Method and system for deriving optimal degree distribution of regular ldpc codes over awgn channels

Abstract

The application belongs to the technical field of communication system, and particularly relates to a method and system for deriving optimal degree distribution of regular LDPC code based on AWGN channel. The method comprises the following steps: S1, taking the maximization of code rate of the regular LDPC code as a criterion and taking the decoding success as a prerequisite, a target function with a constraint condition is constructed; S2, fixing the degree d v of the variable node, based on the fixed point analysis theory, a theoretical analytic expression of the degree d c of the check node in the LDPC code is derived, and an inverse function of the target function in the non-convex optimization problem is solved; S3, based on the mathematical set theory, the uncertainty in the inverse function solving problem in the step S2 is solved, and the optimal degree distribution of the LDPC code under the AWGN channel is obtained. The application has the characteristics of improving the channel coding performance in the communication system.

Description

Technical Field

[0001] This invention belongs to the field of communication system technology, specifically relating to a method and system for deriving the optimal degree distribution of regular LDPC codes based on AWGN channels. Background Technology

[0002] As the most fundamental channel coding technique in the physical layer, LDPC (Low-Density Parity-Check) codes will play a crucial role in future 6G wireless communication systems. In point-to-point communication, performance analysis of LDPC codes can guide the selection of degree distributions and avoid time-consuming simulation evaluations. Based on external information transfer analysis, a full-space search of the LDPC code's degree distribution is used to track whether the mutual information I of the final output is 1 to determine decoding success. Then, from all successfully decoded reliable degree distributions, the degree distribution that maximizes the code rate is identified. However, this full-space search algorithm is too complex and becomes unsuitable as the number of degree distributions increases dramatically.

[0003] Therefore, it is very important to design a method and system for deriving the optimal degree distribution of regular LDPC codes based on AWGN (Additive White Gaussian Noise) channels that can improve the channel coding performance in communication systems.

[0004] For example, Chinese patent document CN201610167517.9 describes a joint search method for optimized LDPC code degree distribution in an asymmetric channel based on OOK modulation. This method calculates the density distribution of the initial variable message space of the LDPC code degree distribution based on the accurate photoelectric conversion model of the photodiode under OOK modulation, and performs iterative density evolution of the LDPC code degree distribution. This allows for a more accurate evaluation of the encoding and decoding performance of LDPC codes in this channel. A joint evolutionary update method is employed, using genetic evolution, differential evolution, particle swarm optimization, and simulated annealing algorithms for optimization. By considering the weight ratios of various algorithms and periodically shuffling the population and inserting superior individuals, the weights of various algorithms are adjusted in terms of both population size and the number of iterations. This leverages the global advantages of joint search, avoids the limitations of single algorithms, and improves search efficiency. While this method enhances the globality and robustness of the search algorithm and limits the search space for effective searching, its drawback is that the complexity of the full-space search algorithm itself leads to low computational efficiency as the number of degree distributions increases, thus limiting its effectiveness. Summary of the Invention

[0005] This invention aims to overcome the problems of existing channel coding techniques, which employ full-space search of the degree distribution of LDPC codes. These problems include excessively high algorithm complexity and unsuitability as the number of degree distributions increases. The invention provides a method and system for deriving the optimal degree distribution of regular LDPC codes based on AWGN channels, which can improve the channel coding performance in communication systems.

[0006] To achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0007] The derivation method of optimal degree distribution of regular LDPC codes based on AWGN channels includes the following steps:

[0008] S1, with the maximization of the code rate of the regular LDPC code as the criterion and successful decoding as the premise, constructs an objective function with constraints;

[0009] S2, the degree d of the fixed variable node. v Based on fixed-point analysis theory, the degree d of the check node in LDPC code is derived. c The theoretical analytical expression is obtained, and the inverse function of the objective function in the nonconvex optimization problem is solved;

[0010] S3, based on mathematical set theory, resolves the uncertainty in finding the inverse function in step S2, and obtains the optimal degree distribution for successful decoding of LDPC codes under the AWGN channel.

