Imitation radio frequency division multiplexing signal detection method based on low complexity optimization expectation propagation

Abstract

The invention discloses an imitated radio frequency division multiplexing signal detection method based on low-complexity optimization expected propagation, and belongs to the signal detection technology of multi-carrier communication. According to the method, information is updated through iteration to approach an ML result, and the complexity is reduced from an exponential level to a cubic level; in order to reduce the complexity of the EP, sparse and quasi-banded channel structures in the DAFT domain are utilized. Through matrix block decomposition, upper and lower triangular decomposition and upper triangular matrix forward substitution, low-complexity matrix inversion is successfully realized, and the complexity of the EP algorithm is reduced to a linear level. In addition, the EP algorithm performance is further improved through a deep learning optimization moment matching strategy, so that the process becomes more adaptive through trainable positive and negative component parameters. In addition, DAFT domain iteration detection and decoding can be achieved based on the method, external information of a decoder is fed back to a detector, and the reliability of the whole system is remarkably improved. According to the invention, effective balance between performance improvement and calculation complexity is realized.

Description

Technical Field

[0001] This invention relates to signal detection technology for multi-carrier communication, and more particularly to a signal detection method based on low-complexity optimized expectation propagation (EP) propagation, which is similar to affine frequency division multiplexing (AFDM). Background Technology

[0002] Orthogonal frequency division multiplexing (OFDM) has become a core technology of 5G due to its high spectral efficiency and multipath resistance. OFDM significantly improves spectral efficiency by dividing a high-speed data stream into multiple slower sub-data streams and performing orthogonal modulation in the frequency domain. However, in high-speed mobile scenarios, OFDM is susceptible to the Doppler effect, resulting in inter-carrier interference (ICI), which degrades system performance. To address this, Hadani et al. proposed orthogonal time-frequency and space (OTFS) technology, which transforms the drastically changing channel in the time-frequency (TF) domain into a stable sparse channel in the delay-Doppler (DD) domain, facilitating channel estimation and equalization. OTFS achieves time-frequency domain dispersion of the signal through inverse symplectic finite Fourier transform (ISFFT), providing full diversity gain. However, the block modulation characteristics of OTFS introduce high complexity, and the leakage caused by fractional Doppler frequency shift leads to the non-sparseness of the equivalent channel in the DD domain, increasing pilot overhead, reducing spectral efficiency, and placing higher demands on the design of detection algorithms, further increasing detection complexity. This has led many studies to design OTFS detection algorithms in the time domain. Although the time-domain channel response exhibits sparsity, time-domain detection algorithms also suffer from performance degradation due to the inability to obtain frequency diversity gain. Furthermore, the high frame delay of OTFS may cause compatibility issues with time division duplex (TDD) systems.

[0003] Affine Frequency Division Multiplexing (AFDM) is a novel multicarrier modulation technique that generates and multiplexes signals in the Discrete Affine Fourier Transform (DAFT) domain. DAFT is a generalization of the Discrete Fourier Transform, and its core advantage lies in its ability to separate all propagation paths, thereby achieving comprehensive capture of time and frequency diversity gain. Unlike Time-Frequency Diversity Shift (OTFS) technology, which relies on two-dimensional modulation to obtain time-frequency diversity gain, AFDM only requires a one-dimensional transformation to achieve complete time-frequency diversity gain, thus avoiding the high latency, high complexity, and high pilot overhead problems caused by the two-dimensional frame structure of OTFS. Furthermore, when dealing with fractional Doppler frequency shift, AFDM's channel response in the DAFT domain still maintains a certain degree of sparsity, contrasting with the fact that the DD domain channel response is no longer sparsity in the fractional Doppler domain. In summary, AFDM not only inherits the capabilities of OTFS in path separation and time-frequency diversity gain but also possesses the advantages of OFDM, such as low complexity and low latency. Therefore, AFDM provides an effective solution for future mobile communication scenarios.

[0004] To fully leverage the advantages of AFDM systems, a robust detection algorithm is crucial. Maximum likelihood (ML) detection algorithms are renowned for their theoretically optimal performance; however, their computational complexity increases exponentially with the number of symbols, making them impractical in real-world applications. Minimum mean square error (MMSE) detection algorithms have cubic complexity and suffer a significant performance loss compared to ML. Furthermore, while maximal ratio combining (MRC) algorithms offer advantages in computational complexity, their bit error rate (BER) performance is less than ideal. MP-based schemes experience severe performance degradation in the presence of fractional Doppler shift and large path numbers. This is because, under these conditions, the sparsity of the channel matrix decreases, limiting the detection performance of MP algorithms and increasing computational complexity. Summary of the Invention

[0005] This invention provides a simulated radio frequency multiplexing signal detection method based on low-complexity optimized expected propagation, aiming to solve the problems of high complexity and large performance loss in existing signal detection algorithms. This invention utilizes the sparsity and quasi-band structure of the DAFT domain equivalent channel matrix to achieve low-complexity signal detection and improve system performance.

