Polarization code construction method suitable for Rician fading channel

Abstract

The invention relates to the technical field of unmanned aerial vehicle communication, particularly discloses a polarization code construction method suitable for a Rician fading channel, and provides two low-complexity polarization code construction methods suitable for a fast Rician fading channel, the first method is a simplified discrete density evolution (DDE) method, and the second method is a low-complexity polarization code construction method suitable for a fast Rician fading channel. According to the method, a probability density function PDF of a received symbol log-likelihood ratio LLR is discretized into a probability quality function PMF, optimization is achieved through PMF evolution based on minimum sum decoding, a numerical result shows that the proposed DDE method can effectively evaluate the polarization code performance and can also be used as a practical construction tool, and the method is suitable for popularization and application. According to the second method, a Rice channel is approximated to be a binary input additive Gaussian white noise B-AWGN channel with equivalent capacity, a Gaussian approximation density evolution (DEGA) method is adopted for optimization, the simulation result shows that the performance of the polarization code constructed by the Gaussian equivalent GE method is equivalent to that of the polarization code constructed by the DDE method, and meanwhile, the construction complexity is reduced.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of unmanned aerial vehicle communication, in particular to a polar code construction method suitable for a Rician fading channel. BACKGROUND

[0002] In recent years, unmanned aerial vehicles have gradually expanded from the military field to the civil and commercial fields due to their high mobility and low cost advantages, and have attracted much attention in the industry. In the application of unmanned aerial vehicles, data transmission is crucial, and direct connection communication between the unmanned aerial vehicle and the ground node is the most commonly used mode. However, it should be noted that the air-ground direct connection communication is usually limited to the line-of-sight condition, and in complex environments, it will significantly affect the transmission effect. For example, in urban areas, the signal is easily blocked by obstacles such as trees and buildings, resulting in reduced reliability. Therefore, the use of efficient channel coding technology plays a key role in improving transmission reliability. It is worth noting that the polar code is the first channel coding that has been proven to achieve channel capacity and has been included in the 5G standard. With its solid theoretical foundation and excellent practical performance, it has become a promising candidate for future unmanned aerial vehicle communication.

[0003] However, existing construction methods are mostly focused on ideal models such as binary erasure channels and binary symmetric channels, or although there are studies on general fading channels such as Rayleigh fading, there is a lack of specialized optimization for fast Rician fading channels, which limits their application in actual scenarios where the line-of-sight and non-line-of-sight of unmanned aerial vehicles are mixed. Early polar code construction for fading channels mainly developed along two paths: one was to estimate the Bhattacharyya parameter of the bit channel by quantization and channel degradation approximation, and the other was to estimate the error probability by channel approximation and diversity analysis. Although these methods provide ideas for construction under fading channels, the original density evolution method has inherent obstacles such as high computational complexity and insufficient practicality when directly applied to Rician channels, making it difficult to meet the balance between reliability and complexity requirements for unmanned aerial vehicle communication. SUMMARY

[0004] (I) Technical problems solved

[0005] The application provides a polar code construction method suitable for a Rician fading channel, which solves the problems mentioned in the background.

[0006] (II) Technical solutions

[0007] To achieve the above purpose, the application is implemented by the following technical solutions: a polar code construction method suitable for a Rician fading channel, comprising the following steps:

[0008] S1: modeling a fast Rician fading channel, establishing a received signal model, wherein a received symbol is composed of a channel gain, a modulated codeword and a Gaussian noise, the channel gain obeys a Rician distribution, a probability density function of which contains a zero-order first-type modified Bessel function, and an average fading power is normalized to a unit value, and a channel attenuation degree is represented by a K-factor of a ratio of a direct path power to a multipath scattering power;

[0009] S2: based on the received signal model established in S1, calculating a log-likelihood ratio (LLR) of the received symbol under a condition that perfect channel state information is known at a receiving end, obtaining statistical characteristics of the LLR, and estimating error probabilities of each bit channel in a polar code by a discrete density evolution method or a Gaussian equivalent method;

[0010] S3: selecting K channels with the smallest error probability values from all N bit channels by sorting comparison according to the error probabilities estimated in S2, to form an information set A for carrying information bits, and setting bit channels not selected into the information set A as fixed values;

[0011] S4: constructing a complete polar code by using the information set A determined in S3, embedding an information sequence to be transmitted into a data carrier with a length of N, and generating a final codeword by a polarization transformation, to complete a polar code construction process with a code length of N and an information bit length of K.

[0012] Further, the discrete density evolution method specifically includes:

[0013] First, a probability density function of the received symbol log-likelihood ratio obtained in S2 is discretized into a probability mass function through a quantization process, then a recursive evolution of the probability mass function sequence is performed on a polarization tree, wherein for a left child node of each node, a convolution operation rule of a corresponding function f is adopted to update the probability mass function, for a right child node, a convolution operation rule of a corresponding function g is adopted to update the probability mass function, and finally, according to the probability mass function corresponding to each bit channel after the evolution is completed, an error probability of the bit channel is calculated by accumulating a probability value of a negative log-likelihood ratio region.