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

[0012] S11 adopts a regular LDPC encoder structure and changes the degree distribution (d v d c The bit rate is adjusted to achieve reliable communication transmission. The specific formula is as follows:

[0013]

[0014] Where, d v It is the degree of the variable node, d c It is the degree of the verification node;

[0015] S12 investigates the fundamental laws governing mutual information at each node in the corresponding factor graph. It assumes an infinite code length and a Gaussian approximation, meaning the variance is twice the mean. For a variable node with degree d, the output mutual information is...

[0016]

[0017] 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;

[0018]

[0019] Formula (3) is the J function, representing the output mutual information, where σ A This represents the variance of the input information;

[0020] For a check node with degree d, the output mutual information is:

[0021] T c (I A,1 ,…,I A,d-1 ) = 1 - T v (1-I A,1 ,…,1-I A,d-1 (4)

[0022] If I A,i =I A If i = 1, ..., w, then the simplified formulas (2) and (4) 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 );

[0023] S13, when the iteration number is l and the root mean square of the channel noise is σ, the output mutual information from channel node S to variable node V is:

[0024]

[0025] Based on external information transfer analysis, the output mutual information from variable node V to check node C is:

[0026]

[0027] in, This represents the output mutual information from check node C to variable node V during the (l-1)th iteration;

[0028] The output mutual information from node C to variable node V is:

[0029]

[0030] The final output mutual information from variable node V to channel node S is obtained as follows:

[0031]

[0032] Reconsidering the final output mutual information at channel node S, we can simplify to obtain...

[0033]

[0034] S14, Substituting formula (9) into formula (8), we finally obtain the objective function.

[0035]

[0036] Among them, I l-1 This represents the final output mutual information between variable node V and channel node S during the (l-1)th iteration.

[0037] Preferably, step S1 further includes the following steps:

[0038] S15, Considering that the objective function is constructed based on external information transfer analysis, the necessary and sufficient condition for successful decoding is:

[0039]

[0040] Ultimately, the objective function with constraints, based on maximizing the LDPC code rate and assuming successful decoding, is obtained as follows:

[0041]

[0042] Preferably, step S2 includes the following steps:

[0043] S21, further exploring the objective function, we set the root mean square of the channel noise as σ and the degree distribution of the LDPC code (d v ,d c When the number of iterations is l, then:

[0044]

[0045] Among them, I * It is a fixed-point equation I = f(I, d) v ,d c The smallest fixed point in σ; the smallest fixed point I. * This corresponds to the convergence point in the factor graph and is used to determine the bit error rate of the user's decoding:

[0046]

[0047] in, It is a complementary error function; as can be seen from formula (13), when I * =1, Pe→0, that is, when the smallest fixed point I * Decoding fails when the value falls within the range [0,1). * When = 1, decoding is successful;

[0048] S22, after observing the output mutual information from variable node V to channel node S on the factor graph, and reconsidering formula (10), when the number of iterations l approaches infinity, we obtain

[0049]

[0050] S23, Assume a given variable, the degree d of the node. v By relaxing the conditions and transforming formula (15), we get

[0051]

[0052] Degree d of the node containing the check node c Move the terms aside and simplify to get

[0053]

[0054] S24, Solve for the degree d of the check node. c The mathematical analytical expression, that is, d c Represented as an explicit function with a fixed point.

[0055]

[0056] Preferably, step S3 includes the following steps:

[0057] S31, put I * Substituting 1 into the formula (18) of the inverse function, we can find the degree d of the reliable check node under successful user decoding. c and the degree d of the given variable node v Matching;

[0058] S32, traversing d v Find the corresponding d c Find the code rate 1-d of the LDPC code. v / d c Maximizing the optimal degree distribution To maximize the system's transmission rate.

