A Low-Complexity Decoding Method for LDPC-Hadamard Codes Based on Prototype Diagrams

Abstract

This invention discloses a low-complexity decoding method for LDPC-Hadamard codes based on a prototype diagram, belonging to the field of digital signal processing. The implementation method is as follows: information bits are LDPC encoded, and then the constraint relationship of the check nodes of the LDPC code is encoded into Hadamard code to obtain the prototype diagram of the LDPC-Hadamard code; the encoded codewords are transmitted and received through a channel, and the PVN and HCN nodes are initialized with the received information; Hadamard decoding is performed using a symbol-by-symbol maximum a posteriori probability decoding algorithm, and DFHT approximation is used to reduce decoding complexity; the decoded information is transmitted to the PVN nodes; the updated HCN node information is summed with the information from the previous iteration, and the summation result is used to determine the decoding; a decoding decision is made on the PVN information, and decoding is performed. This invention combines LDPC encoding, Hadamard encoding, and Max-Log approximation calculation methods to reduce decoding complexity and enhance the stability and reliability of the communication system.

Description

Technical Field

[0001] This invention discloses a low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams, involving encoding and decoding algorithms for LDPC-Hadamard codes and the Dual Fast Hadamard Transform (DFHT) algorithm, belonging to the field of digital signal processing. Background Technology

[0002] In the field of communications, low-bit-rate systems have attracted much attention due to their advantages in improving the reliability of signal transmission. By reducing the number of information bits per data unit, low-bit-rate systems increase redundancy, thereby improving data transmission reliability in environments with signal attenuation, high bit error rates, or unstable channel conditions. This characteristic makes low-bit-rate systems particularly important in fields such as deep space communication, satellite communication, and mobile communication.

[0003] However, one of the main challenges faced by low-code-rate systems is the increased decoding complexity. Due to the increased redundancy, the decoding process needs to process more data, leading to a significant consumption of computational resources. This is particularly pronounced in resource-constrained environments, such as mobile devices or remote communication systems, where high decoding complexity can result in increased power consumption and processing latency. Based on this, this patent proposes a low-complexity decoding algorithm for prototype-based LDPC-Hadamard codes. The core of this algorithm lies in reducing the computational complexity of the decoding process through algorithm optimization and mathematical simplification, while maintaining a high level of error correction performance. By reducing the required computational resources and processing time, this algorithm is suitable for resource-constrained low-code-rate communication systems, improving overall system efficiency without sacrificing reliability. Summary of the Invention

[0004] The purpose of this invention is to provide a low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams. This method leverages the error-correcting capability of LDPC codes (approximating the Shannon limit) and the low-rate structure of Hadamard codes to achieve both low code rate and high coding gain. LDPC-Hadamard codes achieve this by replacing the parity check of LDPC codes with Hadamard constraints and adding additional Hadamard variable nodes. The LLR-BP algorithm is used for decoding the LDPC-Hadamard codes, and the Dual Fast Hadamard Transform (DFHT) algorithm is employed for verification during the decoding process. Approximations are made to the decoding operation to achieve low-complexity decoding.

[0005] The objective of this invention is achieved through the following technical solution.

[0006] This invention discloses a low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams, comprising the following steps:

[0007] Step 1: Encode the data using LDPC-Hadamard code. Encode the information bits using LDPC, and then further encode the constraint relationship of the check nodes of the LDPC code into Hadamard code to obtain the prototype diagram of LDPC-Hadamard code.

[0008] A Hadamard matrix is ​​an orthogonal matrix consisting only of elements {+1, -1}. An n×n Hadamard matrix is ​​constructed recursively, where n = 2^n. r , where r is the order of the Hadamard matrix.

[0009]

[0010] In a Hadamard matrix, both rows and columns are composed of orthogonal code groups. Let H represent a matrix of length 2... r A Hadamard matrix carrying (r+1) bits of information, the codeword set consists of ±H n The column construction uses {±h j ∶j=0,1,...,2 r The codeword is represented by {-1}. In Hadamard coding, the codeword consists of the original information bits, whose indices are {0,1,2,4,...,2}. r -1}.