[0006] The technical solution adopted in this invention is: a simulated radio frequency multiplexing signal detection method based on low-complexity optimized desired propagation, the method comprising:

[0007] Step 1: Obtain the equivalent channel matrix obtained after Discrete Affine Fourier Transform (DAFT) in the AFDM system based on the current transmitted symbol vector. The initial value of the transmission symbol is a preset value, such as a random initialization value;

[0008] Step 2, based on the equivalent channel matrix Due to the sparsity and quasi-band structure of the equivalent channel matrix, it is divided into blocks. The resulting sub-matrices are then decomposed using LU decomposition, decomposing them into products of upper and lower triangular matrices. The inverse of the upper triangular matrix is ​​obtained using a forward substitution algorithm. The equivalent channel matrix is ​​then derived from the inverses of the sub-matrices. The reverse;

[0009] Step 3, based on the equivalent channel matrix With equivalent channel response H A The correspondence is based on the equivalent channel matrix. The inverse and equivalent channel response H A Determine the initial values ​​of the mean matrix μ and standard deviation matrix ∑ of the posterior equivalent Gaussian distribution of the posterior distribution of the transmitted symbols in the optimization expectation EP algorithm;

[0010] The DAFT domain cavity distribution is obtained by removing the prior distribution of the transmitted symbol from the posterior equivalent Gaussian distribution; and the mean matrix and variance matrix of the transmitted symbol are estimated through the DAFT domain cavity distribution.

[0011] The mean matrix of the prior equivalent Gaussian distribution of the prior distribution of the transmission symbols in the EP algorithm is updated based on the mean matrix of each transmission symbol and the mean matrix of the DAFT domain cavity distribution; and the variance matrix of the prior equivalent Gaussian distribution of the transmission symbols in the EP algorithm is updated based on the variance matrix of each transmission symbol and the variance matrix of the DAFT domain cavity distribution; wherein the initial values ​​of the mean matrix and variance matrix of the prior equivalent Gaussian distribution are preset values;

[0012] Step 4: Obtain the transmission symbol vector estimated in the current iteration based on the mean value of each transmission symbol. Based on transmission symbol vector With the updated prior equivalent Gaussian distribution mean and variance matrix, continue with steps 1-3 to obtain the transmission symbol vector for the next iteration. And the mean and variance matrices of the updated prior equivalent Gaussian distribution;

[0013] When the set iterative convergence condition is met, the transmission symbol vector estimated in the last iteration will be... As the final estimated AFDM transmission symbol.

[0014] Furthermore, the prior equivalent Gaussian distribution and the posterior equivalent Gaussian distribution are specifically as follows:

[0015]

[0016] Where q(x) and q(x|y) represent the prior equivalent Gaussian distribution and the posterior equivalent Gaussian distribution of the transmitted symbol vector x, respectively. m γ represents the m-th symbol in the transmission symbol vector x; m , λ m They represent the symbols x respectively m The variance and mean; based on all γ m , λ m They respectively form the variance matrix γ and the parameter matrix λ, that is, γ and λ form the EP parameter matrix, σ 2 Let represent the noise variance, y represent the received symbol in real form, and μ and ∑ represent the mean and variance matrices of the posterior equivalent Gaussian distribution. This indicates that y follows a Gaussian distribution with mean μ and variance ∑.

[0017] Equivalent channel response and These represent taking the real and imaginary parts of the matrix, respectively. Represents the set of real numbers, where N is the number of symbols in a frame. Represents the real number field.

[0018] Furthermore, step 2 specifically includes:

[0019] Define the auxiliary matrix Ψ A for:

[0020] Represent the first half of the elements of matrix λ as The second half of the elements of λ is represented as The auxiliary matrix Ψ A Transformed into:

[0021]

[0022] Where D, B, and C are auxiliary matrices Ψ A The submatrix after partitioning, and symbol Represents the conjugate transpose of a matrix;

[0023] The LU decomposition method is used to decompose submatrices D and C+BD.-1 Inverse B yields Ψ A -1 The value of ∑ is obtained by finding the value of the matrix ∑.

[0024] Furthermore, in step 3, the DAFT domain cavity distribution is specifically as follows:

[0025]

[0026] Where, q \m (x m ) represents the DAFT domain cavity distribution of the prior distribution from the posterior equivalent Gaussian distribution after removing the current transmitted symbol.

[0027] Furthermore, in step 3, the mean matrix and variance matrix of the transmitted symbols are estimated using the DAFT domain cavity distribution as follows:

[0028]

[0029] Where a represents the set of modulation constellations. Any constellation point, These represent the m-th transmission symbol x. m The mean and variance, Modulate constellation set.

[0030] Furthermore, in step 3, the update method for the parameter matrices γ and λ is as follows:

[0031] λ′=(V) -1 -(V obs ) -1

[0032]

[0033] Where λ′ and γ′ represent the EP parameter matrix after this iteration, λ′ is the reciprocal of the variance of q(x), and the mean of q(x) is expressed as... matrix V obs and x obs These are the variance matrix and mean matrix of the DAFT cavity distribution, respectively.

[0034] Furthermore, in step 3, updating the parameter matrices λ and γ in the prior equivalent Gaussian distribution includes:

[0035] Extract positive values ​​from matrix λ and denote them as matrix λ. And extracting negative values ​​as And record the matrix The element index in the mean matrix λ; and the matrix-based index... The variances corresponding to the means in the matrix are obtained. Based on matrix The variances corresponding to the means in the matrix are obtained.

[0036] For matrix Set a damping factor, denoted as W1 and W2 respectively;

[0037] And based on the damping factors W1 and W2, respectively, the matrix Make minor adjustments and updates:

[0038]

[0039] in, These represent the matrices after fine-tuning and updating in the current iteration. They represent The updated value was fine-tuned in the previous iteration; These represent the values ​​of the variance matrix in the updated prior equivalent Gaussian distribution during the current iteration.

[0040] Based on the current results and the recorded matrix The index of each element in the mean matrix λ will be and Map back to the mean matrix λ, and... and Map back to matrix γ.

[0041] Furthermore, when optimizing the damping factors W1 and W2 based on the deep learning model, the input to the model is the transmission symbol vector. and equivalent channel matrix The output is a recovery symbol.