[0059] Preferably, step S31 includes the following steps:

[0060] S311, Considering the case of decoding failure, given the root mean square of the channel noise as σ and the degree d of the variable node. v , put I * Substituting ∈[0,1) into formula (18), we find an unreliable region:

[0061] UR={d c =g(I * ,d v ,σ)|I *∈[0,1)} (19)

[0062] It is concluded that, given d v and UR in d c Under the degree distribution, LDPC code decoding fails;

[0063] S312, based on mathematical set theory, takes the absolute complement of the entire set within the degree distribution range to obtain the reliable region under successful decoding:

[0064]

[0065] It is concluded that, given d v and RR in d c Under the degree distribution, the LDPC code was successfully decoded;

[0066] S313, Select from the reliable region RR the region with the given d v The best match Given d v Bitrate maximum:

[0067]

[0068] Preferably, step S32 includes the following steps:

[0069] S321, traverse d v Find all corresponding And select the optimal degree distribution from them.

[0070]

[0071] This makes the code rate of LDPC codes... Optimization means maximizing the system's transmission rate.

[0072] This invention also provides a system for deriving the optimality distribution of regular LDPC codes based on AWGN channels, comprising:

[0073] The objective function construction module is used to construct an objective function with constraints, based on maximizing the code rate of the regular LDPC code and assuming successful decoding.

[0074] The inverse function solver module is used to fix the degree d of the variable nodes. v Based on fixed-point analysis theory, the degree d of the check node in LDPC code is derived. c The theoretical analytical expression is obtained, and the inverse function of the objective function in the nonconvex optimization problem is solved;

[0075] The optimal degree distribution solution module is used to solve the uncertainty in the problem of finding the inverse function based on mathematical set theory, and obtain the optimal degree distribution for successful decoding of LDPC codes under AWGN channels.

[0076] Compared with the prior art, the beneficial effects of this invention are: (1) Based on the characteristics of external information transfer analysis and fixed-point theory in point-to-point communication, this invention derives the theoretical value of the degree distribution of regular LDPC codes under AWGN channels step by step; (2) This invention first constructs an objective function with constraints, taking the maximization of the code rate of regular LDPC codes as the criterion and successful decoding as the premise; then, it fixes the degree d of the variable node. v Based on fixed-point analysis theory, the degree d of the check node in LDPC code is derived. c The theoretical analytical expression is used to solve for the inverse function of the objective function in the non-convex optimization problem; based on mathematical set theory, the uncertainty in the inverse problem is resolved, and the optimal degree distribution for successful decoding of LDPC codes under AWGN channels is obtained. Thus, the code rate of the LDPC code is optimized. (3) The present invention has the feature of improving channel coding performance in communication systems. Attached Figure Description

[0077] Figure 1 A flowchart illustrating a method for deriving the optimal degree distribution of a regular LDPC code based on an AWGN channel, as provided in an embodiment of the present invention.

[0078] Figure 2 A factor graph of a regular LDPC code provided in an embodiment of the present invention;

[0079] Figure 3 For embodiments of the present invention, when the signal-to-noise ratio (SNR) is 0 dB and σ is 1, given d v =3 most matching A diagram illustrating numerical values. Detailed Implementation

[0080] To more clearly illustrate the embodiments of the present invention, specific implementation methods will be described below with reference to the accompanying drawings. Obviously, the drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings and other implementation methods can be obtained based on these drawings without any creative effort.

[0081] Example:

[0082] like Figure 1 As shown, this invention provides a method for deriving the optimality distribution of regular LDPC codes based on AWGN channels, including the following steps:

[0083] S1, with the maximization of the code rate of the regular LDPC code as the criterion and successful decoding as the premise, constructs an objective function with constraints;

[0084] S2, the degree d of the fixed variable node. v Based on fixed-point analysis theory, the degree d of the check node in LDPC code is derived. c The theoretical analytical expression is obtained, and the inverse function of the objective function in the nonconvex optimization problem is solved;

[0085] S3, based on mathematical set theory, resolves the uncertainty in finding the inverse function in step S2, and obtains the optimal degree distribution for successful decoding of LDPC codes under the AWGN channel.

[0086] Specifically, step S1 includes the following steps:

[0087] S11, for point-to-point communication systems, a regular LDPC encoder structure is adopted, changing the degree distribution (d v d c The bit rate is adjusted to achieve reliable communication transmission. The specific formula is as follows:

[0088]

[0089] Where, d v It is the degree of the variable node, d c It is the degree of the verification node;

[0090] S12, at the receiving end, iterative decoding is performed on a factor graph, specifically as follows: Figure 2 As shown in the figure, letter V represents a variable node, letter C represents a check node, and letter S represents a channel node. The basic rules governing mutual information at each node on the corresponding factor graph are studied. Assuming infinite code length and Gaussian approximation (variance is twice the mean), for a variable node of degree d, the output mutual information is...

