A low complexity multi-user information processing method for an interleaved frequency division multiple access system

Abstract

The application relates to the technical field of communication, in particular to a low-complexity multi-user information processing method for an interleaved frequency division multiple access system, which comprises the following steps: a receiving end receives receiving signals of multiple time slots; for each time slot, an iterative processing is performed on the receiving signal by using a multi-user cross-domain orthogonal approximate message passing receiver to obtain a decoding signal after the iterative processing; and the decoding signal is determined as sending information of each user. The method can significantly reduce the calculation complexity of multi-user detection, improve the reliability of multi-user detection, avoid error propagation, provide a basis for system capacity analysis and optimal coding design, realize a breakthrough in theoretical performance, and further improve the overall transmission performance of the system while maintaining low complexity.

Description

Technical Field

[0001] The embodiments of this application relate to the field of communication technology, and in particular to a low-complexity multi-user information processing method for interleaved frequency division multiple access systems. Background Technology

[0002] In high-speed mobile communication scenarios, supporting reliable access for massive numbers of devices has become an urgent requirement. However, the multipath effect, Doppler shift, and strong inter-user interference caused by high-speed mobility pose challenges to 5G systems based on Orthogonal Frequency Division Multiplexing (OFDM). Therefore, Multicarrier Non-Orthogonal Multiple Access (MC-NOMA) technology has become a research hotspot. By superimposing user signals in the same resource domain (such as power and codeword), it can effectively improve system access capabilities. Typical solutions include Interleave Frequency Division Multiple Access (IFDMA); in addition, there are solutions combining Orthogonal Time Frequency Space (OTFS) modulation with Sparse Code Multiple Access (SCMA), and solutions combining Affine Frequency-Division Multiplexing (AFDM) with SCMA.

[0003] In MC-NOMA systems, receivers require high-precision, low-complexity multi-user detection algorithms to separate superimposed user signals. However, existing solutions suffer from the following problems: Firstly, traditional detection algorithms, such as serial interference cancellation, suffer from error propagation issues, while approximate message-passing algorithms struggle to achieve theoretically optimal performance in complex time-varying channels. Particularly for IFDMA systems, the core linear minimum mean square error detector in the receiver involves large-scale matrix inversion operations, resulting in extremely high computational complexity and making it impractical.

[0004] On the other hand, to achieve transmission close to channel capacity, an optimal channel coding scheme needs to be designed, which relies on a precise analysis of the limits of system information theory. Existing capacity analyses and optimal coding criteria for approximate message-passing receivers are usually based on ideal assumptions such as the channel matrix satisfying a right-unitary invariant distribution and a fixed eigenvalue distribution. However, in actual high-speed moving multipath channels, the channel matrix does not possess these ideal characteristics; its eigenvalue distribution changes rapidly and over a wide range. This makes existing theories unsuitable for direct application to MC-NOMA systems, resulting in a lack of theoretical guidance for the design of optimal coding schemes and hindering further improvements in system performance.

[0005] Therefore, how to design a multi-user detection algorithm that balances low complexity and high reliability, and on this basis, break through the limitations of the ideal assumptions of existing theories, accurately analyze the system capacity to guide the design of the optimal coding scheme, is a technical problem that urgently needs to be solved to realize a high-performance MC-NOMA system. Summary of the Invention

[0006] To address at least one of the aforementioned technical problems, embodiments of this application propose a low-complexity multi-user information processing method for interleaved frequency division multiple access systems. This significantly reduces the computational complexity of multi-user detection, improves the reliability of multi-user detection, avoids error propagation, and provides a foundation for system capacity analysis and optimal coding design, achieving a breakthrough in theoretical performance. Ultimately, this improves the overall transmission performance of the system while maintaining low complexity.

[0007] To achieve the above objectives, embodiments of this application propose a low-complexity multi-user information processing method for interleaved frequency division multiple access systems, the method comprising the following steps: The receiving end receives signals from multiple time slots; wherein, the received signals are obtained by multiple users modulating the signals via interleaved frequency division multiple access and transmitting them through the channel; For the received signal in each time slot, iterative processing is performed using a multi-user cross-domain orthogonal approximation message passing receiver to obtain the decoded signal after iterative processing; The iterative process includes iterative processes of time-domain linear detection, cross-domain transformation, symbol-domain nonlinear detection, and cross-domain inverse transformation; time-domain linear detection is used to perform linear minimum mean square error detection in the time domain, and includes low-complexity calculation of the multi-user joint detection matrix based on block matrix inversion; symbol-domain nonlinear detection includes symbol demodulation and channel decoding; The decoded signal is identified as the information sent by each user.

[0008] To achieve the above objectives, embodiments of this application also propose a low-complexity multi-user information processing system for interleaved frequency division multiple access systems, the system comprising: The receiving module is used to receive signals from multiple time slots at the receiving end; wherein the received signals are obtained by multiple users modulating the signals via interleaved frequency division multiple access and transmitting them through the channel; An iterative processing module is used to iteratively process the received signal in each time slot using a multi-user cross-domain orthogonal approximate message passing receiver to obtain the decoded signal after iterative processing. The iterative processing includes iterative processes of time-domain linear detection, cross-domain transformation, symbol-domain nonlinear detection, and cross-domain inverse transformation. Time-domain linear detection is used to perform linear minimum mean square error detection in the time domain, and includes low-complexity calculations based on block matrix inversion of the multi-user joint detection matrix. Symbol-domain nonlinear detection includes symbol demodulation and channel decoding. The determination module is used to identify the decoded signal as the information sent by each user.

[0009] To achieve the above objectives, embodiments of this application also propose an electronic device, including: a processor and a memory, wherein the memory stores instructions executable by the processor, and the processor is configured to execute the instructions such that the electronic device can implement a low-complexity multi-user information processing method for an interleaved frequency division multiple access system as described above.

[0010] To achieve the above objectives, embodiments of this application also propose a computer-readable storage medium storing a computer program that, when executed by a processor, enables a low-complexity multi-user information processing method for an interleaved frequency division multiple access system as described above.

[0011] This application proposes a low-complexity multi-user information processing method for interleaved frequency division multiple access systems. First, the receiver receives signals from multiple time slots. These signals are obtained by multiple users modulating the signals via interleaved frequency division multiple access and transmitting them through a channel. Then, for each time slot's received signal, iterative processing is performed using a multi-user cross-domain orthogonal approximate message passing receiver to obtain a decoded signal after iterative processing. Finally, the decoded signal is determined as the transmitted information for each user. The iterative processing includes iterative processes of time-domain linear detection, cross-domain transformation, symbol-domain nonlinear detection, and cross-domain inverse transformation. Since time-domain linear detection is used to perform linear minimum mean square error detection in the time domain, it involves low-complexity calculations based on block matrix inversion of the multi-user joint detection matrix. This approach reduces the computational complexity of multi-user detection and improves its practicality. It uses a multi-user cross-domain orthogonal approximation message-passing receiver for iterative processing, ensuring that the input-output errors of time-domain linear detection and symbol-domain nonlinear detection remain uncorrelated in each iteration, thus improving the reliability of multi-user detection and preventing error propagation. Furthermore, the iterative processing using the multi-user cross-domain orthogonal approximation message-passing receiver allows for the design of optimal codes that approximate the receiver's maximum achievable rate, based on channel characteristics and user rate requirements. Therefore, this scheme significantly reduces the computational complexity of multi-user detection, improves its reliability, prevents error propagation, and provides a foundation for system capacity analysis and optimal coding design, achieving a breakthrough in theoretical performance and ultimately improving the overall transmission performance of the system while maintaining low complexity. Attached Figure Description

[0012] To more clearly illustrate the technical solutions in the embodiments or related technologies of this application, the accompanying drawings used in the description of the embodiments or related technologies of this application will be briefly introduced below. Obviously, the following drawings are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. The drawings described herein are only used to explain this application and are not intended to limit this application.

