Derivation method and system of maximum sum rate based on regular ldpc code under two-user gaussian access channel

Abstract

The application discloses a kind of derivation methods and systems of maximum sum rate of two-user Gaussian access channel based on rule LDPC code, method is as follows step: S1, the sum node under two-user Gaussian access channel is analyzed, with the sum rate maximization of two-user rule LDPC code as criterion, with two-user system decoding success as premise, construced target function with constraint condition;S2, the degree of LDPC code variable node is fixed, deduce the theoretical analytic expression of check node degree, solve the inverse function of target function in non-convex optimization problem;S3, optimize the degree distribution of two-user LDPC code of two-user Gaussian access channel under system successful decoding, make system sum rate maximization.The present application will provide important help for the design and optimization of LDPC code, to ensure that high performance and reliability are realized in 6G communication system.

Description

Technical Field

[0001] This invention belongs to the field of wireless communication technology and provides a technical solution to improve the sum rate of two-user regular LDPC codes under a two-user Gaussian access channel. Specifically, based on external information transfer analysis and mathematical set theory, this invention derives a fixed-point analysis method for two-user regular LDPC codes to maximize the sum rate, thereby improving the coding efficiency of the two-user communication system. Background Technology

[0002] LDPC codes, as one of the most commonly used channel codes in current communication systems, possess advantages such as high error correction performance and low complexity. These advantages make LDPC codes play a crucial role in the upcoming 6G communication. Therefore, when utilizing LDPC codes to improve system performance, their parameter design becomes a critical factor. Traditional external information transfer algorithms traverse all possible combinations of system parameters through a full search to find the optimal solution. This method not only consumes significant time and computational resources but may also suffer from problems such as the curse of dimensionality. Therefore, optimizing the degree distribution of LDPC codes is essential. Summary of the Invention

[0003] To address the aforementioned problems in existing technologies, this invention derives a fixed-point analysis scheme for two-user regular LDPC codes under a two-user Gaussian access channel, obtaining the theoretical value of the optimal degree distribution of the regular LDPC code under successful system decoding, thereby maximizing the system's sum rate.

[0004] Application scenario of this invention: In multiple access communication, a regular LDPC code is used to pass through the Gaussian access communication system of two users.

[0005] The present invention adopts the following technical solution:

[0006] The derivation method of the maximum sum rate based on regular LDPC codes in a two-user Gaussian access channel is as follows:

[0007] Step 1: Analyze the sum node under the two-user Gaussian access channel. Based on the maximization of the sum rate of the two-user regular LDPC code and the premise of successful decoding of the two-user system, construct an objective function with constraints.

[0008] Step 2: Fix the degree of the variable nodes of the LDPC code, derive the theoretical analytical expression of the degree of the check node, and solve the inverse function of the objective function in the non-convex optimization problem;

[0009] Step 3: Optimize the degree distribution of the two-user LDPC codes successfully decoded by the system under the two-user Gaussian access channel to maximize the system and rate.

[0010] Preferably, step 1. Analyze the sum node under the two-user Gaussian access channel, take maximizing the sum rate of the two-user regular LDPC code as the criterion, and take the successful decoding of the two-user system as the premise, and construct an objective function with constraints based on the two-user regular LDPC code.

[0011] Based on a two-user Gaussian access channel, under the condition of equal transmission power, the k-th user adopts a rule-based approach. -LDPC code, where It is the degree of the variable node. It is the degree of the verification node, so the code rate of the kth user is:

[0012]

[0013] Therefore, the summation rate of the two user-defined LDPC codes is:

[0014]

[0015] At the receiving end, iterative decoding is performed on the factor graph. Based on the fundamental rules of mutual information at each node in the factor graph, assuming infinite code length and Gaussian approximation (variance is twice the mean), when the degree of the variable node is d, the output mutual information at the variable node is:

[0016]

[0017] Where 0≤I A,i ≤1 represents the input mutual information from check node i to variable node j, J -1 (*) is the inverse function of the J function:

[0018]

[0019] The J function represents the output mutual information, σ A This represents the variance of the input information. When the degree of the check node is d, the output mutual information at the check node is:

[0020] T c (I A,1 ,…,I A,d-1 ) = 1 - T v (1-I A,1 ,…,1-I A,d-1 (5)

[0021] If I A,i =I A i = 1, ..., w, which can be simplified to (3) and (5) as T v (I A ×w,I A,w+1 …,I A,d-1 ) and T c(I A ×w,I A,w+1 …,I A,d-1 ).