[0091]

[0092] 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;

[0093]

[0094] Formula (3) is the J function, representing the output mutual information, where σ A This represents the variance of the input information;

[0095] For a check node with degree d, the output mutual information is:

[0096] T c (I A,1 ,…,I A,d-1 ) = 1 - T v (1-I A,1 ,…,1-I A,d-1 (4)

[0097] If I A,i =I A If i = 1, ..., w, then the simplified formulas (2) and (4) 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 );

[0098] S13, when the iteration number is l and the root mean square of the channel noise is σ, the output mutual information from channel node S to variable node V is:

[0099]

[0100] Based on external information transfer analysis, the output mutual information from variable node V to check node C is:

[0101]

[0102] in, This represents the output mutual information from check node C to variable node V during the (l-1)th iteration;

[0103] The output mutual information from node C to variable node V is:

[0104]

[0105] The final output mutual information from variable node V to channel node S is obtained as follows:

[0106]

[0107] Reconsidering the final output mutual information at channel node S, we can simplify to obtain...

[0108]

[0109] S14, Substituting formula (9) into formula (8), we finally obtain the objective function.

[0110]

[0111] S15, Considering that the objective function is constructed based on external information transfer analysis, the necessary and sufficient condition for successful decoding is:

[0112]

[0113] Ultimately, the objective function with constraints, based on maximizing the LDPC code rate and assuming successful decoding, is obtained as follows:

[0114]

[0115] Specifically, step S2 includes the following steps:

[0116] S21, further exploring the objective function, we set the root mean square of the channel noise as σ and the degree distribution of the LDPC code (d v ,d c When the number of iterations is l, then:

[0117]

[0118] Among them, I * It is a fixed-point equation I = f(I, d) v ,d c The smallest fixed point in σ; the smallest fixed point I. * This corresponds to the convergence point in the factor graph and is used to determine the bit error rate of the user's decoding:

[0119]

[0120] in, It is a complementary error function; as can be seen from formula (13), when I * =1, Pe→0, that is, when the smallest fixed point I * Decoding fails when the value falls within the range [0,1). * When = 1, decoding is successful;

[0121] S22, after... Figure 2 Based on the observation of the output mutual information from variable node V to channel node S on the factor graph, and reconsidering formula (10), when the number of iterations l approaches infinity, we obtain...

[0122]

[0123] S23, Assume a given variable, the degree d of the node. v By relaxing the conditions and transforming formula (15), we get

[0124]

[0125] Degree d of the node containing the check node c Move the terms aside and simplify to get

[0126]

[0127] S24, Solve for the degree d of the check node. c The mathematical analytical expression, that is, d c Represented as an explicit function with a fixed point.

[0128]

[0129] Compared to external information transfer analysis, fixed-point theory does not require iterative and full-space search of the degree distribution of LDPC codes (d). v ,d c This effectively reduces complexity.

[0130] Specifically, step S3 includes the following steps:

[0131] S31, put I * Substituting 1 into the formula (18) of the inverse function, we can find the degree d of the reliable check node under successful user decoding. c and the degree d of the given variable node v Matching;

[0132] S32, traversing d v Find the corresponding d c Find the code rate 1-d of the LDPC code. v / d c Maximizing the optimal degree distribution To maximize the system's transmission rate.

[0133] Furthermore, step S31 includes the following steps:

[0134] S311, Considering the case of decoding failure, given the root mean square of the channel noise as σ and the degree d of the variable node. v , put I * Substituting ∈[0,1) into formula (18), we find an unreliable region:

[0135] UR={d c =g(I * ,d v ,σ)|I * ∈[0,1)} (19)

[0136] It is concluded that, given d v and UR in d c Under the degree distribution, LDPC code decoding fails;

[0137] S312, based on mathematical set theory, takes the absolute complement of the entire set within the degree distribution range to obtain the reliable region under successful decoding:

[0138]

[0139] It is concluded that, given d v and RR in d c Under the degree distribution, the LDPC code was successfully decoded;

[0140] S313, Select from the reliable region RR the region with the given d v The best match Given d v Bitrate maximum:

[0141]

[0142] Furthermore, step S32 includes the following steps:

[0143] S321, traverse d v Find all corresponding And select the optimal degree distribution from them.

