A method for deriving optimal sum-rate of two-user combiner structure RA codes based on fixed point theory

Abstract

The application discloses a two-user Combiner structure RA code optimal sum rate derivation method based on a fixed point theory, and comprises the following steps: firstly, constructing an iterative model of mutual information of the Combiner structure RA code under a two-user Gaussian access channel; then, given the double-user prior mutual information, the number of parallel paths of the check node and the condition of the root mean square of the channel noise, the analytical form of the double-user variable node repetition number is derived; then, the reliable region of the decoding success is determined through the complement operation of the unreliable region, the optimal repetition number combination under the decoding success constraint and the corresponding optimal sum rate are screened, and the global optimal sum rate is obtained by traversing all parallel path combinations. The scheme can improve the parameter optimization and performance analysis efficiency while guaranteeing the global optimality of the solution.

Description

Technical Field

[0001] This invention relates to the field of wireless communication channel coding technology, specifically a method for deriving the optimal sum and rate of a two-user Combiner structure RA code based on fixed-point theory. Background Technology

[0002] Two-user wireless communication architectures have been widely applied in civilian wireless scenarios such as Bluetooth communication, IoT device interconnection, and audio / video data transmission, as well as in military and emergency communication fields such as cooperative relay and terminal direct connection. Repeat-Accumulate (RA) codes, as a type of channel coding scheme with a simple structure and low encoding / decoding complexity, are often chosen as the channel coding scheme for two-user multiple access communication systems due to their excellent performance in approximating channel capacity and low implementation threshold.

[0003] Traditional RA codes employ a single-path check node architecture, where each check node connects to the variable node along a single path. While this is simple to implement and offers low decoding latency, its transmission efficiency and potential for rate improvement are limited, making it difficult to meet the ever-increasing demands for spectral efficiency. To overcome this bottleneck, the industry has introduced a Combiner structure into the check node design of RA codes, such as... Figure 1 As shown, by setting multiple parallel paths at each verification node, mutual information can be transmitted and updated simultaneously along multiple branches in the factor graph, thereby significantly improving the transmission efficiency and decoding convergence performance of the coding system.

[0004] However, the introduction of the Combiner structure also complicates the factor graph topology and node degree distribution of the two-user RA code system. At the receiver, after the two signals are superimposed with noise, they need to undergo joint detection and iterative decoding. The system contains various types of processing nodes, such as channel nodes, variable nodes, check nodes, and sum nodes. Moreover, the mutual information transmission between the nodes has bidirectional coupling characteristics, which makes the mapping relationship between the system and the rate and the coding parameters (especially the number of parallel paths of the check nodes and the number of repetitions of the variable nodes in each branch) extremely complex and difficult to establish directly through intuitive analysis.

[0005] Currently, for parameter optimization and sum rate calculation of two-user Combiner structure RA codes, the field generally adopts numerical iterative methods based on external information transfer (EXIT) analysis or numerical calculation methods based on density evolution (DE) analysis. The basic idea of ​​the above methods is as follows: First, establish the system's factor graph model and mutual information iterative update rules; then, for a given set of encoding parameters (including the number of parallel paths and the number of repetitions), calculate the mutual information values ​​between each node through round-by-round numerical iteration; finally, determine whether the decoding is successful based on whether the mutual information approaches 1 at the time of iteration convergence, and select the optimal sum rate after traversing multiple sets of parameters.

[0006] However, the above-mentioned existing technical solutions have the following significant shortcomings in practical applications: First, it can only obtain numerical solutions and cannot establish closed-form analytical mapping relationships. Existing numerical iterative methods are essentially successive approximation processes for specific input parameters. Each iteration outputs the numerical result of mutual information under given parameters, but cannot mathematically derive an analytical expression between the number of repetitions of variable nodes and the system's mutual information. This makes the inherent correlation between node parameters unclear, preventing system designers from obtaining theoretically guiding closed-form formulas, and resulting in a lack of interpretability and predictability in parameter design.