[0014] Further, the quantization process specifically includes:

[0015] First, an initial quantization interval is determined by using Chebyshev inequality, wherein a lower bound is an expectation of the log-likelihood ratio minus a square root of a preset probability ratio of a variance of the log-likelihood ratio, and an upper bound is the expectation of the log-likelihood ratio plus the square root of the preset probability ratio of the variance of the log-likelihood ratio, then the interval is further refined by a boundary probability constraint to ensure that probabilities outside the left boundary and the right boundary are not more than a preset threshold, and finally, the continuous probability density function is converted into a discrete probability mass function within the refined interval, to complete the quantization process.

[0016] Further, the Gaussian equivalent method specifically comprises:

[0017] First, the Rician fading channel in S1 is approximated as a binary input additive Gaussian white noise channel of equivalent capacity, then the parameters of the equivalent channel are determined through a moment matching equation, wherein the equivalent channel gain is taken as the expectation of the received symbol, and the equivalent noise variance is taken as the sum of the variance of the channel gain and the original noise variance, then the recursive evolution of the mean sequence of the log-likelihood ratio is carried out on the polarization tree, wherein the left child node adopts the mean update rule based on the function φ, and the right child node adopts the addition update rule, and finally the error probability is calculated through the Gaussian Q function according to the mean of the log-likelihood ratio of each bit channel after the evolution is completed.

[0018] Further, the moment matching equation is specifically expressed as:

[0019] The equivalent channel gain h' is equal to the expectation E(y) of the received symbol y, and the equivalent noise variance is equal to the variance D(h) of the channel gain h' plus the original noise variance That is, h' = E(y) and The equation is used to keep the consistency of the statistical characteristics between the equivalent channel and the original Rician channel.

[0020] Further, the error probability calculation specifically comprises:

[0021] According to the theoretical basis of the Gaussian approximation density evolution method, the error probability of each bit channel is calculated through the formula , wherein Q(·) represents a standard Gaussian Q function, represents the mean of the log-likelihood ratio of the t-th bit channel at the bottom layer node after the evolution of the polarization tree.

[0022] (Three) beneficial effects

[0023] The application provides a polar code construction method suitable for a Rician fading channel.

[0024] The polar code construction method suitable for the Rician fading channel provides two polar coding construction methods suitable for fast Rician fading channels, which provides a new idea for the application of polar coding in unmanned aerial vehicle air-ground direct connection communication, wherein the DDE method is an improved scheme of the original DE method, which is optimized for fast Rician fading scenarios, the method not only adopts an efficient quantization algorithm, but also innovatively uses a probability mass function evolution mechanism based on minimum sum decoding, in order to reduce the construction complexity, the GE method is further provided, which is realized by approximating the Rician fading channel as a binary input Gaussian white noise channel, and the numerical simulation results show that the two methods can construct polar codes with excellent performance in fast Rician fading channels. BRIEF DESCRIPTION OF DRAWINGS

[0025] Figure 1 is the method step diagram of the whole application;

[0026] Figure 2 is the schematic diagram of N=8 polar tree structure proposed by the application;

[0027] Figure 3 is the bit channel error rate comparison diagram of the DDE method and the Monte Carlo simulation method of the application;

[0028] Figure 4 is the FER performance diagram of the polar code constructed by the PW method of the application;

[0029] Figure 5 is the SC performance diagram of different polar codes constructed by the PW, DDE and AGE methods of the application;

[0030] Figure 6 is the SCL performance diagram of different polar codes constructed by the PW, DDE and AGE methods of the application;

[0031] Figure 7 is the SC performance diagram of different polar codes constructed by the PW, DDE and AGE methods of the application. DETAILED DESCRIPTION

[0032] The technical solutions in the embodiments of the application will be clearly and completely described below with reference to the drawings in the embodiments of the application. Obviously, the described embodiments are only part of the embodiments of the application, rather than all the embodiments of the application. Based on the embodiments in the application, all other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the application.

[0033] First embodiment: the application provides a technical solution: a polar code construction method based on discrete density evolution, comprising the following steps:

[0034] S101: channel modeling and system parameter initialization:

[0035] First, an accurate mathematical model of fast Rician fading channel is established, and the received signal model is represented as:

[0036]

[0037] where y i represents the i-th received symbol, h i represents the channel gain, c i represents the BPSK modulated code symbol, z i represents a zero-mean Gaussian noise with variance and satisfies: mean: E[zi ] = 0, variance: Each component is independent and identically distributed, and the probability density function of the channel gain h of the Lutz distribution is:

[0038]

[0039] where I0(·) represents the zero-order first-order modified Bessel function, A represents the amplitude of the direct path (specular reflection) component, σ 2 represents the variance of the multipath (diffuse reflection) component, and the average fading power is normalized to a unit value, satisfying:

[0040] E[h 2 ] = A 2 + 2σ 2 = 1

[0041] The severity of channel attenuation is characterized by the Lutz K factor, which is defined as the ratio of direct path power to multipath scattering power:

[0042]

[0043] The K factor is defined as the ratio of direct path power to multipath scattering power, reflecting the relative strength of the direct path in the channel;

[0044] This step establishes a complete fast Lutz fading channel model, providing an accurate description of the channel environment for subsequent polar code construction. The channel model takes into account the effects of direct path and multipath scattering, and can accurately reflect the actual propagation characteristics in unmanned aerial vehicle air-ground direct connection communication;

[0045] S102: Log-likelihood ratio statistical property analysis and calculation:

[0046] Under the condition that the perfect channel state information (CSI) is known at the receiving end, the statistical property analysis and calculation of the log-likelihood ratio (LLR) of the received symbol are performed, and the specific steps are as follows:

[0047] First, based on the Lutz fading channel model established in step S101, the log-likelihood ratio of each received symbol y i is calculated:

[0048]

[0049] Under the condition that the channel state h is known, the log-likelihood ratio L is subject to a Gaussian distribution, and its distribution characteristics are determined by the following parameters:

[0050]

[0051] The specific probability density function expression of this conditional distribution is:

[0052]

[0053] To obtain the complete statistical properties of the log-likelihood ratio, its marginal probability density function needs to be calculated. This is obtained by integrating over the distribution of the channel gain h.