[0013] Figure 1 This is a flowchart of a low-complexity multi-user information processing method for an interleaved frequency division multiple access system provided in one embodiment of this application; Figure 2 This is a framework diagram of an interleaved frequency division multiple access system provided in one embodiment of this application; Figure 3 This is a framework diagram of a low-complexity multi-slot multi-user cross-domain orthogonal approximation message passing receiver provided in one embodiment of this application; Figure 4 This is provided in one embodiment of the present application. Amplitude diagram; Figure 5 This is provided in one embodiment of the present application. Amplitude diagram; Figure 6 This is provided in one embodiment of the present application. Amplitude diagram; Figure 7 This is provided in one embodiment of the present application. Amplitude diagram; Figure 8 This is provided in one embodiment of the present application. Amplitude diagram; Figure 9 This is a comparison chart of the bit error rate performance of a low-complexity multi-slot multi-user cross-domain orthogonal approximation message passing receiver and the original multi-user cross-domain orthogonal approximation message passing receiver provided in one embodiment of this application; Figure 10 This is a schematic diagram of an equivalent multi-user cross-domain orthogonal approximate message passing receiver provided in one embodiment of this application; Figure 11 This is a framework diagram of a variational SE of an equivalent multi-user cross-domain orthogonal approximation message passing receiver provided in one embodiment of this application; Figure 12 This is a schematic diagram of a variational SE transition curve provided in one embodiment of this application; Figure 13 This is a schematic diagram of the MSE curve of a multi-user cross-domain orthogonal approximate message passing receiver and SE analysis provided in one embodiment of this application; Figure 14 This is a schematic diagram illustrating the achievable rate of a multi-user cross-domain orthogonal approximation message passing receiver provided in one embodiment of this application; Figure 15 This is a schematic diagram comparing the bit error rate performance of an interleaved frequency division multiple access system under an optimal coding scheme provided in one embodiment of this application with that of an existing scheme; Figure 16 This is a schematic diagram of the structure of a low-complexity multi-user information processing system for an interleaved frequency division multiple access system provided in another embodiment of this application; Figure 17 This is a schematic diagram of the structure of an electronic device provided in another embodiment of this application. Detailed Implementation

[0014] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the various embodiments of this application will be described in detail below with reference to the accompanying drawings. Those skilled in the art will understand that many technical details have been presented in the embodiments of this application to facilitate better understanding. However, the technical solutions claimed in this application can be implemented even without these technical details and various variations and modifications based on the following embodiments. The division of the following embodiments is for ease of description and should not constitute any limitation on the specific implementation of this application. The following embodiments can be combined with and referenced by each other without contradiction.

[0015] In high-speed mobile communication scenarios, supporting reliable access for massive numbers of devices has become an urgent requirement. However, the multipath effect, Doppler shift, and strong inter-user interference caused by high-speed mobility pose challenges to 5G systems based on Orthogonal Frequency Division Multiplexing (OFDM). Therefore, Multicarrier Non-Orthogonal Multiple Access (MC-NOMA) technology has become a research hotspot. By superimposing user signals in the same resource domain (such as power and codeword), it can effectively improve system access capabilities. Typical solutions include Interleave Frequency Division Multiple Access (IFDMA); in addition, there are solutions combining Orthogonal Time Frequency Space (OTFS) modulation and Sparse Code Multiple Access (SCMA) technology, as well as solutions combining Affine Frequency-Division Multiplexing (AFDM) and SCMA technology.

[0016] In MC-NOMA systems, receivers require high-precision, low-complexity multi-user detection algorithms to separate superimposed user signals. Multi-user detection algorithms improve the receiver's anti-interference performance, which directly determines the communication performance of a practical MC-NOMA system. However, existing detection algorithms used in MC-NOMA systems suffer from problems such as error accumulation, high complexity, and difficulty in achieving optimal performance.

[0017] Furthermore, channel coding is crucial in wireless communication, effectively resisting channel fading and signal impairments, thereby improving system reliability. Especially in MC-NOMA systems, good channel coding helps to effectively separate user signals. However, most encoding and decoding schemes currently used in MC-NOMA systems are based on point-to-point (P2P) channel designs, which can only combat channel noise and ignore interference between multiple users. To further improve system performance, some current research designs optimal channel encoding and decoding schemes based on information matching criteria between detectors and decoders. For example, existing technical solutions are as follows: Multi-user detection algorithms: For power-domain NOMA–OTFS systems, the paper "Nonorthogonal multipleaccess with orthogonal time-frequency space signal transmission" superimposes user symbols at different power levels onto the same delay-Doppler (DD) resource block and uses a Linear Minimum Mean Square Error-Successive Interference Cancellation (LMMSE-SIC) receiver for signal recovery. For OTFS–SCMA systems, the paper "OTFS-SCMA: Acode-domain NOMA approach for orthogonal time-frequency space modulation" introduces two SCMA codeword mapping schemes along the Doppler and delay domains and employs a two-stage receiver, first performing LMMSE channel estimation and then SCMA symbol detection. The paper "Low-complexity memory AMP detector for high-mobilityMIMO OTFS SCMA systems" proposes a low-complexity DD-domain memory approximate message passing (MAMP) receiver for large-scale OTFS–SCMA systems. Furthermore, for AFDM–SCMA systems, the paper "AFDM-SCMA: A promising wave form for massive connectivity overhigh mobility channels" describes how each user's signal is mapped to SCMA codewords, modulated and transmitted via AFDM, and then recovered at the receiver using an orthogonal approximate message passing (OAMP) detector.For IFDMA systems, the paper "IFDMA for massive connectivity over high mobility channels" distinguishes users by assigning different interleavers and proposes a MU-CD-OAMP receiver accordingly. In the time domain, it performs joint multi-user linear minimum mean square error (MU-LMMSE) detection and orthogonality between users, and in the symbol domain, it performs independent posterior probability decoding and orthogonality for each user. It also updates the estimated signals of users through cross-domain iteration, thereby ensuring reliable communication of the system.

