold α are selected from the bit stream groups. The number of bits N contained in each group is the number of columns of the matrix. The bit groups are arranged in rows in sequence, and the number of groups M is the number of rows of the matrix. It is required that the number of rows M of the matrix is ​​greater than the number of columns N. In this way, the bit stream is arranged into a matrix C of size M×N b .

[0054] Furthermore, in step S1, the process of selecting high reliability bits by using log-likelihood ratio is as follows:

[0055] The log-likelihood ratio is constructed from the soft decision data. For BPSK signals, the soft decision data r t It is expressed as:

[0056]

[0057] Where n t is a Gaussian random variable;

[0058] Assuming H0 and H1 represent the transmitted bits as 0 and 1 respectively, the likelihood functions under different assumptions are:

[0059]

[0060] The log-likelihood ratio LLR is constructed from the likelihood function:

[0061]

[0062] The absolute value of the log-likelihood ratio |LLR| is used to measure the reliability of the bit; the reliability of each row of the bitstream matrix is ​​determined by the minimum bit reliability of the row, and the bitstream groups with row reliability greater than the threshold α are selected to form the matrix C to be analyzed. b , in order to reduce the number of bit errors in the matrix.

[0063] The present invention has the following advantages:

[0064] The present invention provides a BCH code fast identification method based on matrix transformation, which adopts a matrix analysis method to simplify the BCH code bit stream matrix into an upper triangular matrix, and estimates the rank and null space of the matrix according to the simplification result. When the matrix is ​​deficient in rank, the code length of the BCH code is estimated by the number of matrix columns, the number of bits of the BCH code information bits is estimated by the matrix rank, and the BCH code check matrix is ​​estimated by the null space of the matrix when the matrix is ​​deficient in rank, and then the generator matrix and the generator polynomial are estimated by using the orthogonality of the generator matrix and the check matrix. The method does not need to perform double traversal detection on the code length and primitive polynomial of the BCH code, and the amount of calculation required for matrix transformation is small, so the calculation complexity is low, and the BCH code can be quickly identified. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 This is a flowchart of a BCH code fast identification method based on matrix transformation described in Example 1 of the present invention.

[0066] Figure 2 This is a flowchart of a BCH code fast identification method based on matrix transformation described in Example 2 of the present invention.

[0067] Figure 3 This is a flowchart of a BCH code fast identification method based on matrix transformation described in Example 3 of the present invention. DETAILED DESCRIPTION

[0068] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention, rather than to limit the present invention.

[0069] Example 1

[0070] like Figure 1 As shown, a BCH code fast recognition method based on matrix transformation includes the following steps:

[0071] S1. Arrange the BCH code bit stream into a matrix C b :

[0072] The coded bit stream is grouped, and the number of bits N contained in each group is the number of columns of the matrix. The bit groups are arranged in rows in sequence, and the number of groups M is the number of rows of the matrix. The number of rows M of the matrix is ​​required to be greater than the number of columns N. In this way, the bit stream is arranged into a matrix C of size M×N b .

[0073] By changing the length of the group, the matrix C can be changed b size.

[0074] S2. Simplify the bitstream matrix into an upper triangular matrix C s :

[0075] The coded bit stream matrix C b Perform elementary row transformations row by row to simplify to the upper triangular matrix C s , if the bitstream matrix C b Full rank, then the upper triangular matrix C s It can be expressed as:

[0076]

[0077] In the formula, Q N×N represents an N×N upper triangular matrix, 0 (M-N)×N Represents a (MN)×N zero matrix.

[0078] S3. Estimate the code length n and check matrix H of the BCH code according to the matrix rank deficiency characteristic:

[0079] If the number of columns N of the matrix is ​​equal to the code length n, the matrix C b There is rank deficiency, and the simplified upper triangular matrix C s It is expressed as:

[0080]

[0081] In the formula, the subscript k is the number of BCH code information bits, which is also the rank of the matrix at this time, and n is the code length of the BCH code. s Simplified to the simplest form, the simplest matrix C s ′ is expressed as:

[0082]

[0083] In the formula, I k represents the unit matrix of size k×k, P k×(n-k) Represents a matrix of size k×(nk) consisting of 0 and 1 bits, and the matrix P k×(n-k) The check matrix H of the BCH code can be constructed and expressed as:

[0084]

[0085] S4. Using the orthogonality of the check matrix H and the generator matrix G, estimate the generator matrix:

[0086] The check matrix H and the generator matrix G of the BCH code satisfy orthogonality, that is:

[0087] H.G. T =0

[0088] Combined with the expression of the check matrix H, the generator matrix G can be obtained as:

[0089] G=[I k P k×(n-k) ]

[0090] S5. Simplify the generator matrix G and estimate the generator polynomial of the BCH code:

[0091] The generating polynomial of BCH code is g(x)=g n-k x n-k +g n-k-1 x n-k-1 +…+g0 and the generator matrix G satisfy the following relationship:

[0092]

[0093] The generating matrix G can be simplified into the above form to obtain the generating polynomial, thus completing the BCH code recognition.