[0007] Second, parameter optimization relies on large-scale numerical traversal, resulting in high computational complexity and low optimization efficiency. Since closed-form solutions do not exist, designers must perform large-scale numerical traversals or heuristic searches across different channel conditions (such as signal-to-noise ratio), different combinations of parallel path numbers, and different repetition counts when optimizing the sum and rate. As the range of possible parallel paths for the verification node increases, the parameter combination space grows exponentially, and the computational load of numerical iterations rises sharply, making it difficult to complete system performance evaluation and parameter optimization within a reasonable timeframe. This is particularly unsuitable for real-world communication scenarios with dynamically changing channel conditions.

[0008] Third, numerical iterative methods are prone to getting trapped in local optima, failing to guarantee the global optimality of the obtained sum rate. Existing iterative algorithms are sensitive to initial values, and the sum rate function of the system may exhibit complex characteristics such as multimodality and non-convexity in the parameter space, causing the iterative solution process to easily converge to local extrema. Even if designers invest significant computational resources in multiple attempts, it is difficult to rigorously prove whether the obtained result is the globally optimal sum rate, thus affecting the reliability and performance ceiling of the system design.

[0009] Fourth, the inverse solution process inherently contains uncertainties, making it difficult to reliably determine the design parameters required for successful decoding. In practical parameter design, designers often need to solve for parameters such as the number of repetitions of variable nodes based on the target rate. However, existing numerical methods often face unstable situations such as multiple solutions, no solutions, or numerical oscillations in nonlinear equations when dealing with such inverse problems. They lack a systematic judgment and screening mechanism, making it difficult to reliably provide a combination of design parameters that satisfies the conditions for successful decoding.

[0010] In summary, for RA code systems with Combiner structures under two-user Gaussian access channels, existing technologies are limited by the inherent defects of the numerical iterative framework. They cannot achieve closed-form analytical derivation of the relationship between the number of repetitions of variable nodes and mutual information, nor can they efficiently solve for the optimal sum rate while ensuring global optimality. Therefore, there is an urgent need in this field to propose a novel analytical derivation method. This method first establishes a closed-form mapping relationship between coding parameters and mutual information through theoretical means, and then uses this as a basis to achieve a global, efficient, and deterministic solution for the optimal sum rate, thereby overcoming all or part of the defects in the existing technologies. Summary of the Invention

[0011] The purpose of this invention is to overcome the shortcomings of existing technologies and propose a method for deriving the optimal sum and rate of a two-user Combiner structure RA code based on fixed-point theory. For RA codes with a Combiner structure under a two-user Gaussian access channel, the analytical expression for the number of repetitions of variable nodes is derived based on fixed-point theory, a closed mapping relationship between system parameters and mutual information is established, and then the optimal sum and rate of the system are obtained.

[0012] To achieve the above objectives, the technical solution specifically adopted by the present invention is as follows: A method for deriving the optimal sum and rate of a two-user Combiner structure RA code based on fixed-point theory includes the following steps: Step 1: Construct a mutual information iterative model of RA codes with Combiner structure under a two-user Gaussian access channel; Step 2: Based on fixed-point theory, determine the number of parallel paths for different verification nodes. Derive the number of repetitions of variable nodes The parsing expression; Step 3: Based on the reliable domain partitioning of set theory, eliminate the uncertainty in the mutual information inverse solution and obtain the system optimality and rate; in, For the first Number of parallel paths for each user's verification node For the first The number of times a user's variable node is repeated. .

[0013] Preferably, in step 1, the mutual information iterative model of the RA code with Combiner structure under the two-user Gaussian access channel is expressed as follows: in, and They represent the first and second users respectively. l -1 iteration variable node V The output mutual information, and They represent the first and second users respectively. l -1 iteration variable node U To the verification node C The output mutual information, The root mean square of the channel noise.

