Method and system for deriving maximum transmission rate of concatenated spread sequence of regular ldpc code under multi-user system

Abstract

This invention discloses a method and system for deriving the maximum transmission rate of a concatenated spread spectrum sequence of regular LDPC codes in a multi-user system. The method is as follows: Step 1: Based on the successful decoding of the concatenated spread spectrum sequence of regular LDPC codes in a multi-user system, construct an objective function with constraints to maximize the transmission rate of the system; Step 2: Based on fixed-point theory, given the degree d of the variable nodes in the LDPC code... v Derive the degree d of the verification node c Using the theoretical analytical expression for the spreading length m, we can solve for the inverse function of the objective function in the non-convex optimization problem; Step 3: Under the premise of successful decoding in the multi-user system, optimize the degree distribution and spreading length m of the LDPC code under the Gaussian multiple access channel. o This maximizes the system's transmission rate.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of wireless communication, and provides a technology for improving the transmission rate of a multi-user system, in particular, a method and system for deriving the maximum transmission rate of a regular LDPC code concatenated with a spread spectrum sequence in a multi-user system. BACKGROUND

[0002] LDPC codes have excellent error correction performance and low complexity, and can achieve high-speed data transmission, which makes them play a crucial role in 6G communication. Therefore, designing the degree distribution of LDPC codes is particularly important in 6G communication. However, the existing technology has a high complexity full-space search, which consumes a lot of time and computing power. Based on this, under the Gaussian multiple access channel, based on the fixed point analysis theory, the application derives the analytical expression of the degree distribution and the spread spectrum length of the LDPC code. SUMMARY

[0003] To solve the above problems existing in the prior art, the application discloses a method and system for deriving the maximum transmission rate of a regular LDPC code concatenated with a spread spectrum sequence in a multi-user system.

[0004] Compared with the traditional extrinsic information transfer analysis method, the application avoids the high complexity full-space search, thereby saving a lot of time and computing power. First, the application constructs a target function with constraints to maximize the transmission rate of the system under the premise of ensuring successful decoding of the multi-user system based on the regular LDPC code concatenated with the spread spectrum sequence. Then, according to the extrinsic information transfer analysis, combined with the fixed point analysis theory, given the total number of users K, the noise root mean square σ and the variable node degree d v , the theoretical analytical expression of the degree of the check node d c and the spread spectrum length m in the LDPC code is derived, and the inverse function of the objective function in the non-convex optimization problem is solved. Finally, according to the mathematical set theory, the uncertainty in the inverse problem is solved, and the optimal degree distribution and spread spectrum length m o of the LDPC code under the Gaussian multiple access channel are selected, so that the transmission rate r of a single user is maximized, i.e. the transmission rate of the multi-user system is optimized.

[0005] The purpose of the application is to derive the analytical expression of the degree distribution and the spread spectrum length of the LDPC code under the Gaussian multiple access channel based on the fixed point analysis method, so as to maximize the transmission rate of the multi-user system.

[0006] Application scenario of this invention: In multiple access communication, in the scenario of a communication system in which a regular LDPC code concatenated spread spectrum sequence is used to access the channel via Gaussian multiple access.

[0007] The specific technical solution of the present invention is as follows:

[0008] The derivation method for the maximum transmission rate of concatenated spread spectrum sequences of regular LDPC codes in a multi-user system is as follows:

[0009] Step 1: Under the premise of ensuring that the multi-user system based on the rule-based LDPC code concatenated spread spectrum sequence can be successfully decoded, construct an objective function with constraints to maximize the transmission rate of the system.

[0010] Step 2: Based on fixed-point theory, analyze the degree d of the variable nodes in the given LDPC code. v Derive the degree d of the verification node c Using the theoretical analytical expression for the spreading length m, the inverse function of the objective function in the non-convex optimization problem is solved.

[0011] Step 3: Optimize the degree distribution of LDPC codes under Gaussian multiple access channels, assuming successful decoding in the multi-user system. and spread spectrum length m o This maximizes the system's transmission rate.