[0094] Example 2

[0095] A method for quickly identifying a BCH code based on matrix transformation comprises the following steps:

[0096] S1. Arrange the BCH code bit stream into a matrix C b :

[0097] In practice, due to the influence of noise and various interferences, the demodulated output bit stream contains bit errors, which will cause the rank deficiency characteristic of the matrix to disappear.

[0098] Therefore, in order to improve the error-resistance capability of the recognition method, the coded bit stream is grouped, and the high-reliability bits are selected using the log-likelihood ratio. Then, the bit stream groups whose row reliability is greater than the threshold α are selected. The number of bits N contained in each group is the number of columns of the matrix. The bit groups are arranged in rows in sequence. The number of groups M is the number of rows of the matrix. It is required that the number of rows M of the matrix is ​​greater than the number of columns N. In this way, the bit stream is arranged into a matrix C of size M×N. b .

[0099] Specifically, the process of selecting high reliability bits using log-likelihood ratio is:

[0100] The log-likelihood ratio is constructed from soft decision data, which refers to the baseband signal sampling value of the demodulated output without decision. For BPSK signals, the soft decision data r t It is expressed as:

[0101]

[0102] Where n t is a Gaussian random variable.

[0103] Assuming H0 and H1 represent the transmitted bits as 0 and 1 respectively, the likelihood functions under different assumptions are:

[0104]

[0105] The log-likelihood ratio LLR is constructed from the likelihood function:

[0106]

[0107] The absolute value of the log-likelihood ratio |LLR| is used to measure the reliability of the bit; the reliability of each row of the bitstream matrix is ​​determined by the minimum bit reliability of the row, and the bitstream groups with row reliability greater than the threshold α are selected to form the matrix C to be analyzed. b , in order to reduce the number of bit errors in the matrix.

[0108] S2. Simplify the bitstream matrix into an upper triangular matrix C s :

[0109] The coded bit stream matrix C b Perform elementary row transformations row by row to simplify to the upper triangular matrix C s , if the bitstream matrix C b Full rank, then the upper triangular matrix C s It can be expressed as:

[0110]

[0111] In the formula, Q N×N represents an N×N upper triangular matrix, 0 (M-N)×N Represents a (MN)×N zero matrix.

[0112] S3. Estimate the code length n and check matrix H of the BCH code according to the matrix rank deficiency characteristic:

[0113] If the number of columns N of the matrix is ​​equal to the code length n, the matrix C b There is rank deficiency, and the simplified upper triangular matrix C s It is expressed as:

[0114]

[0115] In the formula, the subscript k is the number of information bits of the BCH code, which is also the rank of the matrix at this time, and n is the code length of the BCH code. s Simplified to the simplest form, the simplest matrix C s ′ is expressed as:

[0116]

[0117] In the formula, I k represents the unit matrix of size k×k, P k×(n-k) Represents a matrix of size k×(nk) consisting of 0 and 1 bits, and the matrix P k×(n-k) The check matrix H of the BCH code can be constructed and expressed as:

[0118]

[0119] S4. Using the orthogonality of the check matrix H and the generator matrix G, estimate the generator matrix:

[0120] The check matrix H and the generator matrix G of the BCH code satisfy orthogonality, that is:

[0121] H.G. T =0

[0122] Combined with the expression of the check matrix H, the generator matrix G can be obtained as:

[0123] G=[I k P k×(n-k) ]

[0124] S5. Simplify the generator matrix G and estimate the generator polynomial of the BCH code:

[0125] The generating polynomial of BCH code is g(x)=g n-k x n-k +g n-k-1 x n-k-1 +…+g0 and the generator matrix G satisfy the following relationship:

[0126]

[0127] The generating matrix G can be simplified into the above form to obtain the generating polynomial, thus completing the BCH code recognition.

[0128] Example 3

[0129] A method for quickly identifying a BCH code based on matrix transformation comprises the following steps:

[0130] S1. Arrange the BCH code bit stream into a matrix C b :

[0131] The coded bit stream is grouped, and the number of bits N contained in each group is the number of columns of the matrix. The bit groups are arranged in rows in sequence, and the number of groups M is the number of rows of the matrix. The number of rows M of the matrix is ​​required to be greater than the number of columns N. In this way, the bit stream is arranged into a matrix C of size M×N b .