[0022] For the node with degree 3, it is connected to User 1, User 2, and the Gaussian access channel. For the j-th received bit y j Given noise power σ 2 Based on the maximum a posteriori probability criterion, the j-th bit of the k-th user Output log-likelihood ratio for:

[0023]

[0024] in, Let j be the j-th bit of the (3-k)-th user. According to Bayes' theorem, we can obtain:

[0025]

[0026] Divide the numerator and denominator by the same amount. We can obtain:

[0027]

[0028] because The input (prior) log-likelihood ratio is so

[0029] Substituting into formula (8), we get:

[0030]

[0031] To determine the mutual information between the outputs at the nodes, the goal is to find the j-th bit of the k-th user. Lower output log-likelihood ratio The expectation, that is Suppose the k-th user sends a codeword of all 1s, that is... So:

[0032]

[0033] At this point, the j-th bit of the (3-k)-th user There are two situations, one of which is... Another one is

[0034] Scenario 1: When At that time, received bits The noise z follows a mean of 0 and a variance of σ. 2 The Gaussian distribution is z ~ N(0,σ). 2Therefore, y j ~N(2,σ 2 Based on the Gaussian approximation, in It is the mean, noise. so

[0035] For ease of calculation, let parameters be set. Based on the above assumptions, we can obtain Similarly, let's set another parameter. get

[0036] In conclusion, when At that time, the target It can be represented as

[0037]

[0038] At this point, for a Gaussian random variable m with mean μ+a and variance 2μ+b, its probability density function is:

[0039]

[0040] Therefore, we can conclude that:

[0041]

[0042] Perform point replacement, let So Therefore:

[0043]

[0044] Substituting m1 and m2, we get:

[0045]

[0046] Substituting formula (15) into formula (11), we get:

[0047]

[0048] at this time, When the number of iterations is l and the root mean square of the channel noise is σ, the output mutual information from channel node S to the k-th user variable node V is:

[0049]

[0050] Scenario 2: When At that time, received bits Obviously, y j ~N(0,σ 2 ).at this time, so

[0051] Similarly, set parameters achievable

[0052]

[0053] In conclusion, when At that time, the objective was:

[0054]

[0055] At this point, let the mean of the Gaussian random variable ρ be -μ-a and the variance be 2μ+b, because its probability density function is:

[0056]

[0057] Therefore, we get:

[0058]

[0059] Perform point replacement, let So Therefore, there is

[0060]

[0061] Substituting ρ1 and ρ2 into the equations, we get:

[0062]

[0063] Substituting formula (22) into formula (18), we get:

[0064]

[0065] at this time, When the root mean square of the channel noise is σ, the output mutual information from channel node S to the k-th user variable node V is:

[0066]

[0067] In summary, for the k-th user, according to formulas (15) and (22), the output mutual information from channel node S to variable node V is:

[0068]

[0069] Based on external information transfer analysis, for the k-th user, the output mutual information from variable node V to verification node C is:

[0070]

[0071] The output mutual information from node C to variable node V is:

[0072]

[0073] The final output mutual information from the k-th user variable node V to the channel node S is obtained as follows:

[0074]

[0075] Reconsidering the final output mutual information at channel node S, we simplify to obtain:

[0076]

[0077] Substituting formula (29) into formula (28), we finally obtain the objective function for the k-th user:

[0078]

[0079] Since the objective function of the two-user rule LDPC code is constructed based on external information transfer analysis, the necessary and sufficient condition for successful decoding can be determined as follows:

[0080]

[0081] In summary, taking the maximization of the sum rate *r* of the two-user rule LDPC code as the criterion and the successful decoding of the two-user system as the premise, the objective function with constraints is:

[0082]

[0083] Preferably, in step 2, taking the k-th user as an example, the degree of the fixed variable node is... Based on fixed-point analysis theory, the degree of the check node in the LDPC code of the k-th user under a two-user Gaussian access channel is derived. The theoretical analytical expression is used to solve for the inverse function of the objective function in nonconvex optimization problems.

[0084] To further explore the objective function, given the root mean square of the channel noise as σ and the degree distribution of the LDPC code for the k-th user. When the number of iterations is l, we have:

[0085]

[0086] Among them, I *(k) It is a fixed-point equation The smallest fixed point in the interval. The smallest fixed point I. *(k) This corresponds to the convergence point in the factor graph and also determines the bit error rate of the k-th user's decoding:

[0087]

[0088] in, It is a complementary error function.

[0089] As we all know, when I *(k) =1, Pe→0. That is, when the smallest fixed point I... *(k) When the value falls within the range [0,1), the k-th user's decoding fails. *(k) When = 1, the k-th user successfully decodes. Since we are considering a two-user Gaussian access system, the following conditions must be met simultaneously:

[0090] (I *(k) ,I *(3-k) )=(1,1) (35)

[0091] Decoding can only succeed if both user systems are connected. Otherwise, decoding will fail for both user systems.

[0092] After analyzing the output mutual information I from variable node V to channel node S on the factor graph *(k) and I *(3-k) Based on observations, and reconsidering formula (30), when the number of iterations l approaches infinity, we obtain:

[0093]

[0094] Formula (36) is an equation with two unknowns, namely the degree distribution. This makes it impossible to find the inverse function of the objective function for the k-th user. Therefore, we relax the conditions and assume that the degree of the given variable node... To solve for the degree of the check node This prepares us for solving nonconvex optimization problems. Transforming equation (36), we get:

[0095]

[0096] Degree containing check nodes Moving the terms aside, we can simplify to:

[0097]

[0098] Solve for the degree of the check node The mathematical analytical expression, that is, to Represented as a fixed point (I) *(k) ,I *(3-k) The explicit function of )

[0099]

[0100] Compared to external information transfer analysis, fixed-point theory does not require iterative and full-space search of the degree distribution of LDPC codes for each user. This effectively reduces complexity.