[0014] Preferably, the mutual information iterative model is constructed as follows: In a two-user Gaussian access channel, the two users independently encode their signals using RA codes with a Combiner structure and each transmits its own signal symbols to the receiver. The received signal is the sum of the two transmitted signals and zero-mean additive Gaussian noise. The channel reception model is as follows: in, In order to receive signals, and Sending signals to two users respectively. It is zero-mean additive Gaussian noise; For the One user, ,definition To verify the number of parallel paths for this user's node, The coding rate of a single user branch is the number of repetitions of the user variable node. for: Total system rate The branch rates of the two users are jointly determined: The decoding process is based on factor graphs and performs iterative information transfer. The function serves as a standard mapping tool between the log-likelihood ratio (LLR) and mutual information, when the variance of the input information is... hour, The function expression is: for J The inverse function of a function; The receiver uses the Maximum A posteriori (MAP) decoding algorithm at the node, assuming... For sending symbols The corresponding prior LLR, output after MAP decoding for: Introduction and node transfer functions Characterizing the output LLRJ Mutual information of the output after function processing: For degrees The variable nodes output mutual information from the external information transfer function. Characterization: in, Mutual information for inputs at variable nodes; For degrees The verification node outputs mutual information via an external information transfer function. Characterization: in, Verify the input mutual information at the node; In the mutual information transfer model, the 1st... In the next iteration, user 3- j The l -1st iteration, variable node V Output mutual information The output mutual information after processing by the node MAP decoding algorithm is: .

[0015] No. l In the next iteration, the user j Variable Node V To the verification node C and nodes Y The output mutual information are respectively and At the same time, variable nodes U To the verification node C The output mutual information is .

[0016] In the reverse path, the verification node C To the corresponding variable node U and V The output mutual information is respectively and The aforementioned multiple output mutual information can be transformed and represented into each other through the extrinsic information transfer function.

[0017] Finally, in the two-user Gaussian access channel model with Combiner structure RA code, the first user branch... l The next iteration passes to the node. Y The mutual information expression is: Will Substituting into the above formula, we construct an iterative model of mutual information between the two users, and the output mutual information is: .

[0018] in, and They represent the first and second users respectively. l -1 iteration variable node V The output mutual information, and They represent the first and second users respectively. l -1 iteration variable node U To the verification node C The output mutual information, The root mean square of the channel noise. and The number of parallel paths for the verification nodes of user 1 and user 2. and The number of times the variable node is repeated for User 1 and User 2.

[0019] Preferably, the two-user mutual information iterative model uses the sum rate of the two-user rule RA codes. Taking maximization as the criterion and successful system decoding as the premise, the objective function with constraints is established as follows: .

[0020] in, and They represent the first and second users respectively. l -1 iteration variable node V The output mutual information, and They represent the first and second users respectively. l -1 iteration variable node U To the verification node C The output mutual information, The root mean square of the channel noise. and The number of parallel paths for the verification nodes of user 1 and user 2. and The number of times the variable node is repeated for User 1 and User 2.

[0021] As a preferred option, the criterion for successful decoding is that the mutual information approaches 1 when the iteration converges, i.e.: .

[0022] Preferably, in step 2, the analytical expression for the number of repetitions of the variable node is: During the calculation process, intermediate parameters are used for ease of calculation. M and N Used to refer to respectively and .

[0023] Among them, intermediate parameters M and parameters N The expressions are as follows: .

[0024] Preferably, the method for obtaining the analytical expression for the number of repetitions of the variable node is as follows: When the number of iterations approaches infinity, the system iterative process converges to a steady state, and the mutual information between nodes no longer increases with the number of iterations. l The change satisfies the fixed-point condition: Under steady-state conditions, variable nodes U The output mutual information satisfies: For variable nodes U By performing algebraic transformations on the output mutual information, we obtain a preliminary analytical form of the number of repetitions of the variable nodes: For verification nodes C To the corresponding variable node U and V Output mutual information Verification node in C To variable node V The output formula is rearranged. For verification nodes C To the corresponding variable node U and V Output mutual information Verification node in C To variable node U Transform the output formula; Substituting these values ​​into the initial analytical form and simplifying them, we obtain the analytical expression for the number of repetitions of the variable node.

