Power and degree distribution joint optimization method and system for LDPC codes of two users under unequal power

Abstract

The invention discloses a power and degree distribution joint optimization method and system suitable for two-user LDPC codes under unequal powers, and the method comprises the following steps: 1, analyzing the constraint relation between the nodes of the two-user LDPC codes, and on the premise of successful decoding of the two users, taking the maximization of the total rate as the target, and carrying out the optimization of the degree distribution of the two-user LDPC codes; deriving a mutual information transfer objective function of LDPC codes of two users under unequal power; 2, solving the mutual information transfer target function in the step 1 according to a fixed point theory to obtain a check node degree analytical expression; and step 3, according to the check node degree analytic expression obtained in the step 2, dividing a degree distribution value area capable of enabling the LDPC codes of the two users to be successfully decoded in a power and degree distribution joint parameter space, screening out the optimal degree distribution, and solving the maximum total rate of the two users. According to the method, the problem of performance limitation caused by neglecting the coupling relationship between power and degree distribution in a power asymmetric scene in the prior art is solved, and the maximization of the total rate of the system is realized.

Description

A Joint Optimization Method and System for Power and Degree Distribution of Two-User LDPC Codes under Unequal Power Conditions Technical Field

[0001] This invention belongs to the field of wireless communication technology, specifically relating to a method and system for joint optimization of power and degree distribution of low-density parity-check (LDPC) codes in unequal-power two-user communication under Gaussian multiple access channels. More specifically, based on external information transfer analysis and fixed-point theory, this invention derives an analytical expression for the degree distribution of LDPC codes for two users under unequal power conditions, and uses set theory to optimize the system in the joint parameter space of power and degree distribution, thereby maximizing the total system rate. Background Technology

[0002] With the evolution of sixth-generation (6G) communication networks, high-speed and high-reliability multiple access technologies have become a research hotspot. In two-user Gaussian Multiple-Access Channel (GMAC) systems, users share the same radio resources, making the design of efficient multiple access schemes crucial. LDPC codes have attracted widespread attention due to their performance approaching the Shannon limit and low decoding complexity. Existing research mainly focuses on optimizing LDPC code degree distribution under equal power conditions, failing to adequately address user power asymmetry scenarios caused by path loss and near-far effects in real-world systems, thus limiting further exploration of system capacity.

[0003] In unequal-power Gaussian multiple access channels, there is a strong coupling between reasonable power configuration and LDPC code-degree distribution design, making it difficult to achieve system optimization by optimizing a single parameter. Although existing studies have proposed degree distribution optimization methods under the assumption of equal power, their performance degrades significantly in unequal-power scenarios. Therefore, there is an urgent need in this field for a systematic method that can synergistically optimize power allocation and LDPC code-degree distribution to fully unleash the transmission potential of unequal-power multi-user channels. Summary of the Invention

[0004] For Gaussian multiple access channels, this invention proposes a joint optimization method and system for power and degree distribution of two-user LDPC codes under unequal power conditions, in order to solve the performance limitation problem caused by ignoring the coupling relationship between power and degree distribution in existing technologies in power asymmetric scenarios, and to maximize the total system rate.

[0005] Application scenarios of this invention: It is applicable to various systems that use regular LDPC codes for multiple access communication in a two-user Gaussian access channel with unequal power. It is particularly suitable for practical application scenarios such as the Internet of Things, large-scale machine communication, and satellite communication, where there is significant power asymmetry or dynamic power allocation is required. It can provide the system with an efficient and reliable power and coding joint design scheme.

[0006] This invention adopts the following technical solution: a joint optimization method for power and degree distribution of two-user LDPC codes under unequal power conditions, with the following specific steps: Step 1: Node constraint analysis and objective function construction. Analyze the constraint relationships between nodes of the two-user LDPC codes. Under the premise that the two users can successfully decode, derive the mutual information transfer objective function of the two-user LDPC codes under unequal power conditions with the goal of maximizing the total rate; Step 2: Solve the objective function analytical expression using fixed-point theory. Solve the mutual information transfer objective function of Step 1 using fixed-point theory to obtain the analytical expression of the check node degree; Step 3: Decoding reliable region division and maximum total rate solution. Based on the analytical expression of the check node degree obtained in Step 2, divide the degree distribution value region that enables the two-user LDPC codes to be successfully decoded in the joint parameter space of power and degree distribution, select the optimal degree distribution, and obtain the maximum total rate of the two users.