[0012] Preferably, step 1 is as follows: At the sending end, firstly, user k sends an information bit vector of length N. Send to (d v ,d c )-LDPC encoder, where d v It is the degree of the variable node, d c This is the degree of the check node. Its output encoded vector is... Then, a spreading process with a spreading length of m is performed. Next, a vector is generated through an interleaver. Send to the Gaussian Multiple Access Channel.

[0013] Because each user uses the same LDPC code and the same spreading sequence length, each user has the same transmission rate, which is:

[0014]

[0015] At this point, the problem of maximizing the system's transmission rate can be equated to the problem of maximizing the transmission rate of a single user.

[0016] At the receiving end, The received signal y j Represented as:

[0017]

[0018] where z j is a Gaussian noise with mean 0 and variance σ 2 , and K is the total number of users in the system.

[0019] The message passing decoding is performed on the factor graph of the system and is accomplished by effective local decoding at each node. First, a single decoding iteration of the system starts with the local decoding at the and node. Take the kth user as an example, based on the received signal and the prior information from other k-1 users, the user k performs Elementary Signal Estimation (ESE) at the and node. Based on the estimation, the user k performs single user decoding on its own factor graph. The single user decoding includes local decoding of variable node→check node→variable node. Among them, the variable node performs local decoding similar to the decoding of repetition code; the check node performs local decoding similar to the decoding of single parity check code. Finally, the output of the variable node will be fed back to the and node as the prior message of the next iteration of other k-1 users. After a large number of iterations, a decision is made at the variable node to recover the transmitted vector u (k) .

[0020] The present application uses Extrinsic Information Transfer (EXIT) function to describe the local decoding process of each node. Generally, the input and output messages of decoding are represented by Log-likelihood Ratio (LLR). The EXIT function describes the relationship between the mean of input LLR and output LLR. At the same time, the EXIT theory is based on the assumption of infinite code length and Gaussian approximation, in which the input and output LLR are both assumed to be Gaussian variables (the variance of each variable is twice its mean).

[0021] Based on the above conditions, when the degree of the variable node is d, the output mutual information at the variable node is:

[0022]

[0023] where 0≤I A,i ≤1 represents the input mutual information from the check node i to the variable node j, J -1 (*) is the inverse function of J function:

[0024]

[0025] Here, σ A represents the variance of the input information. When the degree of the check node is d, the output mutual information at the check node is:

[0026] Tc (I A,1 ,…,I A,d-1 )=1-T v (1-I A,1 ,…,1-I A,d-1 ) (5)

[0027] In the local decoding process of the and node, let denote the a priori log-likelihood ratio (LLR) of based on the maximum a posteriori (MAP) criterion, the output LLR of

[0028]

[0029] The expectation of the output LLR of the kth user jth bit

[0030]

[0031] When the iteration number is l, the output mutual information of the and node S to the kth user variable node V is:

[0032]

[0033] Based on equation (3), the output mutual information of each user variable node V to the check node C is:

[0034]

[0035] Based on equation (5), the output mutual information of the check node C to the variable node V is:

[0036]

[0037] The final output mutual information of the user variable node V to the and node S can be obtained as:

[0038]

[0039] Substituting equations (8) and (10) into equation (11), the final output mutual information can be expressed as:

[0040]

[0041] According to the relevant knowledge of information theory, it can be obtained that the sufficient and necessary condition for determining the decoding success is:

[0042]

[0043] ​​In summary, to ensure successful decoding of a multi-user system based on concatenated spread spectrum sequences of regular LDPC codes, a constrained objective function is constructed:

[0044]

[0045] This maximizes the transmission rate for a single user. In other words, the system's transmission rate is at its maximum at this point.

[0046] Preferably, in step 2, based on fixed-point theory, the degree d of the variable node in the given LDPC code is determined. v Derive the degree d of the verification node c By using the theoretical analytical expression for the spreading length m, the inverse function of the objective function in the non-convex optimization problem can be obtained.