[0011] According to the Hadamard coding rules, when r=2, the generator matrix of the Hadamard code is:

[0012]

[0013] The prototype graph of the LDPC-Hadamard code consists of a set of variable nodes (PVN) and a set of Hadamard check nodes (HCN), connected by edges. Let n be the number of PVN nodes and m be the number of HCN nodes. The prototype graph is represented by a basis matrix B of size m × n. m×nThe representation is as follows: Each PVN corresponds to a column of the basis matrix, and each HCN corresponds to a row of the basis matrix. The (i,j)th item of the basis matrix represents the number of edges connecting the i-th PVN and the j-th HCN in the prototype graph. Each SPC corresponding to an HCN is further encoded into a Hadamard code to obtain the prototype graph of LDPC-Hadamard. The derived Hadamard parity bit is represented as a Level 1 Hadamard variable node (D1H-VN), and the Level 1 Hadamard variable node is appended to the prototype graph to obtain the final prototype graph of LDPC-Hadamard code.

[0014] Step 2: After sending and receiving the encoded codewords through the channel, initialize the PVN node and HCN node with the received information.

[0015] After completing LDPC-Hadamard encoding in step one, the encoded LDPC-Hadamard codewords are transmitted through the AWGN channel. The received codewords are mapped one-to-one with PVNs and sent to the HCN to which each PVN node is connected. That is:

[0016]

[0017] in The LLR value (β = 0, 1, ..., N) represents the information transmitted by the channel to the β-th PVN. This represents the LLR value (α = 0, 1…M; β = 0, 1…N) sent from the β-th PVN to the α-th HCN.

[0018] The LDPC-Hadamard decoder consists of two parts: a repetition decoder and a Hadamard decoder. The parity node processing uses a Hadamard decoder, employing a symbol-by-symbol maximum a posteriori (MAP) decoding algorithm. The repetition decoder is the same as the variable node processor used in the LDPC decoder.

[0019] Step 3: HCN node information processing. Hadamard decoding is performed using the symbol-by-symbol maximum a posteriori probability decoding algorithm, and DFHT approximation is used to reduce decoding complexity. The decoded information is then transmitted to the PVN node.

[0020] HCN node processing employs a Hadamard decoder, using a symbol-by-symbol maximum a posteriori probability decoding algorithm for Hadamard decoding. When processing information in the LDPC-Hadamard code's HCN node, equation (4) is used for verification:

[0021]

[0022] in A vector (α = 0, 1, ..., M) representing the LLR values ​​of the information of each D1H-VNs connected to the α-th HCN; P(α) represents the LLR value obtained after the verification calculation of the information sent by the α-th HCN to the β-th PVN; P(α) is the set of PVNs connected to the α-th HCN.

[0023] This is a transformation involving FHT and DFHT operations. When performing DFHT processing, the original formula is as follows:

[0024]

[0025] in:

[0026]

[0027] Formula (5) requires multiple multiplication and pairing operations, which can be approximated as follows:

[0028] ln(e a +e b )≈max(a,b)(6)

[0029] The generalized form of equation (6):

[0030]

[0031] According to equations (6) and (7), equation (5) simplifies to the following form:

[0032]

[0033] Update information according to equation (8):

[0034]

[0035] in This represents the posterior LLR value of the HCN information received by the β-th PVN (β = 0, 1, ... N); This represents the LLR value (α = 0, 1…M; β = 0, 1…N) from the α-th HCN to the β-th PVN.

[0036] That is, according to Equation (1), HCN node information is processed to obtain HCN node information used to update PVN node information.

[0037] Step 4: PVN node information processing. The updated HCN node information is summed with the information from the previous iteration, and the summation result is used to determine the decoding.

[0038] Update the HCN node information according to step three, and then return this information to the PVN node.

[0039]

[0040] This information Used for judgment codewords. Updated simultaneously. To continue iterating.

[0041]

[0042] H(β) is the set of HCNs that connect the β-th PVN.

[0043] Step 5: Decode and decide on the PVN information; if... If the value is greater than 0, the code is judged as 1; otherwise, it is judged as 0.

[0044] Step Six: According to HC T The value of 0 or the maximum number of iterations determines whether to continue decoding or end decoding.

[0045] If HC T If the value equals 0 or the maximum number of iterations is reached, the decoding ends. Otherwise, continue iterating from step three.