[0054]

[0055] Where p(h) is the Rice distribution probability density function defined in step S101;

[0056] This step establishes a complete mapping from the channel model to the log-likelihood ratio statistical properties, providing a theoretical basis for the subsequent discretization of the probability density function, determining the probability distribution form of LLR, and providing a probability density function for the quantization process. Through precise analysis of the log-likelihood ratio statistical properties, the effectiveness and accuracy of the discrete density evolution method under Ricean fading channels can be ensured.

[0057] S103: Discretization of the probability density function:

[0058] Based on the received symbol log-likelihood ratio (LLR) probability density function obtained in step S102, discretization is performed to reduce computational complexity. The specific implementation steps are as follows:

[0059] First, a negligible probability threshold ∈>0 is set, which determines the accuracy requirement of the discretization process. The initial quantization interval of the log-likelihood ratio L is determined using Chebyshev's inequality, ensuring that the probability of L falling outside this interval does not exceed ∈. The upper and lower bounds of the initial interval are given by the following formula:

[0060]

[0061] Where E(L) and D(L) represent the expected value and variance of the log-likelihood ratio L, respectively; however, the initial interval [l] determined by Chebyshev's inequality... min ,l max The expression is too broad; to improve computational efficiency, it needs to be further refined into a more compact interval. The refinement process must satisfy the following boundary probability constraints:

[0062]

[0063] This means that outside the refined interval (i.e. )and The sum of the probability masses does not exceed 2∈, where, and Constrained to integer values ​​to facilitate discretization, and finally, within the final determined refinement interval. Inside, sample the continuous probability density function p(L) at integer lattice points to construct a discrete probability mass function

[0064]

[0065] This discrete probability mass function provides a numerical calculation basis for the subsequent evolution of the probability mass function on the polarization tree. Through the above discretization process, the continuous probability distribution is successfully transformed into a discrete probability distribution, significantly reducing the complexity of subsequent processing while ensuring the calculation accuracy, providing key technical support for the efficient construction of polar codes in fast Rayleigh fading channels;

[0066] S104: Recursive evolution of the probability mass function on the polarization tree:

[0067] Perform recursive evolution of the probability mass function (PMF) sequence on the polarization tree to estimate the reliability of each bit channel. The specific steps are as follows:

[0068] Root node initialization:

[0069] Initialize the PMF sequence of the root node (0,0) of the polarization tree:

[0070]

[0071] where is the discrete probability mass function obtained in step S103;

[0072] Recursive update of the PMF sequence of child nodes:

[0073] For each layer 0 ≤ s < n and each node 0 ≤ t < 2 s , recursively update the PMF sequences of the two child nodes according to the coding structure of the polar code:

[0074] PMF update of the left child node:

[0075]

[0076] PMF update of the right child node:

[0077]

[0078] where 0 ≤ i ≤ 2 n-S-1 ;

[0079] Definition of convolution operation:

[0080] Convolution operation and ★ correspond to functions f and g respectively:

[0081] Definition of function f:

[0082]

[0083] Definition of function g:

[0084]

[0085] Due to the high computational complexity of function f, the min-sum approximation is usually adopted in practical applications:

[0086]

[0087] Explanation of the polarization tree structure:

[0088] For a polarization code with code length N = 2 n the corresponding polarization tree has n + 1 layers, where the root node is at the 0th layer, the nodes at the s-th (0 ≤ s < n) layer have two children nodes at the (s + 1)-th layer, and the length of the node (s, t) is l (s,t) = 2 n-s ;

[0089] The PMF evolution process has the following characteristics:

[0090] The computational complexity is O(N·M 2 ), where M is the size of the PMF support set, and as the depth of the polarization tree increases, the PMF support set will expand rapidly. At the same time, in practical implementation, the support set pruning technology can be used to control the computational complexity. Finally, the evolution process assumes that all-zero codewords are transmitted, based on the symmetric channel assumption;

[0091] Through the above recursive evolution process, the reliability of each bit channel can be accurately estimated, providing a basis for subsequent information set selection. This method effectively overcomes the problem of too high computational complexity of the original density evolution method in the Rice fading channel through discretization;

[0092] S105: Bit channel error probability calculation and information set selection:

[0093] Based on the probability mass function corresponding to each bit channel after evolution, calculate the error probability of each bit channel under the genie-aided successive cancellation (GASC) decoding, and complete the selection of the information set. The specific calculation formula is as follows:

[0094]

[0095] where represents the probability mass function corresponding to the t-th bit channel at the bottom layer (the n-th layer) of the polarization tree, is the left boundary of the quantization interval, and the min(·, 1 / 2) operation ensures that the error probability does not exceed 1 / 2. Then, select the K bit channels with the smallest error probability to form the information set Its mathematical expression is:

[0096]

[0097] in Ensure the information set size meets the encoding requirements. Representing all possible K-ary subsets, the argmin operation selects the subset that minimizes the total error probability. The bit channels not selected into the information set constitute the frozen set. These positions are set to a fixed value of 0, that is:

[0098]

[0099] in Information set The complement of, i.e.:

[0100]

[0101] The error probability calculation is based on an important characteristic of polar codes under GASC decoding: the block error probability can be estimated by summing the error probabilities of all bit channels in the information set. This characteristic allows us to optimize the selection of the information set by accurately estimating the error probability of each bit channel, thereby constructing a polar code with excellent performance under fast Ricean fading channels. Through the above steps, the bit channel reliability assessment and information set selection based on the discrete density evolution method were successfully completed, laying a solid foundation for the subsequent construction of polar codes.

[0102] S106: Polar Code Construction and Information Transmission:

[0103] Based on the information set determined in the previous step The final construction of the polar code and its transmission are completed through the following steps:

[0104] The sequence of information to be transmitted Embedded into a data carrier sequence of length N In, the following conditions must be met: in Information set The corresponding position carries information bits. This indicates that the corresponding position in the frozen set is set to a fixed value of 0, and the final codeword is generated through polarization transformation: c = vG N G N It is a polarization transformation matrix of size N×N. The generated codeword c is modulated using binary phase shift keying (BPSK) and transmitted through a fast fading channel, ultimately forming the receive vector. Each received symbol yi Given by the following formula: Where h i Z represents the channel gain. i The variance is Given zero-mean Gaussian noise and perfect channel state information (CSI) at the receiver, to activate the polarity decoder, the log-likelihood ratio (LLR) sequence of the received symbols needs to be calculated. The LLR of each received symbol is referred to in step S102:

[0105]

[0106] Finally, these log-likelihood ratios are fed as input to the polarity decoder, and through successive cancellation (SC) decoding or list decoding (SCL), an estimate of the transmitted sequence is obtained, ultimately outputting the recovered information sequence. The entire communication process is completed, thus successfully achieving a code length of N=2. n The complete construction and transmission process of a polar code with information bit length K in a fast Ricean fading channel.

[0107] Second embodiment: A polar code construction method based on Gaussian equivalence, including the following steps:

[0108] S201: Equivalent approximation of Rice channel to BI-AWGN channel:

[0109] This step approximates the fast Ricean fading channel as an equivalent-capacity binary-input additive white Gaussian noise (BI-AWGN) channel to reduce the complexity of subsequent polar code construction. The specific equivalent process is as follows:

[0110] The original Ricean fading channel: An approximate equivalent BI-AWGN channel is y' = h'·(-1). c +z', where h i For constant equivalent channel gain, z i The variance is Zero-mean Gaussian noise, where c is the transmitted code symbol. This equivalent process is based on the channel capacity preservation principle, that is, by selecting appropriate equivalent parameters h' and This approximation ensures that the equivalent BI-AWGN channel has the same channel capacity as the original Ricean fading channel. This approximation lays the foundation for the subsequent use of the Gaussian approximation density evolution method. The parameter determination in the equivalent channel model depends on the original Ricean fading channel model established in step S101. The original Ricean distribution characteristics reference the probability density function expression in S101, and the channel gain statistical characteristics reference the normalization condition E[h] in S101. 2] = 1, receiving symbol statistical properties referencing the statistical property analysis of the log-likelihood ratio in S102, equivalent parameter h' and The determination of needs to be based on the Ricean channel statistical characteristics established above, including the expected and variance of the received symbols, which can be calculated using the channel model and statistical characteristic formulas in S101 and S102.

[0111] S202: Solving the moment matching equation and determining the equivalent parameters:

[0112] The equivalent channel parameters are determined using the moment matching method, ensuring that the equivalent channel maintains the same statistical characteristics as the original Ricean channel. Based on the Ricean fading channel model established in step S101 and the statistical characteristics of the received symbols in step S102, the moment matching equation is established as follows:

[0113] E(y') = E(y), D(y') = D(y)

[0114] Where E(y') and D(y') represent the expectation and variance of the received symbols in the equivalent channel, respectively, and E(y) and D(y) represent the expectation and variance of the received symbols in the original Ricean channel, respectively.

[0115] Based on the moment matching equation, the equivalent channel parameters are obtained as follows:

[0116]

[0117] Where h' is the equivalent channel gain. D[h] is the equivalent noise variance, and D[h] is the variance of the channel gain in step S101. The variance of the Gaussian noise in step S101;

[0118] In practical calculations, the calculation of E(y) requires the Rice distribution characteristics in step S101 and the received symbol statistical characteristics in step S102. The calculation of D[h] requires the reference to the channel gain statistical characteristics in step S101. All parameter calculations are based on the channel gain normalization condition E[h] in step S101. 2 ] = 1;

[0119] The equivalent parameters obtained through the moment matching equations described above ensure that the equivalent BI-AWGN channel maintains the same mean and variance statistical characteristics as the original Ricean fading channel, providing an accurate parameter basis for the subsequent Gaussian approximation density evolution method.