[0047] Assuming the root mean square noise σ and the total number of users K are known, the degree distribution of the LDPC code (d v ,d c And the spreading length m, according to EXIT analysis, we have:

[0048]

[0049]

[0050]

[0051] in, When the number of iterations approaches infinity The convergence point. It is a fixed-point equation I = f(I, I vc The smallest binary fixed point in (). Simultaneously, the smallest fixed point The convergence point of iterative decoding also determines the error rate of the user's decoding:

[0052]

[0053] in, It is a complementary error function. From formula (17), it can be seen that when I... * =1, Pe→0. That is, when I * Decoding fails when the value falls within the range [0,1). * When the value is 1, the decoding is successful.

[0054] When considering local decoding at variable nodes, the mutual information between variable nodes and check nodes is selected. As an auxiliary variable, it is related to the final mutual information I. l By combining equations (9) and (11) and reconsidering them, we can obtain:

[0055]

[0056] Substitute formula (8) and (10) into formula (19), we have:

[0057]

[0058] where,

[0059]

[0060] If only given the noise root mean square σ and the total number of users K, it is found that the equation group (21) has two equations, but contains three unknown parameters d c ,d v ,m. From the mathematical point of view, the closed-form expression of d c ,d v ,m cannot be solved. Therefore, considering the known degree d v of the variable node, the closed-form expression of the spread spectrum length m and the degree d c of the check node is theoretically solved.

[0061] When l→∞, reconsider formula (19), based on formula (3), take J -1 on both sides of formula (19) and square, we get:

[0062]

[0063] In order to eliminate , formula (22) is transformed, we have:

[0064]

[0065] Subtract the upper and lower equations of equation group (23), we get:

[0066]

[0067] Further simplification, we get:

[0068]

[0069] Finally, the theoretical expression of the spread spectrum length m is obtained:

[0070]

[0071] In order to continue to solve the expression of the degree d c of the check node, substitute the obtained m and into the upper equation of equation group (22):

[0072]

[0073] Based on formula (5), we get:

[0074]

[0075] Based on equation (3), we have

[0076]

[0077] Simplifying, we have

[0078]

[0079] Dividing both sides by d v and square root:

[0080]

[0081] Taking the J function of both sides:

[0082]

[0083] Simplifying, we have

[0084]

[0085] Finally, dividing by and square root, we have the theoretical expression of the check node degree d c :

[0086]

[0087] In summary, given the total number of users K, the root mean square of noise σ and the variable node degree d v , based on equation (8), combined with equations (26) and (34), we have the mathematical expression of the spreading length m and the check node degree d c , i.e. expressing m and d c as the explicit function of the fixed point .

[0088]

[0089] Preferably, step 3, according to mathematical set theory, solves the uncertainty in the inverse problem. Under the premise of successful decoding in a multi-user system, the degree distribution and the spreading length m o of the LDPC code under the Gaussian multiple access channel are optimized to maximize the transmission rate of the system.

[0090] Given the total number of users K, the root mean square of channel noise σ and the degree of variable node d v , considering equation (35), substituting I * and , the degree of check node d cAnd the spreading length m. It is well known that when mutual information I = 1, the bit error rate Pe → 0. Therefore, firstly, let I... * =1 and By substituting the values, we can obtain a reliable solution for successful decoding of multiple user systems. and m re Then, iterate through d. v Find the corresponding results respectively. and m re Finally, from d v Matching and m re The optimal LDPC code-degree distribution under Gaussian multiple access channel was selected. and spread spectrum length m o This maximizes the transmission rate for individual users, meaning the system's transmission rate reaches its maximum. However, because J... -1 The special properties of the () function, J -1 (I=1)=∞, so it is impossible to solve for the reliable decoding under successful multi-user system decoding. and m re .

[0091] Therefore, it is worth considering the opposite of successful decoding in a multi-user system, and first identifying all unreliable methods when the system fails to decode. and m un Therefore, given the degree d of the variable node... v , put I * ∈[0,1) and Substituting into the system of equations (35), we find an unreliable region:

[0092]

[0093] In this situation, the multi-user system decoding fails.