[0144]

[0145] This makes the code rate of LDPC codes... Optimization means maximizing the system's transmission rate.

[0146] like Figure 3 As shown, when the signal-to-noise ratio (SNR) = 0 dB and σ = 1, given d v =3 most matching The value is 5.

[0147] based on Figure 3 The schematic diagram shown yields the data in Table 1 below. Table 1 shows the data when SNR = 0dB, which is σ = 1, after traversing d v The corresponding formula obtained by formula (21) And the optimal degree distribution is selected using formula (22). The optimal code rate of LDPC code is

[0148] Table 1 shows that when SNR = 0 dB and σ = 1, given d v =3 most matching Data table

[0149]

[0150] This invention also provides a system for deriving the optimality distribution of regular LDPC codes based on AWGN channels, comprising:

[0151] The objective function construction module is used to construct an objective function with constraints, based on maximizing the code rate of the regular LDPC code and assuming successful decoding.

[0152] The inverse function solver module is used to fix the degree d of the variable nodes. v Based on fixed-point analysis theory, the degree d of the check node in LDPC code is derived. c The theoretical analytical expression is obtained, and the inverse function of the objective function in the nonconvex optimization problem is solved;

[0153] The optimal degree distribution solution module is used to solve the uncertainty in the problem of finding the inverse function based on mathematical set theory, and obtain the optimal degree distribution for successful decoding of LDPC codes under AWGN channels.

[0154] Based on the characteristics of external information transfer analysis and fixed-point theory in point-to-point communication, this invention derives the theoretical value of the degree distribution of regular LDPC codes under AWGN channels step by step. First, with maximizing the code rate as the criterion and successful decoding as a prerequisite, a constrained objective function is determined. Then, the degree d of the node is fixed as the variable. v Based on fixed-point analysis theory, the degree d of the check node in the regular LDPC code is derived and optimized. c The theoretical analytical expression is used to solve for the inverse function of the objective function in the nonconvex optimization problem. Finally, based on mathematical set theory, the uncertainty in the inverse problem is resolved, and the optimal degree distribution under successful decoding of regular LDPC codes in the AWGN channel is obtained. This leads to the optimal LDPC code rate.

[0155] The above description is merely a detailed explanation of preferred embodiments and principles of the present invention. For those skilled in the art, there may be changes in specific implementation methods based on the ideas provided by the present invention, and these changes should also be considered within the scope of protection of the present invention.