[0120] S203: Log-likelihood ratio mean sequence initialization:

[0121] After completing the equivalent approximation from Rice channel to BI-AWGN channel and determining the equivalent parameters, it is necessary to initialize the log-likelihood ratio (LLR) mean sequence of the root node of the polarization tree to provide the starting conditions for the subsequent recursive evolution based on Gaussian approximation.

[0122] The initialization formula for the LLR mean sequence of the root node (0,0) is as follows:

[0123]

[0124] Where h' is the equivalent channel gain determined in step S202. The equivalent noise variance determined in step S202, N = 2 n This is the code length of the polar code;

[0125] The derivation of this initialization formula is based on the LLR statistical property theory established in step S102. Under the BI-AWGN channel, the log-likelihood ratio follows a Gaussian distribution and satisfies:

[0126]

[0127] Therefore, the mean of LLR is:

[0128]

[0129] That is, the mean of LLR is directly determined by the equivalent channel parameters;

[0130] Characteristics of the initialization process:

[0131] Uniform initialization: The initial value of the mean LLR at all N locations is the same, which reflects the statistical stationarity of the channel;

[0132] Parameter dependency: The initial values ​​are entirely determined by the equivalent channel parameters h' and Decide;

[0133] Computational efficiency: The initialization process is simple to calculate, involving only basic arithmetic operations;

[0134] Comparison with discrete density evolution methods:

[0135] Compared with the discrete density evolution method, the Gaussian equivalent method has significant advantages in the initialization stage, avoiding complex probability density function calculation and discretization process, and does not require numerical integration and quantization processing. At the same time, the computational complexity is significantly reduced, making it particularly suitable for long code construction.

[0136] S204: Recursive evolution of the log-likelihood ratio mean on a polarization tree:

[0137] Recursive evolution of the mean sequence of the log-likelihood ratio (LLR) is performed on the polarization tree to estimate the reliability of each bit channel. This method is based on Gaussian approximation density evolution (DEGA), and the specific process is as follows:

[0138] Based on the initialization result of step S203, mean evolution starts from the root node. The LLR mean sequence of the root node (0, 0) has been initialized as:

[0139]

[0140] For each layer s (0 ≤ s < n) and each node t (0 ≤ t < 2 s ), the LLR mean sequences of the two child nodes are recursively updated according to the coding structure of the polar code:

[0141] Update of the LLR mean of the left child node:

[0142]

[0143] Update of the LLR mean of the right child node:

[0144]

[0145] where 0 ≤ i ≤ 2 n-s-1 , represents the i-th element in the LLR mean sequence of node (s, t);

[0146] The definitions of the functions φ(x) and φ -1 (y):

[0147] Used to approximate the evolution of the LLR mean, and its defined expression is:

[0148]

[0149] Implementation details:

[0150] The evolution process starts from the root node (0, 0), and its LLR mean sequence has been initialized as For the left child node, the update involves the composite operation of the functions φ(x) and φ -1 , which approximates the effect of the function f in the original density evolution. For the right child node, the update is a simple addition, corresponding to the linear property of the function g;

[0151] Evolution characteristics: The computational complexity is O(NlogN), significantly lower than the discrete density evolution method; based on the Gaussian assumption, only the mean evolution needs to be tracked, avoiding complex probability distribution calculations; the piecewise design of the functions φ and φ -1 ensures computational accuracy and numerical stability; and the polarization tree structure adopted in this step is exactly the same as the polarization tree described in step S104;

[0152] Through this recursive evolution, the Gaussian equivalent method significantly reduces the construction complexity while maintaining performance comparable to the discrete density evolution method, making it suitable for polar code construction in fast Ricean fading channels.

[0153] S205: Error Probability Calculation and Information Set Selection:

[0154] The error probability of each bit channel is calculated based on the polarization tree evolution results, and the final selection of the information set is completed. The specific implementation process is as follows:

[0155] The error probability of each bit channel is calculated using the following formula:

[0156]

[0157] Where: Q(·) represents the standard Gaussian Q-function, defined as: This represents the mean log-likelihood ratio of the t-th bit channel at the lowest (n-th) node after polarization tree evolution;

[0158] The Gaussian Q-function has the following important properties:

[0159] Q(x) is a monotonically decreasing function.

[0160] As x→+∞, Q(x)→0.

[0161] As x→-∞, Q(x)→1;

[0162] Information set selection:

[0163] Select the K bits of the channel with the lowest error probability to form the information set.

[0164]

[0165] Where: t1, t2, ..., t K It is the channel index of the first K bits sorted by error probability in ascending order, satisfying and:

[0166]

[0167] Theoretical basis:

[0168] The error probability calculation formula is based on the core assumption of the Gaussian approximate density evolution method. Under the BI-AWGN channel, the log-likelihood ratio follows a Gaussian distribution and satisfies the following:

[0169] D(L)=2E(L)

[0170] This feature enables accurate estimation of the error probability by only tracking the evolution of the LLR mean, greatly reducing the computational complexity;

[0171] Through the above steps, the construction of polar codes under fast Rice fading channels is successfully completed based on the Gaussian equivalent method, achieving a good balance between performance and complexity.