[0025] Preferably, in step 3, the specific method for eliminating uncertainties in the mutual information inverse solution based on reliable domain partitioning in set theory and solving for the system's optimality and rate is as follows: Will and When substituting into the repetition count parsing expression, Since there are no finite values ​​for mutual information inverse solving, it is impossible to directly obtain the values ​​corresponding to each user under the condition of successful decoding. The unreliable region partitioning method is adopted: first, the parameter regions corresponding to all decoding failures are solved, and then the reliable regions of successful decoding are obtained through set complement operations. DD ; Unreliable regions that fail to decode fall into two categories: The first category is the unreliable region corresponding to a single-user decoding failure, where the mutual information of any user satisfies... The system decoding failed, record the first time. The parameter set corresponding to each user's decoding failure is: The union of the unreliable regions of two users yields the unreliable region formed by a single user failure. : ; The second category is the unreliable region corresponding to the failure of decoding by both users, when the mutual information of the two users simultaneously satisfies... The system completely failed to decode, corresponding to an unreliable region. : The union of the two unreliable regions covers all parameter combinations that failed to decode. Taking the complement of this union yields the reliable regions that successfully decoded. in, This represents the set complement operation; In reliable area The inner traversal of parameter combinations is used to select the optimal sum and rate under the condition of successful decoding. and the corresponding optimal number of repetitions : Count of parallel paths for traversing all verification nodes By combining the solutions, the optimal sum rate under the corresponding conditions is solved separately, and finally the global optimal sum rate under a fixed signal-to-noise ratio is obtained. and the corresponding globally optimal combination of repetition counts .

[0026] Preferably, the factor graph includes variable nodes. Variable nodes Verification Node and channel nodes Mutual information passes through nodes in the factor graph sequentially. Variable nodes Verification Node and variable nodes Then propagate back to the node along the reverse path. This completes one full iteration; Functions and The functions together implement bidirectional conversion between LLR and mutual information, which is used for information updates between nodes in the factor graph.

[0027] This invention has the following characteristics and beneficial effects: For RA codes with a Combiner structure under a two-user Gaussian access channel, this paper derives an analytical expression for the number of repetitions of variable nodes based on fixed-point theory, and solves for the optimal sum rate using a reliable domain partitioning method based on set theory. First, a mutual information iterative model of the Combiner structure RA code under a two-user Gaussian access channel is constructed. Then, given the prior mutual information of the two users, and under the conditions of verifying the number of parallel paths of nodes and the root mean square of channel noise, an analytical form for the number of repetitions of variable nodes of the two users is derived. Next, the reliable region of successful decoding is determined through the complement operation of unreliable regions. The optimal combination of repetition counts and the corresponding optimal sum rate under the constraint of successful decoding are obtained. Finally, by traversing all combinations of the number of parallel paths, the globally optimal sum rate is obtained. Attached Figure Description

[0028] Figure 1 This is a schematic diagram of the process for deriving the optimal sum rate of a two-user Combiner structure RA code based on fixed-point theory, according to an embodiment of the present invention. Figure 2 This is a factor diagram of a RA code with Combiner structure under a two-user Gaussian access channel in an embodiment of the present invention. Figure 3 This is a fixed-point curve diagram under the condition of signal-to-noise ratio (SNR) = 2 dB and parallel path number in an embodiment of the present invention. Figure 4 This is an EXIT plot in an embodiment of the present invention with a signal-to-noise ratio (SNR) of 2 dB and the number of times the variable node is repeated. Detailed Implementation

[0029] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0030] A method for deriving the optimal sum and rate of a two-user Combiner structure RA code based on fixed-point theory, such as... Figure 1 As shown, it includes the following steps: Step 1. Construction of mutual information iterative model for Combiner structure RA codes under two-user Gaussian access channel.