[0132] By changing the length of the group, the matrix C can be changed b size.

[0133] S2. Simplify the bitstream matrix into an upper triangular matrix C s :

[0134] The coded bit stream matrix C b Perform elementary row transformations row by row to simplify to the upper triangular matrix C s , if the bitstream matrix C b Full rank, then the upper triangular matrix C s It can be expressed as:

[0135]

[0136] In the formula, Q N×N represents an N×N upper triangular matrix, 0 (M-N)×N Represents a (MN)×N zero matrix.

[0137] S3. Estimate the code length n and check matrix H of the BCH code according to the matrix rank deficiency characteristic:

[0138] If the number of columns N of the matrix is ​​equal to the code length n, the matrix C b There is rank deficiency, and the simplified upper triangular matrix C s It is expressed as:

[0139]

[0140] In the formula, the subscript k is the number of information bits of the BCH code, which is also the rank of the matrix at this time, and n is the code length of the BCH code. s Simplified to the simplest form, the simplest matrix C s ′ is expressed as:

[0141]

[0142] In the formula, I k represents the unit matrix of size k×k, P k×(n-k) Represents a matrix of size k×(nk) consisting of 0 and 1 bits, and the matrix P k×(n-k) The check matrix H of the BCH code can be constructed and expressed as:

[0143]

[0144] If the number of matrix columns is not equal to the code length, then the bitstream matrix is ​​full rank, and the simplified upper triangular matrix C s Still expressed as:

[0145]

[0146] For a bit stream matrix containing bit errors, the simplified matrix may be full rank, but the proportion of 0 elements in the independent columns is large, and the proportion of 0 elements in the non-independent columns is small. Based on this characteristic, the test statistic ρ is constructed:

[0147]

[0148] Among them, N c Represents the upper triangular matrix C s The number of 1 elements in each column, M represents the matrix C s The number of rows, i represents the index value of each column.

[0149] Set the threshold th, when ρ(i) is less than th, that is:

[0150] ρ(i)<th

[0151] Determine whether the column is an independent column.

[0152] When ρ(i) is greater than th, that is:

[0153] ρ(i)>th

[0154] The column is judged as a non-independent column. The threshold value is usually taken as 0.6. When the matrix is ​​rank-deficient, there are non-independent columns, and the rank is equal to the number of independent columns.

[0155] When the matrix is ​​full rank due to bit errors, although the non-independent columns are determined, the BCH code check matrix cannot be correctly estimated from the non-independent columns. In this case, the augmented matrix can be used to estimate the BCH code check matrix. N Perform the same elementary row transformation as the coded bit stream matrix in step S2 to obtain:

[0156] I′ N =D1I N

[0157] Where D1 represents the elementary row transformation matrix.

[0158] In the augmented matrix I′ N In the simplified matrix, the columns corresponding to the non-independent columns in the simplified matrix constitute the check matrix H of the BCH code.

[0159] S4. Using the orthogonality of the check matrix H and the generator matrix G, estimate the generator matrix:

[0160] The check matrix H and the generator matrix G of the BCH code satisfy orthogonality, that is:

[0161] H.G. T =0

[0162] Combined with the expression of the check matrix H, the generator matrix G can be obtained as:

[0163] G=[I k P k×(n-k) ]

[0164] S5. Simplify the generator matrix G and estimate the generator polynomial of the BCH code:

[0165] The generating polynomial of BCH code is g(x)=g n-k x n-k +g n-k-1 x n-k-1 +…+g0 and the generator matrix G satisfy the following relationship:

[0166]

[0167] The generating matrix G can be simplified into the above form to obtain the generating polynomial, thus completing the BCH code recognition.

[0168] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A BCH code fast recognition method based on matrix transformation, characterized in that: The following steps are involved: S1. Arrange the BCH code bit stream into a matrix C b ; S2. Simplify the bitstream matrix into an upper triangular matrix C s ; S3. Estimate the code length n and check matrix H of the BCH code according to the matrix rank deficiency characteristic; S4. Using the orthogonality of the check matrix H and the generator matrix G, estimate the generator matrix G; S5. Simplify the generator matrix G and estimate the generator polynomial of the BCH code.

2. A BCH code fast recognition method based on matrix transformation as claimed in claim 1, characterized in that: The step S1 comprises: The coded bit stream is grouped, and the number of bits N contained in each group is the number of columns of the matrix; each bit group is arranged in rows in sequence, and the number of groups M is the number of rows of the matrix, and the number of rows M of the matrix is ​​greater than the number of columns N, and the bit stream is arranged into a matrix C of size M×N b .

