A MIMO-BP detection method

Abstract

The present invention discloses a MIMO-BP detection method, which relates to the field of communications technology. By selectively calculating factor node messages and utilizing the sparsity of message transmission, the calculation complexity of factor node messages is significantly reduced. The method also improves the parallelism of message calculation and the hardware implementation efficiency of the algorithm. A lattice representation and a pathfinding strategy are used to improve the original serial message update strategy, which is based on the searched path, and parallel message updates are achieved along different paths. The method also improves the parallelism of message calculation and the hardware implementation efficiency of the algorithm. A lattice representation and a pathfinding strategy are used to improve the original serial message update strategy, which is based on the searched path, and parallel message updates are achieved along different paths.

Description

Technical Field

[0001] The present invention relates to the field of communication technology, and in particular to a MIMO-BP detection method. Background Art

[0002] The ever-increasing demand for communication poses increasingly severe challenges to wireless communication technology. To achieve the goals of the fifth and sixth generations of wireless communication (5G) and 6G, multiple-input multiple-output (MIMO) technology, which offers excellent spectrum efficiency (SE) and energy efficiency (EE), has gradually developed antennas with hundreds or even thousands of antennas. However, the expansion of antenna size also poses significant challenges to signal detection.

[0003] Among current detection algorithms, maximum a posteriori probability (MAP) detection is considered to offer the best detection performance. However, the computational complexity of MAP detection grows exponentially with the number of transmit and receive antennas, limiting its widespread application. Consequently, minimum mean square error (MMSE) detection and other inversion-free linear iterative detection algorithms have gained widespread application due to their low complexity. However, these algorithms can suffer significant performance degradation in harsh wireless communication environments.

[0004] Recently, belief propagation (BP) detection algorithms have attracted considerable attention due to their excellent trade-off between error rate performance and computational complexity, as well as their easy-to-implement hardware architecture. In 2008, a fully connected BP detection algorithm proposed by some researchers demonstrated near-optimal detection performance, but its computational complexity increased exponentially. Two improved versions of the BP detection algorithm, BP with Gaussian approximation of interference (GAI-BP) and real-domain GAI-BP (RD-GAI-BP), further reduced the complexity of the BP detection algorithm. However, these algorithms introduce exponential operations and suffer from performance floor effects. To address this issue, researchers have proposed the belief-selective propagation (BsP) detection algorithm. The BsP detection algorithm achieves high error rate performance while effectively reducing computational complexity through a selective message update method.

[0005] The current BP detection algorithm has the following problems:

[0006] 1) BP algorithms that implement iterative message updates based on factor graph models (FGMs) have high message computation complexity at factor nodes (FNs), which is a key factor hindering the widespread application of BP detection algorithms.

[0007] 2) The hardware efficiency of BP detection algorithms is limited by the serial computation of messages at factor nodes. In traditional hardware designs, BP detection algorithms typically use serial computation to update messages at factor nodes. Due to the high complexity of message updates and the nature of serial computation, this leads to significant latency in updating messages at factor nodes. Summary of the Invention

[0008] The technical problem to be solved by the present invention is to overcome the deficiencies of the prior art and provide a large-scale multi-input multi-output belief propagation MIMO-BP detection method, thereby further reducing the computational complexity of BP-type detection algorithms.

[0009] The present invention adopts the following technical solutions to solve the above technical problems:

[0010] A MIMO-BP detection method proposed in the present invention is as follows:

[0011] Step 1: Obtain the received signal vector y, channel matrix H and noise variance according to the received signal, channel information and noise information.

[0012] Step 2: Based on the received signal vector y, channel matrix H and noise variance Calculate the output result of the minimum mean square error (MMSE) method detection and use it to initialize the α message in the Trellis-BP detection algorithm. α is the prior message, and the α message is used to start the iterative message update of the Trellis-BP algorithm.

[0013] Step 3: Update the β message according to the grid representation and the pathfinding strategy. The β message is the LLR format of the posterior message, and update the γ message and α message. The γ message is the soft information of the sent symbol. When the α message, β message and γ message update reaches the set upper limit of the number of iterations Q L When , the iteration stops and the γ message is used to calculate and output the bit soft information;

[0014] The grid representation and pathfinding strategy refer to:

[0015] A corresponding grid is established based on the prior information input to each factor node. Secondly, a path set for each grid is obtained based on the path-finding strategy. Then, the a posteriori information of the constellation points included in the path set is calculated and updated.