[0046] Beneficial effects:

[0047] 1. The present invention discloses a low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams, which utilizes the high error correction capability of LDPC codes, the grid structure of Hadamard codes, and the simplified input DFHT algorithm to significantly reduce the complexity of the decoding algorithm.

[0048] 2. The present invention discloses a low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams. It adopts the DFHT algorithm with simplified input to reduce the terms in the computation that grow exponentially with the coding parameters to constant terms. It also adopts the Max-Log approximation to transform multiplication and exponentiation operations into simple addition and maximum value operations, thereby reducing the decoding complexity and enhancing the stability and reliability of the communication system.

[0049] 3. This invention discloses a low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams. Building upon the beneficial effects 1 and 2, it utilizes the error correction capability of LDPC codes approximating the Shannon limit and the low code rate structure of Hadamard codes to achieve low code rate and high coding gain. LDPC-Hadamard codes are formed by replacing the parity check of LDPC codes with Hadamard constraints and adding additional Hadamard variable nodes. The decoding employs an LLR-BP algorithm similar to that used in LDPC code decoding, replacing the parity check of LDPC codes with a Hadamard code check. The check uses the Dual Fast Hadamard Transform (DFHT) algorithm, with approximations applied to its computational processes to achieve low-complexity decoding. Attached Figure Description

[0050] To more clearly illustrate the technical solutions described in this invention, the accompanying drawings used in this invention will be briefly introduced below. The drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort.

[0051] Figure 1 This is a prototype diagram of an LDPC-Hadamard code.

[0052] Figure 2 This is the decoding structure used for PVN information processing.

[0053] Figure 3 This is the decoding structure used for HCN information processing.

[0054] Figure 4 This is a flowchart of a low-complexity decoding method for LDPC-Hadamard codes based on a prototype diagram, according to the present invention. Detailed Implementation

[0055] Example 1:

[0056] like Figure 1 The example illustrates an LDPC-Hadamard code. This embodiment discloses a low-complexity decoding method for LDPC-Hadamard codes based on a prototype diagram. The specific implementation process is as follows: Figure 4 As shown, the steps are as follows:

[0057] Step 1: Encode the data using LDPC-Hadamard code. Encode the information bits using LDPC, and then further encode the constraint relationship of the check nodes of the LDPC code into Hadamard code to obtain the prototype diagram of LDPC-Hadamard code.

[0058] A Hadamard matrix is ​​an orthogonal matrix consisting only of elements {+1, -1}. An n×n Hadamard matrix is ​​constructed recursively, where n = 2^n. r , where r is the order of the Hadamard matrix.

[0059]

[0060] In a Hadamard matrix, both rows and columns are composed of orthogonal code groups. Let H represent a matrix of length 2... r A Hadamard matrix carrying (r+1) bits of information, the codeword set consists of ±H n The column construction uses {±h j ∶j=0,1,...,2 r The codeword is represented by {-1}. In Hadamard coding, the codeword consists of the original information bits, whose indices are {0,1,2,4,...,2}. r -1}.

[0061] According to the Hadamard coding rules, when r=2, the generator matrix of the Hadamard code is:

[0062]

[0063] The prototype graph of the LDPC-Hadamard code consists of a set of variable nodes (PVN) and a set of Hadamard check nodes (HCN), connected by edges. Let n be the number of PVN nodes and m be the number of HCN nodes. The prototype graph is represented by a basis matrix B of size m × n. m×n The representation is as follows: Each PVN corresponds to a column of the basis matrix, and each HCN corresponds to a row of the basis matrix. The (i,j)th item of the basis matrix represents the number of edges connecting the i-th PVN and the j-th HCN in the prototype graph. Each SPC corresponding to an HCN is further encoded into a Hadamard code to obtain the prototype graph of LDPC-Hadamard. The derived Hadamard parity bit is represented as a Level 1 Hadamard variable node (D1H-VN), and the Level 1 Hadamard variable node is appended to the prototype graph to obtain the final prototype graph of LDPC-Hadamard code.

[0064] Step 2: After sending and receiving the encoded codewords through the channel, initialize the PVN node and HCN node with the received information.

