CN116800376B - Rule RA code optimal code rate deducing method and system based on AWGN channel - Google Patents
Rule RA code optimal code rate deducing method and system based on AWGN channelInfo
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Abstract
本发明属于通信系统技术领域，具体涉及基于AWGN信道的规则RA码最优码率推导方法及系统。方法包括如下步骤：S1，在AWGN信道下推导规则RA码的输出互信息I；S2，基于输出互信息I和不动点理论，推导出RA码的重复次数q的理论解析式，求解非凸优化问题中目标函数的逆函数；S3，基于数学集合论，解决求逆函数问题中的不确定性，得到AWGN信道下规则RA码成功译码的最优重复次数qo。本发明具有能够在通信系统中提高信道编码性能和效率，同时能够有效地求解出最优编码参数闭式表达式的特点。
This invention belongs to the field of communication system technology, specifically relating to a method and system for deriving the optimal code rate of regular RA codes based on AWGN channels. The method includes the following steps: S1, deriving the output mutual information I of the regular RA code under an AWGN channel; S2, based on the output mutual information I and fixed-point theory, deriving the theoretical analytical expression for the number of repetitions 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 problem of finding the inverse function, and obtaining the optimal number of repetitions q<sub>o</sub> for successful decoding of the regular RA code under an AWGN channel. This invention has the characteristics of improving channel coding performance and efficiency in communication systems, while effectively solving for the closed-form expression of the optimal coding parameters.
Description
Technical Field
The invention belongs to the technical field of communication systems, and particularly relates to a method and a system for deriving an optimal code rate of a regular RA code based on an AWGN channel.
Background
At the present time of the popularity of 5G mobile communication, a communication system is measured to be reliable, not only against noise, but also to be successfully decoded. At the same time, the requirements for maximizing the transmission rate are also increasing. In point-to-point channel coding communication, in order to maximize the transmission rate, i.e. the code rate, for each coding parameter, the mutual information transfer analysis needs to track the final output mutual information under the condition of a given signal-to-noise ratio, and then the optimal coding parameter is selected from the final output mutual information. The complexity of this optimization method increases with the increase of the coding parameters, thereby reducing the performance and efficiency of channel coding.
Therefore, it is very important to design a method and a system for deriving the optimal code rate of a regular RA (repeated accumulation) code based on an AWGN (additive white Gaussian noise) channel, which can improve the channel coding performance and efficiency in a communication system and can effectively solve the closed expression of the optimal coding parameters.
For example, the implementation steps of the distributed coding and decoding method of the RA code described in China patent document with the application number of CN201210011026.7 are that (1) the data information of the source node is coded by the RA code and then is respectively transmitted to the relay node and the destination node, (2) the relay node is used for carrying out relay coding on the received coding sequence and transmitting the coded information to the destination node, (3) a joint RA coding line graph is firstly constructed according to the RA code and the relay coding, then a multi-layer RA code bipartite graph is constructed on the basis of the joint RA coding line graph, and (4) the destination node is used for carrying out iterative decoding according to the bipartite graph of the multi-layer RA code, the received coding information of the relay node and the RA coding information transmitted by the source node, and recovering the source data information. Although the coding is simple to implement, the network throughput and the forwarding efficiency of the relay node can be improved, the performance of the destination node can be improved, and the method can be used for a distributed transmission system in relay network transmission, but has the defects that the complexity of the optimization method still increases along with the increase of coding parameters, the method cannot be used for solving a closed expression of the optimal coding parameters, and the channel coding performance and efficiency cannot be improved.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, the prior channel coding technology, under the condition of given signal-to-noise ratio, for each coding parameter, mutual information transfer analysis needs to track the final output mutual information, then an optimal coding parameter is selected from the mutual information, the complexity is increased along with the increase of the coding parameter, and further reduces the performance and efficiency of channel coding, and provides a method and a system for deducing the optimal code rate of the regular RA code based on the AWGN channel, which can improve the performance and efficiency of channel coding in a communication system and can effectively solve the closed expression of the optimal coding parameters.