[0031] A system model with Combiner-structured RA codes is constructed for a two-user Gaussian access channel, and the mapping relationship between core coding parameters and rate is clearly defined, such as... Figure 2 As shown, a complete mutual information iterative update model is established based on the information transmission rules of factor graphs, serving as the mathematical foundation for the fixed-point analytical derivation.

[0032] In this system architecture, two users independently encode using RA codes with a Combiner structure, and each transmits signal symbols to the receiver. The received signal is the sum of the two transmitted signals and zero-mean additive Gaussian noise. The channel reception model is as follows: (1) in, In order to receive signals, and Sending signals to two users respectively. It is zero-mean additive Gaussian noise.

[0033] The system has two users with independent coding branches, for the first... j ( j = 1, 2) users, defined This represents the number of parallel paths for the verification node. This represents the number of repetitions of the user variable node. From this, the coding rate of a single user branch can be obtained. for (2) The total system rate is determined by the branch rates of the two users, and can be adjusted by combining the number of parallel paths of the verification nodes and the number of repetitions of the variable nodes for the two users. For system and rate r Adjustments are needed: (3) The decoding process is based on factor graphs and performs iterative information transfer. J The function serves as a standard mapping tool between the log-likelihood ratio (LLR) and mutual information, when the variance of the input information is... hour, J The function expression is: (4) for J The inverse function of the function, together with the mutual information, realizes the bidirectional conversion between LLR and mutual information, and is the core mathematical mapping for updating node information in the factor graph.

[0034] The receiver employs Maximum A Posteriori (MAP) decoding at the node, combining prior information about the symbols transmitted by another user to complete the estimation of the transmitted symbols. Let... for The corresponding prior LLR, after MAP decoding, is the output LLR. (5) Introduction and node transfer functions Characterizing the output LLR J The output mutual information after function processing is expressed as: (6) Based on the structural symmetry of the two-user RA code, taking a single-user branch as an example, the mutual information passes through the nodes in the factor graph sequentially. Y variable node V Verification node C and variable nodes U Then, propagate back along the reverse path to the node, completing one full iteration. For a degree of... d The variable nodes output mutual information from the external information transfer function. The representation, expressed as: (7) in, This refers to the input mutual information at the variable node. For a degree of... d The verification node outputs mutual information via an external information transfer function. The representation, expressed as: (8) in, This is to verify the input mutual information at the node.

[0035] In the mutual information transfer model, the first l In the next iteration, the mutual information between the processed output of the node is: (9) Variable Node V To the verification node C and nodes Y The output mutual information are as follows: (10) and variable nodes U The output mutual information is: (11) In the reverse path, the verification node C To the corresponding variable node U andV The output mutual information is: (12) Finally, in the two-user Gaussian access channel model with Combiner structure RA code, the first user branch... l The next iteration passes to the node. Y The mutual information expression is: (13) Will j Substituting 1 and 2 into the single-user iterative formula, we construct a two-user mutual information iterative model, and the output mutual information is: (14) The condition for successful decoding is that the mutual information approaches 1 when the iteration converges, that is: (15) The sum rate of two user rule RA codes r Taking maximization as the criterion and successful system decoding as the premise, the objective function with constraints is established as follows: (16) Step 2. Based on fixed-point theory, for different numbers of parallel paths... Derive the number of repetitions The parsing expression.