[0016] As a further optimization scheme of the MIMO-BP detection method described in the present invention, an α message is a message transmitted from a symbol node to a factor node in a factor graph in a BP detection algorithm, and a β message is a message transmitted from a factor node to a symbol node in a factor graph in a BP detection algorithm.

[0017] As a further optimization scheme of the MIMO-BP detection method described in the present invention, the lattice representation method refers to the BPMIMO detection algorithm representing the message passing process based on the factor graph, which is as follows:

[0018] A factor graph is a bipartite graph. It includes two types of nodes: factor nodes and symbol nodes. A grid is established for each factor node, and the path is found based on the grid. The grid of the i-th factor node is N r Represents the number of receiving antennas, each grid consists of Row and 2N t The columns consist of cells, N t is the number of transmitting antennas, each cell includes a constellation point and the prior information corresponding to the constellation point; specifically, The cell in row m and column t is denoted as They represent the constellation point in the cell of the mth row and tth column in the i-th grid and the prior information of the constellation point, respectively. represents the set of constellation points, Indicates the number of constellation points in the constellation point set; in, Represents the constellation point in the grid of the i-th factor node and the set of prior information corresponding to the constellation point; the unit cells in each column of the grid are sorted from large to small according to the size of the prior information of the constellation point in the cell, that is, is the prior information in the cell of the mth row and tth column in the i-th grid; the first row of each grid contains the 2N with the highest confidence t constellation points, the first row includes 2N t The row vector composed of constellation points is called the root path P i 0 ,Right now Select the first n from each column in the grid m cells and the cell containing the constellation point μ1 form a subset of the grid, which is used to update the posterior message. The subsets are represented as follows:

[0019]

[0020] in, express A subset of the tth column, They represent the constellation point in column t and the subset of the prior information corresponding to the constellation point, represents the constellation point μ1 in the tth column of the i-th grid, for The corresponding prior message.

[0021] As a further optimization scheme of the MIMO-BP detection method described in the present invention, the pathfinding strategy is as follows:

[0022] Each path is a 2N t A vector of constellation points, where the constellation points are arranged in columns from Select the obtained, for the i-th factor node, all the found paths are combined into a path set Among them, n c Indicates the offset of each path constellation point in the path set relative to the root path constellation point;

[0023] The path set is represented as follows:

[0024]

[0025] in, represents the path set corresponding to the i-th grid, represents the kth path of the i-th grid, is the t-th element of the k-th path, is the constellation point selected from the t-th column of the grid, 1≤t≤2N t ;n m Represents each column of the grid The minimum number of constellation points included;

[0026] operate As a constraint, used for statistical path Compared to the root path The element offset number, sign represents the Kronecker product, the Operation ensures n c The offset constellation points come from different columns of the grid;

[0027] Select the sending symbol vector s, f i A posteriori message grid Among them, f i is the i-th factor node, is the posterior message of the i-th factor node, and the specific formula for updating the posterior message is as follows:

[0028]

[0029] Furthermore, the β message, i.e. the LLR format of the posterior message, is calculated as follows:

[0030]

[0031] Among them, l represents the current number of iterations, Indicates that in the lth iteration, the calculation from f i To the jth symbol node S j During the message process, the a posteriori message of the constellation point μ1; Indicates that at the lth iteration, the first factor node f i To the jth symbol node S j constellation points β news; represents the posterior message of the constellation point in the cell of the mth row and jth column in the grid corresponding to the i-th factor node in the l-th iteration, Represents the constellation point in the cell of the mth row and jth column in the grid corresponding to the i-th factor node, y i represents the i-th received signal, h i represents the i-th row of the channel matrix, Represents the prior message sent by the kth symbol node to the ith factor node.

[0032] Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0033] (1) This technical solution proposes a Trellis-BP detection algorithm. The design goal of this algorithm is to improve the hardware friendliness of the BP detection algorithm while maintaining good error performance, and further reduce the computational complexity of the BP detection algorithm; low computational complexity: by selectively calculating factor node messages and utilizing the sparsity of message transmission, the computational complexity of factor node messages is significantly reduced; the parallelism of message calculation is improved, and the hardware implementation efficiency of the algorithm is improved: the original serial message update strategy is improved to the path based on the search by using the trellis representation and path finding strategy, and the parallel update of messages is realized according to different paths;

[0034] (2) The Trellis-BP algorithm proposed in this technology can not only improve the parallelism of message calculation at factor nodes, thereby improving the hardware implementation efficiency of the algorithm, but also effectively reduce the complexity of message calculation at factor nodes by selectively calculating the posterior message. Since the pathfinding strategy is specifically developed based on the reliability of the prior message, the Trellis-BP algorithm has almost no performance loss compared with other BP algorithms.