Claims

1. A method for deriving the optimal degree distribution of regular LDPC codes based on AWGN channels, characterized in that, Includes the following steps: S1, with the maximization of the code rate of the regular LDPC code as the criterion and successful decoding as the premise, constructs an objective function with constraints; S2, the degree d of the fixed variable node. v Based on fixed-point analysis theory, the degree d of the check node in LDPC code is derived. c The theoretical analytical expression is obtained, and the inverse function of the objective function in the nonconvex optimization problem is solved; S3, based on mathematical set theory, resolves the uncertainty in finding the inverse function in step S2, and obtains the optimal degree distribution for successful decoding of LDPC codes under the AWGN channel. ; Step S1 includes the following steps: S11 adopts a regular LDPC encoder structure and changes the degree distribution (d v d c The bit rate is adjusted to achieve reliable communication transmission. The specific formula is as follows: Where, d v It is the degree of the variable node, d c It is the degree of the verification node; S12 investigates the fundamental laws governing mutual information at each node in the corresponding factor graph. It assumes an infinite code length and a Gaussian approximation, meaning the variance is twice the mean. For a variable node with degree d, the output mutual information is... (2) 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; (3)； Formula (3) is the J function, which represents the output mutual information, where, This represents the variance of the input information; For a check node with degree d, the output mutual information is: (4) If I A,i =I A If i=1,…w, then simplified formulas (2) and (4) are: and ; S13, when the iteration number is l, the root mean square of the channel noise is... At that time, the output mutual information from channel node S to variable node V is (5) Based on external information transfer analysis, the output mutual information from variable node V to check node C is: (6) in, Indicates the first During the next iteration, the output mutual information from node C to variable node V is verified. The output mutual information from node C to variable node V is: (7) The final output mutual information from variable node V to channel node S is obtained as follows: (8) Reconsidering the final output mutual information at channel node S, we can simplify to obtain... (9) S14, Substituting formula (9) into formula (8), we finally obtain the objective function. (10) in, Indicates the first In the next iteration, the final output mutual information from variable node V to channel node S; S15, Considering that the objective function is constructed based on external information transfer analysis, the necessary and sufficient condition for successful decoding is: (11) Ultimately, the objective function with constraints, based on maximizing the LDPC code rate and assuming successful decoding, is obtained as follows: (12)； Step S2 includes the following steps: S21, further exploring the objective function, setting the root mean square of the channel noise as... Degree distribution of LDPC codes (d) v , d c When the number of iterations is l, then: (13) Among them, I * It is a fixed-point equation I=f(I,d) v ,d c , The smallest fixed point in ) ; the smallest fixed point I * This corresponds to the convergence point in the factor graph and is used to determine the bit error rate of the user's decoding: (14) in, It is a complementary error function; as can be seen from formula (13), when Pe→0, that is, when the smallest fixed point Decoding fails when the value falls within the range [0,1). Decoding was successful. S22, after observing the output mutual information from variable node V to channel node S on the factor graph, and reconsidering formula (10), when the number of iterations l approaches infinity, we obtain (15) S23, Assume a given variable, the degree d of the node. v By relaxing the conditions and transforming formula (15), we get (16) Degree d of the node containing the check node c Move the terms aside and simplify to get (17) S24, Solve for the degree d of the check node. c The mathematical analytical expression, that is, d c Represented as an explicit function with a fixed point. (18)； Step S3 includes the following steps: S31, put Substituting into the formula (18) of the inverse function, we can find the degree d of the reliable verification node under successful user decoding. c and the degree d of the given variable node v Matching; S32, traversing d v Find the corresponding d c Find the code rate of the LDPC code. Maximizing the optimal degree distribution This maximizes the system's transmission rate. Step S31 includes the following steps: S311, Considering the case of decoding failure, given the root mean square of the channel noise is... Degree d of the variable node v , put I * Substituting ∈[0,1) into formula (18), we find an unreliable region: (19) It is concluded that, given d v and UR in d c Under the degree distribution, LDPC code decoding fails; S312, based on mathematical set theory, takes the absolute complement of the entire set within the degree distribution range to obtain the reliable region under successful decoding: (20) It is concluded that, given d v and RR in d c Under the degree distribution, the LDPC code was successfully decoded; S313, Select from the reliable region RR the region with the given d v The best match , such that given d v Bitrate maximum: (21)； Step S32 includes the following steps: S321, traverse d v Find all corresponding And select the optimal degree distribution from them. , (22) This makes the code rate of LDPC codes... Optimization means maximizing the system's transmission rate.

2. A system for deriving the optimal degree distribution of regular LDPC codes based on AWGN channels, used to implement the method for deriving the optimal degree distribution of regular LDPC codes based on AWGN channels as described in claim 1, characterized in that, The regular LDPC code optimality distribution derivation system based on the AWGN channel includes: The objective function construction module is used to construct an objective function with constraints, based on maximizing the code rate of the regular LDPC code and assuming successful decoding. The inverse function solver module is used to fix the degree d of the variable nodes. v Based on fixed-point analysis theory, the degree d of the check node in LDPC code is derived. c The theoretical analytical expression is obtained, and the inverse function of the objective function in the nonconvex optimization problem is solved; The optimal degree distribution solution module is used to solve the uncertainty in the problem of finding the inverse function based on mathematical set theory, and obtain the optimal degree distribution for successful decoding of LDPC codes under AWGN channels. .

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