Rule-based ra code optimal code rate derivation method and system based on awgn channel

Abstract

The present application belongs to the technical field of communication system, and particularly relates to a method and system for deriving optimal code rate of regular RA code based on AWGN channel. The method comprises the following steps: S1, deriving output mutual information I of regular RA code under AWGN channel; S2, based on the output mutual information I and fixed point theory, deriving a theoretical analytic expression of the repetition number q of the RA code, and solving the inverse function of the objective function in the non-convex optimization problem; S3, based on mathematical set theory, solving the uncertainty in the inverse function problem, and obtaining the optimal repetition number q of successful decoding of regular RA code under AWGN channel o The present application has the characteristics of being capable of improving channel coding performance and efficiency in a communication system, and effectively solving the closed expression of the optimal coding parameter.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of communication systems, and particularly relates to a regular RA code optimal code rate derivation method and system based on an AWGN channel. BACKGROUND

[0002] At present, 5G mobile communication is popular, and a communication system is required to resist noise and successfully decode. Meanwhile, the requirement for maximizing transmission rate is also increasing. In point-to-point channel coding communication, in order to maximize the transmission rate, that is, the code rate, under the condition of a given signal-to-noise ratio, for each coding parameter, mutual information transfer analysis needs to track the final output mutual information, and then the optimal coding parameter is selected. The complexity of this optimization method increases with the increase of the coding parameter, thereby reducing the performance and efficiency of channel coding.

[0003] Therefore, it is very important to design a regular RA (Repeat Accumulate) code optimal code rate derivation method and system based on an AWGN (Additive White Gaussian Noise) channel, which can improve the performance and efficiency of channel coding in a communication system, and effectively solve the closed-form expression of the optimal coding parameter.

[0004] For example, the distributed coding and decoding method of the RA code described in the Chinese patent document with the application number CN201210011026.7, the implementation steps are: (1) the data information of the source node is encoded by the RA code, and then the encoded sequence is sent to the relay node and the destination node; (2) the relay node relays the encoding of the received encoded sequence, and sends the encoded information to the destination node; (3) first construct a joint RA coding line graph according to the RA code encoding and relay encoding, and then construct a multi-layer RA code bipartite graph based on the joint RA coding line graph; (4) the destination node iteratively decodes the received relay node encoding information and the RA encoding information sent by the source node according to the bipartite graph of the multi-layer RA code, and restores the source data information. Although the encoding is simple, the network throughput and the forwarding efficiency of the relay node can be improved, and the performance of the destination node can be improved, which can be used in the distributed transmission system in the relay network transmission, but the disadvantage is that the complexity of the above optimization method still increases with the increase of the coding parameter, and the above method cannot be used to solve the closed-form expression of the optimal coding parameter, thereby the performance and efficiency of channel coding cannot be improved. SUMMARY

[0005] The present application provides a regular RA code optimal code rate derivation method and system based on an AWGN channel, which can improve the channel coding performance and efficiency in a communication system and effectively solve the closed-form expression of the optimal coding parameter.

[0006] To achieve the above-mentioned application purposes, the present application adopts the following technical solutions:

[0007] The regular RA code optimal code rate derivation method based on an AWGN channel comprises the following steps:

[0008] S1, deriving the output mutual information I of the regular RA code under the AWGN channel;

[0009] S2, deriving the theoretical analytic expression of the repetition number q of the RA code based on the output mutual information I and the fixed point theory, and solving the inverse function of the objective function in the non-convex optimization problem;

[0010] S3, solving the uncertainty in the inverse function problem based on the mathematical set theory, and obtaining the optimal repetition number q of the regular RA code successful decoding under the AWGN channel o .