[0035] (3) This paper proposes a low-complexity, high-parallelism BP detection algorithm. Compared with the most advanced BSP detection algorithm, in Rayleigh channels, 8×4 antenna ratio uncoded MIMO systems, and modulation orders of 64-QAM and 256-QAM, the algorithm achieves gains of 0.43dB and 0.50dB, respectively, at a bit error rate of 10-3. In this simulation environment, the BSP algorithm reduces its complexity by 77.38%. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 It is the factor graph of BP detection algorithm.

[0037] Figure 2 This is an example diagram of the technical details of the Trellis-BP detection method of the present invention.

[0038] Figure 3 The figures show the bit error rate (BER) variation curves of the Trellis-BP detection method of the present invention and LMMSE, EBRDF, RD-GAI-BP, BsP, and KSD in an uncoded MIMO system with an 8×4 antenna ratio and a {64, 256}-QAM modulation order under a Rayleigh channel; (a) is the bit error rate in an uncoded MIMO system with a 64-QAM modulation order, and (b) is the bit error rate in an uncoded MIMO system with a 256QAM modulation order.

[0039] Figure 4 The BER change curves of the Trellis-BP detection method of the present invention and LMMSE, RD-GAI-BP, BsP, and KSD in an uncoded MIMO system with a 128×64 antenna ratio and a 256-QAM modulation order, where (a) is the Rayleigh channel and (b) is the WINNER-II channel.

[0040] Figure 5 It is the Trellis-BP detection algorithm in the present invention. DETAILED DESCRIPTION

[0041] In order to make the objectives, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0042] Existing BP detection algorithms are limited by the high complexity of factor node message updates and their serial computational nature, resulting in significant latency in factor node message updates. To further reduce computational complexity and hardware design challenges, a Trellis-BP detection algorithm was proposed. This algorithm aims to maintain good bit error performance while improving hardware friendliness and reducing computational complexity.

[0043] The Trellis-BP algorithm primarily utilizes two novel methods to update the message at each factor node: a trellis representation and a pathfinding strategy. Specifically, a corresponding trellis is first established based on the a priori message input to each factor node. Second, the pathfinding strategy is used to derive a path set for each trellis. Finally, the a posteriori message is calculated and updated only for the constellation points included in the path set. Furthermore, the a posteriori message for each path in the path set is calculated independently.

[0044] By implementing the two aforementioned methods, the proposed Trellis-BP algorithm not only increases the parallelism of message computation at factor nodes, thereby improving the algorithm's hardware implementation efficiency, but also effectively reduces the complexity of message computation at factor nodes by selectively calculating posterior messages. Because the pathfinding strategy is specifically based on the reliability of prior messages, the Trellis-BP algorithm exhibits virtually no performance loss compared to other BP-based algorithms.

[0045] A hardware-friendly and low-complexity MIMO-BP detection method comprises the following steps:

[0046] Step 1: Obtain the received signal vector y, channel matrix H and noise variance according to the received signal, channel information and noise information.

[0047] Step 2: Based on the received signal vector y, channel matrix H and noise variance Calculate the output of the MMSE method detection and use it to initialize the α message in the Trellis-BP detection algorithm to start the iterative message update of the Trellis-BP algorithm.

[0048] Step 3: Update the β message according to the grid representation and pathfinding strategy proposed in this scheme. Then, update the γ message and α message according to formulas (5) and (4) respectively. When the message update reaches the set upper limit of the number of iterations Q L The iteration stops when . The bit soft information is calculated and output according to formula (6).

[0049] Consider the sender as N r A single-antenna mobile terminal, the receiving end is equipped with N t A large-scale MIMO communication system with slow fading in the base station uplink with 100 antennas. Assume that the transmitted symbol vector is The average power of each symbol in the vector is 1. represents the set of complex field mapping constellation points and M represents the modulation order. The received signal vector is expressed as The complex domain mathematical model of MIMO system air interface transmission can be expressed as:

[0050]

[0051] in, represents the complex channel matrix subject to Rayleigh fading, where express The i-th row of Each element in is independent and identically distributed, and it obeys a normal distribution with a mean of 0 and a variance of 1, that is, is a noise vector, where all elements have a mean of 0 and a variance of σ n 2 The complex Gaussian distribution of

[0052] It is assumed that the channel state information (CSI) is completely known at the receiving end.