3. A BCH code fast recognition method based on matrix transformation as claimed in claim 1, characterized in that: The step S2 comprises: The BCH code bit stream matrix C b Perform elementary row transformations row by row to simplify to the upper triangular matrix C s , if the bitstream matrix C b Full rank, then the upper triangular matrix C s It is expressed as: In the formula, Q N×N represents an N×N upper triangular matrix, 0 (M-N)×N Represents a (MN)×N zero matrix.

4. A BCH code fast recognition method based on matrix transformation as claimed in claim 1, characterized in that: The step S3 comprises: If the number of columns N of the matrix is ​​equal to the code length n, the matrix C b There is rank deficiency, and the simplified upper triangular matrix C s It is expressed as: In the formula, the subscript k is the number of information bits of the BCH code, which is also the rank of the matrix at this time, and n is the code length of the BCH code; The upper triangular matrix C s Simplified to the simplest form, the simplest matrix C s ′ is expressed as: In the formula, I k represents the unit matrix of size k×k, P k×(n-k) Represents a matrix of size k×(nk) consisting of 0 and 1 bits, and the matrix P k×(n-k) The check matrix H of the BCH code is constructed and expressed as:

5. A BCH code fast recognition method based on matrix transformation as claimed in claim 4, characterized in that: In step S3, if the number of matrix columns is not equal to the code length, then the bitstream matrix is ​​full rank. When the matrix is ​​full rank due to bit errors, a test statistic ρ is constructed: Among them, N c Represents the upper triangular matrix C s The number of 1 elements in each column, M represents the matrix C s The number of rows, i represents the index value of each column; Set the threshold th, when ρ(i) is less than th, that is: ρ(i)<th Determine whether the column is an independent column; When ρ(i) is greater than th, that is: ρ(i)<th Determine whether the column is a non-independent column. When the matrix is ​​rank deficient, there are non-independent columns, and the rank is equal to the number of independent columns. When the matrix affected by the bit error is full rank, the BCH code check matrix is ​​estimated using the augmented matrix. N The same simplification process as the coded bit stream matrix in step S2 is performed to obtain: I′ N =D1I N Where D1 represents the elementary row transformation matrix; In the matrix I′ N In the example, the columns corresponding to the non-independent columns in the simplest matrix constitute the check matrix H of the BCH code.

6. A BCH code fast recognition method based on matrix transformation as claimed in claim 4 or 5, characterized in that: The step S4 comprises: The check matrix H and the generator matrix G of the BCH code satisfy orthogonality, that is: H·G T =0 Combined with the expression of the check matrix H, the generator matrix G is: G=[I k P k×(n-k) ] 7. A method for fast identification of BCH codes based on matrix transformation as claimed in claim 6, characterized in that: The step S5 comprises: The generating polynomial of BCH code is g(x)=g n-k x n-k +g n-k-1 x n-k-1 +…+g0 and the generator matrix G satisfy the following relationship: Simplifying the generator matrix G into the form of the above formula gives the generator polynomial, thus completing the BCH code recognition.

8. A method for fast identification of BCH codes based on matrix transformation as claimed in claim 1, characterized in that: In step S1, the coded bit stream is grouped, and high reliability bits are selected using the log-likelihood ratio, and then the bit stream groups whose row reliability is greater than the threshold α are selected. The number of bits N contained in each group is the number of columns of the matrix, and each bit group is arranged in rows in sequence. The number of groups M is the number of rows of the matrix. It is required that the number of rows M of the matrix is ​​greater than the number of columns N. In this way, the bit stream is arranged into a matrix C of size M×N b .

9. A method for fast identification of BCH codes based on matrix transformation as claimed in claim 8, characterized in that: In step S1, the process of selecting high reliability bits using log likelihood ratio is as follows: The log-likelihood ratio is constructed from the soft decision data. For BPSK signals, the soft decision data r t It is expressed as: Where n t is a Gaussian random variable; Assuming H0 and H1 represent the transmitted bits as 0 and 1 respectively, the likelihood functions under different assumptions are: The log-likelihood ratio LLR is constructed from the likelihood function: The absolute value of the log-likelihood ratio |LLR| is used to measure the reliability of the bit; the reliability of each row of the bitstream matrix is ​​determined by the minimum bit reliability of the row, and the bitstream groups with row reliability greater than the threshold α are selected to form the matrix C to be analyzed. b , in order to reduce the number of bit errors in the matrix.

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