[0011] Preferably, the step S1 comprises the following steps:

[0012] S11, for the variable node with the degree d, the extrinsic information transfer function T v The output mutual information of the variable node is described as follows:

[0013]

[0014] Wherein, 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 the J function;

[0015]

[0016] Wherein, 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 the J function;

[0017] For the check node with the degree d, the extrinsic information transfer function T c The output mutual information of the check node is described as follows:

[0018] T c (I A,1 ,…,I A,d-1 )=1-T v (1-I A,1 ,…,1-I A,d-1 ) (3)

[0019] If I A,i =I A , i = 1, … w, then simplify formula (1) and (3) to T v (I A ×w,I A,w+1 …,I A,d-1 ) and T c (I A ×w,I A,w+1 ,…I A,d-1 );

[0020] S12, from the channel node S to the variable node V, for the lth iteration, the channel noise root mean square is σ, and the output mutual information at the channel node S is:

[0021]

[0022] From the variable node V to the check node C, according to the formula (1) of the output mutual information at the variable node, the output mutual information at the variable node V is obtained as:

[0023]

[0024] Wherein, represents the output mutual information from the check node C to the variable node V at the (l-1)th iteration;

[0025] From the check node C to the variable node U, according to the formula (3) of the output mutual information at the check node, the output mutual information at the check node C is obtained as:

[0026]

[0027] According to formula (1) and (3), the output mutual information at the variable node U and the check node C is obtained as:

[0028]

[0029]

[0030] S13, finally obtain the output mutual information from the variable node V to the channel node S as:

[0031]

[0032] where I l-1 denotes the final output mutual information from variable node V to channel node S at the l-th iteration.

[0033] As a preference, step S2 comprises the following steps:

[0034] S21, for the RA code with code rate 1 / q, when the iteration number l tends to infinity, it is obtained that

[0035]

[0036] where I * is the minimum element in the solution set of the equation, and is the minimum fixed point; the minimum fixed point I * corresponds to the output mutual information I on each edge of the factor graph;

[0037] For the output mutual information I on each edge of the factor graph, the corresponding error probability formula is:

[0038]

[0039] where, is the complementary error function; according to the factor graph, the average decoding error probability of the RA code is P e (I * ), and the average decoding error probability of the information bit is P e (I * uc ); P e (I * uc )→0 if and only if P e (I * )→0.

[0040] As a preference, step S2 further comprises the following steps:

[0041] S22, reconsidering formula (9), when the iteration number l tends to infinity, it is obtained that

[0042]

[0043] Simplifying it obtains:

[0044]

[0045] Considering that

[0046]

[0047] it is obtained that

[0048]

[0049] S23, substituting equation (15) into equation (8), we get

[0050] S24, combining equation (13) and equation (16), we get

[0051] Simplify equation (17), we get

[0052]

[0053] Finally, we get the expression of q as

[0054]

[0055] S25, reconsider the output mutual information I cv Substituting equation (13) into equation (5), we get

[0056]

[0057] S26, substituting equation (20) into equation (19), we get the closed expression of q as

[0058]

[0059] As a preferred, step S3 includes the following steps:

[0060] S31, find all the repetition number under the condition of decoding failure, i.e. substituting the minimum fixed point I∈[0,1) into equation (21), we get an unreliable region:

[0061] UR={q=g(I,σ)|I∈[0,1)} (22)

[0062] It is concluded that the RA code decoding fails when the repetition number is q in UR;

[0063] S32, in the set of repetition number range, take the absolute complement of the full set, we get the reliable region under the condition of decoding success:

[0064]

[0065] It is concluded that the RA code decoding succeeds when the repetition number is q in RR.

[0066] As a preferred, step S3 also includes the following steps:

[0067] S33, select the optimal repetition number q from the reliable region RR o , i.e. the minimum repetition number:

[0068]

[0069] Thus, the code rate r o = 1 / q o is optimized.

[0070] The application also provides a system for deriving an optimal code rate of a regular RA code based on an AWGN channel, comprising:

[0071] a derivation module configured to derive output mutual information I of the regular RA code under the AWGN channel;

[0072] an inverse function solving module configured to derive a theoretical analytic expression of the repetition number q of the regular RA code based on the output mutual information I and the fixed point theorem, and solve an inverse function of an objective function in a non-convex optimization problem;

[0073] an optimal repetition number solving module configured to solve uncertainty in the inverse function problem based on mathematical set theory, and obtain an optimal repetition number q of the regular RA code under the AWGN channel for successful decoding o .