[0042] When further optimizing the damping factors W1 and W2 using a deep learning model, the loss function is set as follows:

[0043]

[0044] in, Represents the L2 loss function. Indicates the size of the batch normalized dimensions; and x d These represent the recovery symbol and the transmission symbol obtained through the d-th iteration, respectively.

[0045] Furthermore, when performing signal decoding processing on the estimated transmitted symbols based on the signal detection method provided by the present invention, information exchange is carried out during the detection and decoding process based on iterative detection decoding. The currently obtained DAFT domain cavity distribution is used as external information provided to the decoder to enhance the error correction capability of the decoding process.

[0046] The technical solution provided by this invention brings at least the following beneficial effects:

[0047] (1) Significantly reduce computational complexity

[0048] This invention fully utilizes the sparsity and quasi-band structure of the DAFT domain channel matrix. After matrix partitioning, it combines LU decomposition and the forward substitution algorithm of the upper triangular matrix, significantly reducing the computational complexity of signal detection algorithms. Specifically, the complexity of traditional matrix inversion is cubic, while the method of this invention reduces the complexity to linear, making it suitable for real-time signal detection requirements in large-scale communication systems, and particularly applicable to systems with numerous antenna arrays and high-dimensional channel matrices.

[0049] (2) High detection accuracy

[0050] Despite a significant reduction in computational complexity, the method of this invention maintains minimal performance loss while ensuring high accuracy in signal detection. Through iterative optimization of the EP algorithm, it effectively approximates the true posterior distribution of the signal, thereby improving detection accuracy. This enables the system to perform stable and efficient signal detection in large-scale communication networks, ensuring the reliability and accuracy of data transmission. Furthermore, by introducing a moment-matching optimization strategy with trainable parameters, the EP algorithm can adaptively adjust the damping factor based on training data, improving overall performance. This optimization not only improves the convergence speed of the EP algorithm but also effectively enhances the system's adaptability to dynamically changing communication environments, further improving detection accuracy and robustness.

[0051] (3) Improve system performance

[0052] By employing a low-complexity detection algorithm, the system maintains high detection accuracy while effectively reducing computational burden and improving overall system processing efficiency. Furthermore, by combining the IDD (Independent Differential Distribution) scheme, this invention enhances the system's robustness and error correction capabilities in complex channel environments. This not only optimizes the performance of the AFDM system in single-user and multi-user environments but also improves signal interference management capabilities in multi-user access scenarios. By effectively acquiring diversity gain, the anti-interference capability and data transmission rate of the AFDM system in complex multipath environments are improved, making it particularly suitable for high-efficiency data transmission wireless communication systems. Attached Figure Description

[0053] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of the embodiments taken in conjunction with the accompanying drawings, wherein:

[0054] Figure 1 This is a block diagram of an AFDM signal detection baseband system based on low-complexity EP, according to an embodiment of the present invention.

[0055] Figure 2 The equivalent channel responses in the DD and DAFT domains are given, where (2a) is a schematic diagram of the structure of matrix D, and (2b) is the equivalent channel response of matrix C+BD. -1 B. Schematic diagram of the structure.

[0056] Figure 3 This represents the block decomposition and LU decomposition of matrix A.

[0057] Figure 4 Simulation graphs of BER performance of the proposed scheme under different approximation orders.

[0058] Figure 5 This is a comparison chart of BER performance for different schemes.

[0059] Figure 6 A comparison of BER performance for different AFDM detection schemes under N=256 and EVA channels.

[0060] Figure 7 To compare the BER performance of different schemes in the AFDM system. Detailed Implementation

[0061] To enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions of the embodiments of the present invention will be described in detail and completely below with reference to the accompanying drawings. Obviously, the embodiments described with reference to the accompanying drawings are exemplary and intended to explain the present invention, but should not be construed as limiting the present invention.

[0062] High-performance and low-complexity detection schemes are crucial for achieving time-frequency diversity gain in AFDM systems. While ML detection is theoretically optimal, its exponential complexity renders it impractical for AFDM. This invention proposes a simulated RF-DDM signal detection method based on low-complexity optimized expectation propagation to improve the detection performance of AFDM systems in high-noise and complex channel environments while reducing computational complexity. In the proposed method, the maximum likelihood probability distribution of the signal is gradually approximated by iteratively refining the transmission process of a set of messages based on optimized expectation, thus approximating the ML result and reducing complexity from exponential to cubic. Optimized expectation is an approximate inference method based on Bayesian theory, which can approximate the true posterior distribution of the signal by iteratively updating the local probability distribution. During the iterative process based on optimized expectation, the method continuously adjusts and updates each local approximate distribution, ensuring that the final posterior distribution accurately reflects the true signal situation. This effectively improves the accuracy of signal detection while significantly reducing computational complexity through its efficient message passing mechanism. Meanwhile, to further reduce the complexity of the optimization expectation, this embodiment of the invention utilizes the sparse and quasi-band channel structure in the DAFT domain. Through matrix block decomposition, upper and lower triangular decomposition, and forward substitution of the upper triangular matrix, low-complexity matrix inversion is successfully achieved, reducing the complexity of the EP algorithm to a linear level. This enables high-precision capture of the underlying statistical characteristics of the signal, resulting in reliable and accurate detection results. Furthermore, this invention further enhances the performance of the optimization expectation by optimizing the moment matching strategy through deep learning. This makes the process more adaptive through trainable positive and negative component parameters, significantly improving the performance of the optimization expectation. In addition, this embodiment of the invention proposes a DAFT domain iterative detection and decoding scheme, in which external information from the decoder is fed back to the detector, significantly improving the overall system reliability. The proposed scheme in this embodiment of the invention effectively achieves a balance between performance improvement and computational complexity. That is, by introducing a low-complexity matrix inversion scheme, a deep learning-based moment matching optimization strategy, and an iterative detection and decoding (IDD) scheme, this method not only effectively improves the detection accuracy of the system under high noise conditions but also significantly reduces computational complexity, making it suitable for efficient and robust communication systems.