[0036] When the number of iterations approaches infinity, the system iterative process converges to a steady state, and the mutual information between nodes no longer increases with the number of iterations. l The change satisfies the fixed-point condition: (17) Under steady-state conditions, variable nodes U The output mutual information satisfies: (18) By performing algebraic transformations on formula (18), we obtain a preliminary analytical form of the number of repetitions of the variable nodes: (19) To solve this problem, we first need to derive the steady-state solution. and The expression is given by: Taking the inverse transformation of formula (13), we get: (20) Substitute the variable nodes into formula (10) V To the verification node C The output mutual information expression is obtained as follows: (twenty one) For the verification nodes in formula (12) C To variable node V Rearranging the output formula, we get: (twenty two) For the verification nodes in formula (12) C To variable node U Transforming the output formula, we get: (twenty three) Substituting formulas (20)-(23) into formula (19), and simplifying, we obtain the analytical expression for the number of repetitions of the variable node: (twenty four) Among them, intermediate parameters M and parameters N The expressions are as follows: Take respectively j = 1 and j = 2, which gives the number of repetitions of the two user variable nodes. and The expressions, when combined, form a system of two user-parameter equations: (25) Step 3. Based on the reliable domain partitioning of set theory, eliminate the uncertainty in the mutual information inverse solution and realize the optimal system and rate design.

[0037] directly and When substituting into the repetition count parsing expression, Since there are no finite values ​​for mutual information inverse solving, it is impossible to directly obtain the values ​​corresponding to each user under the condition of successful decoding. This inverse problem is non-convex, and direct optimization easily leads to multiple solutions and invalid solutions. Therefore, an unreliable region partitioning method is adopted: first, the parameter regions corresponding to all decoding failures are solved, and then the reliable regions of successful decoding are obtained through set complement operations. DD Unreliable regions that fail to decode fall into two categories: The first category represents the unreliable region corresponding to a single-user decoding failure. The mutual information of any user satisfies... At that time, the system decoding failed. (Note: The last part appears to be a fragment and doesn't translate directly.) j The parameter set corresponding to each user's decoding failure is: The union of the unreliable regions of two users yields the unreliable region formed by a single user failure. DS : (26) The second category is the unreliable region corresponding to the failure of both users to decode. This occurs when the mutual information of the two users simultaneously satisfies... At that time, the system completely failed to decode, corresponding to the unreliable region. DR : (27) The union of the two unreliable regions covers all parameter combinations that failed to decode. Taking the complement of this union yields the reliable regions that successfully decoded. (28) In reliable area DD The inner traversal of parameter combinations is used to select the optimal sum and rate under the condition of successful decoding. and the corresponding optimal number of repetitions : (29) Count of parallel paths for traversing all verification nodes By combining the solutions, the optimal sum rate under the corresponding conditions is solved separately, and finally the global optimal sum rate under a fixed signal-to-noise ratio is obtained. and the corresponding optimal number of repetitions .

[0038] In this embodiment, as Figure 1 The diagram shows a flowchart illustrating the derivation method for the optimal sum and rate of a two-user Combiner structure RA code based on fixed-point theory.

[0039] In this embodiment, Figure 2 The factor diagram of the RA code with Combiner structure under a two-user Gaussian access channel and the first l The iteration process represents the mutual information transfer process between nodes in the factor graph.

[0040] In this embodiment, the signal-to-noise ratio (SNR) is 2 dB, and the number of parallel paths is... Under certain conditions, such as Figure 3 As shown, where DD This represents the reliable decoding region, corresponding to the optimal sum rate. Combination of the number of repetitions of variable nodes The value is (4,6).

[0041] In this embodiment, the signal-to-noise ratio (SNR) is 2 dB, and the number of parallel paths is... Under certain conditions, such as Figure 4 As shown, the number of repetitions of a variable node can be verified using an EXIT graph. Decoding can succeed when the sum rate is optimal.