In order to achieve the aim of the invention, the invention adopts the following technical scheme:
the method for deducing the optimal code rate of the regular RA code based on the AWGN channel comprises the following steps:
S1, deducing output mutual information I of a regular RA code under an AWGN channel;
s2, deriving a theoretical analysis formula of the repetition number q of the RA code based on the output mutual information I and the fixed point theory, and solving an inverse function of an objective function in the non-convex optimization problem;
And S3, based on a mathematical set theory, solving the uncertainty in the problem of an inversion function, and obtaining the optimal repetition number q o of successful decoding of the regular RA code under the AWGN channel.
Preferably, the step S1 includes the steps of:
S11, for a variable node with the degree d, the external information transfer function T v describes the output mutual information of the variable node:
Wherein 0≤I A,i≤1 represents the input mutual information from check node I to variable node J, J -1 is the inverse of the J function;
Wherein 0≤I A,i≤1 represents the input mutual information from check node I to variable node J, J -1 is the inverse of the J function;
For a degree d check node, the extrinsic information transfer function T c describes the output mutual information of the check node:
Tc(IA,1,…,IA,d-1)＝1-Tv(1-IA,1,…,1-IA,d-1) (3)
If I A,i＝IA, i=1, the combination of w and, then formulas (1) and (3) are simplified to T v(IA×w,IA,w+1…,IA,d-1) and T c(IA×w,IA,w+1,…IA,d-1);
s12, from the channel node S to the variable node V, for the first iteration, the root mean square of the channel noise is sigma, 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 mutual information output from the variable node, the mutual information output from the variable node V is obtained as follows:
wherein, the Representing the output mutual information from the check node C to the variable node V in the first-1 iteration;
From the check node C to the variable node U, according to the formula (3) of the mutual information output at the check node, the mutual information output at the check node C is obtained as follows:
and then according to formulas (1) and (3), respectively obtaining the output mutual information of the variable node U and the check node C as follows:
s13, finally obtaining the output mutual information from the variable node V to the channel node S as follows:
wherein I l-1 represents the final output mutual information from variable node V to channel node S at the first-1 iteration.
Preferably, step S2 includes the steps of:
S21, for RA codes with the code rate of 1/q, when the iteration times l tend to infinity, obtaining
The minimum fixed point I * corresponds to the output mutual information I on each side of the factor graph;
For the output mutual information I on each side of the factor graph, the corresponding error probability formula is:
wherein, the Is a complementary error function, and 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);Pe(I* uc) to 0 and only P e(I*) to 0.
Preferably, the step S2 further includes the steps of:
s22, reconsidering the formula (9), when the iteration number l approaches infinity, obtaining
Simplifying to obtain:
Taking into account that
Deriving
S23, substituting the formula (15) into the formula (8) to obtain:
S24, combining the formula (13) and the formula (16) to obtain:
Simplifying the formula (17) to obtain:
The expression of q is finally obtained as:
S25, reconsidering the output mutual information I cv, substituting the formula (13) into the formula (5) to obtain:
S26, substituting the formula (20) into the formula (19), and solving the closed expression of q as follows:
preferably, step S3 includes the steps of:
S31, all repetition times under the condition of decoding failure are obtained, namely, the minimum fixed point I E [0,1 ] is substituted into a formula (21), and an unreliable area is obtained:
UR={q=g(I,σ)|I∈[0,1)} (22)
obtaining that when the repetition number in UR is q, RA code decoding fails;
S32, taking the absolute complement of the complete set in the set of the repetition frequency range to obtain a reliable region under successful decoding:
It is obtained that RA code decoding is successful when the repetition number in RR is q.
Preferably, the step S3 further includes the steps of:
S33, selecting the optimal repetition number q o from the reliable region RR, namely the minimum repetition number:
Thereby optimizing the code rate r o＝1/qo.
The invention also provides a rule RA code optimal code rate deducing system based on the AWGN channel, which comprises the following steps:
the deduction module is used for deducting the output mutual information I of the regular RA codes under the AWGN channel;
The inverse function solving module is used for deducing theoretical analysis type of repetition number q of the RA code based on the output mutual information I and the fixed point theory and solving an inverse function of an objective function in the non-convex optimization problem;
And the optimal repetition number solving module is used for solving the uncertainty in the inversion function problem based on the mathematical set theory and obtaining the optimal repetition number q o for successfully decoding the regular RA code under the AWGN channel.