[0018] Optimal channel coding design: For right-unitary invariant channel matrices and arbitrary input signals, the paper "On capacityoptimality of OAMP: Beyond IID sensing matrices and gaussian signaling" presents the optimal coding principle based on the achievable rate analysis of OAMP receivers and proves that OAMP can achieve the constrained capacity of P2P generalized multiple input multiple output (GMIMO) systems. For generalized multi-user multiple input multiple output (GMU-MIMO) systems, the paper "Constrained capacity optimal generalized multi-user MIMO: Atheoretical and practical framework" groups users according to different communication rates, analyzes the achievable rate of multi-user OAMP receivers under asymmetric group constraints, and provides corresponding multi-user optimal coding design criteria. The paper "Memory AMP for generalized MIMO: Coding principle and information-theoretic optimality" presents the achievable rate analysis and optimal coding principle of low-complexity MAMP in GMIMO and proves the information-theoretic optimality of MAMP.

[0019] However, the existing solutions have the following problems: For existing multi-user detection methods, in power-domain MC-NOMA systems, the SIC receiver eliminates inter-user interference through user-specific detection. However, this method is prone to error propagation problems, and there is currently no effective solution. For OTFS / AFDM-SCMA systems, the equivalent channel matrix does not satisfy right-unitary invariance, causing approximate message-passing algorithms (such as OAMP and MAMP) to fail to achieve Bayesian optimal or capacity-optimal detection performance. For IFDMA systems, although the MU-CD-OAMP receiver can achieve good performance, its MU-LMMSE detector involves a large number of high-dimensional matrix inversion operations, making it difficult to fully utilize the sparse structure of the multi-user time-domain channel, resulting in extremely high computational complexity for the MU-CD-OAMP receiver.

[0020] Existing optimal coding schemes, based on OAMP and MAMP algorithms, typically rely on the assumptions that the channel matrix satisfies right-unitary invariance and that the eigenvalue distribution is uniquely invariant. However, in real-world high-speed, time-varying multipath channels, the channel matrix does not satisfy the right-unitary invariance assumption, and the eigenvalue distribution changes rapidly and over a wide range. Therefore, existing achievable rate analyses and optimal coding design criteria cannot be directly applied. Furthermore, MC-NOMA systems still lack in-depth analysis of their theoretical performance limits, and the design problem of optimal coding schemes remains unsolved.

[0021] Therefore, how to design a multi-user detection algorithm that balances low complexity and high reliability, and on this basis, break through the limitations of the ideal assumptions of existing theories, accurately analyze the system capacity to guide the design of the optimal coding scheme, is a technical problem that urgently needs to be solved to realize a high-performance MC-NOMA system.

[0022] In view of this, embodiments of this application provide a low-complexity multi-user information processing method for interleaved frequency division multiple access systems. Specifically, the method relates to a low-complexity multi-user cross-domain orthogonal approximate message passing (MU-CD-OAMP) receiver; furthermore, embodiments of this application also relate to an information theory limit analysis of the multi-slot MU-CD-OAMP receiver and the corresponding optimal coding principle.

[0023] In summary, based on the shortcomings of the prior art, the embodiments of this application are used to achieve the following objectives: First, we design a low-complexity, high-reliability MU-CD-OAMP detection algorithm that can make full use of the quasi-band structure of the time-domain channel, simplify the complex calculation problem of MU-LMMSE, and solve the interference problem faced by IFDMA system in scenarios with massive user access, thus ensuring the actual reliable communication requirements.

[0024] Second, this study analyzes the information theory limits of the IFDMA system, providing theoretical support for the design of the optimal coding scheme and filling the theoretical gap in the performance limit analysis of existing MC-NOMA schemes.

[0025] Third, we study the optimal coding criteria for multiple users, decoupling the complex multi-user joint capacity analysis problem into an independent single-user problem. Based on this, we study the optimal coding criteria for multiple users applicable to high-speed mobile non-stationary channels, overcoming the dependence of existing analysis methods on the ideal assumption that the channel matrix satisfies right unitary invariance and the eigenvalue distribution is unique and invariant. This enables us to optimize and approximate capacity coding schemes for users or user groups with different rate requirements.

[0026] Fourth, by using a multi-user optimal coding scheme, the technical problem of poor convergence in traditional point-to-point channel design is solved, and the robustness to different speed settings for multiple users is improved.

[0027] One embodiment of this application proposes a low-complexity multi-user information processing method for interleaved frequency division multiple access systems, applied to an electronic device, wherein the electronic device can be a terminal or a server. This embodiment and the following embodiments will use a server as an example for description. The implementation details of the low-complexity multi-user information processing method for interleaved frequency division multiple access systems proposed in this embodiment will be described in detail below. The following implementation details are provided for ease of understanding and are not necessary for implementing this solution.

[0028] The specific process of the low-complexity multi-user information processing method for interleaved frequency division multiple access systems proposed in this embodiment can be described as follows: Figure 1 As shown, it includes: Step 101: The receiving end receives signals from multiple time slots.

[0029] The received signal is obtained by multiple users modulating the signal via interleaved frequency division multiple access and transmitting it through the channel.

[0030] like Figure 2 As shown, Figure 2 A multi-user, multi-slot IFDMA system is provided as an embodiment of this application. The system includes... Single antenna user and The receiving end of a receiving antenna.

[0031] At the sending end, each user message vector After channel coding, a codeword vector is generated. And obtain the symbol vector through constellation mapping. Next, the symbol vector Time-domain signals in different time slots are obtained by modulation using a target modulation matrix. See the following examples for details.

[0032] The target modulation matrix is ​​any one of the following: an interleaved frequency division multiplexing modulation matrix, an interleaved block sparse transformation matrix, a random multiplexing modulation matrix, a random unitary matrix, or a modulation matrix obtained by combining a universal class matrix with a sensing matrix.

[0033] If the target modulation matrix is ​​an interleaved frequency division multiplexing modulation matrix Then the user time domain signal The calculation formula is as follows: On the sending end, the user time domain signal From symbol vector After user-specific interleaved frequency division multiplexing modulation matrix The modulation is obtained, and its calculation formula is as follows: ; in, Indicates user The interlacing matrix; Represents the inverse fast Fourier transform matrix; Indicates user index, .

[0034] Ultimately, all users' time-domain signals pass through the channel. Then, the receiving end signal Represented as: ; in, Indicates the time slot index. ; ,express The receiving antenna is at the first Received signals in each time slot; Indicates the first Channel matrix for each time slot, ; express The transmitting antenna is at the ... User signals transmitted in each time slot; , representing additive white Gaussian noise in the channel.

[0035] Step 102: For the received signal in each time slot, perform iterative processing using a multi-user cross-domain orthogonal approximation message passing receiver to obtain the decoded signal after iterative processing.

[0036] The iterative process includes: iterative processes of time-domain linear detection, cross-domain transformation, symbol-domain nonlinear detection, and cross-domain inverse transformation; time-domain linear detection is used to perform linear minimum mean square error detection in the time domain, and time-domain linear detection includes low-complexity calculation based on block matrix inversion of the multi-user joint detection matrix; symbol-domain nonlinear detection includes symbol demodulation and channel decoding.

[0037] For example, such as Figure 3 As shown, Figure 3 A low-complexity multi-slot MU-CD-OAMP receiver is proposed for embodiments of this application. The receiver includes a time-domain linear detection module. Cross-domain transformation module, symbolic domain nonlinear detection module And a cross-domain inverse transform module. Among them, the time-domain linearity detection module. It includes a multi-slot MU-LMMSE detector and a user-specific orthogonalization module, while the symbol-domain nonlinear detection module is composed of... It consists of an independent posterior probability decoder and an orthogonalization module, and the estimated signals of the two modules are iteratively updated through cross-domain operations. For a detailed description of the low-complexity multi-slot MU-CD-OAMP receiver, please refer to the following embodiment.

