A Design Method of Dual-iterative Turbo Equalizer for OTFS-MIMO-ISAC System

Abstract

The present invention discloses a design method for a dual-iterative Turbo equalizer in an OTFS-MIMO-ISAC system. By sparsifying the inverse of the covariance matrix, the matrix inversion overhead of traditional MMSE equalization is reduced through linear complexity derivation. The TFQMR algorithm is used to solve the sparse linear system to reduce the complexity of the first iteration, and the FSPAI algorithm is used to generate a sparse approximate inverse matrix in subsequent iterations to further achieve a balance between computational efficiency and performance. An iterative channel estimation method is adopted, and MMSE equalization feedback is utilized to dynamically optimize channel information. Through outer iteration and inner iteration, the present invention gradually improves the symbol estimation accuracy while reducing the computational complexity, providing a low-complexity and high-precision equalization and channel estimation solution for high-mobility wireless communication systems.

Description

Technical Field

[0001] The present invention relates to wireless communication technology, and specifically to a design method for a dual-iteration Turbo equalizer in an OTFS-MIMO-ISAC system. Background Art

[0002] With the continuous evolution of 5G and future 6G communication systems, how to achieve high-reliability, low-latency, and high-data-rate wireless communication in a high-speed mobile environment has become a core research topic. To meet these performance requirements, key technologies such as orthogonal frequency division multiplexing (OFDM) and massive multiple-input multiple-output (MIMO) have been widely applied. However, in scenarios with high Doppler spread, OFDM modulation is vulnerable to severe inter-carrier interference (ICI), and its performance significantly degrades, making it difficult to meet the communication requirements of high mobility.

[0003] In recent years, as a new modulation method, orthogonal time-frequency-space (OTFS) multiplexes symbols in the delay-Doppler (DD) domain, demonstrating better robustness and performance than traditional OFDM. OTFS spreads each symbol over the entire time-frequency (TF) domain through a two-dimensional transformation, enabling all symbols to obtain a unified channel gain and significantly improving the channel diversity utilization efficiency. Compared with the TF domain, the DD domain channel exhibits stronger sparsity and lower time-variability, providing the possibility for high-precision and low-overhead channel estimation. In addition, massive MIMO has become an important enabling technology for 5G systems due to its great potential in spectral efficiency and system capacity. Combining OTFS with massive MIMO is expected to further enhance the system's performance in high-speed and high-frequency transmissions. However, this integration also brings new challenges: on the one hand, the global channel coupling brought by OTFS modulation significantly increases the detection complexity; on the other hand, the channel estimation accuracy directly affects the overall system performance, which is particularly crucial in a high-dimensional MIMO environment.

[0004] In current research, although various equalization and detection methods have been proposed, such as Turbo detectors based on the MPA (Message Passing Algorithm) and SPA (Sum Product Algorithm), these methods generally face problems such as high complexity and performance bottlenecks caused by short loops. Therefore, how to reduce complexity while ensuring performance has become a key topic in the design of Turbo equalizers for OTFS-MIMO systems. Summary of the Invention

[0005] Aiming at the problem of system performance degradation caused by high detection complexity and insufficient channel estimation accuracy in existing OTFS large-scale MIMO communication systems, the present invention provides a design method for a dual-iteration Turbo equalizer in an OTFS-MIMO-ISAC system, designs a dual-iteration sparse minimum mean square error (MMSE) Turbo equalizer, and combines an improved channel estimation method to achieve a good trade-off between complexity and performance. Compared with traditional methods, the present invention provides a low-complexity and high-precision equalization and channel estimation solution for high-mobility wireless communication systems.

[0006] To achieve the above technical objectives, the technical solutions adopted by the present invention are as follows:

[0007] A design method for a dual-iteration Turbo equalizer in an OTFS-MIMO-ISAC system, the method comprising the following steps:

[0008] S1, encoding the input bit stream of each user equipment through an encoder, and mapping the obtained encoded vector to a symbol sequence after interleaving;

[0009] S2, converting the symbol sequence to the time-frequency domain through an inverse symplectic finite Fourier transform to obtain a time-frequency domain matrix;

[0010] S3, applying a Heisenberg transform to convert the time-frequency domain matrix to a time-domain signal;