[0074] Compared with the prior art, the application has the following beneficial effects: (1) the application derives the theoretical value of the repetition number of the regular RA code under the AWGN channel in steps according to the characteristics of the point-to-point communication external information transfer analysis and the fixed point theorem; (2) the application first derives the output mutual information I of the regular RA code under the AWGN channel, then effectively derives the theoretical analytic expression of the repetition number of the regular RA code based on the output mutual information I and the fixed point theorem, and solves the inverse function of the objective function in the non-convex optimization problem; finally, the application solves the uncertainty in the inverse function problem based on the mathematical set theory, and obtains the optimal repetition number for successful decoding; (3) the application has the characteristics of improving the channel coding performance and efficiency in the communication system, and effectively solving the closed-form expression of the optimal coding parameter. BRIEF DESCRIPTION OF DRAWINGS

[0075] Figure 1 a flow chart of the method for deriving an optimal code rate of a regular RA code based on an AWGN channel according to an embodiment of the application;

[0076] Figure 2 a factor graph of the regular RA code according to an embodiment of the application;

[0077] Figure 3 a schematic diagram of the optimal repetition number and the optimal code rate matched when the signal-to-noise ratio SNR = -2 dB and sigma = 1.2589 according to an embodiment of the application. DETAILED DESCRIPTION

[0078] In order to more clearly illustrate the embodiments of the present application, the specific embodiments of the present application will be described below with reference to the accompanying drawings. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor on the basis of these drawings.

[0079] Embodiments:

[0080] As shown in the formula (1), the present application provides a method for deriving the optimal code rate of regular RA code based on AWGN channel, which comprises the following steps: Figure 1

[0081] S1, deriving the output mutual information I of regular RA code under AWGN channel;

[0082] S2, based on the output mutual information I and the fixed point theory, deriving the theoretical analytical expression of the repetition number q of RA code, and solving the inverse function of the objective function in the non-convex optimization problem;

[0083] S3, based on mathematical set theory, solving the uncertainty in the inverse function problem, and obtaining the optimal repetition number q of regular RA code under AWGN channel o .

[0084] Specifically, the step S1 comprises the following steps:

[0085] S11, generally, the decoding of input and output information is represented by log likelihood ratio (LLR). The extrinsic information transfer function describes the relationship between the mean values of input LLR and output LLR, or the relationship between their mutual information forms. In addition, the extrinsic information transfer function is based on the assumption of infinite code length and Gaussian approximation, that is, the input LLR and the output LLR are both Gaussian variables, and the variance of each variable is twice the mean value; for a variable node with degree d, the extrinsic information transfer function T v describes the output mutual information of the variable node:

[0086]

[0087] wherein, 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;

[0088]

[0089] The formula (2) is the J function, which represents the output mutual information, wherein, σ A represents the variance of input information;

[0090] ​For a check node with degree d, the external information transfer function T c The output mutual information of the check node is described:

[0091] T c (I A,1 ,…,I A,d-1 ) = 1 - T v (1-I A,1 ,…,1-I A,d-1 (3)

[0092] If I A,i =I A If i = 1, ..., w, then the simplified formulas (1) and (3) are T v (I A ×w,I A,w+1 ,I A,d-1 ) and T c (I A ×w,I A,w+1 ,I A,d-1 );

[0093] S12, as Figure 2 As shown, letters U and V represent variable nodes, letter C represents a check node, and letter S represents a channel node. From channel node S to variable node V, for the l-th iteration, the root mean square of the channel noise is σ, and the output mutual information at channel node S is:

[0094]

[0095] From variable node V to check node C, according to formula (1) for the output mutual information at variable node V, the output mutual information at variable node V is obtained as follows:

[0096]

[0097] in, This represents the output mutual information from check node C to variable node V during the (l-1)th iteration;

[0098] From check node C to variable node U, according to formula (3) for the output mutual information at check node C, the output mutual information at check node C is obtained as follows:

[0099]

[0100] Based on formulas (1) and (3), the output mutual information at variable node U and check node C are obtained as follows:

[0101]

[0102]

[0103] S13, the final output mutual information from the variable node V to the channel node S is obtained:

[0104]

[0105] where I l-1 is the final output mutual information from the variable node V to the channel node S at the (l-1)-th iteration.