[0007] This invention also discloses a joint optimization system for power and degree distribution of two-user LDPC codes under unequal power conditions, used to execute the above method. It includes the following modules: a node constraint analysis and objective function construction module: used to analyze the constraint relationships between nodes of the two-user LDPC codes, and derive the mutual information transfer objective function of the two-user LDPC codes under unequal power conditions with the goal of maximizing the total rate, assuming successful decoding by both users; a fixed-point theory objective function solution module: based on fixed-point theory, the mutual information transfer objective function is solved to obtain the analytical expression of the check node degree; a reliable decoding region partitioning and maximum total rate solution module: based on the obtained analytical expression of the check node degree, the degree distribution value region that enables successful decoding of the two-user LDPC codes is partitioned within the joint parameter space of power and degree distribution, the optimal degree distribution is selected, and the maximum total rate of the two users is obtained.

[0008] Compared with the prior art, the present invention has the following significant advancements.

[0009] This invention combines external information transfer and fixed-point analysis to propose a joint optimization method and system for two-user Gaussian access channels under unequal power conditions, specifically for LDPC codes. Under unequal power conditions, this invention derives a closed-form expression for the LDPC code degree distribution, achieving coordinated optimization of power ratio and degree distribution parameters, thereby maximizing the overall system rate.

[0010] The technical solution adopted in this invention first analyzes the mutual information transfer relationship between nodes under unequal power conditions based on external information transfer, aiming to maximize the total rate of the two-user LDPC codes. Under the premise of ensuring successful system decoding, a constrained objective function is constructed. Next, using fixed-point analysis theory, given the power ratio and variable node degrees, a closed-form expression for the corresponding check node degree is derived, transforming the original optimization problem into solving the inverse function of the objective function. Then, addressing the uncertainty caused by the divergence of the inverse function at the boundary, set theory is used to delineate a reliable parameter region that guarantees successful decoding under the current parameters, and locally optimal degree distribution pairs are selected from this region. Finally, by traversing the variable node degree combinations and optimizing the power ratio, the globally optimal LDPC code degree distribution and power configuration that maximizes the total system rate are selected, completing the joint optimization design of the two-user system under unequal power conditions. Attached Figure Description

[0011] Figure 1 is a schematic diagram of the factor graph structure of a regular LDPC code under a two-user Gaussian access channel.

[0012] Figure 2 shows the SNR = 4 dB and power ratio A1:A2 = 1.22:1 at the variable node degree. A schematic diagram of the reliable region of the verification node under the given conditions.

[0013] Figure 3 shows the degree distribution when SNR = 4 dB and power ratio A1:A2 = 1.22:1. and The EXIT chart below.

[0014] Figure 4 is a flowchart of a preferred embodiment of the present invention for a joint optimization method of power and degree distribution of two-user LDPC codes under unequal power conditions.

[0015] Figure 5 is a block diagram of a preferred embodiment of the present invention for a joint optimization system of power and degree distribution of two-user LDPC codes under unequal power conditions. Detailed Implementation

[0016] The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0017] As shown in Figure 4, this embodiment discloses a joint optimization method for power and degree distribution of two-user LDPC codes under unequal power conditions. The specific steps are as follows: Step 1: Node constraint analysis and objective function construction. Analyze the constraint relationship between each node of the two-user LDPC codes. Under the premise of ensuring successful decoding of the two-user system, and taking the maximization of the total system rate as the criterion, derive the mutual information transfer objective function of the two-user LDPC codes under unequal power conditions.

[0018] At the sending end, the two users each send a data bit sequence of length N. Send in Encoder, where, This represents the degree of the message node for user k. This represents the degree of the check node for user k. After encoding, the codeword sequence is obtained. Where M is of length The codewords are then modulated using BPSK (Binary Phase Shift Keying) to obtain each modulation symbol: in, The two signals are transmitted to the receiving device via a Gaussian multiple access channel according to a preset unequal power distribution scheme.

[0019] At the receiving end, the two user signals are received synchronously. Let their transmit power amplitudes be A1 and A2, respectively, transmitted via a Gaussian multiple access channel, and assuming ideal frame synchronization, then the received signal model can be expressed as: Among them, y i For the i-th received signal, the channel noise n i It follows a pattern with a mean of 0 and a variance of σ. 2 The signal is distributed according to a Gaussian distribution. The receiver performs iterative decoding based on the received signal, a process that can be simulated on a factor graph. Based on the iterative decoding process, the mutual information between nodes on the factor graph can be tracked using the Extrinsic Information Transfer (EXIT) function. Under the conditions of infinite code length and Gaussian approximation (i.e., assuming that the probability density function of each message satisfies a variance twice the mean), the EXIT function of each node is expressed as follows: For a degree of d v The variable node outputs mutual information as follows: Where fv(•) represents the mathematical relationship between the output mutual information of the variable nodes and their input mutual information. These are the prior mutual information from other check nodes and sum nodes input to this variable node, respectively; I AV,j I represents the mutual information input to the j-th verification node. Es The mutual information between the node inputs and the variable is represented; J(•) is a function that maps the LLR variance to mutual information, and its expression is: Here, x represents the integration variable, σ A J represents the standard deviation of the input LLR. -1 (•) is the inverse function of J(•). When the output mutual information under consideration is the transfer of variable node to sum node, its expression simplifies to: For a degree of d c The verification node outputs the following mutual information: Where fc(•) represents the mathematical relationship between the output mutual information of the verification node and the input mutual information. This represents the prior mutual information input from other nodes to the verification node.