[0101] Preferably, step 3. Based on mathematical set theory, the uncertainty in the inverse problem is resolved to obtain the maximum sum rate of successful decoding of the two-user system under the two-user Gaussian access channel, that is, the degree distribution of the optimal two-user LDPC code.

[0102]

[0103] From the perspective of the inverse function alone, given the root mean square of the channel noise σ and the degree of the corresponding variable nodes for two users... and Will (I *(1) ,I *(2) Substituting (1,1) into formula (39), we can solve for the degree of the reliable check node under successful decoding of the two-user system. and Then, by traversing separately and Find the corresponding and And find the optimal degree distribution that maximizes the sum of the two-user system and the rate r. However, due to J in the inverse function -1 (I *(1) =1)=∞ and J -1 (I *(2) =1) =∞, which makes it impossible to directly solve for the degree of the reliable check node corresponding to each user under the condition that the two-user system is successfully decoded. and

[0104] The aforementioned problem is a core issue in solving non-convex optimization problems. The main reason for this core problem lies in the uncertainty of the inverse function of the objective function, and the lack of a universally accepted solution. However, optimizing channel codes is an unavoidable task in future communication applications. Therefore, there is an urgent need to find an efficient method to address non-convex optimization problems.

[0105] Because the inverse function of the objective function is uncertain, it is difficult to find the degree of the corresponding reliable check node assuming successful decoding. and In other words, I *(1) =1 and and I *(2) =1 and None of them are one-to-one mappings.

[0106] First, consider the case of decoding failure in a two-user system. There are three main scenarios: the first is that both users fail to decode; the second is that the first user fails to decode regardless of whether the second user succeeds; and the third is that the second user fails to decode regardless of whether the first user succeeds.

[0107] Case 1: When both users fail to decode, given the root mean square of the channel noise σ and the degree of the two user variable nodes. and Put I *(1) ∈[0,1),I *(2) Substituting ∈[0,1) into formula (39), we find an unreliable region:

[0108]

[0109] Among them, C 12 ={(I *(1) ,I *(2) )|I *(1) ∈[0,1) and I *(2) ∈[0,1)}. In other words, for a given And σ, select unreliable region UR 12 any and The first user -LDPC code and the second user - LDPC codes will fail to decode, resulting in decoding failure for both user systems.

[0110] Case 2: When the first user fails to decode, regardless of whether the second user's decoding is successful, the two-user system will be judged as having failed to decode because it cannot correctly decode the first user's signal. In this case, formula (39) degenerates into:

[0111]

[0112] Therefore, given σ and the degree of the first user variable node. Will I *(1) Substituting ∈[0,1) into formula (41), we find an unreliable region:

[0113]

[0114] Where, C1={I *(1) |I *(1) ∈[0,1)} and N + This represents the set of all positive integers. That is to say, given σ and... In the case of selecting any unreliable region UR1 Regardless of what LDPC code the second user uses, and regardless of whether the second user can successfully decode it, the first user's... The code will fail to be decoded, thus indicating that the decoding of both user systems has failed.

[0115] Case 3: Similarly, when the second user fails to decode, based on formula (41), an unreliable region is obtained:

[0116]

[0117] Where, C2={I *(2) |I *(2) ∈[0,1)}. Given σ and Select any unreliable region UR2 The second user -LDPC code decoding failed, resulting in decoding failure for both user systems.

[0118] Based on the above three scenarios, the unreliable regions where decoding fails in the two-user system are identified:

[0119] UR=UR 12 ∪UR1∪UR2 (44)

[0120] Using set theory, by taking the absolute complement of UR, we obtain the reliable region for successful decoding of the two-user system:

[0121]

[0122] In other words, given σ, and RR If both users' rule LDPC codes can be successfully decoded, then the two-user system has successfully decoded the code.

[0123] After obtaining the reliable region RR, given σ, d v (1) and d v (2) Select the best match from RR. To maximize the sum rate r of the two-user system:

[0124]

[0125] Finally, traverse all and Solve for all corresponding And select the optimal degree distribution of the two-user LDPC codes from them. for:

[0126]

[0127] Maximize the sum rate of the two-user system

[0128] This invention also discloses a system for deriving the maximum sum rate based on regular LDPC codes in a two-user Gaussian access channel, which is based on the above method and includes the following modules:

[0129] Objective function construction module: Analyze the sum node under a two-user Gaussian access channel, take maximizing the sum rate of the two-user regular LDPC code as the criterion, and assume successful decoding of the two-user system as the premise, and construct an objective function with constraints.

[0130] Inverse function solving module: Fix the degree of the variable nodes of LDPC code, derive the theoretical analytical expression of the degree of the check node, and solve the inverse function of the objective function in non-convex optimization problems;

[0131] Optimization module: Optimizes the degree distribution of the two-user LDPC codes successfully decoded by the system under a two-user Gaussian access channel, thereby maximizing the system and rate.