[0011] S4, distributing the time-domain signal to multiple transmit antennas through a MIMO architecture for multi-antenna transmission, and adding a cyclic prefix to each transmit signal to combat multi-path delay; when the transmit signal is transmitted through a time-varying channel, it suffers from inter-symbol interference caused by multi-path effects, inter-carrier interference caused by Doppler frequency shift, and the influence of additive white Gaussian noise, forming a received signal with delay-Doppler two-dimensional coupling characteristics;

[0012] S5, remapping the received signal back to the DD domain through a Wigner transform and a symplectic finite Fourier transform;

[0013] S6, inputting the received signal remapped to the DD domain into the constructed dual-iteration sparse MMSE Turbo equalizer to estimate the transmit symbols of all user equipments; the dual-iteration sparse MMSE Turbo equalizer includes an MMSE estimator and a soft input soft output decoder, and the dual-iteration sparse MMSE Turbo equalizer performs an outer iteration by iteratively passing soft information between the MMSE estimator and the soft input soft output decoder, and performs an inner iteration within the MMSE estimator;

[0014] Specifically, when performing the outer iteration, the MMSE estimator estimates the transmitted symbols from the received sequence based on the prior information of the current outer iteration, and then maps the estimated transmitted symbols as the prior information of the soft input soft output decoder to the corresponding log-likelihood ratio of the soft input soft output decoder. The decoder feeds back the calculated extrinsic information to the MMSE estimator, from which new mean, variance, and sparse covariance matrix variance are obtained as the prior information for starting the next outer iteration;

[0015] When performing the inner iteration, the TFQMR algorithm is used to sparsify the problem of calculating the inverse matrix of the covariance matrix into solving an equivalent sparse linear system, and then the FSPAI algorithm is used to derive an approximation of;

[0016] S7. Based on the estimated transmitted symbols, channel estimation is performed to obtain channel information.

[0017] Step S1 further includes:

[0018] Multiple user devices are distributed around the base station, and each user device has its own antenna array. The downlink transmission in the MIMO-OTFS system is adopted between the base station and each user device; The base station is deployed with antennas to serve user devices; The base station is equipped with a uniform linear array, and the antenna spacing is set to half wavelength;

[0019] The input bit stream of each user device is encoded by an encoder to obtain a coding vector , , is the generating matrix of the encoder, is the length of the input bit stream of user device , and its relationship with the length of the encoded bit stream and the code rate is: ; The coding vector is interleaved to obtain , and mapped to a symbol sequence , where is the length of the symbol sequence of user device , is the symbol set of user device .

[0020] Furthermore, in step S2, for user device , a fast Fourier transform of points is performed in the delay dimension, and a fast Fourier transform of Inverse fast Fourier transform of points, and each user equipment is assigned to the specified DD resource blocks to ensure that signals between user equipments do not interfere with each other; the mathematical expression is:

[0021] ;

[0022] Among them, is the DFT matrix of points, which is used to perform fast Fourier transform in the delay dimension; is the IDFT matrix of points, which is used to perform inverse fast Fourier transform in the Doppler dimension; represents the time-frequency domain matrix; represents the symbol matrix, which is the matrix obtained after the symbol sequence is mapped in the delay-Doppler domain.

[0023] Step S3 further includes:

[0024] Perform inverse fast Fourier transform of points on each column of the time-frequency domain matrix and apply the pulse shaping matrix to obtain the time-domain transmission matrix : ;

[0025] ;

[0026] In the formula, is the DFT matrix of points, represents the conjugate transpose of the matrix;

[0027] Vectorize the time-domain transmission matrix into the time-domain signal :

[0028] ;

[0029] Among them, , represents the Kronecker product, represents the vectorization function, is the IDFT matrix of points, represents the symbol matrix.