[0053] In large-scale MIMO detection, the real value decomposition (RVD) method is usually used to rewrite the MIMO system complex model. The complex domain model (1) is rewritten as:

[0054] y=Hs+n (2)

[0055] in, and Represent the operations of taking the real part and imaginary part respectively, is the set of all real numbers, is the set of constellation points mapped to the actual transmitted symbols. The Trellis-BP detection method proposed in this invention is an improvement on the MIMOBP detection method. Therefore, this method first introduces the algorithmic principles of the BP detection method. The BP detection algorithm is a detection algorithm based on FGM for iterative message updates. Figure 1 The FGM of BP detection algorithm in the real field is presented.

[0056] Usually, an FGM contains two types of nodes: factor nodes and symbol nodes (SN), which are represented by f i and S j Specifically, the message transmitted by the i-th FN to the j-th SN is denoted as β ij , and its specific update formula is:

[0057]

[0058] Where l represents the number of iterations, μ k Represents a set of constellation points The kth constellation point in ; Indicates that at the lth iteration, the tth factor node f i To the jth symbol node S j Constellation point μ k β message; y i represents the i-th element in y, h i represents the i-th row of H; σ n 2 represents the noise variance; s represents the transmitted symbol vector.

[0059] The message sent by the jth SN to the ith FN is recorded as The update formula is:

[0060]

[0061] The messages passed from symbolic nodes to factor nodes are collectively referred to as α, where Indicates the constellation point μ from the jth symbol node to the ith factor node at the Ith iteration k Corresponding α message. Indicates that at the lth iteration, the tth factor node f i To the jth symbol node S j Constellation point μ k β news;

[0062] When the BP detection method reaches the maximum number of iterations Q L Stop message β ij and α ji Then the BP detection algorithm is updated according to the β ij Message, calculate the soft information of each transmitted symbol according to the following formula:

[0063]

[0064] γ represents the overall soft information of the transmitted symbol. Specifically, γ j (μ k ) indicates that the jth transmitted symbol constellation point is μ k Corresponding soft information. tj (μ k ) represents the number of nodes f from the tth factor i To the jth symbol node S j Constellation point μ k Beta news.

[0065] Finally, based on the calculated symbol soft information, that is, the symbol log-likelihood ratios (LLRs), the bit soft information in each symbol is output. The specific calculation is as follows:

[0066]

[0067] Among them, b i Represents the i-th bit in the transmitted symbol.

[0068] In existing hardware implementations of BP detection algorithms, the processing elements (PEs) of the FN must perform complex posterior message calculations serially, severely reducing hardware implementation efficiency and throughput. To address this issue, the Trellis-BP algorithm was proposed to increase the parallelism of FN message calculations. Furthermore, Trellis-BP further improves hardware implementation throughput by reducing the computational complexity of the posterior messages.

[0069] The Trellis-BP hardware-friendly low-complexity detection method includes:

[0070] a) Grid representation: In the Trellis-BP detection algorithm proposed in this scheme, in order to more intuitively display the prior information input by each factor node, a grid is established for each factor node. The grid of the i-th factor node is like Figure 2 As shown, each grid consists of Row and 2N t Each cell contains a constellation point and its corresponding prior information. Specifically, The cell in row m and column t is denoted as Therefore, the i-th grid can be expressed as in and Represents the set of constellation points and their corresponding prior messages. The cells in each column of the grid are sorted from large to small according to the size of their prior messages, that is, Therefore, the first row of each grid contains the 2N most confident t constellation points. The row vector formed by them is called the root path (rootpath), recorded as Right now Furthermore, select the first n m The cells and the cells containing the constellation point μ1 form a subset of the grid, which is used to update the posterior message. It is expressed as follows:

[0071]

[0072] in, express A subset of the tth column, that is, Next, the proposed pathfinding strategy will be applied on each established grid. The details of the pathfinding strategy will be given in the next section.