[0038] Understandably, the time-domain linearity detection module The received mixed signals can be initially separated and estimated in the time domain to reduce interference between users.

[0039] In the time-domain linear detection module In this process, the input signal for each iteration can be the nonlinear detection module in the symbol domain from the previous round. The output and the time-domain estimated signal after cross-domain inverse transform In the first iteration, the input signal is initialized to zero, and its formula is as follows: ; in, This indicates the length of the signal sent by each user in each time slot. Indicates iterative index, from Start, Initialization . Indicates the Kronecker product. It is an orthogonalized coefficient vector. . express An identity matrix of dimension 1 .

[0040] The expression for the multi-slot MU-LMMSE detector is as follows: ; in, and Let these represent the prior and posterior covariance matrices of the MU-LMMSE detection, respectively. Represents the reciprocal of noise power. The conjugate transpose of . The inverse matrix of . Where, The calculation method can be found in the following examples.

[0041] First, the original MU-LMMSE calculation scheme is as follows.

[0042] Calculate directly using matrix inversion. This is the computation scheme in the original MU-LMMSE.

[0043] ; ; in, , express The average estimated variance.

[0044] Understandably, traditional MU-LMMSE detectors need to adjust their settings based on the current situation during each iteration. Calculating the matrix inverse for each time slot requires involving... Finding the inverse of a large matrix has a time complexity of O(n log n). Therefore, users and the signal length for each user The increase of will greatly increase the algorithm complexity of the MU-CD-OAMP receiver.

[0045] Note that in the matrix middle, It is a quasi-banded matrix and It is a diagonal matrix, therefore It still has a quasi-banded structure (such as Figure 4 (As shown). To address this characteristic, the block matrix inversion theorem based on Schur's complement is used, enabling recursive computation of matrix inversion on smaller block matrices, thereby significantly reducing computational complexity while maintaining good detection performance. This method, a low-complexity MU-LMMSE detector based on the block matrix inversion theorem, will be described in detail below.

[0046] In one possible embodiment of this application, in temporal linear detection, the low-complexity computation based on block matrix inversion of the multi-user joint detection matrix includes: Get the The next iteration, the... Multi-user joint detection matrix for each time slot Among them, the multi-user joint detection matrix The calculation formula is as follows: ; ; in, Represents the reciprocal of noise power. The conjugate transpose of; ; This represents the prior estimate of variance; , Indicates user The prior estimate of variance; It represents the Kronecker product.

[0047] First, the principle of the block matrix inversion theorem based on Schur's complement will be illustrated by an example.

[0048] The block matrix inversion theorem based on Schur's complement can be stated as: [The matrix is ​​then inverted...] Divided into Four block matrices: ; Its inverse can be calculated using the following formula: ; in: ; yes middle Schur supplement.

[0049] For simplicity, we can initialize ,like Figure 5 As shown, the matrix The matrix is ​​divided into four blocks along the user dimension, and its inverse is recursively solved using the block matrix inversion theorem based on Schur's complement. Among them, the four block matrices include . The inverse matrix is ​​approximated by further dividing it into block matrices and utilizing the attenuation characteristics of its quasi-band structure; .

[0050] In one possible embodiment of this application, the inverse matrix is ​​recursively solved using the block matrix inversion theorem based on Schur's complement. ,include: For the first recursion, the current matrix is ​​obtained as follows: After determining the block matrix, Schur supplement At the same time Truncation to bandwidth The quasi-banded matrix, and use it as In the second recursion, the matrix Divided into four block matrices along the user dimension; among them, The preset cutoff length; where, the cutoff length The preset cut-off length; optionally, It can also be obtained based on the number of channel taps. It should be noted that the truncation length... It can be set based on the channel multipath delay spread, and since the energy of out-of-band elements is negligible (on the order of e⁻⁴), the energy concentration is still within the band, so the truncation operation does not affect the receiver performance.

[0051] Schur supplement The calculation formula is as follows: ; For the The recursion will turn the current matrix into a single recursive function. Divided into: ; in, ; ; calculate The inverse matrix and its Schur complement ;Schur supplement The calculation formula is as follows: ; Schur supplement Truncation to bandwidth quasi-banded matrix; Will Determined as Execute the first The recursion continues until the first recursion is completed. After the recursion, the calculated Substitute back into the block matrix and invert it to obtain the final inverse matrix. .

[0052] For example: The four block matrices include The expression for each matrix block is as follows: ; ; ; ; in: ; User The time-domain channel, based on the block matrix inversion theorem, The inverse mainly depends on and The reverse.

[0053] for The reverse: such as Figure 6 As shown, Figure 6 Showing The amplitude. Because It is a quasi-banded matrix, and its inverse matrix can be calculated by inverting the block matrix. For example... Figure 6 As shown, by setting the truncation length to , It can be divided into .in It is a banded matrix whose inverse satisfies the decay property and can be decomposed based on the lower-upper triangular matrix (Lower–Upper, LU). Only the calculation is required. of Central diagonal and surrounding The value of the diagonal of the strip can be approximated. The reverse, that is .for Although this is a dense matrix, its dimensions are... Therefore, the cost of inversion is low, only [amount missing]. .based on , , , , It can be calculated using the inverse theorem of the block matrix; such as Figure 7 As shown, Figure 7 Showing The amplitude.

[0054] for The reverse: due to , , , They are all quasi-zonal, resulting in Schur supplements. Only a slight spread occurred, such as Figure 8 As shown. Utilizing this attenuation characteristic, It can be truncated to a bandwidth of The quasi-banded matrix, retaining the With the same structural characteristics and negligible performance loss, low-complexity recursive inversion can be achieved. According to the block matrix inversion theorem, In the next recursive call, it will be considered as needing to be inverted in blocks. ,in This indicates a recursive index.

[0055] ; The four block matrices are: ; ; ; ; and Schur supplement .

[0056] After completing all After the recursion, by supplementing Schur Substituting back into the block matrix and finding its inverse yields the result. . Figure 9 It can be seen that the low-complexity MU-CD-OAMP receiver achieves lossless system performance.

[0057] It should be noted that, in order to more clearly illustrate the advantages of the low-complexity MU-LMMSE detector based on the block matrix inverse theorem provided by the embodiments of this application, an analysis of the complexity of this low-complexity MU-CD-OAMP receiver will be given here.

[0058] In each iteration, the complexities of cross-domain transformation and constellation symbol mapping are respectively... and For LDPC encoding and decoding operations, a decoding scheme based on the min-sum algorithm can achieve a code length of [missing information]. The linear complexity. In the low-complexity MU-LMMSE, directly... matrix The inverse operation is decomposed into... An independent Inverting quasi-banded submatrices significantly reduces the computational burden of each iteration. Furthermore, according to the block matrix inverse theorem, the sparsity and conjugate symmetry of the submatrices further simplify multiplication between them. Table 1 summarizes the main computational operations and their computational complexity for each iteration and time slot in the recursive steps. It can be clearly seen that the computational complexity of MU-LMMSE has decreased from the original... Reduced to ,in .