[0020] In a two-user communication system, the number of edges connected to the sum node is typically 3, corresponding to user 1, user 2, and the Gaussian access channel, respectively. In message passing decoding, the sum node performs joint estimation of the transmitted symbols based on the prior information of each user; that is, the symbol of any user can be determined by the prior information of the other user. Therefore, the mutual information output by the sum node to the two users can be expressed as: in, This represents the mathematical relationship between the mutual information of node output and the mutual information of user k input. This indicates taking the expected function. This represents the prior mutual information between the input of the k-th user and the node. Let LLR be the external log-likelihood ratio (LLR) output by the node to the k-th user.

[0021] Taking user 1 as an example, with a channel noise variance of σ 2 Given that the transmit powers of the two users are A1 and A2 respectively, the receiver targets the i-th received signal y. i The calculated value corresponds to the i-th bit of user 1. The output LLR information is: Where p(•) represents the probability density function, Let represent the i-th bit of user 2. Based on Bayes' theorem, the above expression can be transformed into the following form: Since the channel model is an additive white Gaussian noise channel, therefore, for a given transmitted bit... Under the condition of receiving signal y i Follow the mean The variance is Substituting the conditional probability density function into the LLR expression (10) derived from Bayes' theorem, we obtain the log-likelihood ratio of the i-th bit for user 1 as: Because the prior log-likelihood ratio information of the user's two-way input and the node input is Therefore, there is Substituting into formula (11), we get: To facilitate numerical calculation, both the numerator and denominator on the right side of formula (12) are divided by 1 / 2. We can obtain: Similarly, the i-th bit of user 2 can be derived. The output LLR is: Based on the iterative detection structure defined in equations (7) and (8), after obtaining the LLR of each bit, it is necessary to further calculate the expected value of the product of this information and the corresponding transmitted bit for subsequent variance propagation analysis. Define the i-th bit of user k (k = 1, 2). The expected value is ,Right now To simplify the analysis without loss of generality, we make the following assumptions: User 1 sends a completely one-codeword, that is, for all i, we have User 2 sends binary codewords with equal probability, that is It takes either +1 or -1 with equal probability.

[0022] User 1: Substitute the log-likelihood ratio expression (13) of User 1 into its expected value. In the definition. Considering the assumption The expected value is simplified to Substituting the values, we get: Equation (15) contains The expectation of the form , where e is the natural constant and t is a Gaussian random variable. Analytical closed-form solutions to such expectations are difficult to obtain, but can be efficiently and accurately calculated using Gauss-Hermitian numerical integration. Specifically, if Where a and b are parameters, then: Where w is the Gaussian Hermitian integral weighting coefficient.

[0023] To apply this numerical integration method, the distributions of the random variables within the two logarithmic terms in formula (15) need to be determined. For this purpose, we define: Next, the mean and variance of t1 and t2 need to be derived separately before they can be substituted into the numerical integration formula above for calculation.

[0024] Case 1: When the i-th transmitted bit of both User 1 and User 2 is +1, i.e. At that time, the prior LLR of the user's two-way input and the node input can be modeled as its expected value. With Gaussian noise n i,A The sum of According to the Gaussian approximation, this noise Therefore, there is Due to the received signal ,in Therefore Based on this, the distributions of the random variables t1 and t2 defined in equation (17) can be derived. Note that t1 and t2 are both Gaussian random variables y i and They are linear combinations, therefore they also follow a Gaussian distribution. By calculating their mean and variance, we can obtain: .

[0025] Substituting the distribution parameters of t1 and t2 into the Gauss-Hermitian integral shown in equation (16), we can obtain the following results: In conclusion, Under these circumstances, a functional relationship can be obtained: this function takes the expected value of LLR input by user 2 as the independent variable and the expected value of LLR output by user 1 as the dependent variable. The expression of this function is: Therefore, in the i-th bit sent by user 1 User 2's i-th transmitted bit Under the given conditions, the mutual information output by node 1 to user 1 can be obtained by processing the above expected value function, the expression of which is: Scenario 2: When At that time, receive signal ,so At this point, User 2's prior LLR , where n A This represents the Gaussian noise component in the prior LLR, therefore Based on the above distributions, the distributions of -t1 and -t2 can be obtained as follows: , .