[0042] Table 1

[0043] Table 1 lists the number of parallel paths for each group under different signal-to-noise ratio conditions. The corresponding optimal combination of repetition counts With optimal sum rate And the globally optimal combination of repetition counts obtained by traversing all paths under each signal-to-noise ratio. With global optimal sum rate Different signal-to-noise ratios and different numbers of parallel paths correspond to different combinations of optimal sum rate and variable node repetition counts, and the globally optimal sum rate varies for each signal-to-noise ratio. The corresponding number of parallel paths differs. Therefore, after verification, it has been shown that by calculating the analytical expression, the globally optimal sum and rate can be obtained intuitively and quickly under a given signal-to-noise ratio and parameters. .

[0044] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely preferred examples and are not intended to limit the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for deriving the optimal sum rate of a two-user Combiner structure RA code based on fixed-point theory, characterized in that, Includes the following steps: Step 1: Construct a mutual information iterative model of RA codes with Combiner structure under a two-user Gaussian access channel; Step 2: Based on fixed-point theory, determine the number of parallel paths for different verification nodes. Derive the number of repetitions of variable nodes The parsing expression; Step 3: Based on the reliable domain partitioning of set theory, eliminate the uncertainty in the mutual information inverse solution and obtain the system optimality and rate; in, For the first Number of parallel paths for each user's verification node For the first The number of times a user's variable node is repeated. .

2. The method according to claim 1, characterized in that, In step 1, the mutual information iterative model of RA codes with Combiner structure under a two-user Gaussian access channel is expressed as follows: in, and They represent the first and second users respectively. l -1 iteration variable node V The output mutual information, and They represent the first and second users respectively. l -1 iteration variable node U To the verification node C The output mutual information, The root mean square of the channel noise.

3. The method according to claim 1, characterized in that, The mutual information iterative model is constructed as follows: In a two-user Gaussian access channel, the two users independently encode their signals using RA codes with a Combiner structure and each transmits its own signal symbols to the receiver. The received signal is the sum of the two transmitted signals and zero-mean additive Gaussian noise. The channel reception model is as follows: in, In order to receive signals, and Sending signals to two users respectively. It is zero-mean additive Gaussian noise; For the One user, ,definition To verify the number of parallel paths for this user's node, The coding rate of a single user branch is the number of repetitions of the user variable node. for: Total system rate The branch rates of the two users are jointly determined: The decoding process is based on factor graphs and performs iterative information transfer. The function serves as a standard mapping tool between the log-likelihood ratio (LLR) and mutual information, when the variance of the input information is... hour, The function expression is: for J The inverse function of a function; The receiver uses the Maximum A posteriori (MAP) decoding algorithm at the node, assuming... For sending symbols The corresponding prior LLR, output after MAP decoding for: Introduction and node transfer functions Characterizing the output LLR J Mutual information of the output after function processing: For degrees The variable nodes output mutual information from the external information transfer function. Characterization: in, Mutual information for inputs at variable nodes; For degrees The verification node outputs mutual information via an external information transfer function. Characterization: in, Verify the input mutual information at the node; In the mutual information transfer model, the 1st... In the next iteration, user 3- j The l -1st iteration, variable node V Output mutual information The output mutual information after processing by the nodes is: ; No. l In the next iteration, the user j Variable Node V To the verification node C and nodes Y The output mutual information are respectively and Meanwhile, variable nodes U The output mutual information is ; In the reverse path, the verification node C To the corresponding variable node U and V The output mutual information are respectively and The aforementioned multiple output mutual information can be transformed and represented by each other through the extrinsic information transfer function; Finally, in the two-user Gaussian access channel model with Combiner structure RA code, the first user branch... l The next iteration passes to the node. Y The mutual information expression is: Will Substituting into the above formula, we construct an iterative model of mutual information between the two users, and the output mutual information is: in, and They represent the first and second users respectively. l -1 iteration variable node V The output mutual information, and They represent the first and second users respectively. l -1 iteration variable node U To the verification node C The output mutual information, The root mean square of the channel noise. and The number of parallel paths for the verification nodes of user 1 and user 2. and The number of times the variable node is repeated for User 1 and User 2.