[0030] Step S4 further includes:

[0031] Add a length of to each OTFS frame of the time-domain signal The cyclic prefix is then converted into an analog signal to obtain the transmitted signal; wherein, the length of the cyclic prefix satisfies the following conditions: , where is the maximum delay of the user equipment , represents the bandwidth, and

[0032] is the subcarrier spacing; The received signal of the user equipment is expressed as:

[0033] ;

[0034] where is the additive white Gaussian noise in the time domain; represents the channel impulse response of the user equipment , where and represent the time delay and Doppler frequency shift of the signal respectively, and t is the time;

[0035] After removing the cyclic prefix, the received signal is sampled at an interval , and the discrete received signal is expressed as:

[0036] ;

[0037] where , represents the modulo operation on , is the delay index of the th path of the user equipment , is the Doppler frequency shift index of the th path of the user equipment , is the number of paths of the user equipment , is the separated time point;

[0038] Starting from the discrete received signal, the received signal vector of the user equipment is expressed as:

[0039] ;

[0040] where is the additive Gaussian white noise vector, is the time-domain effective channel matrix of the user equipment , , is the user equipment The channel gain of the th path,

[0041] Step S5 further includes:

[0042] S51, for the user equipment , rearrange the received signal vector into a matrix , and perform a sparse fast Fourier transform, including performing an inverse fast Fourier transform of points on the columns and a fast Fourier transform of points on the rows; specifically, using a rectangular received pulse shaping waveform , apply a fast Fourier transform of points to each column of the matrix to obtain the time-frequency domain received signal , ; then represent the time-frequency domain received signal as , where is the identity matrix of , perform a fast Fourier transform of points on the rows to obtain the DD domain received matrix ; in the formula, is the DFT matrix of points, is the IDFT matrix of

[0043] S52, vectorize the DD domain received matrix to obtain the received sequence , and the relationship between the transmitted symbol sequence and the received received sequence is expressed as follows:

[0044] ;

[0045] In the formula, is the time-domain effective channel matrix of the user equipment , is the additive white Gaussian noise vector;

[0046] Simplify the relationship formula to , where is the DD domain noise vector, represents the effective DD domain channel matrix of the user equipment .

[0047] Furthermore, in step S6, the external iteration includes the following steps:

[0048] S61, for a user equipment , the transmitted symbol is estimated from the received sequence using a MMSE estimator, where the input of the MMSE estimator includes the mean and variance of each symbol ; the transmitted symbol estimated by the MMSE estimator is: ;

[0049] In the formula,

[0050] is the covariance formula, denotes the discrete-valued signal after sampling of the time-domain signal ;

[0051] Let , , and the covariance matrix of the received sequence is , then the transmitted symbol is expressed as:

[0052] ;

[0053] where denotes the mean vector, denotes the th column of the effective DD-domain channel matrix of the user equipment ;

[0054] The above formula is modified to:

[0055] ;

[0056] where the initial values of the mean and variance are taken as 0 and 1 respectively;

[0057] Using the Woodbury matrix identity, the estimated transmitted symbol is expressed as:

[0058] ;

[0059] S62, map the estimated transmitted symbol to the corresponding log-likelihood ratio of the soft input soft output decoder, and calculate the mean and variance of the estimated symbol:

[0060] ;​​

[0061] ;

[0062] where is obtained from a Hermitian matrix, and the symbol set is , and each symbol represents the th constellation, , which corresponds to two bits, namely [0,0], [0,1], [1,0], and [1,1]; is a real number, denotes the complex conjugate of ;

[0063] By substituting the mean and variance into the corresponding formula for the log-likelihood ratio, the extrinsic LLR of is derived as:

[0064] ;

[0065] ;

[0066] where and represent the real and imaginary parts of the corresponding symbol. After de-interleaving into , it is used as the prior information for the decoder;

[0067] S63. The extrinsic information output by the decoder is obtained through the interleaver as , and is fed back to the MMSE estimator, and it is derived that:

[0068] ;

[0069] where , the new mean and variance are expressed as: from the symbol set

[0070]

[0071] ;

[0072] S64. The covariance matrix is sparsified using the following formula:

[0073] ;

[0074] where represents the sparse pattern of the matrix, Denote the user equipment 's symbol covariance matrix, denote the noise power, denote the identity matrix, denote the path and path 's difference in Doppler shift index, denote the path and path 's difference in delay index, is the set of path pairs of the user equipment .