[0026] Similarly, substituting the distribution parameters of -t1 and t2 into the Gauss-Hermitian integral shown in equation (16), we obtain: Therefore, we can obtain the following: The expected function expression under the following conditions: At this point, the mutual information between the output from node 1 and node 2 can be calculated using the corresponding expected value, and its expression is: In summary, according to equations (20) and (23), the output mutual information from node to user 1 variable node is the average of the mutual information under two conditions: User 2: Similarly, the expected value of the output log-likelihood ratio for User 2's i-th bit can be derived. Considering the bits sent by User 1... With both positive and negative cases, User 2's expectation function can be expressed in the following form: Similarly, the random variables within the logarithmic term in the definition are q1 and q2: The next step is to find the mean and variance of q1 and q2.

[0027] Scenario 1: When At that time, receive signal At this point, User 1's prior LLR Therefore, the distributions of q1 and q2 can be obtained: , Correspondingly, substituting the distribution parameters of q1 and q2 into the Gauss-Hermitian integral shown in equation (16), we can obtain: Substituting the result of equation (27) into equation (25), we can obtain the result when... At that time, User 2 outputs the function expression expected by LLR: Accordingly, user 2 can be obtained in Output mutual information under certain conditions: Scenario 2: When At that time, receive signal User 1's prior LLR Therefore, the distributions of q1 and q2 are as follows: , Substituting the above distribution parameters into the Gauss-Hermitian integral formula, we get: Substituting the result of equation (30) into equation (25), we obtain the result in... At that time, the LLR expectation function of user 2 is: Therefore, User 2 in The output mutual information under the given conditions is: Thus, the average output mutual information from node 1 to user 2 variable node is obtained. Combining equations (29) and (32), and considering the assumption that user 1 sends an all-1 codeword, the average mutual information is: After obtaining the mutual information transfer function between all nodes, iterative mutual information transfer between the nodes in the system is required. Assume two users each use a degree distribution of... The rule-based LDPC code. Taking user 1's variable node as the starting point for iteration, its complete mutual information update order can be represented as: Based on the EXIT property of the variable node (Formula (3)), the variable node of user 1 outputs external information to its connected verification node during the r-th iteration. It is calculated from its input information. Assuming it comes from the rest... The prior mutual information of each verification node is The prior mutual information from the nodes is Then we have: Based on the EXIT property of the verification node (Formula (6)), the external information output by the verification node of user 1 to the variable node is... It is determined by its input information. Assuming it comes from the rest... The prior mutual information of each variable node is 1. Then we have: During the iteration process, the current input mutual information of the verification node is checked. That is, it is equal to the external information output by the variable node. The variable nodes derive their input mutual information from the check nodes. This is equal to the external information output by the verification node. Therefore, by combining equations (34) and (35), we can obtain the iterative relationship for the external information of the user 1 variable node: The mutual information function between the user variable node and the output node is used as the analysis target. According to formula (5), the expression of this mutual information in the r-th iteration is: Substituting formula (37) into formula (38), we obtain the iterative relationship that depends only on the information from the previous round: To eliminate intermediate variables in the iterative equation, we invert equation (38) to obtain: Substituting formula (40) and the mutual information transfer function of the nodes (formula (24)) into formula (39), we finally obtain the objective function expression for user 1: Similarly, the objective function expression for user 2 can be obtained: At this point, the complete mutual information iteration process between the two users is simplified to only relating to I. (1) and I (2) A set of coupled iterative equations for two target parameters.

[0028] In summary, this section proposes a derivation method for the maximum total rate based on regular LDPC codes in unequal-power two-user Gaussian multiple access channels. Typically, the rate of a regular LDPC code can be completely determined by its degree distribution, i.e., 1-d... v / d c Therefore, the total system rate for two-user rule LDPC codes is: in, These represent the rates of user 1 and user 2, respectively.

[0029] According to EXIT analysis theory, the necessary and sufficient condition for successful iterative decoding of the system is: as the number of iterations approaches infinity, the mutual information between the outputs of the two user variable nodes and the sum node converges to 1, that is: Therefore, under the premise of ensuring successful decoding of the system, the following constrained optimization problem can be established with the objective of maximizing the total rate R of the two users. The channel model and the unequal power condition between users are implicitly contained in the mutual information transfer functions F1 and F2: Step 2: Solving the objective function using fixed-point theory. Based on fixed-point theory, the optimization problem defined by formula (45) is analyzed. Given the transmit powers A1 and A2 of the two users and the standard deviation of the channel noise σ, the objective is to establish the user LDPC code distribution parameters through mathematical derivation. The theoretical relationship that must be satisfied between them transforms the optimization problem into a form that can be solved analytically or through efficient numerical search.