[0172] The third embodiment: Provide a detailed implementation method of the polarization tree structure, and elaborate on the specific implementation method of the polarization tree structure in the construction of polar codes. This structure is a core component in the encoding and decoding process of polar codes, providing a computational framework for the reliability evaluation of bit channels. The specific steps are as follows;

[0173] S301: Establish the basic structure of the polarization tree:

[0174] Determine the tree structure parameters:

[0175] For a polar code with a code length of N = 2 n , establish the corresponding polarization tree structure. The total number of layers of this polarization tree is:

[0176] Total number of layers = n + 1

[0177] Define the node levels: The root node is at the 0th layer; the leaf nodes are at the nth layer; each node at the s-th layer (where 0 ≤ s < n) has two child nodes at the s + 1-th layer;

[0178] Node identification system:

[0179] Each node is uniquely identified by a pair (s, t), where s represents the layer where the node is located, and t represents the node number from left to right within that layer, satisfying 0 ≤ t < 2 s ;

[0180] Calculate the node length:

[0181] The sequence length corresponding to the node (s, t) is:

[0182] l (s,t) = 2 n-s

[0183] This length formula is exactly the same as the node length defined in step S104, ensuring the unity of the computational framework;

[0184] Properties of the tree structure:

[0185] There are 2 S nodes in the s-th layer; the total number of paths from the root node to the leaf nodes is N = 2 n ; Among them, the tree structure is completely symmetric, reflecting the recursive construction characteristics of polar codes;

[0186] Reference for structure visualization:

[0187] For easier understanding, please refer to the document. Figure 2 The diagram shown is a schematic of a polarization tree structure with code length N=8. This diagram clearly shows the connection relationships and length variation patterns between nodes at each level.

[0188] The polarization tree structure established through the above steps provides a complete framework for the subsequent evolution of the probability mass function and the calculation of the mean log-likelihood ratio, ensuring the systematicness and computability of the polar code construction process.

[0189] S302: Node data structure definition:

[0190] In the polarization tree structure, each node (s,t) contains two key data sequences, used to store log-likelihood ratio information and hard bit estimation information, respectively, as defined below:

[0191] Log-likelihood ratio sequence definition:

[0192] Each node (s,t) contains a log-likelihood ratio sequence of length l(s,t), defined as:

[0193]

[0194] in: This represents the log-likelihood ratio at the i-th position of the node; l (s,t) =2 n-s This represents the length of node (s,t); this sequence is used for reliability calculations and bit estimation during decoding.

[0195] Hard bit estimation sequence definition:

[0196] Each node (s,t) also contains a hard bit estimation sequence of length l(s,t), defined as:

[0197]

[0198] in: This represents the hard bit estimate of the i-th position of the node; This represents a binary field, with a value of 0 or 1; this sequence is used to store intermediate decoding results and final bit decisions during the decoding process.

[0199] Sequence length relationship:

[0200] The sequence length of node (s,t) satisfies the following relationship:

[0201] l (s,t) =2 n-s

[0202] This relationship ensures the regularity of the polarization tree structure, with the sequence length decreasing exponentially from the root node to the leaf node;

[0203] Data storage structure:

[0204] In practical implementation, these two sequences can be stored using data structures such as arrays or linked lists. The specific storage format is as follows:

[0205] LLR sequence:

[0206] HBE sequence:

[0207] This data structure design provides the foundation for subsequent sequence updates and recursive calculations, ensuring the efficient implementation of the polar code encoding and decoding process;

[0208] S303: Sequence initialization:

[0209] In the construction and decoding of polar codes, sequence initialization is a crucial starting step, and the specific initialization process is as follows:

[0210] The LLR sequence initialization formula for the root node (0,0) is:

[0211]

[0212] Where L i Let be the log-likelihood ratio of the i-th received symbol, and its calculation formula is:

[0213]

[0214] The meanings of the parameters in the formula are as follows: y i h is the i-th received symbol; i Let the channel gain be i, which follows a Rice distribution. c is the variance of the Gaussian noise; i This refers to the i-th transmission code character;

[0215] During initialization, the following conditions must also be met: all N LLR values ​​constitute the LLR sequence of the root node; the hard bit estimation sequence β (0,0) Initialize to an all-zero sequence; assume the receiver has perfect channel state information (CSI);

[0216] This initialization process provides the foundation for subsequent recursive calculations of the polar tree, ensuring the correct starting point for polar code construction. Through precise LLR value initialization, the reliability of each bit channel can be accurately evaluated, laying the foundation for the selection of the optimal information set.

[0217] S304: LLR sequence recursive update rule:

[0218] Recursive updates of the log-likelihood ratio (LLR) sequence on the polarization tree are the core computational process for polar code construction and decoding. The specific update rules are as follows:

[0219] Left child node LLR update:

[0220] For the left child node, its LLR sequence is updated according to the following rules:

[0221]

[0222] Where: 0≤s <n,0≤t<2 s , 0≤i≤2 n-s-1

[0223] Right child node LLR update:

[0224] For the right child node, its LLR sequence is updated according to the following rules:

[0225]

[0226] Where: 0≤s <n,0≤t<2 s , 0≤i≤2 n-s-1 For the specific definitions of functions f and g, please refer to step S104;

[0227] Computational complexity analysis:

[0228] The computational complexity of this recursive update process is: each node requires 2... n-s-1 Calculation of the quadratic function; total processing required. The computation is performed in 2^n times; the overall complexity is O(NlogN), where N = 2^n. n ;

[0229] This recursive update process makes full use of the structural characteristics of polar codes and achieves channel polarization by gradually merging channels, laying the foundation for subsequent reliability estimation and information set selection.