[0075] Furthermore, in step S6, the internal iteration includes the following steps:

[0076] Transform the problem of calculating the original into solving an equivalent sparse linear system , and use the TFQMR algorithm to solve the equation. The solving process includes:

[0077] For the linear system , starting from the initial guess , construct the Krylov subspace , where is the initial residual; through multiple iterations, gradually reduce the residual ; at each iteration, generate two orthogonal sequences, one is the sequence composed of , and the other is the sequence composed of , expressed as:

[0078] ;

[0079] ;

[0080] where, and are orthogonal sequences generated by the double Lanczos process, calculate is the new vector generated after the matrix acts on the current vector , calculate then is generated by the conjugate transpose of the matrix ; and are scalar coefficients calculated at each iteration, is the number of iterations;

[0081] After each iteration, according to the current residual and the updated solution to update the approximate solution, and the solution update formula is:

[0082] ;

[0083] where, is the increment obtained from the Krylov subspace ;

[0084] Calculate the new residual , and check whether the stopping condition is satisfied; if the residual norm is less than the preset tolerance , the algorithm stops and outputs ; if convergence is not achieved, continue the iteration and further optimize the solution by updating and ;

[0085] Select a predefined sparse pattern , calculate the matrix , represents the symbol covariance matrix of the user equipment ; the calculation process includes:

[0086] Based on , initialize the sparse approximate Cholesky factor such that , where is a lower triangular matrix, and it is obtained by minimizing the Frobenius norm with respect to the specified sparse pattern of ; dynamically capture the sparse pattern of by minimizing the Kaporin condition number :

[0087] ;

[0088] where, tr(·) and det(·) represent the trace and determinant of the matrix respectively, ; represents the noise power, represents the identity matrix;

[0089] For each column of , independently calculate its non-zero element index set and the corresponding values, and iteratively improve the approximation by selecting a new index and adding it to ; specifically, for each new index , update by minimizing the new Kaporin condition number :

[0090] ;

[0091] where is the th column of the identity matrix, denotes the index corresponding to the numerical value of the th column of the sparse approximate Cholesky factor; select the index from where represents the operation of the th row of the matrix and the set ; add the index with the largest to and to form a new index set and where is used to measure the reduction in the condition number due to adding the new index

[0092] ;

[0093] Continue the iteration until or the maximum value of is less than the preset tolerance

[0094] Furthermore, in step S7, the new estimate of the channel gain is expressed as:

[0095] ;

[0096] where denotes the conjugate complex number of denotes the effective DD-domain channel matrix of the user equipment denotes the estimated transmitted symbol, denotes the received sequence, denotes the expected value function.

[0097] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0098] First, the dual-iteration Turbo equalizer design method of the OTFS-MIMO-ISAC system of the present invention makes full use of the statistical channel state information, and significantly reduces the computational complexity of symbol estimation and channel estimation in the multi-user equipment scenario through the combination of the TFQMR and FSPAI algorithms.

[0099] ​Second, for the dual-iteration Turbo equalizer design method of the OTFS-MIMO-ISAC system of the present invention, the designed Turbo equalizer gradually improves the accuracy of symbol estimation and approaches the performance of the optimal detector through a dual-iteration structure of outer iteration and inner iteration.

[0100] Third, for the dual-iteration Turbo equalizer design method of the OTFS-MIMO-ISAC system of the present invention, considering that the complexity of the traditional MMSE equalizer mainly comes from the inversion calculation of the covariance matrix, the matrix is sparsified, and the linear complexity derivation of the inverse of the covariance matrix is realized by using the sparse structure, effectively reducing the overall computational burden of the system. Brief Description of the Drawings

[0101] Figure 1 is the model diagram of the OTFS-MIMO-ISAC system of the present invention;

[0102] Figure 2 is the flowchart of the dual-iteration Turbo equalizer design method for the OTFS-MIMO-ISAC system of the present invention;

[0103] Figure 3 is the schematic diagram of the comparison of bit error rates under different algorithms;

[0104] Figure 4 is the schematic diagram of the comparison of channel estimation MSE under different algorithms;

[0105] Figure 5 is the schematic diagram of the comparison of detection probabilities under different numbers of user equipment;

[0106] Figure 6 is for the number of outer iterations schematic diagram of the influence relationship of sparsity. Detailed Embodiments

[0107] The following further describes the embodiments of the present invention in detail with reference to the drawings.