[0030] Specifically, let's analyze this using User 1 as an example. According to fixed-point theory, as the number of iterations r approaches infinity, its mutual information sequence will converge to a steady-state value: Among them, I *(1) Let F1 be a fixed point of the mutual information iterative function, corresponding to the convergence state in the factor graph decoding process, and directly determining the decoding performance of User 1. Specifically, if If I..., then user 1's decoding fails; if I... *(1) =1, then user 1 has successfully decoded. For a two-user Gaussian multiple access system, the necessary and sufficient condition for the overall system decoding success is that the iterative processes of both users converge to 1 simultaneously, that is, the following must be satisfied: If the above conditions are not met, the entire system decoding will fail.

[0031] Therefore, reconsidering formula (41), when the number of iterations r approaches infinity, we get: As can be seen from formula (48), the equation contains two unknown degree distribution parameters. In order to analytically solve for the objective function (i.e., the inverse function of the objective function), and thereby address the non-convexity of the original optimization problem, adopting the following strategy: fix the degree of the variable nodes. ,Will As the variable to be solved. By performing algebraic transformations on formula (48), we can obtain: Formula (49) contains the degree of the check node. By moving the term to one side of the equation and simplifying it, we can obtain the result. The parsing expression: This analytical expression is denoted as: This indicates that user 1 is verifying the node degree. The parsing expression function.

[0032] At this point, User 1's verification node degree... The mathematical expression for can be obtained. This expression shows that This can be represented as the system fixed point (I). *(1) ,I *(2) ), given the node degree of the variable and channel parameters . .

[0033] Similarly, the degree of the verification node for user 2 can be derived. Parsing expression function: Step 3: Decoding Reliable Region Partition and Maximum Total Rate Solution. This step utilizes set theory to address the problem caused by... The uncertainty in solving the problem arises from the divergence of the function when the input is 1. By systematically dividing the degree distribution range within the joint parameter space of power and degree distribution to enable successful decoding of the two-user LDPC code, and further selecting the optimal degree distribution, the maximum total rate of the two-user system is finally obtained.

[0034] For the derived degree d of the two user verification nodes c The analytical expressions (Equations (50) and (52)) explicitly depend on (i.e., are directly included in the formulas) the node degree distribution variable, and implicitly depend on (i.e., indirectly affect) the transmit power of the two users through the mutual information transfer function of the nodes. Given the root mean square σ of the channel noise, the power ratio of the two users will directly affect the convergence characteristics of the mutual information iteration process. Here, it is assumed that the sum of the power coefficients of the two users is constant. For a given variable node degree and power ratio If we need to obtain the degree of verification nodes that simultaneously satisfy the decoding success of two users, In theory, it is only necessary to Substitute into formulas (50) and (52) to solve. However, due to The functional characteristics, namely It is impossible to directly obtain a reliable check node degree. .

[0035] Therefore, we will conduct a reverse analysis starting from the case of decoding failure in a two-user system, and discuss the following three scenarios: 1. Both users fail to decode; 2. Only user 1 fails to decode; 3. Only user 2 fails to decode.

[0036] Case 1: Both users fail to decode. When both users fail to decode, the minimum convergence value of the objective function of their mutual information iteration satisfies... At this point, for a given variable node degree ,Will Substituting into equations (50) and (52), the degree of the corresponding check node can be obtained. The set of infeasible values ​​is denoted as: Therefore, for a given If a set is selected Any set of check node degrees Both will cause the LDPC code decoding of both users to fail.

[0037] Scenario 2: Decoding Failure Only for User 1. When user 1 fails to decode, regardless of whether user 2's decoding is successful, the entire system is considered to have failed. In this case, the system degenerates into a single-user scenario, and the received signal model can be simplified as follows: Correspondingly, the output mutual information of User 1 variable nodes Updated to the following format: in, This represents the mutual information of the AWGN channel output to user 1 during the (r-1)th iteration. According to... The transformation relationship, if we want to solve First, it is necessary to calculate the known transmitted bits. conditional variance under condition Given the channel noise variance Given user 1's transmit power A1, for the transmitted bits The channel log-likelihood ratio (LLR) is defined as: Substituting the probability density function of the Gaussian distribution, we get: Therefore, in sending bits Given the conditions, the conditional variance of the channel LLR is: This indicates taking the variance function.