The invention has the advantages that (1) according to the characteristics of external information transfer analysis and fixed point theory in point-to-point communication, the theoretical value of the repetition number of the regular RA codes under the AWGN channel is deduced step by step, (2) the invention firstly deduces the output mutual information I of the regular RA codes under the AWGN channel, then effectively deduces the theoretical analysis formula of the repetition number in the RA codes based on the output mutual information I and the fixed point theory, solves the inverse function of the objective function in the non-convex optimization problem, finally solves the uncertainty in the inversion problem based on the mathematical set theory, and obtains the optimal repetition number under successful decoding, and (3) the invention has the characteristics of improving the channel coding performance and efficiency in a communication system and effectively solving the closed expression of the optimal coding parameters.
Drawings
Fig. 1 is a flowchart of a method for deriving an optimal code rate of a regular RA code based on an AWGN channel according to an embodiment of the present invention;
FIG. 2 is a factor graph of a regular RA code provided in an embodiment of the present invention;
Fig. 3 is a schematic diagram of the optimal repetition number and optimal code rate of matching when the snr= -2dB and σ= 1.2589 according to the embodiment of the present invention.
Detailed Description
In order to more clearly illustrate the embodiments of the present invention, specific embodiments of the present invention will be described below with reference to the accompanying drawings. It is evident that the drawings in the following description are only examples of the invention, from which other drawings and other embodiments can be obtained by a person skilled in the art without inventive effort.
Examples:
as shown in fig. 1, the invention provides a method for deriving an optimal code rate of a regular RA code based on an AWGN channel, comprising the following steps:
S1, deducing output mutual information I of a regular RA code under an AWGN channel;
s2, deriving a theoretical analysis formula of the repetition number q of the RA code based on the output mutual information I and the fixed point theory, and solving an inverse function of an objective function in the non-convex optimization problem;
And S3, based on a mathematical set theory, solving the uncertainty in the problem of an inversion function, and obtaining the optimal repetition number q o of successful decoding of the regular RA code under the AWGN channel.
Specifically, step S1 includes the steps of:
S11, the decoding of the input and output information is typically represented by log-likelihood ratios (Loglikelihood ratio, LLRs). The extrinsic information transfer function describes the relationship between the mean of the input LLR and the output LLR, or their mutual information form. Furthermore, the extrinsic information transfer function is based on the assumption that infinite code length and Gaussian approximation are used, that is, input LLR and output LLR are Gaussian variables, and the variance of each variable is twice the mean value, and for a variable node with degree d, the extrinsic information transfer function T v describes the output mutual information of the variable node:
Wherein 0≤I A,i≤1 represents the input mutual information from check node I to variable node J, J -1 is the inverse of the J function;
Equation (2) is a J function, representing the output mutual information, wherein σ A represents the variance of the input information;
For a degree d check node, the extrinsic information transfer function T c describes the output mutual information of the check node:
Tc(IA,1,…,IA,d-1)＝1-Tv(1-IA,1,…,1-IA,d-1) (3)
if I A,i＝IA, i=1, the combination of w and, then formulas (1) and (3) are simplified to T v(IA×w,IA,w+1,IA,d-1) and T c(IA×w,IA,w+1,IA,d-1);
S12, as shown in FIG. 2, the letters U and V represent variable nodes, the letter C represents check nodes, and the letter S represents channel nodes. From channel node S to variable node V, for the first iteration, the root mean square of the channel noise is σ, and the output mutual information at channel node S is:
from the variable node V to the check node C, according to the formula (1) of the mutual information output from the variable node, the mutual information output from the variable node V is obtained as follows:
wherein, the Representing the output mutual information from the check node C to the variable node V in the first-1 iteration;
From the check node C to the variable node U, according to the formula (3) of the mutual information output at the check node, the mutual information output at the check node C is obtained as follows:
and then according to formulas (1) and (3), respectively obtaining the output mutual information of the variable node U and the check node C as follows:
s13, finally obtaining the output mutual information from the variable node V to the channel node S as follows:
wherein I l-1 represents the final output mutual information from variable node V to channel node S at the first-1 iteration.