[0108] The present invention discloses a dual-iteration Turbo equalizer design method for an OTFS-MIMO-ISAC system, and the method includes the following steps:

[0109] S1, encoding the input bit stream of each user equipment through an encoder, and mapping the obtained encoded vector to a symbol sequence after interleaving;

[0110] S2, converting the symbol sequence to the time-frequency domain through the inverse symplectic finite Fourier transform to obtain a time-frequency domain matrix;

[0111] S3, applying the Heisenberg transform to convert the time-frequency domain matrix into a time-domain signal;

[0112] S4. The time-domain signal is distributed to multiple transmit antennas through the MIMO architecture for multi-antenna transmission. A cyclic prefix is added to each transmitted signal to combat multipath delay. When the transmitted signal passes through the time-varying channel, it suffers from inter-symbol interference caused by multipath effects, inter-carrier interference caused by Doppler frequency shift, and the influence of additive white Gaussian noise, forming a received signal with two-dimensional coupling characteristics of delay and Doppler.

[0113] S5. The received signal is remapped back to the DD domain through the Wigner transform and the symplectic finite Fourier transform.

[0114] S6. The received signal remapped to the DD domain is input into the constructed double-iterative sparse MMSE Turbo equalizer to estimate the transmitted symbols of all user equipments. The double-iterative sparse MMSE Turbo equalizer includes an MMSE estimator and a soft-input soft-output decoder. This double-iterative sparse MMSE Turbo equalizer performs external iterations by iteratively passing soft information between the MMSE estimator and the soft-input soft-output decoder, and performs internal iterations within the MMSE estimator.

[0115] Specifically, when performing external iterations, the MMSE estimator estimates the transmitted symbols from the received sequence based on the prior information of the current external iteration, and then maps the estimated transmitted symbols to the corresponding log-likelihood ratios of the soft-input soft-output decoder as the prior information of the soft-input soft-output decoder. The decoder feeds back the calculated external information to the MMSE estimator, obtains the new mean and variance and the variance of the sparse covariance matrix from it, and uses them as the prior information for starting the next external iteration.

[0116] When performing internal iterations, the TFQMR algorithm is used to sparsify the problem of calculating the inverse matrix of the covariance matrix into the problem of solving an equivalent sparse linear system, and then the FSPAI algorithm is used to deduce the approximation value of

[0117] S7. Based on the estimated transmitted symbols, channel estimation is performed to obtain channel information.

[0118] The system designed by the present invention is as Figure 1 shown. Multiple user equipments (UEs) are distributed around the base station (BS). Each user equipment has its own antenna array, considering the downlink transmission in the MIMO-OTFS system. The BS deploys antennas for Services are provided to user equipment. In centralized massive MIMO, a uniform linear array (ULA) is equipped at the base station, and the antenna spacing is set to half a wavelength. Without loss of generality, for simplicity, it is assumed that the impacts of other user equipment have been effectively isolated or eliminated. Only one user equipment is concerned, and the dependence of the channel on the user equipment index is ignored.

[0119] The channel equalization process of the present invention is as follows Figure 2 As shown, after the transmitted signal is modulated by OTFS, it is first mapped from the DD-domain symbols to the time-frequency domain through ISFFT, and then the time-domain waveform is generated through the Heisenberg transform. In MIMO transmission, the base station synchronously transmits the OTFS waveform through multiple transmitting antennas of the uniform linear array, and a cyclic prefix is added to each antenna signal to combat multipath delay. When this signal is transmitted through a time-varying channel, it will suffer from inter-symbol interference caused by multipath effects, inter-carrier interference caused by Doppler frequency shift, and the influence of additive white Gaussian noise, forming a received signal with delay-Doppler two-dimensional coupling characteristics. Finally, the time-domain received signal is remapped back to the DD domain through the Wigner transform and the symplectic finite Fourier transform (SFFT) for subsequent processing. The received signal passes through the double-iterative sparse MMSE Turbo equalizer designed by the present invention, which mainly consists of an MMSE estimator and a soft input soft output (SISO) decoder. This equalizer has a double-iterative structure, namely an outer iteration and an inner iteration: the outer iteration is operated by iteratively passing soft information between the MMSE estimator and the decoder; the inner iteration is performed within the MMSE estimator to reduce the computational cost through two iterative algorithms, namely the TFQMR algorithm for the initial iteration and the FSPAI algorithm for subsequent iterations. Step 6 mainly focuses on the outer iteration. The first step in this part is to estimate the transmitted symbols from the received sequence using the MMSE estimator, then map the estimated symbols to the corresponding log-likelihood ratio (LLR) of the decoder, and finally the estimator outputs the extrinsic information as the prior information of the decoder. Thereafter, the decoder further reduces interference and feeds back the extrinsic information to the estimator to obtain new mean and variance. Before starting another outer iteration, the covariance matrix is sparsified to reduce the detection complexity. After obtaining relatively accurate symbols through this equalizer, conjugate matching calculation is performed based on the recovered expected user equipment signal.