[0073] b) Pathfinding strategy: The core idea of ​​the pathfinding strategy is to find the path for updating the posterior message based on the grid established on the factor nodes according to certain rules. The pathfinding strategy aims to improve the parallelism of the posterior message update and reduce the message calculation complexity of the factor nodes to a certain extent. Each path is a 2N t A vector of constellation points, where the constellation points are arranged in columns from For the i-th factor node, all the found paths are combined into a path set, recorded as where n c Indicates the offset of each path constellation point relative to the root path constellation point in the path set. Therefore, the path set has the following representation:

[0074]

[0075] Among them, the operation As a constraint, used for statistical path Compared to the root path The number of elements offset by . represents the Kronecker product. The purpose of this operation is to ensure that n c The offset constellation points are from different columns of the grid.

[0076] c) Calculation of a posteriori information: Similar to the grid construction of a priori information, a posteriori information can also be displayed through the grid. i The posterior message grid is denoted as Depend on Figure 2 "Establishing a grid" and "Message calculation" correspond to the grid. and The structure is the same, the only difference is Each cell stores the posterior message Rather than prior information In the specific update of the a posteriori message, the constellation points not included in the path set are not calculated for the a posteriori message, and their initial values ​​are set to 0. According to formula (3) and the pathfinding strategy proposed by the present invention, the specific formula for updating the a posteriori message is as follows:

[0077]

[0078] Furthermore, the β message, i.e. the LLR format of the posterior message, is calculated as follows:

[0079]

[0080] Where l represents the current number of iterations. Indicates that in the lth iteration, the calculation from f i to S j During the message process, the a posteriori message of constellation point μ1 is calculated. According to formula (9), it can be inferred that the updates to the a posteriori message by different paths are independent of each other. If multiple paths pass through a cell in the grid, the a posteriori message value contained in the cell may undergo multiple updates until its maximum value is obtained. Therefore, the processing unit of the factor node can calculate the a posteriori messages of different paths in parallel, thereby improving the overall hardware implementation efficiency of the algorithm.

[0081] Figure 2 This paper shows how the Trellis-BP detection algorithm can update the posterior message and gives a specific example. In this scheme, the update of the posterior message includes the following three steps: grid establishment, path finding and message calculation. These three steps have been Figure 2 First, in the figure of the "Building a Grid" step, an N t=2 when f i Grid It can be seen is Each circle in the figure represents a cell point, and different colors are used to distinguish cells with different meanings. A dotted rounded rectangle frame frames the cells in the first two rows, indicating that this example sets n m =2. Figure 2 The "Find Path" step shows three paths, one root path and two deviation paths. It can be seen that the number of deviation constellation points between the two deviation paths and the root path is 1, that is, n c = 1. Therefore, in this example, the set of paths used is Figure 2 The grid corresponding to “Message Calculation” shows the specific a posteriori message calculation. i The a posteriori information establishes the grid The cell (square) contains the currently calculated a posteriori message value. You can see that only The constellation points are included. And the cells in the path b are offset The calculation of the posterior message includes the calculation of the likelihood message and the calculation of the prior message. As you can see, the calculation results of these two parts are -55.3 and 79.7 respectively. Therefore, in the example

[0082] Simulation results show that the proposed Trellis-BP detection algorithm shows near-optimal performance in uncoded large-scale and medium-scale MIMO systems in Rayleigh and WINNER II channels. Figure 3 and Figure 4 Specifically, in a Rayleigh channel, 8×4 antenna ratio uncoded MIMO system, when the modulation order is 64-QAM and 256-QAM respectively, the Trellis-BP detection algorithm of the present invention has a gain of 1.5dB and 1.2dB respectively when compared with LMMSE at a bit error rate of 10-4; compared with the BSP detection algorithm, it has a gain of 0.43dB and 0.50dB respectively when the bit error rate is 10-3. The specific simulation results are shown in Figure 3 , Figure 3 (a) is the bit error rate of the uncoded MIMO system with 64-QAM modulation order. Figure 3 (b) in the figure is the bit error rate of the uncoded MIMO system with 256-QAM modulation order.

[0083] The above content not only introduces the technical details of this solution in detail, but also describes the implementation steps of this solution in detail. The overall description of the algorithm proposed in this solution is as follows Figure 5 shown.

[0084] Under 128×64 antenna ratio, for 256-QAM uncoded MIMO system, when the channels are Rayleigh channels (such as Figure 4 (a) in the figure) and the WINNER-II channel (as shown in Figure 4 As shown in (b) in the figure), the Trellis-BP detection algorithm of the present invention achieves gains of 2.9dB and 3.3dB respectively when the bit error rate is 10-4 compared to the LMMSE method. For specific simulation results, please refer to Figure 4 .