[0132] This invention, based on external information transfer analysis, proposes a fixed-point analysis method for two-user regular LDPC codes under a two-user Gaussian access channel. It derives the theoretical value of the optimal degree distribution of the regular LDPC code under successful decoding of the two-user system step by step, thereby maximizing the system's sum rate. First, the sum nodes under a two-user Gaussian access channel are analyzed. Taking the maximization of the sum rate of the two-user regular LDPC code as the criterion and the premise of successful decoding of the two-user system, a constrained objective function is constructed. Then, based on fixed-point analysis theory, the degree of the variable nodes of the LDPC code is fixed, and the theoretical analytical expression (closed-form expression) of the degree of the check node is derived, solving for the inverse function of the objective function in the non-convex optimization problem. Finally, based on mathematical set theory, the uncertainty of the inverse function is resolved, and the optimal degree distribution of the two-user LDPC code successfully decoded under a two-user Gaussian access channel is obtained. That is, maximizing the system and the rate. Attached Figure Description

[0133] Figure 1 This is a factor diagram of a two-user regular LDPC code under a preferred embodiment of a two-user Gaussian access channel. The letter V represents a variable node, the letter C represents a check node, and the letter S represents a sum node.

[0134] Figure 2 When SNR = 4dB, which is σ = 0.8923, given The most matching

[0135] Figure 3This is a flowchart illustrating a preferred embodiment of a method for deriving the maximum sum rate based on a regular LDPC code in a two-user Gaussian access channel.

[0136] Figure 4 This is a block diagram illustrating the derivation of the maximum sum rate based on regular LDPC codes in a preferred embodiment of a two-user Gaussian access channel. Detailed Implementation

[0137] The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0138] The application scenario of this embodiment is: in multiple access communication, the regular LDPC code is used to pass through the Gaussian access communication system of two users.

[0139] like Figure 1-3 As shown in this embodiment, the derivation method of the maximum sum rate based on regular LDPC codes under a two-user Gaussian access channel follows the steps below:

[0140] Step 1. Analyze the sum node under the two-user Gaussian access channel. Taking the maximization of the sum rate of the two-user regular LDPC code as the criterion and the successful decoding of the two-user system as the premise, construct an objective function with constraints based on the two-user regular LDPC code.

[0141] Based on a two-user Gaussian access channel, under the condition of equal transmission power, the k-th user adopts a rule-based approach. -LDPC code, where It is the degree of the variable node. It is the degree of the verification node, so the code rate of the kth user is:

[0142]

[0143] Therefore, the summation rate of the two user-defined LDPC codes is:

[0144]

[0145] At the receiving end, iterative decoding is performed on the factor graph. Based on the fundamental rules of mutual information at each node in the factor graph, assuming infinite code length and Gaussian approximation (variance is twice the mean), when the degree of the variable node is d, the output mutual information at the variable node is:

[0146]

[0147] Where 0≤I A,i ≤1 represents the input mutual information from check node i to variable node j, J -1 (*) is the inverse function of the J function:

[0148]

[0149] Here, the J function represents the output mutual information, σ A This represents the variance of the input information. When the degree of the check node is d, the output mutual information at the check node is:

[0150] T c (I A,1 ,…,I A,d-1 ) = 1 - T v (1-I A,1 ,…,1-I A,d-1 (5)

[0151] Of course, if I A,i =I A i = 1, ..., w, which can be simplified to (3) and (5) as T v (I A ×w,I A,w+1 …,I A,d-1 ) and T c (I A ×w,I A,w+1 …,I A,d-1 ).

[0152] For the node with degree 3, it is connected to User 1, User 2, and the Gaussian access channel. For the j-th received bit y j Given noise power σ 2 Based on the maximum a posteriori probability criterion, the j-th bit of the k-th user Output log-likelihood ratio for:

[0153]

[0154] in, Let j be the j-th bit of the (3-k)-th user. According to Bayes' theorem, we can obtain:

[0155]

[0156] Divide the numerator and denominator by the same amount. We can obtain:

[0157]

[0158] because The input (prior) log-likelihood ratio is so Substituting into formula (8), we get:

[0159]

[0160] To determine the mutual information between the outputs at the nodes, the goal is to find the j-th bit of the k-th user. Lower output log-likelihood ratio The expectation, that is Suppose the k-th user sends a codeword of all 1s, that is... So:

[0161]

[0162] At this point, the j-th bit of the (3-k)-th user There are two situations, one of which is... Another one is

[0163] Scenario 1: When At that time, received bits The noise z follows a mean of 0 and a variance of σ. 2 The Gaussian distribution is z ~ N(0,σ). 2 Therefore, y j ~N(2,σ 2 Based on the Gaussian approximation, in It is the mean, noise. so

[0164] For ease of calculation, let parameters be set. Based on the above assumptions, we can obtain Similarly, let's set another parameter. get

[0165] In conclusion, when At that time, the target It can be represented as

[0166]

[0167] At this point, for a Gaussian random variable m with mean μ+a and variance 2μ+b, its probability density function is:

[0168]

[0169] Therefore, we can conclude that:

[0170]

[0171] Perform point replacement, let So Therefore:

[0172]

[0173] Substituting m1 and m2, we get:

[0174]

[0175] Substituting formula (15) into formula (11), we get:

[0176]

[0177] at this time, When the number of iterations is l and the root mean square of the channel noise is σ, the output mutual information from channel node S to the k-th user variable node V is:

[0178]

[0179] Scenario 2: When At that time, received bits Obviously, y j ~N(0,σ 2 ).at this time, so

[0180] Similarly, set parameters and achievable

[0181]

[0182] In conclusion, when At that time, the objective was:

[0183]

[0184] At this point, let the mean of the Gaussian random variable ρ be -μ-a and the variance be 2μ+b, because its probability density function is:

[0185]

[0186] Therefore, we get:

[0187]

[0188] Perform point replacement, let So Therefore, there is

[0189]

[0190] Substituting ρ1 and ρ2 into the equations, we get:

[0191]

[0192] Substituting formula (22) into formula (18), we get:

[0193]

[0194] at this time, When the root mean square of the channel noise is σ, the output mutual information from channel node S to the k-th user variable node V is:

[0195]

[0196] In summary, for the k-th user, according to formulas (15) and (22), the output mutual information from channel node S to variable node V is:

[0197]

[0198] Based on external information transfer analysis, for the k-th user, the output mutual information from variable node V to verification node C is:

[0199]

[0200] The output mutual information from node C to variable node V is:

[0201]

[0202] The final output mutual information from the k-th user variable node V to the channel node S is obtained as follows:

[0203]

[0204] Reconsidering the final output mutual information at channel node S, we simplify to obtain:

[0205]

[0206] Substituting formula (29) into formula (28), we finally obtain the objective function for the k-th user:

[0207]

[0208] Since the objective function of the two-user rule LDPC code is constructed based on external information transfer analysis, the necessary and sufficient condition for successful decoding can be determined as follows:

[0209]

[0210] In summary, taking the maximization of the sum rate *r* of the two-user rule LDPC code as the criterion and the successful decoding of the two-user system as the premise, the objective function with constraints is:

[0211]

[0212] Step 2. Taking the k-th user as an example, fix the degree of the node. Based on fixed-point analysis theory, the degree of the check node in the LDPC code of the k-th user under a two-user Gaussian access channel is derived. The theoretical analytical expression is used to solve for the inverse function of the objective function in nonconvex optimization problems.

[0213] To further explore the objective function, given the root mean square of the channel noise as σ and the degree distribution of the LDPC code for the k-th user. When the number of iterations is l, we have:

[0214]

[0215] Among them, I *(k) It is a fixed-point equation The smallest fixed point in the interval. The smallest fixed point I. *(k) This corresponds to the convergence point in the factor graph and also determines the bit error rate of the k-th user's decoding:

[0216]

[0217] in, It is a complementary error function.

[0218] As we all know, when I *(k) =1, Pe→0. That is, when the smallest fixed point I... *(k) When the value falls within the range [0,1), the k-th user's decoding fails. *(k) When = 1, the k-th user successfully decodes. Since we are considering a two-user Gaussian access system, the following conditions must be met simultaneously:

[0219] (I *(k) ,I *(3-k) )=(1,1) (35)

[0220] Decoding can only succeed if both user systems are connected. Otherwise, decoding will fail for both user systems.

[0221] After analyzing the output mutual information I from variable node V to channel node S on the factor graph *(k) and I *(3-k) Based on observations, and reconsidering formula (30), when the number of iterations l approaches infinity, we obtain:

[0222]

[0223] Formula (36) is an equation with two unknowns, namely the degree distribution. This makes it impossible to find the inverse function of the objective function for the k-th user. Therefore, we relax the conditions and assume that the degree of the given variable node... To solve for the degree of the check node This prepares us for solving nonconvex optimization problems. Transforming equation (36), we get:

[0224]

[0225] Degree containing check nodes Moving the terms aside, we can simplify to:

[0226]

[0227] Solve for the degree of the check node The mathematical analytical expression, that is, to Represented as a fixed point (I) *(k) ,I *(3-k) The explicit function of )

[0228]

[0229] Compared to external information transfer analysis, fixed-point theory does not require iterative and full-space search of the degree distribution of LDPC codes for each user. This effectively reduces complexity.

[0230] Step 3. Based on mathematical set theory, resolve the uncertainty in the inverse problem to obtain the maximum sum rate of successful decoding for a two-user system under a two-user Gaussian access channel, i.e., the degree distribution of the optimal two-user LDPC code.

[0231] From the perspective of the inverse function alone, given the root mean square of the channel noise σ and the degree of the corresponding variable nodes for two users... and Will (I *(1) ,I *(2) Substituting (1,1) into formula (39), we can solve for the degree of the reliable check node under successful decoding of the two-user system. and Then, by traversing separately and Find the corresponding and And find the optimal degree distribution that maximizes the sum of the two-user system and the rate r. However, due to J in the inverse function -1 (I *(1) =1)=∞ and J -1 (I *(2) =1) =∞, which makes it impossible to directly solve for the degree of the reliable check node corresponding to each user under the condition that the two-user system is successfully decoded. and

[0232] The aforementioned problem is a core issue in solving non-convex optimization problems. The main reason for this core problem lies in the uncertainty of the inverse function of the objective function, and the lack of a universally accepted solution. However, optimizing channel codes is an unavoidable task in future communication applications. Therefore, there is an urgent need to find an efficient method to address non-convex optimization problems.