[0120] The experimental process of this embodiment is as follows:

[0121] Step 1, bitstream coding and mapping: First, the input bitstream of each user equipment is encoded by the encoder to obtain the coding vector , is the generating matrix of the encoder, is the length of the input bitstream of user equipment , and it is related to the length of the encoded bitstream ​ And the code rate The relationship is as follows: ; The encoding vector is interleaved to obtain and mapped to the symbol sequence where is the symbol sequence length of the user equipment , and is the symbol set of the user equipment .

[0122] Step 2, Convert the symbol sequence to the time-frequency domain through the inverse symplectic finite Fourier transform (ISFFT transform) to obtain the time-frequency domain matrix: For the user equipment , perform the fast Fourier transform (FFT) of points on the delay dimension, and perform the inverse fast Fourier transform (IFFT) of points on the Doppler dimension. Each user equipment is assigned a specific DD resource blocks to ensure that the signals between user equipments do not interfere with each other. The mathematical expression is:

[0123] (1);

[0124] where is the point DFT matrix for performing FFT on the delay dimension; is the point IDFT matrix for performing IFFT on the Doppler dimension; represents the time-frequency domain matrix; represents the symbol matrix, which is the matrix obtained after the symbol sequence is mapped in the delay-Doppler domain.

[0125] Step 3, After the ISFFT transform, apply the Heisenberg transform to convert the time-frequency domain matrix to a time-domain signal: The Heisenberg transform performs the inverse fast Fourier transform (IFFT) of points on each column of the time-frequency domain matrix , and apply the pulse shaping matrix to obtain the time-domain transmission matrix . After the Heisenberg transform, the obtained time-domain transmission matrix can be expressed as:

[0126] (2);

[0127] where is the pulse shaping matrix of the user equipment , and its diagonal elements are For a sample, the duration of the pulse shaping waveform is , and the sampling interval is , that is:

[0128] (3).

[0129] To simplify the processing, it is assumed here that a rectangular waveform is used. At this time can be simplified to the identity matrix ; Other well-designed windows (such as raised cosine windows) can also be used to optimize the signal transmission performance. The time-domain transmission matrix can be vectorized into the time-domain signal :

[0130] (4);

[0131] where , represents the Kronecker product, represents the vectorization function.

[0132] Step 4. To further ensure the robustness of the signal, a cyclic prefix (CP) with a length of is added to each OTFS frame of the time-domain signal . After converting it into an analog signal, the transmitted signal is obtained. The transmitted signal is transmitted into the time-varying channel.

[0133] The channel impulse response can be expressed as a combination of independent delays and Doppler frequency shifts for each user equipment. For user equipment , the channel impulse response can be expressed as:

[0134] (5);

[0135] where is the channel gain of the th path of user equipment ; is the delay of the th path of user equipment ; is the Doppler frequency shift of the th path of user equipment ; is the number of paths of user equipment ; The delay and Doppler frequency shift can be expressed as:

[0136] (6);

[0137] where is the delay index of the th path of the user equipment, is the Doppler shift index of the th path of the user equipment, is the fractional Doppler shift of the th path of the user equipment. For a traditional wideband system, the sampling time is small enough so that the delay taps can be approximated as integers. However, due to the limitation of the delay, cannot be arbitrarily large. Therefore, fractional Doppler may exist in an actual system. This embodiment assumes that is large enough to neglect the fractional Doppler, which helps to understand the sparse characteristics of the channel matrix.

[0138] The above CP length should be greater than the maximum delay, i.e., , where is the maximum delay of the user equipment , represents the bandwidth, is the subcarrier spacing. According to Equation (5), the received signal of the user equipment can be expressed as:

[0139] (7);

[0140] where is the additive white Gaussian noise (AWGN) in the time domain, and its distribution is .