[0063] In one embodiment, the simulated radio frequency multiplexing signal detection method based on low-complexity optimized desired propagation provided by this invention includes the following steps:

[0064] Step 1: Obtain the equivalent channel matrix after Discrete Affine Fourier Transform (DAFT) in the AFDM system based on the current transmitted symbol vector. The initial value of the transmission symbol is a preset value, such as a random initialization value;

[0065] Step 2, based on the equivalent channel matrix The sparsity and quasi-band structure of the DAFT domain channel matrix are used to divide the equivalent channel matrix into blocks. That is, by utilizing the high sparsity and quasi-band structure of the DAFT domain channel matrix, unnecessary calculations are reduced by effectively dividing the matrix into blocks, making full use of the structural characteristics of the matrix and significantly reducing the computational complexity.

[0066] Step 3: For the sub-matrices obtained after block division, the LU decomposition method is used to decompose the sub-matrices into the product of upper and lower triangular matrices. The forward substitution algorithm is then used to invert the upper triangular matrix to obtain the inverse of the sub-matrices. Based on the inverses of the sub-matrices after each block, the equivalent channel matrix is ​​obtained. The inverse of the matrix; based on this processing step, the high computational complexity involved in traditional matrix inversion methods can be effectively avoided. Especially for large-scale matrix inversion, the computational load can be reduced to linear order, thereby significantly reducing the computational burden of the system.

[0067] Step 4, based on the equivalent channel matrix With equivalent channel response H A The correspondence is based on the equivalent channel matrix. The inverse and equivalent channel response H A Determine the initial values ​​of the mean matrix μ and variance matrix ∑ of the posterior equivalent Gaussian distribution of the posterior distribution of the transmitted symbols in the optimization expectation EP algorithm;

[0068] The DAFT domain cavity distribution is obtained by removing the prior distribution of the transmitted symbol from the posterior equivalent Gaussian distribution; and the mean matrix and variance matrix of the transmitted symbol are estimated through the DAFT domain cavity distribution.

[0069] The mean matrix of the prior equivalent Gaussian distribution of the prior distribution of the transmission symbols in the EP algorithm is updated based on the mean matrix (estimated) of each transmission symbol and the mean matrix (true) of the DAFT domain cavity distribution; and the variance matrix of the prior equivalent Gaussian distribution of the transmission symbols in the EP algorithm is updated based on the variance matrix of each transmission symbol and the variance matrix of the DAFT domain cavity distribution; wherein the initial values ​​of the mean matrix and variance matrix of the prior equivalent Gaussian distribution are preset values;

[0070] Step 5: Obtain the transmission symbol vector estimated in the current iteration based on the mean value of each transmission symbol. Based on transmission symbol vector With the updated prior equivalent Gaussian distribution mean and variance matrix, continue with steps 1-4 to obtain the transmission symbol vector for the next iteration. And the mean and variance matrices of the updated prior equivalent Gaussian distribution;

[0071] When the set iterative convergence conditions are met (such as the number of iterations reaching the upper limit or the estimation accuracy reaching the preset condition), the transmission symbol vector estimated in the last round of iteration will be... As the final estimated AFDM transmission symbol.

[0072] Furthermore, in step 4, updating the parameter matrices λ and γ in the prior equivalent Gaussian distribution includes:

[0073] Extract positive values ​​from matrix λ and denote them as matrix λ. And extracting negative values ​​as And record the matrix The element index in the mean matrix λ; and the matrix-based index... The variances corresponding to the means in the matrix are obtained. Based on matrix The variances corresponding to the means in the matrix are obtained.

[0074] For matrix Set a damping factor, denoted as W1 and W2 respectively;

[0075] And based on the damping factors W1 and W2, respectively, the matrix Make minor adjustments and updates:

[0076]

[0077] in, These represent the matrices after fine-tuning and updating in the current iteration. They represent The updated value was fine-tuned in the previous iteration; These represent the values ​​of the parameter matrices in the updated prior equivalent Gaussian distribution during the current iteration.

[0078] Based on the current results and the recorded matrix The index of each element in the mean matrix λ will be and Map back to matrix λ, and... and Map back to matrix γ.

[0079] Furthermore, the damping factors W1 and W2 can be optimized based on deep learning.

[0080] Specifically, during the iterative process of the EP algorithm in this embodiment of the invention, a moment matching optimization strategy based on deep learning (DL) is also introduced. This strategy further improves the overall performance of the EP algorithm by analyzing the mathematical meaning of key parameters in the EP algorithm and assigning different damping factors to their positive and negative components. By introducing trainable parameters and adaptively updating these damping factors during training, the convergence speed and detection accuracy of the EP algorithm can be significantly improved during the iterative process, thereby enhancing the overall signal detection capability of the system.

[0081] Furthermore, the DAFT symbol detection method provided in this embodiment can be extended to an EP-based AFDM detection scheme for coded AFDM systems. Specifically, an iterative detection decoding (IDD) scheme is introduced into the DAFT domain. By exchanging information between the detection and decoding processes, the accuracy of symbol recovery is further improved, and the error correction capability is enhanced. Through iterative information exchange in the IDD scheme, the method proposed in this embodiment improves the robustness and error correction performance of the system in complex channel environments, thereby providing higher system reliability and anti-interference capability.