[0233] Because the inverse function of the objective function is uncertain, it is difficult to find the degree of the corresponding reliable check node assuming successful decoding. and In other words, I *(1) =1 and and I *(2) =1 and None of them are one-to-one mappings.

[0234] First, consider the case of decoding failure in a two-user system. There are three main scenarios: the first is that both users fail to decode; the second is that the first user fails to decode regardless of whether the second user succeeds; and the third is that the second user fails to decode regardless of whether the first user succeeds.

[0235] Case 1: When both users fail to decode, given the root mean square of the channel noise σ and the degree of the two user variable nodes. and Put I *(1) ∈[0,1),I *(2) Substituting ∈[0,1) into formula (39), we find an unreliable region:

[0236]

[0237] Among them, C 12 ={(I *(1) ,I *(2) )|I *(1) ∈[0,1) and I *(2) ∈[0,1)}. In other words, for a given And σ, select unreliable region UR 12 any and The first user -LDPC code and the second user - LDPC codes will fail to decode, resulting in decoding failure for both user systems.

[0238] Case 2: When the first user fails to decode, regardless of whether the second user's decoding is successful, the two-user system will be judged as having failed to decode because it cannot correctly decode the first user's signal. In this case, formula (39) degenerates into:

[0239]

[0240] Therefore, given σ and the degree of the first user variable node. Will I *(1) Substituting ∈[0,1) into formula (41), we find an unreliable region:

[0241]

[0242] Where, C1={I *(1) |I *(1) ∈[0,1)} and N + This represents the set of all positive integers. That is to say, given σ and... In the case of selecting any unreliable region UR1 Regardless of what LDPC code the second user uses, and regardless of whether the second user can successfully decode it, the first user's... - LDPC codes will fail to decode, thus indicating that the decoding of both user systems has failed.

[0243] Case 3: Similarly, when the second user fails to decode, based on formula (41), an unreliable region is obtained:

[0244]

[0245] Where, C2={I *(2) |I *(2) ∈[0,1)}. Given σ and Select any unreliable region UR2 The second user -LDPC code decoding failed, resulting in decoding failure for both user systems.

[0246] Based on the above three scenarios, the unreliable regions where decoding fails in the two-user system are identified:

[0247] UR=UR 12 ∪UR1∪UR2 (44)

[0248] Using set theory, by taking the absolute complement of UR, we obtain the reliable region for successful decoding of the two-user system:

[0249]

[0250] In other words, given σ, and RR If both users' rule LDPC codes can be successfully decoded, then the two-user system has successfully decoded the code.

[0251] After obtaining the reliable region RR, given σ, dv (1) and d v (2) Select the best match from RR. To maximize the sum rate r of the two-user system:

[0252]

[0253] Finally, traverse all and Solve for all corresponding And select the optimal degree distribution of the two-user LDPC codes from them. for:

[0254]

[0255] Maximize the sum rate of the two-user system

[0256] Table 1 shows the results when SNR = 4dB, or σ = 0.8923, after traversing all... and All corresponding equations can be solved using formula (46). Then, the optimal degree distribution is selected using formula (47). Maximize the sum rate of the two-user LDPC codes

[0257] Table 1

[0258]

[0259] like Figure 4 As shown, this embodiment also discloses a system for deriving the maximum sum rate based on regular LDPC codes under a two-user Gaussian access channel, used to execute the above method embodiment, which includes the following modules:

[0260] Objective function construction module: Analyze the sum node under a two-user Gaussian access channel, take maximizing the sum rate of the two-user regular LDPC code as the criterion, and assume successful decoding of the two-user system as the premise, and construct an objective function with constraints.

[0261] Inverse function solving module: Fix the degree of the variable nodes of LDPC code, derive the theoretical analytical expression of the degree of the check node, and solve the inverse function of the objective function in non-convex optimization problems;

[0262] Optimization module: Optimizes the degree distribution of the two-user LDPC codes successfully decoded by the system under a two-user Gaussian access channel, thereby maximizing the system and rate.

[0263] Other aspects of this embodiment can be found in the embodiments described above.

[0264] In summary, compared with the external information transfer algorithm based on full-space search, the fixed-point analysis method for two-user regular LDPC codes in this invention can efficiently and quickly find the closed-form expression of the optimal degree distribution of two-user regular LDPC codes, ensuring optimal coding efficiency while saving time and computing resources. This invention will provide significant assistance in the design and optimization of LDPC codes, ensuring high performance and reliability in 6G communication systems. This invention first constructs an objective function with constraints, based on maximizing the sum rate of two-user regular LDPC codes and assuming successful decoding by the two-user system. Then, based on fixed-point analysis theory, the degree of the variable nodes corresponding to the two users is fixed. Derive the degree of the check node corresponding to the two-user LDPC code The theoretical analytical expression is derived to solve for the inverse function of the objective function in the nonconvex optimization problem. Finally, based on mathematical set theory, the uncertainty of the inverse function is resolved, and the optimal degree distribution of the two-user LDPC code successfully decoded under a two-user Gaussian access channel is obtained. This allows the sum rate of the two-user system to reach its maximum.