Specifically, step S2 includes the following steps:
S21, for RA codes with the code rate of 1/q, when the iteration times l tend to infinity, obtaining
The minimum fixed point I * corresponds to the output mutual information I on each side of the factor graph;
For the output mutual information I on each side of the factor graph, the corresponding error probability formula is:
wherein, the Is a complementary error function, the average decoding error probability of the RA code is P e(I*, the average decoding error probability of the information bits is P e(I* uc);Pe(I* uc) to 0 and only if P e(I*) to 0, i.e. when the information bits are decoded without errors, the encoded bits of the RA code can also be decoded without errors, and vice versa.
S22, reconsidering the formula (9), when the iteration number l approaches infinity, obtaining
Simplifying to obtain:
Taking into account that
Deriving
S23, substituting the formula (15) into the formula (8) to obtain:
S24, combining the formula (13) and the formula (16) to obtain:
Simplifying the formula (17) to obtain:
The expression of q is finally obtained as:
S25, reconsidering the output mutual information I cv, substituting the formula (13) into the formula (5) to obtain:
S26, substituting the formula (20) into the formula (19), and solving the closed expression of q as follows:
After solving the inverse function of the objective function in the non-convex optimization problem, the optimal repetition number q o under the practical condition i=1 where the decoding is successful is then solved. However, since the equation (21) of the inverse function has J -1 (i=1) = infinity, the analysis solution of the inverse function is not suitable for the condition that the decoding is successful.
Specifically, step S3 includes the following steps:
S31, all repetition times under the condition of decoding failure are obtained, namely, the minimum fixed point I E [0,1 ] is substituted into a formula (21), and an unreliable area is obtained:
UR={q=g(I,σ)|I∈[0,1)} (22)
obtaining that when the repetition number in UR is q, RA code decoding fails;
S32, taking the absolute complement of the complete set in the set of the repetition frequency range to obtain a reliable region under successful decoding:
It is obtained that RA code decoding is successful when the repetition number in RR is q.
S33, selecting the optimal repetition number q o from the reliable region RR, namely the minimum repetition number:
Thereby optimizing the code rate r o＝1/qo.
As shown in fig. 3, when the snr= -2dB, i.e. σ= 1.2589, the optimal repetition number is q o =4, and the optimal code rate is r o＝1/qo =0.25.
The invention also provides a rule RA code optimal code rate deducing system based on the AWGN channel, which comprises the following steps:
the deduction module is used for deducting the output mutual information I of the regular RA codes under the AWGN channel;
The inverse function solving module is used for deducing theoretical analysis type of repetition number q of the RA code based on the output mutual information I and the fixed point theory and solving an inverse function of an objective function in the non-convex optimization problem;
And the optimal repetition number solving module is used for solving the uncertainty in the inversion function problem based on the mathematical set theory and obtaining the optimal repetition number q o for successfully decoding the regular RA code under the AWGN channel.
According to the characteristics of external information transfer analysis and fixed point theory in point-to-point communication, the invention deduces the theoretical value of the repetition times of the regular RA codes under the AWGN channel in steps. First, based on the external information transfer analysis, the output mutual information I of the regular RA code under the AWGN channel is derived. And then, based on the output mutual information and the fixed point theory, deducing a theoretical analysis formula of the repetition number q in the RA code, and solving an inverse function of the objective function in the non-convex optimization problem. Finally, based on mathematical set theory, the uncertainty in the inversion problem is solved, the optimal repetition number q o under the successful decoding of the regular RA code under the AWGN channel is obtained, and the optimal code rate r o＝1/qo is further obtained. The invention has the characteristics of improving the channel coding performance and efficiency in a communication system and effectively solving the optimal coding parameter closed expression.
The foregoing is only illustrative of the preferred embodiments and principles of the present invention, and changes in specific embodiments will occur to those skilled in the art upon consideration of the teachings provided herein, and such changes are intended to be included within the scope of the invention as defined by the claims.