[0082] In one embodiment, the proposed RF multiplexing signal detection method based on low-complexity optimized expected propagation is based on AFDM signal detection. By fully utilizing the sparsity and quasi-band structure of the DAFT domain channel matrix, combined with LU decomposition and forward replacement algorithms, the computational complexity in the signal detection process is significantly reduced, thereby improving the system's real-time performance and processing efficiency. The specific implementation process is as follows:

[0083] Figure 1 This is a block diagram of an AFDM signal detection baseband system based on a low-complexity EP, designed for an embodiment of the present invention. Let... m = 0, ..., N-1 represents discrete transmission symbols in the DAFT domain, where N is the number of symbols in a frame. The AFDM modulation module uses inverse DAFT (IDAFT) to... Transform to the time domain:

[0084]

[0085] in, It is the kernel function of IDAFT, c1 and c2 are AFDM system parameters, e is the natural base, and n is the symbol index.

[0086] Formula (1) can be expressed in vector form:

[0087]

[0088] Where A represents the transformation matrix of DAFT. Where the subscript separator c∈{c1,c2}, diag(a) represents a diagonal matrix with diagonal elements a, and F is the discrete Fourier transform (DFT) matrix, with the symbol... This represents the calculation of the conjugate transpose of a matrix.

[0089] In AFDM systems, a prefix is ​​needed to combat multipath propagation and convert linear convolutions into circular convolutions. Unlike OFDM, which uses a cyclic prefix (CP), AFDM employs a chirp-periodic prefix (CPP) due to the inherent periodicity of DAFT.

[0090]

[0091] Among them, L cpp The length of CPP.

[0092] After passing through the wireless channel, discrete-time domain received symbols It can be represented as:

[0093]

[0094] Among them, h i ,f i and l i Let l represent the gain, Doppler frequency shift index, and time delay index of the i-th path, respectively, where l represents the discrete time delay index, and P is the number of paths. It is discrete additive white Gaussian noise (AWGN). δ() represents the unit impulse function.

[0095] Then AFDM demodulation was performed using DAFT:

[0096]

[0097] in(·) * This is a conjugate operation.

[0098] The input-output relationship of AFDM can be expressed as:

[0099]

[0100] in

[0101]

[0102] The equivalent channel matrix in the DAFT field is given by:

[0103]

[0104] in, For the Doppler effect, II is the positive cyclic shift matrix, expressed as:

[0105]

[0106] The real-number form of the AFDM system receiver symbol can be represented as:

[0107] y = H A x+w #(11)

[0108] Where y,H A x, w are the received symbols in real form, the DAFT equivalent channel response, the transmitted symbols, and additive white Gaussian noise, respectively.

[0109]

[0110] in and Let these represent the real and imaginary parts of the matrix, respectively. It represents the set of real numbers.

[0111] The posterior probability distribution of the transmitted symbol x is expressed as:

[0112]

[0113] Where σ 2 Let I be the noise variance. 2N Let p(x) be a 2N×2N dimensional identity matrix, and let p(x) be the prior distribution of x, expressed as:

[0114]

[0115] in Only when When the value is 1, all others are 0. To modulate the constellation set.

[0116] As can be seen from the above formula, p(x) is a discrete function and does not satisfy a Gaussian distribution. Direct calculation requires traversing all x. i The value of is exponential in complexity. The core of the EP algorithm is to construct a Gaussian distribution q(x) to replace p(x). This algorithm uses an iterative optimization approach, continuously adjusting the mean and variance of q(x) to match the moments of p(x). Ultimately, q(x) converges to a high-precision approximation of the true distribution p(x), thus transforming the original problem of calculating posterior probabilities involving complex polynomial multiplication into efficient Gaussian distribution operations, greatly simplifying the computation, as expressed in:

[0117]

[0118] Among them, I 2N Let q(x) be a 2N×2N dimensional identity matrix, and let q(x) be a Gaussian distribution, expressed as:

[0119]

[0120] Where, γ m With λ m There are two trainable parameters, representing the parameter vectors respectively. and The m-th value is obtained and updated in each iteration to approximate the true distribution.

[0121] Both p(y|x) and q(x) are Gaussian distributed, therefore their product also follows a Gaussian distribution. This can be derived from the above. satisfy

[0122]

[0123] Since (17) involves matrix inversion, the EP algorithm has a cubic complexity. This embodiment considers using the sparsity of the DAFT domain channel matrix to reduce the complexity of the EP algorithm for AFDM detection. According to (17), let the auxiliary quantity Ψ A for:

[0124]

[0125] Real-valued channel matrix The value is

[0126]

[0127] Represent the first half of λ as The second half of the elements of λ is represented as Ψ A It can be represented as:

[0128]

[0129] in, This indicates taking the imaginary part. This indicates taking the real part.

[0130] According to the derivation, we can obtain...

[0131]

[0132] According to (21), regarding Ψ A The process of finding the inverse involves reversing matrix D and matrix C+BD. -1 Find the inverse of B, and the relationship between D and C+BD. -1 B are both sparse banded matrices (i.e., D and C+BD)-1 The matrix results of B are the same, but the element values ​​are different, such as... Figure 2 As shown:

[0133] Taking matrix D as an example, the matrix is ​​divided into... in Let loc be one of the four parts of matrix D. max For all paths, the corresponding loc i The maximum value in, ε = N - loc max LU decomposition of it as follows Figure 3 As shown, it can be represented as

[0134]

[0135] The inverse matrix of D can be represented as:

[0136]

[0137] In equation (23), matrices K and Q are both The matrix, due to the normal system loc max Since K and Q are relatively small, they can be directly inverted. In addition, we need to obtain... and The inverse matrix of the given matrix is ​​obtained by directly performing matrix inversion, which has high complexity. This embodiment is based on... and Given the characteristics of an upper triangular (or lower triangular) matrix, a forward substitution algorithm is used to approximate its inversion. The precision of the approximate inversion is set as θ = zloc. max z is a positive integer.