[0085] The above description is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by a technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

Claims

1. A MIMO-BP detection method, characterized in that: The details are as follows: Step 1: Obtain the received signal vector y, channel matrix H and noise variance according to the received signal, channel information and noise information. Step 2: Based on the received signal vector y, channel matrix H and noise variance Calculate the output result of the minimum mean square error (MMSE) method detection and use it to initialize the α message in the Trellis-BP detection algorithm. α is the prior message, and the α message is used to start the iterative message update of the Trellis-BP algorithm. Step 3: Update the β message according to the grid representation and the pathfinding strategy. The β message is the LLR format of the posterior message, and update the γ message and α message. The γ message is the soft information of the sent symbol. When the α message, β message and γ message update reaches the set upper limit of the number of iterations Q L When , the iteration stops and the γ message is used to calculate and output the bit soft information; The grid representation and pathfinding strategy refer to: A corresponding grid is established based on the prior information input to each factor node. Secondly, a path set for each grid is obtained based on the path-finding strategy. Then, the a posteriori information of the constellation points included in the path set is calculated and updated.

2. A MIMO-BP detection method according to claim 1, characterized in that: α message is the message passed from symbol node to factor node in the factor graph of BP detection algorithm, and β message is the message passed from factor node to symbol node in the factor graph of BP detection algorithm.

3. A MIMO-BP detection method according to claim 1, characterized in that: The lattice representation method refers to the BP MIMO detection algorithm showing the message passing process based on the factor graph, as follows: A factor graph is a bipartite graph. It includes two types of nodes: factor nodes and symbol nodes. A grid is established for each factor node, and the path is found based on the grid. The grid of the i-th factor node is N r Represents the number of receiving antennas, each grid consists of Row and 2N t The columns consist of cells, N t is the number of transmitting antennas, each cell includes a constellation point and the prior information corresponding to the constellation point; specifically, The cell in row m and column t is denoted as They represent the constellation point in the cell of the mth row and tth column in the i-th grid and the prior information of the constellation point, respectively. represents the set of constellation points, Indicates the number of constellation points in the constellation point set; in, Represents the constellation point in the grid of the i-th factor node and the set of prior information corresponding to the constellation point; the unit cells in each column of the grid are sorted from large to small according to the size of the prior information of the constellation point in the cell, that is, is the prior information in the cell of the mth row and tth column in the i-th grid; the first row of each grid contains the 2N with the highest confidence t constellation points, the first row includes 2N t The row vector of constellation points is called the root path Right now Select the first n from each column in the grid m cells and the cell containing the constellation point μ1 form a subset of the grid, which is used to update the posterior message. The subsets are represented as follows: in, express A subset of the tth column, They represent the constellation point in column t and the subset of the prior information corresponding to the constellation point, represents the constellation point μ1 in the tth column of the i-th grid, for The corresponding prior message.

4. A MIMO-BP detection method according to claim 3, characterized in that: The pathfinding strategy is as follows: Each path is a 2N t A vector of constellation points, where the constellation points are arranged in columns from Select the obtained, for the i-th factor node, all the found paths are combined into a path set Among them, n c Indicates the offset of each path constellation point in the path set relative to the root path constellation point; The path set is represented as follows: in, represents the path set corresponding to the i-th grid, represents the kth path of the i-th grid, is the t-th element of the k-th path, is the constellation point selected from the t-th column of the grid, 1≤t≤2N t ;n m Represents each column of the grid The minimum number of constellation points included; operate As a constraint, used for statistical path Compared to the root path The element offset number, sign represents the Kronecker product, the Operation ensures n c The offset constellation points come from different columns of the grid; Select the sending symbol vector s, f i A posteriori message grid Among them, f i is the i-th factor node, is the posterior message of the i-th factor node, and the specific formula for updating the posterior message is as follows: Furthermore, the β message, i.e. the LLR format of the posterior message, is calculated as follows: Among them, l represents the current number of iterations, Indicates that in the lth iteration, the calculation from f i To the jth symbol node S j During the message process, the a posteriori message of the constellation point μ1; Indicates that at the lth iteration, the first factor node f i To the jth symbol node S j constellation points β news; represents the posterior message of the constellation point in the cell of the mth row and jth column in the grid corresponding to the i-th factor node in the l-th iteration, Represents the constellation point in the cell of the mth row and jth column in the grid corresponding to the i-th factor node, y i represents the i-th received signal, h i represents the i-th row of the channel matrix, Represents the prior message sent by the kth symbol node to the ith factor node.

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