[0230] S305: Hard Bit Estimation Sequence Update Rule:

[0231] The update rules for the hard bit estimation sequence are divided into two cases based on the node's position in the polarization tree: leaf nodes and non-leaf nodes. The specific update rules are as follows:

[0232] Leaf node (nth level) update rules:

[0233] For the leaf nodes at the bottom level of the polarization tree (s=n, 0≤t<2), n The update formula for the hard-bit estimated sequence is:

[0234]

[0235] Where: denotes the hard bit estimate of the \(t\)-th leaf node, is the information set, which contains the channel indices for transmitting information bits, is the log-likelihood ratio of the corresponding node;

[0236] Update rule for non-leaf nodes (layer \(s\), \(0\leq s\lt n\)):

[0237] For non-leaf nodes (\(0\leq s\lt n\), \(0\leq t\lt 2 s ), the update of the hard bit estimate sequence adopts the combination rule:

[0238]

[0239] Where: \(0\leq i\leq 2 n-s-1 ; denotes the modulo-2 addition operation; \((\cdot,\cdot)\) denotes the sequence concatenation operation; and are the hard bit estimate sequences from the left and right child nodes respectively;

[0240] Explanation of the update process:

[0241] The update process of the hard bit estimate sequence has the following characteristics: it proceeds in reverse from the leaf nodes to the root node; the hard bit estimate of the leaf nodes is determined based on the sign of the log-likelihood ratio; the hard bit estimate of the non-leaf nodes is obtained by combining the estimates of the child nodes; the whole process forms a complete two-way calculation with the update of the log-likelihood ratio sequence;

[0242] Mathematical expression form:

[0243] The above update rules can be uniformly expressed as:

[0244] For \(s = n\):

[0245]

[0246] For \(0\leq s\lt n\):

[0247]

[0248] Where is the indicator function;

[0249] This update rule of the hard bit estimate sequence and the update rule of the log-likelihood ratio sequence together constitute the core algorithm of the polarization code SC decoding, providing a theoretical basis for the construction and performance evaluation of the polarization code;

[0250] S306: Calculation of the final estimate value:

[0251] Based on the hard bit estimation sequence of each node in the polarization tree, the final estimate of the transmission sequence is calculated. The specific calculation process is as follows:

[0252] Leaf node hard bit estimation sequence extraction:

[0253] Extract hard bit estimates from all leaf nodes at the bottom level (nth level) of the polarization tree to form a complete estimation sequence:

[0254]

[0255] in: This represents the hard bit estimate of the t-th leaf node in the n-th layer, where t = 0, 1, ..., N-1, corresponding to all n-bit channels. The value can be 0 or 1;

[0256] Information sequence estimate extraction:

[0257] From the complete estimated sequence Extract information set The corresponding bit positions yield an estimate of the transmitted information sequence:

[0258]

[0259] in: Ensure that the information bit length meets the encoding requirements. Represents a sequence In the index set Projection on;

[0260] Mathematical properties of the estimated value:

[0261] The final estimated value satisfies the following mathematical relationship:

[0262]

[0263] in: It is the polarization transformation matrix G N The inverse matrix of the polar code, due to the linearity of the polar code, ensures the consistency of encoding and decoding;

[0264] Estimation error analysis:

[0265] The estimation error of the transmitted sequence can be evaluated by comparing the original sequence and the estimated sequence:

[0266]

[0267] in It is an indicator function that takes the value 1 when the condition is true and 0 otherwise.

[0268] Information bit estimation error:

[0269] Specifically, the estimation error for the information bits is:

[0270]

[0271] This error directly reflects the decoding performance of polar codes under given channel conditions;

[0272] Through the above steps, the complete decoding process from the received signal to the final information sequence estimation is completed, providing technical support for reliable communication under fast Ricean fading channels;

[0273] S307: Fundamentals of Error Probability Estimation

[0274] Under Genie-Aided Successive Cancellation (GASC) decoding, the frame rate error probability of the polar code can be estimated by summing the error probabilities of all bit channels in the information set. Specifically, the upper bound of the frame rate error probability (FER) is given by the following formula:

[0275]

[0276] in: The information set, i.e., the set of indices of the K optimal bit channels carrying information bits, is Pr{E t} represents the probability of a decoding error occurring on the t-th bit channel. Furthermore, the lower bound of the frame rate error probability can be estimated using the minimum error probability in the information set.

[0277]

[0278] In the process of constructing polarization coding, information sets are selected. The core objective is to minimize the upper bound of the frame rate error probability given the information length K:

[0279]

[0280] This theoretical basis comes from the successive cancellation decoding characteristic of polar codes: under the GASC decoding assumption, the decoding error events of each information bit can be approximated as independent of each other, so the error probability of the entire codeword can be estimated by the sum of the error probabilities of each information bit.

[0281] The error probability estimation framework established in this step provides a theoretical basis for the optimized construction of polar codes under fast Ricean fading channels, ensuring that the constructed polar codes can achieve optimal reliability performance in UAV air-to-ground direct communication environments.

[0282] Fourth embodiment: Simulation results:

[0283] like Figure 3 As shown, the coding length in this simulation is N = 256. The fast Ricean fading channel is characterized by the signal-to-noise ratio ES / N0 = 2.5dB and the Ricean factor κ = 1. Experimental results show that the DDE method can accurately simulate the bit channel error rate obtained from the Monte Carlo simulation, and the error between the two is minimal.