4. The method according to claim 3, characterized in that, The two-user mutual information iterative model uses the sum rate of the two-user rule RA codes. Taking maximization as the criterion and successful system decoding as the premise, the objective function with constraints is established as follows: in, and They represent the first and second users respectively. l -1 iteration variable node V The output mutual information, and They represent the first and second users respectively. l -1 iteration variable node U To the verification node C The output mutual information, The root mean square of the channel noise. and The number of parallel paths for the verification nodes of user 1 and user 2. and The number of times the variable node is repeated for User 1 and User 2.

5. The method according to claim 4, characterized in that, The condition for successful decoding is that the mutual information approaches 1 when the iteration converges, that is: 。 6. The method according to claim 1, characterized in that, In step 2, the analytical expression for the number of repetitions of the variable node is: During the calculation process, intermediate parameters are used for ease of calculation. M and N Used to refer to respectively and .

7. The method according to claim 6, characterized in that, intermediate parameters M and parameters N The expressions are as follows: 。 8. The method according to claim 6, characterized in that, The method for obtaining the parsing expression for the number of repetitions of the variable node is as follows: When the number of iterations approaches infinity, the system iterative process converges to a steady state, and the mutual information between nodes no longer increases with the number of iterations. l The change satisfies the fixed-point condition: Under steady-state conditions, variable nodes U The output mutual information satisfies: For variable nodes U By performing algebraic transformations on the output mutual information, we obtain a preliminary analytical form of the number of repetitions of the variable nodes: For verification nodes C To the corresponding variable node U and V Output mutual information Verification node in C To variable node V The output formula is rearranged. For verification nodes C To the corresponding variable node U and V Output mutual information Verification node in C To variable node U Transform the output formula; Substituting these values ​​into the initial analytical form and simplifying them, we obtain the analytical expression for the number of repetitions of the variable node.

9. The method according to claim 1, characterized in that, In step 3, the specific method for eliminating uncertainties in the mutual information inverse solution based on reliable domain partitioning using set theory and solving for the system's optimality and rate is as follows: Will and When substituting into the repetition count parsing expression, Since there are no finite values ​​for mutual information inverse solving, it is impossible to directly obtain the values ​​corresponding to each user under the condition of successful decoding. The unreliable region partitioning method is adopted: first, the parameter regions corresponding to all decoding failures are solved, and then the reliable regions of successful decoding are obtained through set complement operations. DD ; Unreliable regions that fail to decode fall into two categories: The first category is the unreliable region corresponding to a single-user decoding failure, where the mutual information of any user satisfies... The system decoding failed, record the first time. The parameter set corresponding to each user's decoding failure is: The union of the unreliable regions of two users yields the unreliable region formed by a single user failure. : ； The second category is the unreliable region corresponding to the failure of decoding by both users, when the mutual information of the two users simultaneously satisfies... The system completely failed to decode, corresponding to an unreliable region. : The union of the two unreliable regions covers all parameter combinations that failed to decode. Taking the complement of this union yields the reliable regions that successfully decoded. in, This represents the set complement operation; In reliable area The inner traversal of parameter combinations is used to select the optimal sum and rate under the condition of successful decoding. and the corresponding optimal number of repetitions : Count of parallel paths for traversing all verification nodes By combining the solutions, the optimal sum rate under the corresponding conditions is solved separately, and finally the global optimal sum rate under a fixed signal-to-noise ratio is obtained. and the corresponding globally optimal combination of repetition counts .

10. The method according to claim 8, characterized in that, The factor graph contains variable nodes. Variable nodes Verification Node and channel nodes Mutual information passes through nodes in the factor graph sequentially. Variable nodes Verification Node and variable nodes Then propagate back to the node along the reverse path. This completes one full iteration; Functions and The functions together implement bidirectional conversion between LLR and mutual information, which is used for information updates between nodes in the factor graph.

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