[0094] Then, based on mathematical set theory, by taking the absolute complement of UR, we obtain the reliable region for successful decoding in a multi-user system:

[0095]

[0096] At this point, the multi-user system successfully decoded the code.

[0097] Next, from the reliable region RR, find the region that corresponds to the given d. v corresponding and m * :

[0098]

[0099] This enables the transmission rate of a single user maximum.

[0100] Finally, iterate through d. v Select the optimal LDPC code-degree distribution and spread spectrum length m o :

[0101]

[0102] Make Maximum, meaning the system transmission rate reaches its maximum.

[0103] This invention also discloses a derivation system for the maximum transmission rate of a regular LDPC code concatenated spread spectrum sequence in a multi-user system, based on the above method, and including the following modules:

[0104] Objective function construction module: Based on the successful decoding of a multi-user system with concatenated spread spectrum sequences of regular LDPC codes, construct an objective function with constraints to maximize the transmission rate of the system;

[0105] Inverse function solution module: Based on fixed-point theory, given the degree d of the variable node in the LDPC code... v Derive the degree d of the verification node c Using the theoretical analytical expression for the spreading length m, the inverse function of the objective function in the non-convex optimization problem is solved.

[0106] Transmission rate maximization module: Optimizes the degree distribution of LDPC codes under Gaussian multiple access channels, provided that decoding is successful in a multi-user system. and spread spectrum length m o This maximizes the system's transmission rate.

[0107] This invention combines external information transfer analysis and fixed-point analysis to derive the optimal degree distribution and spreading length of LDPC codes under Gaussian multiple access channels step by step. First, under the premise of ensuring successful decoding of the multi-user system, a constrained objective function is constructed to maximize the system's transmission rate. Then, based on external information transfer analysis and fixed-point analysis theory, given the total number of users K, the root mean square noise σ, and the variable node degree d... v Derive the degree d of the check node in the LDPC code. c By using the theoretical analytical expression for the spreading length *m*, the inverse function of the objective function in the non-convex optimization problem is obtained. Finally, based on set theory, the uncertainty in the inverse problem is resolved, and the optimal LDPC code-degree distribution is selected. and spread spectrum length m o This maximizes the transmission rate r of a single user, thus optimizing the transmission rate of a multi-user system. Attached Figure Description

[0108] Figure 1A factor graph of a regular LDPC code. The letter V represents a variable node, the letter C represents a check node, and the letter S represents a sum node.

[0109] Figure 2 SNR = 0 dB, i.e. σ = 3.1623, the total number of users K = 10, the degree of variable node d v = 3, and the matched and m re and m . *

[0110] Figure 3 A flow chart of a method for deriving the maximum transmission rate of a concatenated spreading sequence of a regular LDPC code in a multi-user system according to a preferred embodiment of the present application.

[0111] Figure 4 A block diagram of a system for deriving the maximum transmission rate of a concatenated spreading sequence of a regular LDPC code in a multi-user system according to a preferred embodiment of the present application. DETAILED DESCRIPTION

[0112] The preferred embodiment of the present application will be described in detail below with reference to the accompanying drawings.

[0113] As shown in Figures 1-3 , the present embodiment discloses a method for deriving the maximum transmission rate of a concatenated spreading sequence of a regular LDPC code in a multi-user system, and the specific steps are as follows:

[0114] Step 1, under the premise that the multi-user system based on the concatenated spreading sequence of the regular LDPC code can be successfully decoded, a target function with a constraint condition is constructed to maximize the transmission rate of the system.

[0115] At the sending end, first, user k sends a length-N information bit vector to a (d v , d c )-LDPC encoder, where d v is the degree of the variable node, and d c is the degree of the check node. The output of the encoder is a coded vector Subsequently, a spreading process with a spreading length of m is performed. Then, a vector is generated by an interleaver and sent to a Gaussian multiple access channel.