[0059] Table 1. Computational complexity of MU-LMMSE for each iteration and each time slot

[0060] Understandably, the core function of the linear detection module is to utilize the optimized, low-complexity MU-LMMSE algorithm to perform preliminary separation and estimation of superimposed user signals in the time domain, and to create favorable conditions for subsequent iterations through user-specific orthogonalization processing. The embodiments of this application utilize the quasi-band structure of the channel matrix and significantly reduce the computational complexity of the core matrix operations through a recursive block inversion method based on Schur's complement.

[0061] In one possible embodiment of this application, after performing temporal linearity detection, the method provided by the embodiments of this application further includes: To cope with drastic changes in the singular values ​​of the channel matrix, the receiver averages the posterior variances of multiple time slots to obtain the posterior estimated variance vector: ; To fully utilize the random unitary characteristics of each user's IFDM modulation matrix, the multi-slot MU-CD-OAMP receiver performs decoupling orthogonalization on the estimation results for each user separately, ensuring that the input and output errors are uncorrelated during the iteration process.

[0062] The orthogonalized coefficient vector is calculated based on the prior and posterior estimated variance vectors; where the orthogonalized coefficient vector... The calculation formula is as follows: ; in, This indicates element-wise division. , express The mean posterior variance. After orthogonalization, As the estimated signal The variance.

[0063] The cross-domain transformation module will be described next.

[0064] To correspond with the transmitter, each user's time-domain estimation result is mapped to the symbol domain through a cross-domain transformation to obtain a symbol-domain detection signal. In one possible embodiment of this application, the cross-domain transformation specifically includes: The current time-domain detection signal of each user is mapped to the symbol domain through a cross-domain transformation, resulting in the symbol-domain detection signal and its variance; symbol-domain detection signal and its variance The calculation formula is as follows: ; ; in, ; They represent The inverse matrix and its conjugate transpose; Indicates user The interleaved frequency division multiplexing modulation matrix.

[0065] Next, we will discuss the symbolic domain nonlinearity detection module. Describe the symbolic domain nonlinear detection module. It may include an orthogonalization module and a set of APP decoders. ,Right now Each has an independent posterior probability decoder. The decoder... This includes symbol-by-symbol MMSE demodulation and channel decoding.

[0066] In one possible embodiment of this application, Indicates user The signal; For users Independently performing the symbol-domain nonlinear detection specifically includes: The output of each decoder can be represented as: ; in, Indicates user The nonlinear terminal function, .

[0067] Detection signal in the symbol field Perform posterior probability decoding to obtain the posterior decoded signal. The calculation formula is as follows: ; in, The set of transmitted codewords represents the encoding constraints; Indicates user The minimum mean square error function.

[0068] Based on the variance of the signal detected in the symbol domain, the orthogonalization coefficients are calculated. The calculation formula is as follows: ; ; in, This represents the variance of the detected signal in the symbol field. This represents the posterior variance.

[0069] At the same time, the orthogonal signals Estimated variance Represented as: .

[0070] The cross-domain inverse transform module is described next. This module performs a cross-domain inverse transform on the estimated signal output by the nonlinear detection to obtain the time-domain detection signal for the next iteration.

[0071] In one possible embodiment of this application, the cross-domain inverse transform specifically includes: The current symbol domain detection signal of each user is mapped to the time domain through an inverse cross-domain transform to obtain the time domain detection signal for the next iteration. and its variance The calculation formula is as follows: ; .

[0072] Understandably, multi-user decoupling orthogonalization ensures that the input and output errors of time-domain linear detection and symbol-domain nonlinear detection remain uncorrelated in each iteration, thus enabling State Evolution (SE) to accurately predict the asymptotic mean squared error (MSE) performance of the MU-CD-OAMP receiver.

[0073] Optionally, in addition to the IFDM detection algorithm (i.e. the detection algorithm applied in the CD-MAMP detector), embodiments of this application may also employ expectation propagation detection algorithms, approximate message passing (AMP) detection algorithms, generalized approximate message passing (GAMP) algorithms, or variational message passing algorithms, etc., and embodiments of this application do not impose specific limitations on this.

[0074] It should be noted that the standard I-MMSE theorem is often used to transform system capacity calculation into detector achievable rate analysis, but this standard I-MMSE theorem requires the detector to be MMSE optimal. However, as... Figure 3 As shown, the nonlinear detection in the receiver includes orthogonalization operations and MMSE symbol demodulation, and therefore is not MMSE optimal. This means that I-MMSE cannot be directly used for achievable rate analysis of MU-CD-OAMP.

[0075] Existing MC-NOMA schemes still lack theoretical performance limit analysis, failing to guide the design of optimal coding schemes. Furthermore, mutual interference among multiple users and varying transmission rate requirements further complicate information theory limit analysis. Accurately characterizing the information theory limits of MC-NOMA systems remains a crucial issue and the foundation for achieving optimal transceiver design.

[0076] Furthermore, most existing multi-user coding optimization designs are based on the ideal assumptions that the channel matrix satisfies right-unitary invariance and has a fixed eigenvalue distribution. However, in high-speed, time-varying multipath channels, the channel matrix does not satisfy the right-unitary invariance assumption, and its eigenvalue distribution changes rapidly and over a wide range. Directly applying existing methods would require designing a separate coding and decoding scheme for each channel matrix, leading to excessive overhead in practical systems. Therefore, how to design the optimal coding scheme for multi-slot MC-NOMA systems remains an unsolved problem.

[0077] Therefore, as Figure 11 As shown, embodiments of this application further incorporate the orthogonalization operation in the symbol domain into the linear detection end in the time domain through structural equivalence transformation, thereby constructing a functionally equivalent MU-CD-OAMP receiver model. In this equivalent model, linear and nonlinear detection can be reformulated, making the entire receiver framework more suitable for subsequent theoretical analysis, as detailed in the following embodiments.

[0078] In one possible embodiment of this application, the method provided by the embodiments of this application further includes: Obtain the variational state evolution function and nonlinear detection transfer function of the receiver; construct the constraint relationship of the estimation variance among users based on the target constraints among users; based on the constraint relationship of the estimation variance among users, decouple the multi-input multi-output state evolution function of the multi-user cross-domain orthogonal approximate message passing receiver into multiple independent single-input single-output state evolution functions; determine the achievable rate of the receiver based on the multiple independent single-input single-output state evolution functions after decoupling; and determine the optimal coding criterion that enables the receiver to achieve the maximum achievable rate based on the achievable rate of the receiver.

[0079] Among them, the target constraints include transmission rate constraints or power constraints; the optimal coding criterion for the maximum achievable rate requires that, under a given signal-to-noise ratio, the nonlinear detection transfer function approximates its upper bound, thereby achieving function decoupling to simplify capacity analysis.

[0080] Linear and nonlinear detection in a MU-CD-OAMP receiver can be expressed as follows: ; ; The corresponding variational SE can be expressed as: ; ; in, Indicates and Independent additive white Gaussian noise vectors; It is a multiple-input multiple-output function. The coupling between users complicates reachability analysis.