[0265]

[0266] The above description is merely a detailed explanation of preferred embodiments and principles of the present invention. For those skilled in the art, there may be changes in specific implementation methods based on the ideas provided by the present invention, and these changes should also be considered within the scope of protection of the present invention.

Claims

1. A method for deriving the maximum sum rate based on regular LDPC codes in a two-user Gaussian access channel, characterized by: Follow these steps: S1. Analyze the sum node under a two-user Gaussian access channel. Based on the maximization of the sum rate of the two-user regular LDPC code and the premise of successful decoding of the two-user system, construct an objective function with constraints. S2, fix the degree of the variable node of LDPC code, derive the theoretical analytical expression of the degree of the check node, and solve the inverse function of the objective function in the non-convex optimization problem; S3 optimizes the degree distribution of the two-user LDPC codes successfully decoded by the system under a two-user Gaussian access channel, thereby maximizing the system's efficiency and rate.

2. The method for deriving the maximum sum rate based on regular LDPC codes in a two-user Gaussian access channel as described in claim 1, characterized in that, Step S1 is as follows: Based on a two-user Gaussian access channel, under the condition of equal transmission power, the k-th user adopts a rule-based approach. -LDPC code, where It is the degree of the variable node. It is the degree of the verification node, so the code rate of the kth user is: Therefore, the summation rate of the two user-defined LDPC codes is: At the receiving end, iterative decoding is performed on the factor graph. Based on the fundamental rules of mutual information at each node in the factor graph, assuming infinite code length and Gaussian approximation, when the degree of the variable node is d, the output mutual information at the variable node is: Where 0≤I A,i ≤1 represents the input mutual information from check node i to variable node j, J -1 (*) is the inverse function of the J function: Where J represents the output mutual information, σ A The variance of the input information is represented by d; when the degree of the check node is d, the output mutual information at the check node is: T c (I A,1 ,…,I A,d-1 )=1-T v (1-I A,1 ,…,1-I A,d-1 ) (5) If I A,i =I A ,i=1,…,w, simplify equations (3) and (5) to T respectively v (I A ×w,I A,w+1 …,I A,d-1 ) and T c (I A ×w,I A,w+1 …,I A,d-1 ); For the node with degree 3, it is connected to User 1, User 2, and the Gaussian access channel, respectively; for the j-th received bit y j Given noise power σ 2 Based on the maximum a posteriori probability criterion, the j-th bit of the k-th user Output log-likelihood ratio for: in, For the j-th bit of the (3-k)-th user; according to Bayes' theorem, we get: Divide the numerator and denominator by the same amount. have to: because The input log-likelihood ratio is so Substituting into formula (8), we get: To determine the output mutual information at node k, the goal is to find the j-th bit of the k-th user. Lower output log-likelihood ratio The expectation, that is Suppose the k-th user sends a codeword of all 1s, that is So: At this moment, the j-th bit of the (3-k)-th user There are two situations, one of which is... Another one is 3. The method for deriving the maximum sum rate based on regular LDPC codes in a two-user Gaussian access channel as described in claim 2, characterized in that, In step S1, case 1: when At that time, received bits The noise z follows a mean of 0 and a variance of σ. 2 The Gaussian distribution is z ~ N(0,σ). 2 Therefore, y j ~N(2,σ 2 Based on the Gaussian approximation, in It is the mean, noise. so Set parameters Based on the above assumptions, we get Let another parameter get In conclusion, when At that time, the target Represented as At this point, for a Gaussian random variable m with mean μ+a and variance 2μ+b, its probability density function is: Therefore, we get: Perform point replacement, let So Therefore: Substituting m1 and m2 into the equations, we get: Substituting formula (15) into formula (11), we get: at this time, When the number of iterations is l and the root mean square of the channel noise is σ, the output mutual information from channel node S to the k-th user variable node V is:

4. The method for deriving the maximum sum rate based on regular LDPC codes under a two-user Gaussian access channel as described in claim 3, characterized in that the steps are as follows: In S1, case 2: when At that time, received bits y j ~N(0,σ 2 );at this time, so Set parameters and have to In conclusion, when At that time, the objective was: At this point, let the mean of the Gaussian random variable ρ be -μ-a and the variance be 2μ+b, because its probability density function is: Therefore, we get: Perform point replacement, let So Therefore, there is Substituting ρ1 and ρ2 into the equations, we get: Substituting formula (22) into formula (18), we get: at this time, When the root mean square of the channel noise is σ, the output mutual information from channel node S to the k-th user variable node V is: In summary, for the k-th user, according to formulas (15) and (22), the output mutual information from channel node S to variable node V is: Based on external information transfer analysis, for the k-th user, the output mutual information from variable node V to verification node C is: The output mutual information from node C to variable node V is: The final output mutual information from the k-th user variable node V to the channel node S is obtained as follows: Reconsidering the final output mutual information at channel node S, we simplify to: Substituting formula (29) into formula (28), we finally obtain the objective function for the k-th user: Since the objective function of the two-user rule LDPC code is constructed based on external information transfer analysis, the necessary and sufficient condition for successful decoding is: In summary, taking the maximization of the sum rate *r* of the two-user rule LDPC code as the criterion and the successful decoding of the two-user system as the premise, the objective function with constraints is:

5. The method for deriving the maximum sum rate based on regular LDPC codes in a two-user Gaussian access channel as described in claim 4, characterized in that, Step 2 is as follows: Further refine the objective function, given the root mean square of the channel noise as σ and the degree distribution of the LDPC code for the k-th user. When the number of iterations is l, we have: Among them, I *(k) It is a fixed-point equation The smallest fixed point in the middle; the smallest fixed point I. *(k) This corresponds to the convergence point in the factor graph and also determines the bit error rate of the k-th user's decoding: in, It is a complementary error function; when I *(k) =1, Pe→0; that is: when the smallest fixed point I *(k) When the value falls within the range [0,1), the decoding by the k-th user fails; *(k) When = 1, the kth user successfully decodes; since two users access the system via Gaussian connections, the following conditions must be met simultaneously: (I *(k) ,I *(3-k) )=(1,1) (35) Decoding can only be successful with the help of two user systems; otherwise, decoding will fail for both user systems. After analyzing the output mutual information I from variable node V to channel node S on the factor graph *(k) and I *(3-k) Based on observations, and reconsidering formula (30), when the number of iterations l approaches infinity, we obtain: Formula (36) is an equation with two unknowns, namely the degree distribution. Assuming a given variable, the degree of the node To solve for the degree of the check node To prepare for solving nonconvex optimization problems; by transforming formula (36), we get: Degree containing check nodes Moving the terms aside, we simplify to: Solve for the degree of the check node The mathematical analytical expression, that is, to Represented as a fixed point (I) *(k) ,I *(3-k) The explicit function of ) 6. The method for deriving the maximum sum rate based on regular LDPC codes in a two-user Gaussian access channel as described in claim 5, characterized in that, Step 3 is as follows: Considering the decoding failure in a two-user system, there are three scenarios: first, both users fail to decode; second, the first user fails to decode regardless of whether the second user succeeds; and third, the second user fails to decode regardless of whether the first user succeeds. Case 1: When both users fail to decode, given the root mean square of the channel noise σ and the degree of the two user variable nodes. and Put I *(1) ∈[0,1),I *(2) Substituting ∈[0,1) into formula (39), we find an unreliable region: Among them, C 12 ={(I *(1) ,I *(2) )|I *(1) ∈[0,1) and I *(2) ∈[0,1)}; that is, for a given And σ, select unreliable region UR 12 any and The first user Code and the second user Both codes will fail to be decoded, resulting in decoding failure for both user systems. Case 2: When the first user fails to decode, regardless of whether the second user's decoding is successful, the two-user system will be judged to have failed to decode because the first user's signal cannot be correctly decoded; at this time, formula (39) degenerates into: Therefore, given σ and the degree of the first user variable node. Will I* (1) Substituting ∈[0,1) into formula (41), we find an unreliable region: Where, C1={I *(1) |I *(1) ∈[0,1)} and N + Represents the set of all positive integers; that is, given σ and In the case of selecting any unreliable region UR1 Regardless of what LDPC code the second user uses, and regardless of whether the second user can successfully decode it, the first user's... The code will fail to be decoded, thus determining that the decoding of both user systems has failed; Case 3: Similarly, when the second user fails to decode, based on formula (41), an unreliable region is obtained: Where, C2={I *(2) |I *(2) ∈[0,1)}; given σ and Select any unreliable region UR2 The second user The code decoding failed, resulting in decoding failure for both user systems. Based on the above three scenarios, the unreliable regions where decoding fails in the two-user system are identified: CLOCK=CLOCK 12 ∪UR1∪UR2 (44) Using set theory, by taking the absolute complement of UR, we obtain the reliable region for successful decoding of the two-user system: That is, given σ, and RR If both users' rule LDPC codes can be successfully decoded, then the two-user system decoding is successful. After obtaining the reliable region RR, given σ, and Select the best match from RR. To maximize the sum rate r of the two-user system: Finally, traverse all and Solve for all corresponding And select the optimal degree distribution of the two-user LDPC codes from them. for: Maximize the sum rate of the two-user system 7. A system for deriving the maximum sum rate based on regular LDPC codes in a two-user Gaussian access channel, wherein the system is based on the method described in any one of claims 1-6, characterized in that... Includes the following modules: Objective function construction module: Analyze the sum node under a two-user Gaussian access channel, take maximizing the sum rate of the two-user regular LDPC code as the criterion, and assume successful decoding of the two-user system as the premise, and construct an objective function with constraints. Inverse function solving module: Fix the degree of the variable nodes of LDPC code, derive the theoretical analytical expression of the degree of the check node, and solve the inverse function of the objective function in non-convex optimization problems; Optimization module: Optimizes the degree distribution of the two-user LDPC codes successfully decoded by the system under a two-user Gaussian access channel, thereby maximizing the system and rate.

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