[0116] Because each user adopts the same regular LDPC code and the same length of spreading sequence, the transmission rate of each user is the same, i.e.:

[0117]

[0118] ​At this time, the system transmission rate maximization problem can be equivalent to a single user transmission rate maximization problem.

[0119] At the receiving end, where the received signal y j is expressed as:

[0120]

[0121] where z j is a Gaussian noise with mean 0 and variance σ 2 , and K is the total number of users of the system.

[0122] Message passing decoding is performed on the factor graph of the system, and is completed by effective local decoding at each node. First, a single decoding iteration of the system starts from the local decoding at the node. Taking the kth user as an example, based on the received signal and the prior information from the other k-1 users, the user k performs elementary signal estimation (ESE) at the node. Based on the estimation, the user k performs single user decoding on its own factor graph. Single user decoding includes local decoding of variable node→check node→variable node. Among them, the variable node performs local decoding, which is similar to the decoding of repetition code; the check node performs local decoding, which is similar to the decoding of single parity check code. Finally, the output of the variable node will be fed back to the node as the prior message of the next iteration of the other k-1 users. After a large number of iterations, a decision is made at the variable node to recover the transmitted vector u (k) .

[0123] The application uses extrinsic information transfer (EXIT) function to describe the local decoding process of each node. Generally, the input and output messages of decoding are represented by log-likelihood ratio (LLR). The EXIT function describes the relationship between the mean of the input LLR and the output LLR. At the same time, the EXIT theory is based on the assumption of infinite code length and Gaussian approximation, in which the input and output LLR are both assumed to be Gaussian variables (the variance of each variable is twice its mean).

[0124] Based on the above conditions, when the degree of the variable node is d, the output mutual information at the variable node is:

[0125]

[0126] where 0≤I A,i ≤1 represents the input mutual information from the check node i to the variable node j, and J -1 (*) is the inverse function of the J function:

[0127]

[0128] Here, σ A denotes the variance of the input information. When the degree of the check node is d, the output mutual information at the check node is:

[0129] T c (I A,1 ,…,I A,d-1 )=1-T v (1-I A,1 ,…,1-I A,d-1 ) (44)

[0130] In the local decoding process of the sum node, let denote the a priori log-likelihood ratio about based on the maximum a posteriori probability criterion, the output log-likelihood ratio of

[0131]

[0132] The expectation of the output log-likelihood ratio of the kth user jth bit at the (l+1)th iteration can be obtained as:

[0133]

[0134] When the iteration number is l, the output mutual information from the sum node S to the kth user variable node V is:

[0135]

[0136] Based on formula (3), the output mutual information from each user variable node V to the check node C is:

[0137]

[0138] Based on formula (5), the output mutual information from the check node C to the variable node V is:

[0139]

[0140] The final output mutual information from the user variable node V to the sum node S can be obtained as:

[0141]

[0142] By substituting formula (8) and (10) into formula (11), the final output mutual information can be expressed as:

[0143] ​

[0144] Based on relevant knowledge of information theory, the necessary and sufficient condition for successful decoding is:

[0145]

[0146] In summary, to ensure successful decoding of a multi-user system based on concatenated spread spectrum sequences of regular LDPC codes, a constrained objective function is constructed:

[0147]

[0148] This maximizes the transmission rate for a single user. In other words, the system's transmission rate is at its maximum at this point.

[0149] Step 2: Based on fixed-point theory, analyze the degree d of the variable nodes in the given LDPC code. v Derive the degree d of the verification node c By using the theoretical analytical expression for the spreading length m, the inverse function of the objective function in the non-convex optimization problem can be obtained.