[0138] Table 1. Forward Replacement Algorithm Flow

[0139]

[0140]

[0141] For lower triangular matrices The method for finding the inverse is similar; simply... Invert the upper triangular matrix obtained after transposing it, and then transpose the result to obtain the final product. When obtained K -1 Q -1 Then, D can be obtained from (27). -1 C+BD -1 B has a similar structure to D, and (C+BD) can be obtained using the same method. -1 B) -1 Then, based on (25), we obtain Ψ. A-1 That is, ∑, and after calculating μ, q(x|y) can be obtained.

[0142] Since the transmitted symbols are independent of each other, that is, the distribution of a single transmitted symbol is the corresponding component of the joint distribution:

[0143]

[0144] Next, from q(x) m Remove q(x) from |y) m Obtain the DAFT cavity distribution:

[0145]

[0146] Estimating the mean of transmitted symbol x using DAFT domain cavity distribution and variance

[0147]

[0148] The parameter γ is then updated based on the estimated distribution and the true posterior distribution. m With λ m :

[0149]

[0150] in V obs and x obs The cavity distribution q \m (x m The variance and mean matrices of are given, and λ′ and γ′ are the parameters updated in this iteration for the next iteration. This process is repeated until the number of iterations N is reached. iter Once the upper limit or accuracy requirement is met, output the result of the last iteration. As an estimated AFDM transmission symbol.

[0151] In this embodiment of the invention, a deep learning-based moment matching strategy is used to fine-tune parameters λ′ and γ′ during each iteration. Specifically, this deep learning-based moment matching strategy is configured as follows:

[0152] For the EP algorithm, this embodiment proposes a moment-matching optimization strategy based on deep learning, jointly optimizing moment matching and damping factors. Specifically, this embodiment assigns different damping factors to the positive and negative values ​​of λ, and replaces these damping factors by introducing trainable parameters. This method fully utilizes the mathematical meaning of λ, avoiding information loss caused by ignoring updates of negative values ​​in traditional moment-matching strategies. This enhances the self-correction capability of the EP algorithm and optimizes its performance. The positive and negative values ​​of λ are denoted as follows: and The element positions are recorded so that the corresponding updated values ​​can be mapped back to the original matrix, and the γ corresponding to λ is denoted as... and The moment matching strategy designed in this embodiment can be expressed as:

[0153]

[0154] W1 and W2 are two trainable damping factors that control convergence.

[0155] The values ​​of W1 and W2 were determined through offline training. During training, these damping parameters are iteratively updated to find the optimal parameter combination, thereby improving the overall performance of the EP algorithm. The L2 loss function was applied during training, with the following expression:

[0156]

[0157] in, It is the size of the batch normalized dimensions; and x d Let represent the recovered symbol and the transmitted symbol obtained through the d-th iteration, respectively. The proposed deep learning-based moment matching optimization strategy is trained using the end-to-end error as the loss. The input to the network model is the received symbol and . The output is the recovered symbol. During training, the two trainable parameters are adaptively and iteratively updated. After offline training is completed, the optimized parameters W1 and W2 are obtained. Using these trained parameters, this embodiment of the invention can effectively apply the proposed moment matching optimization strategy to the framework of the EP algorithm, thereby effectively improving the performance of the EP algorithm and accurately recovering the transmitted symbol.

[0158] In one embodiment, when implementing DAFT domain iterative detection and decoding, this embodiment extends the proposed detection algorithm based on a low-complexity improved EP to the coded AFDM system, realizing a DAFT domain IDD scheme. Taking low-density parity check (LDPC) codes as an example, the DAFT domain IDD scheme is introduced. Both EP-based detection and LDPC decoding algorithms (such as belief propagation) can output soft information, enabling them to exchange information and form an IDD system. Unlike traditional single detection and decoding systems, the detection algorithm in the IDD scheme can iteratively improve its performance by feeding back the output of the channel decoder.

[0159] DAFT cavity distribution q (t)\m (x m The cavity distribution q is used as external information provided to the decoder. Since the input to the LDPC decoder is bit information, the cavity distribution q needs to be... (t)\m (x m Convert to bit log-likelihood ratio (LLR):

[0160]

[0161] Where L (t) (b n ) represents the bit LLR, where t represents the IDD iteration round.

[0162] L (t) (b n Input the data into the LDPC decoder to obtain the output decoding result. Then Convert to bit discrete probability distribution

[0163]

[0164] Then convert to a symbolic probability distribution p (t) (x m ):

[0165]

[0166] Where a c As one of the modulation constellation points, M is the number of constellation points, d j (a c ) is a c The j-th bit.

[0167] Next, calculate q. (t+1) (x m variance and mean

[0168]

[0169] Where, q (t+1) (x m This is used for the next iteration. Repeat the above process to reach the maximum number of iterations.

[0170] Example

[0171] This embodiment presents a simulation analysis of the BER performance of a proposed RF demultiplexing signal detection method based on low-complexity optimized desired propagation, considering QPSK modulation and N. iter =10. ML, MMSE, MRC, and time-domain EP schemes were used as baseline schemes. This embodiment considers channel models for four paths, with a delay index of l. i In [0, l max The Doppler index f takes a random value within the range of [[l1 is fixed at 0]]. i In [-f max ,f max The value is randomly selected from the range of f, where f is a random value. max =l max =2, the Doppler index can take the value of a fraction while the time delay index is an integer, c1 is set to c2 is set as In addition, the size of the training dataset and The values ​​are 10000 and 128 respectively. The optimizer is Adam, and the initial learning rate is 0.0001. LDPC coding is selected for the channel.