[0065] After completing LDPC-Hadamard encoding in step one, the encoded LDPC-Hadamard codewords are transmitted through the AWGN channel. The received codewords are mapped one-to-one with PVNs and sent to the HCN to which each PVN node is connected. That is:

[0066]

[0067] in The LLR value (β = 0, 1, ..., N) represents the information transmitted by the channel to the β-th PVN. This represents the LLR value (α = 0, 1…M; β = 0, 1…N) sent from the β-th PVN to the α-th HCN.

[0068] The LDPC-Hadamard decoder consists of two parts: a repetition decoder and a Hadamard decoder. The parity node processing uses a Hadamard decoder, employing a symbol-by-symbol maximum a posteriori (MAP) decoding algorithm. The repetition decoder is the same as the variable node processor used in the LDPC decoder.

[0069] Step 3: HCN node information processing. Hadamard decoding is performed using the symbol-by-symbol maximum a posteriori probability decoding algorithm, and DFHT approximation is used to reduce decoding complexity. The decoded information is then transmitted to the PVN node.

[0070] HCN node processing employs a Hadamard decoder, using a symbol-by-symbol maximum a posteriori probability decoding algorithm for Hadamard decoding. When processing information in the LDPC-Hadamard code's HCN node, equation (4) is used for verification:

[0071]

[0072] in A vector (α = 0, 1, ..., M) representing the LLR values ​​of the information of each D1H-VNs connected to the α-th HCN; P(α) represents the LLR value obtained after the verification calculation of the information sent by the α-th HCN to the β-th PVN; P(α) is the set of PVNs connected to the α-th HCN.

[0073] This is a transformation involving FHT and DFHT operations. When performing DFHT processing, the original formula is as follows:

[0074]

[0075] in:

[0076]

[0077] Formula (5) requires multiple multiplication and pairing operations, which can be approximated as follows:

[0078] ln(e a +e b )≈max(a,b)(6)

[0079] The generalized form of equation (6):

[0080]

[0081] According to equations (6) and (7), equation (5) simplifies to the following form:

[0082]

[0083] Update information according to equation (8):

[0084]

[0085] in This represents the posterior LLR value of the HCN information received by the β-th PVN (β = 0, 1, ..., N); This represents the LLR value (α = 0, 1…M; β = 0, 1…N) sent from the α-th HCN to the β-th PVN.

[0086] That is, according to Equation (1), HCN node information is processed to obtain HCN node information used to update PVN node information.

[0087] Step 4: PVN node information processing. The updated HCN node information is summed with the information from the previous iteration, and the summation result is used to determine the decoding.

[0088] Update the HCN node information according to step three, and then return this information to the PVN node.

[0089]

[0090] This information Used for judgment codewords. Updated simultaneously. To continue iterating.

[0091]

[0092] H(β) is the set of HCNs that connect the β-th PVN.

[0093] Step 5: Decode and decide on the PVN information; if... If the value is greater than 0, the code is judged as 1; otherwise, it is judged as 0.

[0094] Step Six: According to HCT The value of 0 or the maximum number of iterations determines whether to continue decoding or end decoding.

[0095] If HC T If the value equals 0 or the maximum number of iterations is reached, the decoding ends. Otherwise, continue iterating from step three.

[0096] The above detailed description further illustrates the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams, characterized in that: Includes the following steps, Step 1: Encode the data using LDPC-Hadamard code. Encode the information bits using LDPC, and then further encode the constraint relationship of the check nodes of the LDPC code into Hadamard code to obtain the prototype diagram of LDPC-Hadamard code. The prototype graph of the LDPC-Hadamard code consists of a set of variable nodes (PVN) and a set of Hadamard check nodes (HCN), connected by edges. Let n be the number of PVNs and m be the number of HCNs. The prototype graph is represented by a basis matrix B of size m × n. m×n The representation is as follows: each PVN corresponds to a column of the basis matrix, and each HCN corresponds to a row of the basis matrix; the (i,j)th item of the basis matrix represents the number of edges connecting the i-th PVN and the j-th HCN in the prototype graph; each SPC corresponding to an HCN is further encoded into a Hadamard code to obtain the prototype graph of LDPC-Hadamard; the derived Hadamard parity bit is represented as a level 1 Hadamard variable node D1H-VN, and the level 1 Hadamard variable node is appended to the prototype graph to obtain the final prototype graph of LDPC-Hadamard code; Step 2: After sending and receiving the encoded codewords through the channel, initialize the PVN node and HCN node with the received information; Step 3: HCN node information processing. Hadamard decoding is performed using the symbol-by-symbol maximum a posteriori probability decoding algorithm, and DFHT approximation is used to reduce decoding complexity. The decoded information is then transmitted to the PVN node. Step 4: PVN node information processing. The updated HCN node information is summed with the information from the previous iteration, and the summation result is used to determine the decoding. Step 5: Decode and decide on the PVN information; if... If the value is greater than 0, the code is judged as 1; otherwise, it is judged as 0. Step Six: According to HC T The value of 0 or the maximum number of iterations determines whether to continue decoding or end decoding.