[0150] Assuming the root mean square noise σ and the total number of users K are known, the degree distribution of the LDPC code (d v ,d c And the spreading length m, according to EXIT analysis, we have:

[0151]

[0152]

[0153]

[0154] in, These are the cases when the number of iterations approaches infinity. The convergence point. It is a fixed-point equation I = f(I, I vc The smallest binary fixed point in (). Simultaneously, the smallest fixed point The convergence point of iterative decoding also determines the error rate of the user's decoding:

[0155]

[0156] in, It is a complementary error function. From formula (17), it can be seen that when I... * =1, Pe→0. In other words, when I * Decoding fails when the value falls within the range [0,1). * When the value is 1, the decoding is successful.

[0157] When considering the local decoding at the variable nodes, the mutual information between the variable node and the check node is selected As an auxiliary variable, the final mutual information I l By solving (9) and (11) simultaneously, we have

[0158]

[0159] Substituting (8) and (10) into (19), we have

[0160]

[0161] where

[0162]

[0163] If only the noise root mean square σ and the total number of users K are given, it is found that the equation group (21) has two equations, but contains three unknown parameters d c ,d v ,m. From the mathematical point of view, the closed-form expression of d c ,d v ,m cannot be solved. Therefore, considering the known degree d v of the variable node, the closed-form expression of the spread spectrum length m and the degree d c of the check node is theoretically solved.

[0164] When l→∞, reconsidering formula (19), based on formula (3), first take J -1 on both sides of formula (19) and square, we have

[0165]

[0166] In order to eliminate , formula (22) is transformed, and we have

[0167]

[0168] Subtracting the upper and lower equations of the equation group (23), we have

[0169]

[0170] Further simplification gives

[0171]

[0172] Finally, the theoretical expression of the spread spectrum length m is obtained:

[0173]

[0174] To continue solving for the degree d of the verification node c The expression will yield m and Substituting into the above equation of system of equations (22):

[0175]

[0176] Based on formula (5), we get:

[0177]

[0178] Based on formula (3), we get:

[0179]

[0180] Simplifying, we get:

[0181]

[0182] Divide both sides by d v And take the square root:

[0183]

[0184] Take the J function from both sides simultaneously:

[0185]

[0186] Simplifying, we get:

[0187]

[0188] Finally, divide by Take the square root to obtain the degree d of the check node. c Theoretical expression:

[0189]

[0190] In summary, given the total number of users K, the root mean square noise σ, and the variable node degree d... v Based on formula (8), combined with formulas (26) and (34), the spreading length m and the check node degree d are obtained. c The mathematical analytical expression, that is, combining m and d c Represented as a fixed point . .

[0191]

[0192] Step 3: Based on set theory, resolve the uncertainties in the inverse problem. Under the premise of successful decoding in a multi-user system, optimize the degree distribution of the LDPC code under a Gaussian multiple access channel. and spread spectrum length m oThis maximizes the system's transmission rate.

[0193] Given the total number of users K, the root mean square of channel noise σ, and the degree d of the variable nodes. v Consider formula (35), and put I * and Substituting these values, the degree d of the verification node can theoretically be calculated. c And the spreading length m. It is well known that when mutual information I = 1, the bit error rate Pe → 0. Therefore, firstly, let I... * =1 and By substituting the values, we can obtain a reliable solution for successful decoding of multiple user systems. and m re Then, iterate through d. v Find the corresponding and m re Finally, from d v Matching and m re The optimal LDPC code-degree distribution under Gaussian multiple access channel was selected. and spread spectrum length m o This maximizes the transmission rate for individual users, meaning the system's transmission rate reaches its maximum. However, because J... -1 The special properties of the () function, J -1 (I=1)=∞, so it is impossible to solve for the reliable decoding under successful multi-user system decoding. and m re .

[0194] Therefore, it is worth considering the opposite of successful decoding in a multi-user system, and first identifying all unreliable methods when the system fails to decode. and m un Therefore, given the degree d of the variable node... v , put I * ∈[0,1) and Substituting into the system of equations (35), we find an unreliable region:

[0195]

[0196] In this situation, the multi-user system decoding fails.

[0197] Then, based on mathematical set theory, by taking the absolute complement of UR, we obtain the reliable region for successful decoding in a multi-user system:

[0198]

[0199] At this point, the multi-user system successfully decoded the code.