[0106] Specifically, the step S2 comprises the following steps:

[0107] S21, for the RA code with the code rate of 1 / q, when the iteration number l tends to infinity, the following equation is obtained:

[0108]

[0109] where I * is the minimum element in the solution set of the equation, and is the minimum fixed point; the minimum fixed point I * corresponds to the output mutual information I on each edge of the factor graph.

[0110] For the output mutual information I on each edge of the factor graph, the corresponding error probability formula is:

[0111]

[0112] where is the complementary error function; according to the factor graph, the average decoding error probability of the RA code is P e (I * ), and the average decoding error probability of the information bit is P e (I * uc ); P e (I * uc )→0 if and only if P e (I * )→0, that is, when the information bit is decoded without error, the encoding bit of the RA code can also be decoded without error, and vice versa.

[0113] S22, reconsidering the formula (9), when the iteration number l tends to infinity, the following equation is obtained:

[0114]

[0115] Simplifying the above equation, the following equation is obtained:

[0116]

[0117] Considering that

[0118]

[0119]

[0120]

[0121] S23, substituting formula (15) into formula (8), we get:

[0122] S24, combining formula (13) and formula (16), we get:

[0123] Simplifying formula (17), we get:

[0124]

[0125] Finally, the expression of q is:

[0126]

[0127] S25, re-considering the output mutual information I cv Substituting formula (13) into formula (5), we get:

[0128]

[0129] S26, substituting formula (20) into formula (19), we get the closed expression of q:

[0130]

[0131] After solving the inverse function of the objective function in the non-convex optimization problem, then the optimal repetition number q under the actual condition of successful decoding I = 1 is solved o However, because there is J -1 (I = 1) = ∞, so the analytical solution of the inverse function is not applicable to the condition of successful decoding.

[0132] Specifically, step S3 includes the following steps:

[0133] S31, find all repetition numbers under the condition of decoding failure, that is, substitute the minimum fixed point I ∈ [0, 1) into formula (21) to solve an unreliable region:

[0134] UR = {q = g (I, σ) |I ∈ [0, 1)} (22)

[0135] It is obtained that the repetition number q in UR is under the condition of decoding failure of the RA code;

[0136] S32, in the set of repetition number range, take the absolute complement of the full set, and get the reliable region under the condition of successful decoding:​

[0137]

[0138] It is concluded that the RA code is successfully decoded when the repetition number q in the RR is obtained.

[0139] S33, the optimal repetition number q is selected from the reliable region RR o , that is, the minimum repetition number:

[0140]

[0141] So that the code rate r o = 1 / q o is optimized.

[0142] After the above steps, the results shown in Figure 3 are obtained, and it is known from Figure 3 that when SNR = -2dB, that is, sigma = 1.2589, the optimal repetition number is q o = 4, and the optimal code rate is r o = 1 / q o = 0.25.

[0143] The application also provides a system for deriving the optimal code rate of a regular RA code based on an AWGN channel, comprising:

[0144] A derivation module is configured to derive the output mutual information I of the regular RA code under the AWGN channel.

[0145] An inverse function solving module is configured to derive a theoretical analytical expression of the repetition number q of the RA code based on the output mutual information I and the fixed point theory, and solve the inverse function of the objective function in the non-convex optimization problem.

[0146] An optimal repetition number solving module is configured to solve the uncertainty in the inverse function problem based on mathematical set theory, and obtain the optimal repetition number q o of the regular RA code under the AWGN channel for successful decoding.

[0147] According to the characteristics of the extrinsic information transfer analysis and the fixed point theory in point-to-point communication, the application derives the theoretical value of the repetition number of the regular RA code under the AWGN channel in steps. First, the output mutual information I of the regular RA code under the AWGN channel is derived based on the extrinsic information transfer analysis. Then, the theoretical analytical expression of the repetition number q of the RA code is derived based on the output mutual information and the fixed point theory, and the inverse function of the objective function in the non-convex optimization problem is solved. Finally, the uncertainty in the inverse problem is solved based on the mathematical set theory, and the optimal repetition number q o of the regular RA code under the AWGN channel for successful decoding is obtained, and the optimal code rate r o = 1 / q is obtained.o The present application has the characteristics of improving the channel coding performance and efficiency in the communication system, and effectively solving the optimal coding parameter closed expression.