[0081] Understandably, in multi-user scenarios, the signals between users interfere with each other, resulting in the state evolution equation describing the evolution of the detector's input and output errors being a coupled multi-input multi-output function, making the analysis extremely complex.

[0082] Therefore, embodiments of this application propose a user variance decoupling scheme. By utilizing the inherent constraints among users within the system (e.g., different transmission powers or target bit rates for each user), a mathematical relationship can be established between the estimated variances of different users.

[0083] In one possible embodiment of this application, the constraint relationship of the estimated variance between users is expressed as follows: ; in, Indicates the first Estimated variance for each user Indicates the first Variance constraint coefficients for each user.

[0084] Through the above constraints, the originally coupled multi-user state evolution equations were successfully decoupled into... This provides an independent, single-input, single-output scalar function relationship, which greatly simplifies the theoretical analysis process of system capacity.

[0085] Decoupled as A single-input, single-output function, specifically represented as follows: ; ; in, , represents the MSE function of the multi-slot MU-LMMSE estimator. This indicates that the inverse is taken. It is important to note that the variational SE converges to the same fixed point as the SE, which can be used to analyze the achievable rate of the receiver and guide the design of the optimal coding scheme.

[0086] After decoupling, the variational state evolution equation can be used to accurately characterize the performance evolution of the receiver in each iteration. This equation describes the recursive update relationship of the estimation error between the linear detection end and the nonlinear detection end (i.e., the decoding end).

[0087] like Figure 12 As shown, the transition curve of variational SE is a key analytical tool. The figure contains two key curves: the detection transition curve. and decoding transfer curve . Figure 12 Demodulation transfer curve It should be the decoding transfer curve. The upper bound, namely: ; Assuming in and There is a unique fixed point between them. In order to achieve error-free decoding, and There should be a decoding channel between them, that is: ; This allows us to obtain the decoding curve. The upper bound is: ; Based on the I-MMSE theorem, a fixed decoding transfer function can be calculated. The achievable rate of the MU-CD-OAMP receiver is determined based on the decoupled multiple independent single-input single-output state evolution functions. The calculation formula is as follows: ; ; in, Indicates the achievable speed of the receiver; for The x-coordinate of the intersection point with the horizontal axis is represented as follows: ; in, yes The inverse function of .

[0088] Based on the above analysis and derivation, it can be determined that when When the maximum achievable data rate is reached, the MU-CD-MAMP receiver can reach this value, so this is used as the optimal coding criterion for the MU-CD-MAMP receiver.

[0089] Under the constraint of variance estimation among users, the IFDMA system can achieve communication at unequal user rates. At this point, the multi-user capacity and overall system performance are improved. It is expressed as follows: ; in, express The fixed point of each user; ; Indicates the first indivual A diagonal submatrix of dimension ; , , .

[0090] It is understandable that by calculating the capacity domain of an IFDMA system, one can reflect the sum of the ultimate performance limits that an IFDMA system can achieve when supporting multiple users communicating at different rates, using a receiver and combining user variance decoupling methods with optimal coding criteria.

[0091] Furthermore, considering the grouped heterogeneous IFDMA system, which is about to Users were divided into Groups, each group contains There are multiple users in the same group. Because users in the same group share the same channel characteristics and variance constraints, they exhibit uniform rate characteristics. At this time, the group... The maximum reachable rate for each user can be expressed as: ; in, For the optimal decoding MSE function based on state evolution; for the th... Group (whose user index set is denoted as) Any user Its individual state evolution function They are all the same, therefore they are collectively called group functions. .

[0092] Based on the above analysis, it can be determined that the MU-CD-OAMP receiver possesses a set of fixed points in each variational state evolution. The time-limited group rate region. For any subset of user groups. The total rate of its group Satisfies the following information theory boundaries: ; in, Representing a subset The complement; It represents mutual information.

[0093] Understandably, this formula, from an information theory perspective, gives the upper limit of the group rate that any receiver can achieve under given channel conditions.

[0094] The maximum achievable total rate and maximum achievable group rate of the MU-CD-OAMP receiver can reach the corresponding total capacity and group capacity range of the IFDMA system under time-varying multipath channels, respectively.

[0095] like Figure 13 As shown, Figure 13 This diagram illustrates the MSE curves of a MU-CD-OAMP and SE analysis provided for embodiments of this application. The MU-CD-OAMP simulation MSE curve is obtained through computer simulation, simulating the performance of the proposed low-complexity multi-slot MU-CD-OAMP receiver in actual operation. The SE analysis MSE curve is theoretically calculated using the established state evolution equations to reflect the theoretical prediction of the receiver's asymptotic performance. By comparing the simulation results with the theoretical predictions, the correctness of the state evolution theory established for the MU-CD-OAMP receiver can be proven.

[0096] Understandably, the above analytical derivation framework breaks away from the idealized assumptions of existing theories that "the channel matrix satisfies right unitary invariance and fixed eigenvalue distribution," making it applicable to the actual performance analysis and coding design of IFDMA systems under high-speed mobile time-varying channels.

[0097] like Figure 14 As shown, Figure 14 The diagram illustrates a comparison of the achievable rates of the proposed MU-CD-OAMP receiver and a discrete MU-CD-OAMP receiver under different load levels. It is understood that the discrete MU-CD-OAMP receiver (i.e., a separable MU-CD-MAMP receiver) performs a complete iterative process between linear detection and nonlinear constellation mapping, followed by independent APP decoding. Figure 14The maximum transmission rates achievable by the two receivers were evaluated by employing different signal modulation schemes, such as Quadrature Phase Shift Keying (QPSK), 8-Phase Shift Keying (8PSK), 16-Quadrature Amplitude Modulation (16QAM), and Gaussian signals. Analysis shows that under full load, the achievable rate of the MU-CD-OAMP receiver of this application is slightly higher than that of the discrete MU-CD-OAMP receiver; while under overload conditions, the achievable rate of the MU-CD-OAMP receiver of this application is significantly higher than that of the discrete MU-CD-OAMP receiver, thus verifying the superiority of the receiver structure provided in this application.

[0098] Understandably, this optimal coding criterion provides a clear objective and optimization direction for the design of practical channel coding (such as LDPC codes and polar codes). Figure 15 As shown, under the same signal-to-noise ratio conditions, compared with existing schemes, the coding scheme optimized according to the criteria provided by the embodiments of this application has a significantly better bit error rate (BER) performance than traditional coding schemes designed for point-to-point (P2P) channels. This demonstrates that the optimal coding scheme of this application can more effectively combat channel noise and inter-user interference, thereby improving the reliability of the system.

[0099] Step 103: Determine the transmitted information of each user from the decoded signal.