[0172] First, let's discuss the low-complexity EP algorithm proposed in this embodiment, specifically θ = zloc. max The value of θ is discussed and selected, and the value of θ determines the accuracy of the approximate inversion. Considering N=64 and QPSK modulation, the number of iterations N of the EP algorithm is... iter =10. Simulation results are as follows: Figure 4 As shown, when z ≤ 3, the error in approximate inversion is relatively large, resulting in a performance loss for the low-complexity EP algorithm compared to the traditional EP algorithm. However, when z = 4, θ = 4loc. max The BER performance of the method proposed in this embodiment is close to that of the traditional EP algorithm, so z=4 is used in the subsequent simulation.

[0173] Next, the BER performance of various schemes was compared. AFDM and OFDM have 64 subcarriers per frame (N=64), while OTFS has 8 subcarriers per frame (K) and 8 symbols per frame (L). Figure 5As shown, the BER performance of AFDM based on DAFT domain MMSE detection and OTFS based on DD domain detection is similar, and both significantly outperform OFDM based on frequency domain MMSE. This is because AFDM and OTFS can identify all paths and obtain the complete time-frequency diversity gain, while OFDM based on frequency domain MMSE can only obtain frequency domain diversity gain, resulting in performance loss. Among the remaining schemes, ML has the best performance; however, its exponential complexity makes it difficult to apply in practice. The OTFS system based on DD domain EP has performance close to the scheme proposed in this invention, with BER = 10. -4 Compared to AFDM and OTFS systems using MMSE detection, the time-domain EP system offers a performance gain of nearly 5 dB. However, due to leakage caused by fractional Doppler, the sparsity of the equivalent channel in the DD domain is poor, resulting in high detection complexity for DD-domain EP, and the low-complexity scheme proposed in this embodiment cannot be applied. Another approach is the proposed time-domain EP scheme, which utilizes the sparsity of the time-domain equivalent channel for low-complexity processing. Its complexity is close to that of the scheme proposed in this embodiment. Nevertheless, the time-domain detection scheme cannot fully utilize the frequency-domain diversity gain, thus resulting in a lower BER (5 × 10⁻⁶) compared to the scheme proposed in this embodiment. -5 At that time, there is approximately a 2dB performance loss. This invention utilizes the sparsity and quasi-band properties of the DAFT domain equivalent channel to design a low-complexity EP for AFDM signal detection, achieving both high performance and low complexity. Furthermore, the moment-matching optimization strategy proposed in this invention significantly improves performance. Compared to EP detection algorithms using traditional moment-matching strategies, the improved EP detection scheme proposed in this invention achieves a BER of 1×10⁻⁶. -5 At that time, the performance improvement is approximately 1dB.

[0174] Subsequently, the performance of multiple AFDM detection algorithms under different channel conditions and AFDM system parameters was simulated, considering N=256 and a more realistic EVA channel model. For non-loss of generality, the carrier frequency was set to 4 GHz, the subcarrier spacing to 15 kHz, the speed to 500 km / h, and K=L=16. Figure 6 As shown, the method proposed in this embodiment of the invention still exhibits good performance under such conditions, significantly outperforming various baseline schemes for AFDM detection. BER=10 -4 At the same time, the method proposed in this embodiment of the invention has a performance gain of 5dB, 4dB and 2dB compared with MRC, MMSE and time-domain EP, respectively.

[0175] Finally, the DAFT domain IDD scheme proposed in the embodiments of this invention was simulated. N = 256, N... iter=1, 16-quadrature amplitude modulation (QAM), code rate 1 / 3, IDD iteration rounds 6. In Figure 7 This embodiment demonstrates an LDPC-coded AFDM system under different schemes. Clearly, the proposed IDD scheme offers a significant improvement in BER performance compared to traditional independent detection and decoding schemes. Furthermore, this performance improvement increases progressively with the modulation order: approximately 1.5 dB for QPSK; approximately 3.5 dB for 16-QAM; and up to 8 dB for 64-QAM at a BER of 10⁻³. This improvement can be attributed to the increased significance of AWGN (Additive White Noise) with increasing modulation order, making detection more difficult. Therefore, the feedback from the decoder to the detector becomes increasingly effective in improving the accuracy of symbol estimation, thereby mitigating the impact of noise and improving overall system performance. The advantage of the IDD scheme lies in its ability to utilize the interaction between the detection and decoding processes. By integrating these two traditionally separate modules, the system can better utilize the decoder's feedback information, which contains valuable information about the transmitted symbols.