[0038] Accordingly, formula (50) can be rewritten in a form that depends only on the parameters of user 1: This represents the degree of the check node when user 1 fails to decode itself. The function for parsing expressions. In Given the facts, for a given ,Will Substituting into formula (59), we can obtain the result. The unreliable set of values: Therefore, in a given In the case of verifying node degree Taken from set Any value in the range will result in user 1's [data being affected] regardless of whether user 2 can successfully decode the code. The code will fail to decode, thus causing the entire two-user system to fail to decode.

[0039] Case 3: Decoding failure only for User 2 Similarly, when decoding fails for User 2, formula (52) can be simplified to a form that only relates to the parameters of User 2: at this time, The unreliable set of values ​​is: Similarly, for a given If the node degree is verified Taken from set If so, user 2 will inevitably fail to decode, which will cause the entire system to fail to decode.

[0040] In summary, the complete set of infeasible parameters that caused the decoding failure of the two-user system was obtained: According to set theory, take The absolute complement of the entire parameter space yields the reliable parameter region that guarantees successful decoding for the two-user system, denoted as . .

[0041] Based on this, given the power ratio A1, A1 and variable node degree Under the condition of reliable area The degree of the verification node that maximizes the total system rate R is selected. That is, to solve: At this point, the maximum total rate under the corresponding current parameters is .

[0042] Next, iterate through all possible combinations of variable node degrees. The above optimization process is repeated for each set of parameters to obtain the corresponding values ​​for each combination. By comparing the rates R of each combination * The optimal degree distribution of the two-user LDPC codes under the current power ratio is selected: The corresponding local maximum total rate at this time is denoted as . .

[0043] Finally, by comparing the local optimum distributions under different power ratios A1 and A2 The corresponding total rate The optimal combination of degree distribution and power ratio is selected globally, thereby achieving joint optimization of power and degree distribution for two user systems with unequal power, and obtaining the maximum total system rate.

[0044] The technical advantages of the present invention will be verified through experiments below.

[0045] As shown in Figures 1-3, Figure 1 is a schematic diagram of the factor graph structure of a regular LDPC code under a two-user Gaussian access channel. In the figure, VNs represents variable nodes, CNs represents check nodes, and SNs represents sum nodes.

[0046] Figure 2 shows the power ratio when the SNR is 4 dB. At that time, at the degree of the variable node A schematic diagram of the reliable region of the verification node under the given conditions.

[0047] Figure 3 shows the power ratio when SNR = 4 dB. At that time, in degree distribution The EXIT chart below.

[0048] Figure 1 shows the factor graph structure of the LDPC code under two-user GMAC, illustrating the message passing mechanism of this invention in multi-user detection and decoding.

[0049] Figure 2 shows the reliable region of the verification node degree under a given signal-to-noise ratio and power ratio, illustrating that the system of the present invention has divided the feasible parameter space.

[0050] The EXIT plot in Figure 3 shows that the optimized degree distribution makes the EXIT trajectories of the two users non-overlapping, indicating that the decoding can converge and verifying the effectiveness of the method of the present invention.

[0051] Table 1 shows the power ratio at SNR = 4 dB. At that time, different variable node degrees Corresponding verification node degree distribution The study compared the changes in the total system rate under various parameters.

[0052] Table 2 shows the power ratio at SNR = 4 dB. At that time, different variable node degrees Corresponding verification node degree distribution And the changes in its total rate.

[0053] Table 3 shows the power ratio at SNR = 4 dB. At that time, different variable node degrees Corresponding verification node degree distribution And the changes in its total rate.

[0054] As shown in Tables 1-3, the degree distribution and corresponding total rate obtained through the optimization method of this invention demonstrate that, under the same signal-to-noise ratio, joint optimization of power and degree distribution improves the overall system rate. Therefore, this invention can effectively adapt to unequal power scenarios, enhancing system performance and robustness.

[0055] Table 1 Table 2 Table 3 As shown in Figure 5, this embodiment discloses a joint optimization system for power and degree distribution of two-user LDPC codes under unequal power conditions, used to execute the above method. It includes the following modules: a node constraint analysis and objective function construction module: used to analyze the constraint relationships between nodes of the two-user LDPC codes, and derive the mutual information transfer objective function of the two-user LDPC codes under unequal power conditions with the goal of maximizing the total rate, assuming successful decoding by both users; a fixed-point theory objective function solution module: based on fixed-point theory, the mutual information transfer objective function is solved to obtain the analytical expression of the check node degree; a reliable decoding region partitioning and maximum total rate solution module: based on the obtained analytical expression of the check node degree, the degree distribution value region that enables successful decoding of the two-user LDPC codes is partitioned within the joint parameter space of power and degree distribution, the optimal degree distribution is selected, and the maximum total rate of the two users is obtained.