[0148] The above only describes the preferred embodiments and principles of the present application in detail. For those skilled in the art, the specific embodiments can be changed according to the ideas provided by the present application, and these changes should be considered as the protection scope of the present application.

Claims

1. A method for deriving the optimal code rate of a regular RA code based on an AWGN channel, characterized in that, It comprises the following steps: S1, Deriving output mutual information of regular RA code under AWGN channel I ; S2, based on output mutual information I and fixed point theory, deduce the repetition number of RA code q theoretical analysis of the inverse function of the objective function in non-convex optimization problems; S3, based on mathematical set theory, solving the uncertainty in the inverse function problem, get AWGN channel under the regular RA code decoding success of the optimal number of repetitions q0; Step S1 comprises the following steps: S11, for a variable node of degree d the extrinsic information transfer function T v The output mutual information of a variable node is described: where 0≤ I A,i ≤1 represents the input mutual information from check nodes i to variable nodes j , J -1 (*) is the inverse function of the J function; Equation (2) is the J function, which represents the output mutual information, where represents the variance of the input information; For check nodes of degree d the extrinsic information transfer function T c The output mutual information of a check node is described: If I A,i = I A , i =1,…, w , then simplify equations (1) and (3) to and ; S12, from the channel node S to the variable node V , for the first l iteration, the channel noise root mean square is , and the output mutual information at the channel node S is: From the variable node V to the check node C , according to the formula (1) of the output mutual information at the variable node, the output mutual information at the variable node V is obtained as: （5） wherein, represents the output mutual information at the check node C at the variable node V at the variable node From the check node C to the variable node U , the output mutual information at the check node C is obtained according to the formula (3) of the output mutual information at the check node: According to the formulas (1) and (3), the output mutual information at the variable node U and the check node C is respectively obtained as follows: S13, the output mutual information from the variable node V to the channel node S is: where I l-1 represents the final output mutual information from the variable node V to the check node S at the l-1th iteration. Step S2 comprises the following steps: S21, for a bit rate of 1 / q The RA code, when the number of iterations... l As it approaches infinity, we get: wherein, I * is the smallest element of the solution set of the equation, the smallest fixed point; the smallest fixed point I * corresponding to the output mutual information on each edge of the factor graph I ; For the output mutual information on each edge of the factor graph I The corresponding error probability formula is: where is the complementary error function; according to the factor graph, the average decoding error probability of the RA code is P e ( I * ), and the average decoding error probability of the information bits is P e ( I * uc ); P e ( I * uc )→0 if and only if P e ( I * )→0; Step S2 also comprises the following steps: S22, reconsidering formula (9), when the number of iterations l tends to infinity, we obtain Simplify to get: Consider: Get: S23, put formula (15) into formula (8), get: S24, merge formula (13) and formula (16), get: Simplify formula (17), get: Finally, the expression for q is S25, reconsidering output mutual information I cv Substituting equation (13) into equation (5) gives S26, substituting equation (20) into equation (19), solving out q The closed-form expression of is: Step S3 comprises the following steps: S31, find all the number of repetitions under the condition of decoding failure, that is, put the minimum fixed point I ∈[0,1) into equation (21) to solve an unreliable area: It is derived that the number of repetitions in UR is q When the RA code decoding fails, the RA code is retransmitted. S32, in the set of the range of the number of repetitions, take the absolute complement of the universal set, get the reliable area under the decoding success: It is concluded that the number of repetitions in RR is q When the RA code is decoded successfully, the time is S33, from the reliable region RR, select the optimal repetition number q o i.e. the smallest repetition number: thereby enabling the bit rate to be optimized.

2. A system for AWGN channel based regular RA code optimal rate derivation, for implementing the method for AWGN channel based regular RA code optimal rate derivation of claim 1, characterized in that, The system comprises: a derivation module configured to derive output mutual information of a regular RA code under an AWGN channel I ; An inverse function solving module is configured to solve the inverse function of the objective function based on the output mutual information I and fixed point theory, deduce the theoretical expression of the repetition number of RA code q of a non-convex optimization problem; The optimal repetition number solving module is used for solving the uncertainty in the inverse function problem based on mathematical set theory, and obtaining the optimal repetition number for successful decoding of the regular RA code under the AWGN channel .

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