[0100] This application proposes a low-complexity multi-user information processing method for interleaved frequency division multiple access systems. First, the receiver receives signals from multiple time slots. These signals are obtained by multiple users modulating the signals via interleaved frequency division multiple access and transmitting them through a channel. Then, for each time slot's received signal, iterative processing is performed using a multi-user cross-domain orthogonal approximation message passing receiver to obtain an iteratively processed symbol-domain estimated signal. Finally, the iteratively processed symbol-domain estimated signal is decoded to recover the transmitted information of each user. The iterative processing includes iterative processes of time-domain linear detection, cross-domain transformation, symbol-domain nonlinear detection, and cross-domain inverse transformation. Since time-domain linear detection is used to perform linear minimum mean square error detection in the time domain, it includes low-complexity multi-user joint detection matrix based on block matrix inversion. The complexity calculation is optimized to reduce the computational complexity of multi-user detection and improve its practicality. Iterative processing using a multi-user cross-domain orthogonal approximation message-passing receiver ensures that the input-output errors of time-domain linear detection and symbol-domain nonlinear detection remain uncorrelated in each iteration, thereby improving the reliability of multi-user detection and preventing error propagation. Furthermore, iterative processing using a multi-user cross-domain orthogonal approximation message-passing receiver allows for the design of optimal codes that approximate the receiver's maximum achievable rate, based on channel characteristics and user rate requirements. Based on these principles, this scheme significantly reduces the computational complexity of multi-user detection, improves its reliability, prevents error propagation, and provides a foundation for system capacity analysis and optimal code design, achieving a breakthrough in theoretical performance. Ultimately, it improves the overall transmission performance of the system while maintaining low complexity.

[0101] In summary, compared with the prior art, the embodiments of this application have the following advantages: First, compared with existing solutions, the low-complexity MU-CD-OAMP receiver can achieve a significant reduction in computational complexity, approaching the performance of the original MU-CD-OAMP receiver and exceeding the performance of existing multi-user OAMP receivers.

[0102] Second, by comparing the bit error rate of the coding scheme of traditional point-to-point channel design with that of the proposed optimal coding scheme (multi-user LDPC code) in the IFDMA system, the proposed optimal coding scheme has significant gains. That is, it can solve the technical problem of difficult convergence of coding in traditional point-to-point channel design under heavy load scenarios, improve the robustness to different speed settings of multiple users, and thus achieve higher reliability communication in high-mobility and high-volume access scenarios.

[0103] Third, compared with existing schemes, by introducing multi-user variance constraint relationships, the needs of multiple users with different transmission code rates can be met, and the optimal coding design scheme under asymmetric channel conditions and unequal transmission rates can be achieved.

[0104] Fourth, compared to the discrete MU-CD-OAMP receiver, the MU-CD-OAMP receiver can achieve a higher achievable data rate.

[0105] The steps described above are for clarity only. In implementation, they can be combined into one step, or some steps can be broken down into multiple steps, as long as they involve the same logical relationship, they are all within the scope of protection of this application. Adding insignificant modifications or introducing insignificant designs to the algorithm or process, without changing the core design of the algorithm and process, are also within the scope of protection of this application.

[0106] Another embodiment of this application proposes a low-complexity multi-user information processing system for interleaved frequency division multiple access systems. The details of this low-complexity multi-user information processing system for interleaved frequency division multiple access systems are described below. The following implementation details are provided for ease of understanding and are not essential for implementing this example. Figure 16 This is a schematic diagram of the structure of a low-complexity multi-user information processing system for an interleaved frequency division multiple access system proposed in this embodiment, including: The receiving module 210 is used to receive signals from multiple time slots at the receiving end; wherein the received signals are obtained by multiple users modulating the signals via an interleaved frequency division multiple access method and transmitting them through a channel; The iterative processing module 220 is used to iteratively process the received signal in each time slot using a multi-user cross-domain orthogonal approximate message passing receiver to obtain the decoded signal after iterative processing. The iterative processing includes iterative processes of time-domain linear detection, cross-domain transformation, symbol-domain nonlinear detection, and cross-domain inverse transformation. Time-domain linear detection is used to perform linear minimum mean square error detection in the time domain, and includes low-complexity calculations based on block matrix inversion of the multi-user joint detection matrix. Symbol-domain nonlinear detection includes symbol demodulation and channel decoding. The determination module 230 is used to determine the decoded signal as the transmission information of each user.

[0107] It is not difficult to see that this embodiment is a system embodiment corresponding to the above method embodiments, and this embodiment can be implemented in conjunction with the above method embodiments. The relevant technical details and technical effects mentioned in the above method embodiments are still valid in this embodiment, and will not be repeated here to reduce repetition. Accordingly, the relevant technical details mentioned in this embodiment can also be applied to the above method embodiments.

[0108] It is worth mentioning that all modules and units involved in this embodiment are logical modules. In practical applications, a logical unit can be a physical unit, a part of a physical unit, or a combination of multiple physical units. Furthermore, to highlight the innovative aspects of this application, this embodiment does not introduce units that are not closely related to solving the technical problems proposed in this application; however, this does not mean that other units do not exist in this embodiment.

[0109] Another embodiment of this application provides an electronic device, such as Figure 17 As shown, it includes a processor 31 and a memory 32. The memory 32 stores instructions that the processor 31 can execute. When the processor 31 is configured to execute the instructions, the electronic device can implement a low-complexity multi-user information processing method for an interleaved frequency division multiple access system as described in the above method embodiment.

[0110] The memory and processor are connected via a bus, which includes any number of interconnecting buses and bridges, connecting various circuits of one or more processors and the memory. The bus can also connect various other circuits such as peripheral devices, voltage regulators, and power management circuits, which are well known in the art and will not be described further herein. The bus interface provides an interface between the bus and the transceiver. The transceiver can be a single component or multiple components, such as multiple receivers and transmitters, providing a unit for communicating with various other devices over a transmission medium. Data processed by the processor is transmitted over the wireless medium via an antenna, which further receives data and transmits it to the processor.

[0111] The processor manages the bus and general processing, and also provides various functions, including timing, peripheral interfaces, voltage regulation, power management, and other control functions. Memory is used to store data used by the processor during operation.

[0112] Another embodiment of this application proposes a computer-readable storage medium storing a computer program that, when executed by a processor, can implement a low-complexity multi-user information processing method for an interleaved frequency division multiple access system as described in the above method embodiments.

[0113] That is, those skilled in the art will understand that all or part of the steps in the above method embodiments can be implemented by a program instructing related hardware. The program is stored in a storage medium and includes several instructions to cause a device (such as a microcontroller, chip, etc.) or processor to execute all or part of the steps of the method described in the method embodiments of this application. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory, random access memory, magnetic disks, or optical disks.

[0114] Those skilled in the art will understand that the above embodiments are specific implementations of this application, and in practical applications, various changes can be made in form and detail without departing from the spirit and scope of this application. For those skilled in the art, several improvements and modifications can be made without departing from the principles of this application, and these improvements and modifications are also considered to be within the scope of protection of this application.

Claims

1. A low-complexity multi-user information processing method for interleaved frequency division multiple access systems, characterized in that, The method includes: The receiving end receives signals from multiple time slots; wherein, the received signals are obtained by multiple users modulating the signals via interleaved frequency division multiple access and transmitting them through the channel; For the received signal in each time slot, iterative processing is performed using a multi-user cross-domain orthogonal approximation message passing receiver to obtain the decoded signal after iterative processing; The iterative process includes iterative processes of time-domain linear detection, cross-domain transformation, symbol-domain nonlinear detection, and cross-domain inverse transformation; time-domain linear detection is used to perform linear minimum mean square error detection in the time domain, and includes low-complexity calculation of the multi-user joint detection matrix based on block matrix inversion; symbol-domain nonlinear detection includes symbol demodulation and channel decoding; The decoded signal is identified as the information sent by each user.