[0056] Other aspects of this embodiment can be found in the above method embodiments.

[0057] In summary, this invention proposes a joint optimization method and system for power and degree distribution of two-user LDPC codes under unequal power conditions. By establishing an analytical relationship between power and degree distribution, it achieves collaborative design of the two within a unified framework, thereby significantly improving the overall system speed and robustness. Specifically, this invention combines external information transfer and fixed-point theory to derive an analytical expression for the degree distribution of the check node; then, it uses set theory methods to systematically search and optimize in the joint parameter space of power and degree distribution. Simulation results show that, under the condition of reasonable power allocation, the two-user LDPC code system designed using this method has a higher overall speed than the traditional equal-power optimization scheme, achieving efficient and reliable transmission under two-user GMAC under unequal power conditions.

[0058] 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 joint optimization method for power and degree distribution of two-user LDPC codes under unequal power conditions, characterized in that, The specific steps are as follows: Step 1: Node constraint analysis and objective function construction. Analyze the constraint relationships between nodes of the two-user LDPC codes. Under the premise that the two users can decode successfully, derive the mutual information transfer objective function of the two-user LDPC codes under unequal power with the goal of maximizing the total rate. Step 2: Solve the objective function analytical expression using fixed-point theory. Solve the mutual information transfer objective function of Step 1 using fixed-point theory to obtain the analytical expression of the check node degree. Step 3: Decoding reliable region division and maximum total rate solution. Based on the analytical expression of the check node degree obtained in Step 2, divide the degree distribution value region that enables the two-user LDPC codes to decode successfully in the joint parameter space of power and degree distribution. Select the optimal degree distribution and obtain the maximum total rate of the two users.

2. The method for joint optimization of power and degree distribution of LDPC codes for two users under unequal power conditions as described in claim 1, characterized in that, Step 1 is as follows: At the sending end, the two users each send a data bit sequence of length N. Send in Encoder, where, This represents the degree of the message node for user k. This represents the degree of the check node for user k; after encoding, it yields the codeword sequence. Where M is of length The codewords are then BPSK modulated to obtain each modulation symbol: in, The two signals are transmitted to the receiving end via a Gaussian multiple access channel according to a preset unequal power allocation scheme. At the receiving end, the two user signals are received synchronously. Assuming the transmit power amplitudes of the two users are A1 and A2 respectively, the received signal model via the Gaussian multiple access channel is represented as follows: Among them, y i For the i-th received signal, the channel noise n i It follows a mean of 0 and a variance of . The signal is Gaussian distributed; the receiver performs iterative decoding simulation on the factor graph based on the received signal; based on the iterative decoding process, the mutual information between nodes on the factor graph is tracked through the external information transfer EXIT function.

3. The method for joint optimization of power and degree distribution of LDPC codes for two users under unequal power conditions as described in claim 2, characterized in that, In step 1, under the conditions of infinite code length and Gaussian approximation, the EXIT function of each node is expressed as follows: 1) For a variable node with degree d v The variable nodes output mutual information as follows: in, These are the prior mutual information from other check nodes and sum nodes input to this variable node, respectively; I AV,j I represents the mutual information input to the j-th verification node. Es Represents the mutual information between the node inputs; J(•) is a function that maps the LLR variance to mutual information, expressed as: in, J represents the standard deviation of the input LLR. -1 (•) is the inverse function of J(•); when the output mutual information under consideration is the transmission from variable node to sum node, the expression simplifies to: 2) Verify the node for degree d c The verification node outputs the following mutual information: in, 3) In the communication between two users, the number of edges connected to the node is 3, corresponding to user 1, user 2, and the Gaussian access channel, respectively; During the message passing decoding process, the node performs joint estimation of the transmitted symbols based on the prior information of each user. Therefore, the mutual information output by the node to the two users is represented as follows: in, This represents the prior mutual information between the input of the k-th user and the node. Let LLR be the external log-likelihood value output by node to the k-th user; for user 1, the channel noise variance is... Given that the transmit powers of the two users are A1 and A2 respectively, the receiver targets the i-th received signal y. i The calculated value corresponds to the i-th bit of user 1. The output LLR information is: in, Let represent the i-th bit of user 2; based on Bayes' theorem, the above expression is transformed into the following form: In a given transmit bit Under the condition of receiving signal y i Follow the mean The variance is ,Right now: Substituting the conditional probability density function into equation (10), we obtain the log-likelihood ratio of the i-th bit for user 1 as: Because the prior log-likelihood ratio information of the user's two-way input and the node input is Therefore, there is Substituting into formula (11), we get: Divide both the numerator and denominator on the right side of formula (12) by... ,have to: Similarly, the i-th bit of user 2 can be derived. The output LLR is: Define the i-th bit of user k The expected value is ,Right now Let k = 1, 2; Suppose: User 1 sends a completely unique codeword, that is, for all i, we have User 2 sends binary codewords with equal probability, that is Take +1 or -1 with equal probability; For user 1: Substitute the log-likelihood ratio expression (13) of user 1 into its expected value. In the definition; let The expected value is simplified to Substituting the values, we get: Among them, equation (15) contains The expectation of the form, where t is a Gaussian random variable; if Where a and b are parameters, then: definition: Based on different scenarios, the mean and variance of t1 and t2 are derived separately, and substituted into the above formula for calculation; the total rate of the two user-defined LDPC codes is: According to EXIT analysis theory, the necessary and sufficient condition for successful iterative decoding is: as the number of iterations approaches infinity, the mutual information between the outputs of the two user variable nodes and the sum node converges to 1, i.e.: Therefore, under the premise of ensuring successful decoding, and with the goal of maximizing the total rate R of the two users, the following mutual information transfer objective function is established; The channel model and the unequal power conditions between users are implicitly contained in the mutual information transfer functions F1 and F2.