Claims (2)
1. The method for deriving the optimal code rate of the regular RA code based on the AWGN channel is characterized by comprising the following steps:
S1, deducing output mutual information I of a regular RA code under an AWGN channel;
s2, deriving a theoretical analysis formula of the repetition number q of the RA code based on the output mutual information I and the fixed point theory, and solving an inverse function of an objective function in the non-convex optimization problem;
s3, based on a mathematical set theory, the uncertainty in the problem of an inversion function is solved, and the optimal repetition number q 0 of successful decoding of the regular RA code under the AWGN channel is obtained;
step S1 comprises the steps of:
S11, for a variable node with the degree d, the external information transfer function T v describes the output mutual information of the variable node:
Wherein 0≤I A,i≤1 represents the input mutual information from check node I to variable node J, J -1 is the inverse of the J function;
Equation (2) is a J function, representing the output mutual information, wherein, Representing the variance of the input information;
For a degree d check node, the extrinsic information transfer function T c describes the output mutual information of the check node:
If I A,i=IA, i=1, the combination of the first and second components, w, then the formulas (1) and (3) are simplified to And;
S12, from the channel node S to the variable node V, for the first iteration, the root mean square of the channel noise isThe 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 mutual information output from the variable node, the mutual information output from the variable node V is obtained as follows:
(5)
wherein, the Representing the output mutual information from the check node C to the variable node V in the first-1 iteration;
From the check node C to the variable node U, according to the formula (3) of the mutual information output at the check node, the mutual information output at the check node C is obtained as follows:
and then according to formulas (1) and (3), respectively obtaining the output mutual information of the variable node U and the check node C as follows:
s13, finally obtaining the output mutual information from the variable node V to the channel node S as follows:
wherein, I l-1 represents the final output mutual information from the variable node V to the channel node S in the first-1 iteration;
Step S2 includes the steps of:
s21, for RA codes with code rate of 1/q, when the iteration times l tend to infinity, obtaining:
the minimum fixed point I * corresponds to the output mutual information I on each side of the factor graph;
For the output mutual information I on each side of the factor graph, the corresponding error probability formula is:
wherein, the The average decoding error probability of the RA code is P e(I* according to a factor graph, and the average decoding error probability of the information bit is P e(I* uc);Pe(I* uc) to 0 and only P e(I*) to 0;
step S2 further comprises the steps of:
s22, reconsidering the formula (9), when the iteration number l approaches infinity, obtaining
Simplifying to obtain:
consider that:
the following steps are obtained:
S23, substituting the formula (15) into the formula (8) to obtain:
S24, combining the formula (13) and the formula (16) to obtain:
Simplifying the formula (17) to obtain:
The expression of q is finally obtained as:
S25, reconsidering the output mutual information I cv, substituting the formula (13) into the formula (5) to obtain:
S26, substituting the formula (20) into the formula (19), and solving the closed expression of q as follows:
Step S3 includes the steps of:
S31, all repetition times under the condition of decoding failure are obtained, namely, the minimum fixed point I E [0,1 ] is substituted into a formula (21), and an unreliable area is obtained:
obtaining that when the repetition number in UR is q, RA code decoding fails;
S32, taking the absolute complement of the complete set in the set of the repetition frequency range to obtain a reliable region under successful decoding:
obtaining that when the repetition number in RR is q, RA code decoding is successful;
S33, selecting the optimal repetition number q o from the reliable region RR, namely the minimum repetition number:
Thereby enabling a code rate And (5) optimizing.
2. The AWGN channel-based regular RA code optimal code rate derivation system for implementing the AWGN channel-based regular RA code optimal code rate derivation method according to claim 1, is characterized in that the AWGN channel-based regular RA code optimal code rate derivation system includes:
the deduction module is used for deducting the output mutual information I of the regular RA codes under the AWGN channel;
The inverse function solving module is used for deducing theoretical analysis type of repetition number q of the RA code based on the output mutual information I and the fixed point theory and solving an inverse function of an objective function in the non-convex optimization problem;
The optimal repetition number solving module is used for solving the uncertainty in the inversion function problem based on the mathematical set theory to obtain the optimal repetition number of successful decoding of the regular RA code under the AWGN channel 。
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