2. The method according to claim 1, characterized in that, The receiving end receives signals from multiple time slots, including: Received signal The calculation formula is as follows: ； in, Indicates the time slot index. ; ,express The receiving antenna is at the first Received signals in each time slot; Indicates the first Channel matrix for each time slot, ; express The transmitting antenna is at the ... User signals transmitted in each time slot; , representing additive white Gaussian noise in the channel; On the sending end, the user time domain signal From symbol vector The target modulation matrix is ​​obtained by modulation through a target modulation matrix; wherein the target modulation matrix is ​​any one of the following: an interleaved frequency division multiplexing modulation matrix, an interleaved block sparse transform matrix, a random multiplexing modulation matrix, a random unitary matrix, or a modulation matrix obtained by combining a universal class matrix with a sensing matrix; If the target modulation matrix is ​​an interleaved frequency division multiplexing modulation matrix Then the user time domain signal The calculation formula is as follows: ； in, Indicates user The interlacing matrix; This represents the inverse fast Fourier transform matrix; Indicates user index, .

3. The method according to claim 2, characterized in that, In temporal linear detection, the low-complexity computation based on block matrix inversion of the multi-user joint detection matrix includes: Get the The next iteration, the... Multi-user joint detection matrix for each time slot Among them, the multi-user joint detection matrix The calculation formula is as follows: ； ； in, Represents the reciprocal of noise power. The conjugate transpose of; ; This represents the prior estimate of variance; , Indicates user The prior estimate of variance; express An identity matrix of dimensionality; Indicates the Kronecker product; initialization and the matrix The matrix is ​​divided into four blocks along the user dimension, and its inverse is recursively solved using the block matrix inversion theorem based on Schur's complement. Among them, the four block matrices include ; The inverse matrix is ​​approximated by further dividing it into block matrices and utilizing the attenuation characteristics of its quasi-band structure; The expression for each matrix block is as follows: ； ； ； ； Elements of a block matrix It is given by the following formula: 。 4. The method according to claim 3, characterized in that, The method utilizes the block matrix inversion theorem based on Schur's complement to recursively solve for its inverse matrix. ,include: For the first recursion, the current matrix is ​​obtained as follows: After determining the block matrix, Schur supplement At the same time Truncation to bandwidth The quasi-banded matrix, and use it as In the second recursion, the matrix Divided into four block matrices along the user dimension; among them, The preset cutoff length; Schur complement The calculation formula is as follows: ； For the The recursion will turn the current matrix into a single recursive function. Divided into: ;in, ; The dimensions of the four block matrices are as follows: ； ； ； ； calculate Schur supplement ;Schur supplement The calculation formula is as follows: ； Schur supplement Truncation to bandwidth The quasi-banded matrix, and use it as Execute the first The recursion continues until the first recursion is completed. After the recursion, the calculated Substitute back into the block matrix and invert it to obtain the final inverse matrix. .

5. The method according to claim 4, characterized in that, After performing block matrix inversion on the multi-user joint detection matrix, the method further includes: The posterior variance vector is obtained by averaging the posterior variances of multiple time slots. The orthogonalized coefficient vector is calculated based on the prior and posterior estimated variance vectors; where the orthogonalized coefficient vector... The calculation formula is as follows: ； in, , express The mean posterior variance; This indicates element-wise division.

6. The method according to claim 5, characterized in that, Perform time-domain linear detection to obtain the time-domain detection signal Its expression is as follows: ； ； in, ;from Start, Initialization ; ; This represents the joint multi-user linear minimum mean square error detector function; The cross-domain transformation specifically includes: The current time-domain detection signal of each user is mapped to the symbol domain through a cross-domain transformation, resulting in the symbol-domain detection signal and its variance; symbol-domain detection signal and its variance The calculation formula is as follows: ； ； in, ; They represent The inverse matrix and its conjugate transpose; Indicates user The interleaved frequency division multiplexing modulation matrix.

7. The method according to claim 6, characterized in that, Indicates user Signals to users; Independently performing the symbol-domain nonlinear detection specifically includes: Detection signal in the symbol field Perform posterior probability decoding to obtain the posterior decoded signal. The calculation formula is as follows: ； in, The set of transmitted codewords represents the encoding constraints; Indicates user The minimum mean square error function; Based on the variance of the signal detected in the symbol domain, the orthogonalization coefficients are calculated. The calculation formula is as follows: ； ； in, This represents the variance of the signal detected in the symbol field. Indicates the posterior variance; The updated symbol-domain estimated signal is calculated using the orthogonalization coefficients; where the updated symbol-domain estimated signal... The calculation formula is as follows: ； in, Indicates user The nonlinear terminal function, .

8. The method according to claim 7, characterized in that, The cross-domain inverse transform specifically includes: The current symbol domain detection signal of each user is mapped to the time domain through an inverse cross-domain transform to obtain the time domain detection signal for the next iteration. and its variance The calculation formula is as follows: ； 。 9. The method according to any one of claims 1 to 8, characterized in that, The method further includes: Obtain the variational state evolution function and nonlinear detection transfer function of the receiver; Based on the target constraints among users, a constraint relationship for estimating the variance among users is constructed; where the target constraints include transmission rate constraints or power constraints. Based on the constraint relationship of the estimated variance among users, the multi-input multi-output state evolution function of the multi-user cross-domain orthogonal approximation message passing receiver is decoupled into multiple independent single-input single-output state evolution functions; The achievable rate of the receiver is determined based on the decoupled multiple independent single-input single-output state evolution functions; Based on the receiver's achievable rate, the optimal coding criterion that enables the receiver to reach its maximum achievable rate is determined; wherein, the optimal coding criterion for the maximum achievable rate requires that, under a given signal-to-noise ratio, the nonlinear detection transfer function approximates its upper bound.

10. The method according to claim 9, characterized in that, The constraint relationship between the estimated variances among users is expressed as follows: ； in, Indicates the first Estimated variance for each user Indicates the first Variance constraint coefficients for each user; The achievable rate of the receiver is determined based on the decoupled multiple independent single-input single-output state evolution functions, and the calculation formula is as follows: ； ； in, Indicates the achievable speed of the receiver; for The x-coordinate of the intersection point with the x-axis is represented as follows: ； in, yes The inverse function; Under the constraint of variance estimation among users, the multi-user and capacity of interleaved frequency division multiple access systems. It is expressed as follows: ； in, express The fixed point of each user; ; Indicates the first indivual A diagonal submatrix of dimension ; , , ; Will Users were divided into Groups, each group contains Individual users; groups The maximum reachable rate for each user is expressed as: ； in, For the optimal decoding MSE function based on state evolution; for the th... Any user in the group Its individual state evolution function They are the same, and therefore collectively referred to as group functions. ; For any subset of user groups Its total group rate Satisfies the following information theory boundaries: ； in, Representing a subset The complement; It represents mutual information.

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