2. The low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams as described in claim 1, characterized in that: The implementation method for step one is as follows: A Hadamard matrix is ​​an orthogonal matrix consisting only of elements {+1, -1}. An n×n Hadamard matrix is ​​constructed recursively, where n = 2^n. r , where r is the order of the Hadamard matrix; In a Hadamard matrix, both rows and columns are composed of orthogonal code groups; H represents a code group of length 2. r A Hadamard matrix carrying (r+1) bits of information, the codeword set consists of ±H n The column construction uses {±h j :j=0,1,...,2 r -1} represents the index {0, 1, 2, 4, ..., 2} in Hadamard coding, where the codeword consists of the original information bits. r -1}; According to the Hadamard coding rules, when r=2, the generator matrix of the Hadamard code is:

3. The low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams as described in claim 1, characterized in that: The second step is implemented as follows: After completing LDPC-Hadamard encoding according to step one, the encoded LDPC-Hadamard codewords are sent through the AWGN channel; the received codewords are matched one-to-one with PVNs and sent to the HCN to which each PVN node is connected. Right now: in The LLR value (β = 0, 1, ..., N) represents the information transmitted by the channel to the β-th PVN. This represents the LLR value (α = 0, 1…M; β = 0, 1…N) sent from the β-th PVN to the α-th HCN. The LDPC-Hadamard decoder consists of two parts: a repetition decoder and a Hadamard decoder. The parity node processing uses a Hadamard decoder and employs the symbol-by-symbol maximum a posteriori probability decoding algorithm for Hadamard decoding. The repetition decoder is the same as the variable node processor used in the LDPC decoder.

4. The low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams as described in claim 3, characterized in that: The method for implementing step three is as follows: HCN node processing employs a Hadamard decoder, using the symbol-by-symbol maximum a posteriori probability decoding algorithm for Hadamard decoding; when processing information in the LDPC-Hadamard code HCN node, equation (4) is used for verification: in A vector (α = 0, 1, ..., M) representing the LLR values ​​of the information of each D1H-VNs connected to the α-th HCN; P(α) represents the LLR value obtained after the verification calculation of the information sent by the α-th HCN to the β-th PVN; P(α) is the set of PVNs connected to the α-th HCN. It is a transformation involving FHT and DFHT operations; the original formula is as follows when performing DFHT processing: in: Formula (5) requires multiple multiplication and pairing operations, which can be approximated as follows: ln(e a +e b )≈max(a,b)(6) gives the generalized form of equation (6): According to equations (6) and (7), equation (5) simplifies to the following form: Update information according to equation (8): in This represents the posterior LLR value of the HCN information received by the β-th PVN (β = 0, 1, ... N); This represents the LLR value (α = 0, 1…M; β = 0, 1…N) sent from the α-th HCN to the β-th PVN. That is, according to Equation (10), HCN node information is processed to obtain HCN node information used to update PVN node information.

5. The low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams as described in claim 4, characterized in that: Step four is implemented as follows: Update the HCN node information according to step three, and then return this information to the PVN node; This information Used for judgment codewords; updated simultaneously. To continue iterating; H(β) is the set of HCNs that connect the β-th PVN.

6. The low-complexity decoding method for LDPC-Hadamard codes based on prototype diagrams as described in claim 5, characterized in that: Step six is ​​implemented as follows: If HC T If the value equals 0 or the maximum number of iterations is reached, the decoding ends; otherwise, continue iterating from step three.

Images