[0141] After removing the cyclic prefix, the received signal is sampled at an interval . By substituting Equation (5), the discrete received signal can be expressed as:

[0142] (8);

[0143] where , represents the modulo operation on.

[0144] Starting from the discrete received signal, the time-domain effective channel matrix of the user equipment can be expressed as: , where is the permutation matrix (forward cyclic shift), representing the delay effect:

[0145] (9);

[0146] is a diagonal matrix representing the Doppler effect and additional frequency offsets: , where , and thus, the received signal vector of the user equipment can be expressed as:

[0147] (10);

[0148] where is the AWGN noise vector.

[0149] Step 5, for the user equipment , the received signal vector is rearranged into a matrix , arranged by columns. Considering the use of a rectangular received pulse shaping waveform, apply the point fast Fourier transform (FFT) to each column of the matrix to obtain the time-frequency domain received signal . The specific formula is as follows:

[0150] (11).

[0151] After FFT processing, the time-frequency domain received signal can be expressed as , where is the identity matrix. Then perform SFFT, including performing an inverse fast Fourier transform (IFFT) of points on the columns and a fast Fourier transform (FFT) of points on the rows.

[0152] By substituting the time-frequency domain received signal , the DD domain received matrix is obtained: , vectorize it to obtain the received sequence , so the relationship between the transmitted DD domain vector (symbol sequence) and the received DD domain vector (received sequence) is expressed as follows:

[0153] (12);

[0154] Simplify Equation (12) to , where is the DD domain noise vector, so the effective DD domain channel matrix of the user equipment ​ It can be expressed as:

[0155] (13);

[0156] where is the component caused by the path. According to previous studies, it can be derived that:

[0157] (14);

[0158] where are the Doppler index and delay index of the th element of the user equipment respectively, and represent the Doppler shift index and delay index of the th path of the user equipment respectively, is the phase factor, which is used to represent the Doppler effect.

[0159] Obviously, contains only one non-zero element in each row and each column. Therefore is a sparse matrix, and there are non-zero elements in each row and each column. In subsequent processing, the received sequence is fed into the Turbo equalizer to recover the information bits. In practical applications, the size of the OTFS frame (i.e., ) is usually very large. This frame structure is designed to adapt to the complex characteristics of high-speed mobile environments and multipath channels, thereby improving the robustness and data transmission efficiency of the system. However, due to the large size of the OTFS frame, traditional equalizers often face high computational complexity when processing these frames, which not only increases the power consumption of the system but also may affect the real-time performance and response speed of the system. Therefore, in practical deployments, it is particularly important to design a low-complexity equalizer.

[0160] Step 6, the dual-iteration sparse MMSE Turbo equalizer designed by the present invention aims to provide a lower-complexity equalizer for the channel when the OTFS frame is large. Pass the received signal through the equalizer, as shown in the channel equalization processing flow Figure 2 . The outer iteration of the designed equalizer consists of three steps: performing MMSE estimation using the prior information; calculating the extrinsic information of the decoder; after decoding, updating the mean and variance, and the covariance matrix of the sparse MMSE estimator. The design uses quadrature phase shift keying (QPSK) modulation, and the symbol set is , and each symbol represents the A constellation, which corresponds to two bits, namely [0,0], [0,1], [1,0] and [1,1].

[0161] For the user equipment , the first step is to estimate the transmitted symbols from the received sequence using the MMSE estimator. The inputs to the MMSE estimator include the mean and variance of each symbol , where , represents the discrete-valued signal after sampling of the time-domain signal . To minimize the mean square error (MMSE) , the MMSE estimator derives the estimated transmitted symbol from Equation (12) as follows:

[0162] (15).

[0163] For the convenience of notation, the covariance matrix of the received sequence is denoted as , and calculating its inverse matrix largely determines the complexity of the equalizer.

[0164] Map the estimated transmitted symbol to the corresponding log-likelihood ratio (LLR) of the decoder. To obtain the potential full-channel diversity gain, the posterior probability of each bit with respect to the entire received sequence is desirable, that is , where represents the interleaved bits corresponding to the symbols and . However, directly calculating is very time-consuming, especially when and are large. Therefore, the posterior probability based on the estimated transmitted symbol is introduced to approximate . The definition of the LLR is as follows:

[0165] (16);

[0166] where the first term is defined as the extrinsic LLR , and the second term is defined as the prior LLR , which is derived by the decoder, and and should be strictly independent.