[0200] Next, from the reliable region RR, find the region that corresponds to the given d.v corresponding and m * :

[0201]

[0202] such that the transmission rate of a single user is maximized.

[0203] Finally, the optimal LDPC code degree distribution v and the spreading length m o are selected by traversing d :

[0204]

[0205] such that is maximized, i.e., the system transmission rate is maximized.

[0206] Table 1 is the SNR = 0 dB, that is, σ = 3.1623, the total number of users K = 10, traversing d v , the corresponding and m re are obtained by formula (35), and the optimal LDPC code degree distribution and the spreading length or such that the transmission rate of a single user is maximized, i.e., the system transmission rate

[0207] Table 1

[0208]

[0209] As shown in Figure 4 , the embodiment discloses a derivation system of the maximum transmission rate of a regular LDPC code concatenated spread spectrum sequence in a multi-user system, and based on the above method embodiment, includes the following modules:

[0210] Objective function construction module: based on the successful decoding of the multi-user system of the regular LDPC code concatenated spread spectrum sequence, an objective function with a constraint condition is constructed, so that the transmission rate of the system is maximized;

[0211] Inverse function solving module: according to the fixed point theory analysis, given the degree d v of the variable node in the LDPC code, the theoretical analysis formula of the degree d c of the check node and the spreading length m is derived, and the inverse function of the objective function in the non-convex optimization problem is solved;

[0212] Transmission rate maximization module: under the premise of the successful decoding of the multi-user system, the degree distribution of the LDPC code in the Gaussian multiple access channel is optimized.and spreading length m o so that the transmission rate of the system is maximized.

[0213] Other aspects of the present embodiment can refer to the above method embodiments.

[0214] The above is only a preferred embodiment of the present application, and does not limit the present application in any way. Any simple modification, change, and equivalent structural change made according to the technical essence of the present application to the above embodiments are still within the protection scope of the present technical solution.

Claims

1. A method for deriving the maximum transmission rate of concatenated spread spectrum sequences of regular LDPC codes in a multi-user system, characterized by: Follow these steps: Step 1: Successfully decode the multi-user system based on the concatenated spread spectrum sequence of the rule LDPC code, and construct an objective function with constraints to maximize the transmission rate of the system. Step 2: Based on fixed-point theory, analyze the degree d of the variable nodes in the given LDPC code. v Derive the degree d of the verification node c Using the theoretical analytical expression for the spreading length m, the inverse function of the objective function in the non-convex optimization problem is solved. Step 3: Optimize the degree distribution of LDPC codes under Gaussian multiple access channels, assuming successful decoding in the multi-user system. and spread spectrum length m o This maximizes the system's transmission rate.

2. The derivation method as described in claim 1, characterized in that, Step 1 is as follows: At the sending end, user k sends an information bit vector of length N. Send to (d v ,d c )-LDPC encoder, where d v It is the degree of the variable node, d c It is the degree of the check node; the output encoded vector is Then, a spreading process with a spreading length of m is performed; subsequently, a vector is generated through an interleaver. Send to the Gaussian multiple access channel; Since each user uses the same LDPC code and the same spreading sequence length, the transmission rate is the same for each user, i.e.: At the receiving end, Among them, the received signal y j Represented as: Among them, z j The mean is 0 and the variance is σ. 2 The noise is Gaussian, and K is the total number of users in the system; When the degree of the variable node is d, the output mutual information at the variable node is: Where, 0≤I A,i ≤1 represents the input mutual information from check node i to variable node j, J -1 (*) is the inverse function of the J function: Where, σ A The variance of the input information is represented by d; when the degree of the check node is d, the output mutual information at the check node is: T c (I A,1 ,…,I A,d-1 )=1-T v (1-I A,1 ,…,1-I A,d-1 ) (5) During the local decoding process of a node, let... Indicates about The prior log-likelihood ratio, based on the maximum a posteriori probability criterion. The output log-likelihood ratio is expressed as: Get the j-th bit of the k-th user Lower output log-likelihood ratio Expectations: When the iteration number is l, the output mutual information from node S to the k-th user variable node V is: Based on formula (3), the output mutual information from each user variable node V to the verification node C is: Based on formula (5), the output mutual information from check node C to variable node V is: The final output mutual information from user variable node V to node S is obtained as follows: Substituting equations (8) and (10) into equation (11), the final output mutual information is expressed as: The necessary and sufficient condition for a successful decoding is: In summary, to ensure successful decoding of a multi-user system based on concatenated spread spectrum sequences of regular LDPC codes, a constrained objective function is constructed:

3. The derivation method as described in claim 2, characterized in that, Step 2 is as follows: Assuming the root mean square noise σ and the total number of users K are known, the degree distribution of the LDPC code (d v ,d c And the spreading length m, according to EXIT analysis, we have: in, When the number of iterations approaches infinity The convergence point; It is a fixed-point equation I = f(I, I vc The smallest binary fixed point in ); at the same time, the smallest fixed point The convergence point of iterative decoding also determines the error rate of the user's decoding: in, It is a complementary error function; from formula (17), we know that when I * =1, Pe→0; in other words, when I * Decoding fails when the value falls within the range [0,1); * When = 1, decoding is successful; When considering local decoding at variable nodes, the mutual information between variable nodes and check nodes is selected. As an auxiliary variable, it is related to the final mutual information I. l Combining equations (9) and (11), we get: Substituting equations (8) and (10) into equation (19), we get: in, When l→∞, reconsider formula (19). Based on formula (3), first take J on both sides of formula (19). -1 And by taking the square root, we get: In order to eliminate Transforming formula (22), we get: Subtracting the two equations from the system of equations (23), we get: Further simplification yields: Finally, the theoretical expression for the spreading length m is obtained: The obtained m and Substituting into the above equation of system of equations (22): Based on formula (5), we get: Based on formula (3), we get: Simplifying, we get: Divide both sides by d v And take the square root: Take the J function from both sides simultaneously: Simplifying, we get: Finally, divide by Take the square root to obtain the degree d of the check node. c Theoretical expression: In summary, given the total number of users K, the root mean square noise σ, and the variable node degree d... v Based on formula (8), combined with formulas (26) and (34), the spreading length m and the check node degree d are obtained. c The mathematical analytical expression, that is, combining m and d c Represented as a fixed point The explicit function; 4. The derivation method as described in claim 3, characterized in that, Step 3 is as follows: Given the degree d of a node v , put I * ∈[0,1) and Substituting into the system of equations (35), we find an unreliable region: In this situation, the multi-user system decoding fails; Then, based on mathematical set theory, by taking the absolute complement of UR, we obtain the reliable region for successful decoding in a multi-user system: At this point, the multi-user system successfully decoded the code; Next, from the reliable region RR, find the region that corresponds to the given d. v corresponding and m * : This enables the transmission rate of a single user maximum; Finally, iterate through d. v Select the optimal LDPC code-degree distribution and spread spectrum length m o : Make Maximum, meaning the system transmission rate reaches its maximum.

5. A system for deriving the maximum transmission rate of concatenated spread spectrum sequences of regular LDPC codes in a multi-user system, based on the method described in any one of claims 1-4, characterized in that: Includes the following modules: Objective function construction module: Based on the successful decoding of a multi-user system with concatenated spread spectrum sequences of regular LDPC codes, construct an objective function with constraints to maximize the transmission rate of the system; Inverse function solution module: Based on fixed-point theory, given the degree d of the variable node in the LDPC code... v Derive the degree d of the verification node c Using the theoretical analytical expression for the spreading length m, the inverse function of the objective function in the non-convex optimization problem is solved. Transmission rate maximization module: Optimizes the degree distribution of LDPC codes under Gaussian multiple access channels, provided that decoding is successful in a multi-user system. and spread spectrum length m o This maximizes the system's transmission rate.

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