[0176] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the application, and are not intended to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for detecting simulated radio frequency multiplexing signals based on low-complexity optimized desired propagation, characterized in that, Includes the following steps: Step 1: Obtain the equivalent channel matrix obtained after Discrete Affine Fourier Transform (DAFT) in the AFDM system based on the current transmitted symbol vector. The initial value of the transmission symbol is a preset value; Step 2, for the equivalent channel matrix The process involves segmenting the matrix into blocks. The resulting sub-matrices are then decomposed using LU decomposition, decomposing them into products of upper and lower triangular matrices. A forward substitution algorithm is then used to invert the upper triangular matrix to obtain the inverse of each sub-matrix. The equivalent channel matrix is ​​then derived from the inverses of the sub-matrices in each segment. The reverse; Step 3, based on the equivalent channel matrix With equivalent channel response H A The correspondence is based on the equivalent channel matrix. The inverse and equivalent channel response H A Determine the initial values ​​of the mean matrix μ and standard deviation matrix ∑ of the posterior equivalent Gaussian distribution of the posterior distribution of the transmitted symbols in the optimization expectation EP algorithm; The DAFT domain cavity distribution is obtained by removing the prior distribution of the transmitted symbol from the posterior equivalent Gaussian distribution; and the mean matrix and variance matrix of the transmitted symbol are estimated through the DAFT domain cavity distribution. The mean matrix of the prior equivalent Gaussian distribution of the prior distribution of the transmission symbols in the EP algorithm is updated based on the mean matrix of each transmission symbol and the mean matrix of the DAFT domain cavity distribution; and the variance matrix of the prior equivalent Gaussian distribution of the transmission symbols in the EP algorithm is updated based on the variance matrix of each transmission symbol and the variance matrix of the DAFT domain cavity distribution; wherein the initial values ​​of the mean matrix and variance matrix of the prior equivalent Gaussian distribution are preset values; Step 4: Obtain the transmission symbol vector estimated in the current iteration based on the mean value of each transmission symbol. Based on the estimated transmission symbol vector With the updated prior equivalent Gaussian distribution mean and variance matrix, continue with steps 1-3 to obtain the estimated transmission symbol vector for the next iteration. And the mean and variance matrices of the updated prior equivalent Gaussian distribution; When the set iterative convergence condition is met, the transmission symbol vector estimated in the last iteration will be... As the final estimated AFDM transmission symbol.

2. The method as described in claim 1, characterized in that, The prior equivalent Gaussian distribution and the posterior equivalent Gaussian distribution are as follows: Where q(x) and q(x|y) represent the prior equivalent Gaussian distribution and the posterior equivalent Gaussian distribution of the transmitted symbol vector x, respectively. m γ represents the m-th symbol in the transmission symbol vector x; m , λ m They represent the symbols x respectively m The variance and mean; based on all γ m , λ m Construct the variance matrix γ and the mean matrix λ, σ respectively. 2 Let represent the noise variance, y represent the received symbol in real form, and μ and ∑ represent the mean and variance matrices of the posterior equivalent Gaussian distribution. This indicates that y follows a Gaussian distribution with mean μ and variance ∑. Equivalent channel response and These represent taking the real and imaginary parts of the matrix, respectively. Represents the set of real numbers, where N is the number of symbols in a frame. Represents the real number field.

3. The method as described in claim 2, characterized in that, Step 2 specifically includes: Define auxiliary matrix Ψ A for: Represent the first half of the elements of matrix λ as The second half of the elements of λ are represented as The auxiliary matrix Ψ A Transformed into: Where D, B, and C are auxiliary matrices Ψ A The submatrix after partitioning, and symbol Represents the conjugate transpose of a matrix; The LU decomposition method is used to decompose submatrices D and C+BD. -1 Inverse B yields Ψ A -1 The value of ∑ is obtained by finding the value of the matrix ∑.

4. The method as described in claim 2, characterized in that, In step 3, the DAFT domain cavity distribution is specifically as follows: Where, q \m (x m ) represents the DAFT domain cavity distribution of the prior distribution from the posterior equivalent Gaussian distribution after removing the current transmitted symbol.

5. The method as described in claim 4, characterized in that, In step 3, the mean matrix and variance matrix of the transmitted symbols are estimated using the DAFT domain cavity distribution as follows: Where a represents the set of modulation constellations. Any constellation point, These represent the m-th transmission symbol x. m The mean and variance, Modulate constellation set.

6. The method as described in claim 4, characterized in that, In step 3, the parameter matrices γ and λ are updated as follows: λ′=(V) -1 -(V obs ) -1 Where λ′ and γ′ represent the parameter matrices after this round of iteration, respectively. V obs and x obs These are the variance matrix and mean matrix of the DAFT cavity distribution, respectively.

7. The method as described in claim 2, characterized in that, In step 3, updating the parameter matrices λ and γ in the prior equivalent Gaussian distribution includes: Extract positive values ​​from matrix λ and denote them as matrix λ. And extracting negative values ​​as And record the matrix The element index in the mean matrix λ; and the matrix-based index... The variances corresponding to the means in the matrix are obtained. Based on matrix The variances corresponding to the means in the matrix are obtained. For matrix Set a damping factor, denoted as W1 and W2 respectively; And based on the damping factors W1 and W2, respectively, the matrix Make minor adjustments and updates: in, These represent the matrices after fine-tuning and updating in the current iteration. They represent The updated value was fine-tuned in the previous iteration; These represent the values ​​of the parameter matrices in the updated prior equivalent Gaussian distribution during the current iteration. Based on the current results and the recorded matrix The index of each element in the mean matrix λ will be and Map back to the mean matrix λ, and... and Map back to matrix γ.

8. The method as described in claim 7, characterized in that, In step 3, the damping factors W1 and W2 are optimized based on a deep learning model; the input of the model is the transmission symbol vector. and equivalent channel matrix The output is a recovery symbol.

9. The method as described in claim 8, characterized in that, When optimizing damping factors W1 and W2 using a deep learning model, the loss function is set as follows: in, Represents the L2 loss function. Indicates the size of the batch normalized dimensions; and x d These represent the recovery symbol and the transmission symbol obtained through the d-th iteration, respectively.

10. The method according to any one of claims 1 to 9, characterized in that, When performing signal decoding based on the final estimated AFDM transmission symbols, information is exchanged during the detection and decoding process based on iterative detection decoding, and the currently obtained DAFT domain cavity distribution is used as external information provided to the decoder.

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