4. The method for joint optimization of power and degree distribution of two-user LDPC codes under unequal power conditions as described in claim 3, characterized in that, In step 2, the optimization problem defined by formula (45) is analyzed according to fixed-point theory; given the transmit powers A1 and A2 of the two users, and the standard deviation of the channel noise. Under these conditions, the goal is to establish the user LDPC code-degree distribution parameters through mathematical derivation. The theoretical relationship that must be satisfied between them transforms the optimization problem into a form that can be solved analytically or through efficient numerical search.

5. The method for joint optimization of power and degree distribution of LDPC codes for two users under unequal power conditions as described in claim 4, characterized in that, Step 2 is as follows: For user 1, according to fixed-point theory, as the number of iterations r approaches infinity, its mutual information sequence will converge to a steady-state value: Among them, I *(1) Let F1 be a fixed point of the mutual information iterative function F1. If I..., then user 1's decoding fails; if I... *(1) If the value is 1, then user 1 has successfully decoded. For a two-user Gaussian multiple access system, the necessary and sufficient condition for the overall system decoding success is that the iterative processes of both users converge to 1 simultaneously, i.e., satisfying: If the above conditions are not met, the entire system decoding will fail; therefore, when the number of iterations r approaches infinity, we get: Formula (48) contains two unknown degree distribution parameters. Degree of fixed variable nodes ,Will As the variable to be solved; by algebraically transforming formula (48), we obtain: Formula (49) contains the degree of the check node. Move the term to one side of the equation separately and simplify to obtain... The parsing expression: This analytical expression is denoted as: At this point, User 1's verification node degree... The mathematical expression can be obtained; this expression shows that, Represented as a fixed point (I) *(1) ,I *(2) ), given the node degree of the variable and channel parameters The explicit function; similarly, the degree of the verification node for user 2 is derived. Fixed-point dominant functions: 。 6. The method for joint optimization of power and degree distribution of LDPC codes for two users under unequal power conditions as described in claim 5, characterized in that, Step 3 is as follows: Using the ideas of mathematical set theory, process the... The uncertainty in solving the problem caused by the divergence of the function when the input is 1; by dividing the degree distribution value region in the joint parameter space of power and degree distribution so that the two-user LDPC code can be successfully decoded, the optimal degree distribution is selected, and finally the maximum total rate of the two-user system is obtained.

7. A joint optimization system for power and degree distribution of two-user LDPC codes under unequal power conditions, used to perform the method as described in any one of claims 1-6, characterized in that, The system includes the following modules: Node constraint analysis and objective function construction module: This module analyzes the constraint relationships between nodes of the two-user LDPC codes. Under the premise of successful decoding by both users, it derives the mutual information transfer objective function of the two-user LDPC codes under unequal power conditions, with the goal of maximizing the total rate. Fixed-point theory objective function solution module: Based on fixed-point theory, this module solves the mutual information transfer objective function to obtain the analytical expression of the check node degree. Decoding reliable region partitioning and maximum total rate solution module: Based on the obtained analytical expression of the check node degree, this module partitions the degree distribution value region within the joint parameter space of power and degree distribution, enabling successful decoding of the two-user LDPC codes. It then selects the optimal degree distribution and calculates the maximum total rate for both users.

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