[0284] like Figure 4 As shown, the coding length in this simulation is N=256 and the information length is K=128, demonstrating the performance of polar codes. It can be observed that in the high signal-to-noise ratio region, the calculated upper bound is in high agreement with the actual FER performance curve.

[0285] Figure 5 and Figure 6 The performance of polar codes constructed using the PW, DDE, and GE methods in SC and SCL is compared. All codes maintain a fixed length N=256, but the dimensions are different, K∈{32,64,96,128,160,192}. It is observed that the DDE method performs best overall. It is worth noting that as the list size in SCL decoding increases, its performance advantage becomes more and more significant. The GE method performs on par with the DDE method and is always better than the PW method.

[0286] Figure 7 The SC performance of polar codes with code length N=256 and dimension K=128 in Rice fading channels (Rice factors κ=0, 1 and 10) is shown. It can be found that the DDE method maintains the best performance under all parameter values, and the GE method can approach the performance of DDE in all scenarios and outperform the PW method.

[0287] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0288] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for constructing a polar code suitable for a Ricean fading channel, characterized in that: The method comprises the following steps: S1: modeling a fast Rician fading channel, and establishing a received signal model, wherein a received symbol is composed of a channel gain, a modulated code word and Gaussian noise, the channel gain obeys a Rician distribution, a probability density function of the Rician distribution comprises a zero-order first-type modified Bessel function, and an average fading power is normalized to a unit value, and a channel attenuation degree is represented by a K factor of a ratio of a direct path power to a multipath scattering power; S2: based on the received signal model established in S1, calculating a log-likelihood ratio (LLR) of the received symbol under the condition that perfect channel state information is known at a receiving end, obtaining statistical characteristics of the LLR, and estimating error probabilities of bit channels in a polar code by using a discrete density evolution method or a Gaussian equivalent method; S3: selecting K channels with minimum error probabilities from all N bit channels by sorting and comparing according to the error probabilities estimated in S2, and constructing an information set A for carrying information bits, wherein bit channels not selected into the information set A are set to fixed values; S4: constructing a complete polar code by using the information set A determined in S3, embedding an information sequence to be transmitted into a data carrier with a length of N, and generating a final code word by a polarization transformation, so as to complete a polar code construction process with a code length of N and an information bit length of K. 2.The method for constructing a polar code suitable for a Ricean fading channel according to claim 1, wherein: The discrete density evolution method specifically comprises: First, the probability density function of the received symbol log-likelihood ratio obtained in S2 is discretized into a probability mass function by a quantization process, then a recursive evolution of the probability mass function sequence is performed on a polarization tree, wherein for a left child node of each node, a convolution operation rule of a corresponding function f is used for probability mass function updating, and for a right child node, a convolution operation rule of a corresponding function g is used for probability mass function updating, and finally, according to the probability mass function corresponding to each bit channel after the evolution is completed, the error probability of the bit channel is calculated by accumulating the probability value of the negative log-likelihood ratio region. 3.The method of claim 2, wherein: The quantization process specifically comprises: First, the initial quantization interval is determined by using Chebyshev inequality, wherein the lower bound is the expectation of the log-likelihood ratio minus the square root of the variance of the log-likelihood ratio and a preset probability ratio, and the upper bound is the expectation of the log-likelihood ratio plus the square root of the variance of the log-likelihood ratio and the preset probability ratio, then the interval is further refined by boundary probability constraint to ensure that the probability outside the left boundary and the right boundary is not more than a preset threshold, and finally, the continuous probability density function is converted into a discrete probability mass function in the refined interval, and the quantization process is completed. 4.The method of claim 1, wherein: The Gaussian equivalent method specifically comprises: First, the Rician fading channel in S1 is approximated as a binary input additive white Gaussian noise channel with equivalent capacity, then the parameters of the equivalent channel are determined by a moment matching equation, wherein the equivalent channel gain is taken as the expectation of the received symbol, and the equivalent noise variance is taken as the sum of the channel gain variance and the original noise variance, then a recursive evolution of a log-likelihood ratio mean value sequence is performed on the polarization tree, wherein the left child node adopts a mean value updating rule based on the function φ, and the right child node adopts an additive updating rule, and finally, according to the log-likelihood ratio mean value of each bit channel after the evolution is completed, the error probability of the bit channel is calculated by using a Gaussian Q function.

5. The method of claim 4, wherein the method is applied to a polar code for a Ricean fading channel. The moment matching equation is specifically expressed as: The equivalent channel gain h ' is equal to the expectation E(y) of the received symbol y, the equivalent noise variance is equal to the variance D(h) of the channel gain h ' plus the original noise variance i.e. h ' =E(y) and The equation preserves the statistical identity of the equivalent channel and the original Rician channel. 6.The method for constructing a polar code suitable for a Ricean fading channel according to claim 4, characterized in that: The error probability calculation specifically includes: According to the theoretical basis of the Gaussian approximation density evolution method, the error probability of each bit channel is calculated by the formula where Q(·) represents a standard Gaussian Q function, represents the log-likelihood ratio mean value of the tth bit channel at the bottom layer node after the polarization tree evolution.

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