[0167] Finally, the MMSE estimator outputs extrinsic information as the prior information for the decoder. Thereafter, the decoder further reduces interference and feeds back the extrinsic information to the MMSE estimator to obtain new mean and variance. Before starting another extrinsic iteration, the covariance matrix A is sparsified to reduce the detection complexity. These three steps will be described in detail below.

[0168] Step 1, MMSE estimation based on prior information: For the user equipment , in the initial iteration, assume the mean and variance . In subsequent iterations, the mean and variance are provided by the decoder. By substituting into the formula v, we have:

[0169] (17);

[0170] (18);

[0171] (19);

[0172] Equation (17) is due to the assumption that the transmitted symbols are independent, which approximately holds under the action of the interleaver. Equation (19) is derived from and . In addition, it can be derived that , where represents the -th column of . By substituting equations (18) and into equation (15), the estimated symbol can be expressed as:

[0173] (20);

[0174] where represents the mean vector. Since the derivation of depends on the prior knowledge , the result derived from is independent of . Therefore, the above equation is modified to:

[0175] (21);

[0176] where initially and , thus excluding the prior knowledge of itself. To avoid repeatedly calculating the inverse matrix in the above equation, the Woodbury matrix identity is used:

[0177] (22);

[0178] Among them, , by substituting into Equation (22), the estimated symbol can be expressed as:

[0179] (23);

[0180] It can be seen from the above formula that is used twice, that is and , which largely determines the complexity of the MMSE estimation.

[0181] Step 2, calculate the extrinsic information of the decoder: To calculate , assume that follows a Gaussian distribution, that is

[0182] , where Among them, and , according to the estimated transmitted symbol , the mean and variance of the estimated symbol can be calculated as follows:

[0183] (24);

[0184] (25);

[0185] Among them, is obtained from which is a Hermitian matrix, so is a real number. By substituting the mean and variance of the above estimated symbol into the calculation formula of the extrinsic LLR (Equation (16)), the of can be deduced as

[0186] (26);

[0187] (27);

[0188] Then, is deinterleaved into , and used as the prior information of the decoder.

[0189] Step 3, update the mean and variance and sparsify the covariance matrix: For the user equipment , the decoder outputs the new extrinsic information . Through the interleaver, And feed it back to the MMSE estimator. According to the definition of LLR in Equation (16), it can be derived that:

[0190] (28);

[0191] where , from the symbol set , the new mean and variance can be expressed as:

[0192] (29);

[0193] (30).

[0194] In the last step of the outer iteration, the covariance matrix needs to be sparsified to facilitate the calculation of its inverse matrix . To this end, some sparsification criteria are proposed with the help of graph theory, and the specific criteria are as follows:

[0195] Without loss of generality, first an arbitrary random sparse matrix is introduced, and two definitions are given respectively to consider its sparsity and randomness. Then, two sparsity criteria are proposed using graph theory to sparsify

[0196] Criterion 1: (Sparsify from edge constraints): Normalize the diagonal elements of to by Jacobi scaling, where is a diagonal matrix, for , , then, if an edge satisfies , ignore the edge and set to zero. Among them, is an appropriate threshold, and in the case of a significant performance drop, try to make .

[0197] Criterion 2 (Sparsify from node constraints): If the degree of a node is less than the threshold , that is , then the node can be disconnected from other nodes. To avoid too much inaccuracy, the threshold should be much smaller than the maximum degree.

[0198] The first criterion eliminates the edges whose magnitudes are negligible compared to the corresponding diagonal elements, because these edge pairs have little impact on matrix operations. Use Jacobi scaling to normalize the diagonal elements to facilitate the selection of . When ​When the magnitude of the diagonal elements is greater than that of most non - diagonal elements, Criterion 1 can effectively sparsify . In addition, the second criterion further improves the sparsity level by isolating nodes to a lesser extent, because these nodes are almost independent of other nodes. According to the criterion, smaller and larger will result in a sparser . Once is well - sparsified, the computational complexity of calculating can be greatly reduced. However, this reduction in complexity comes at the cost of a performance degradation. Therefore, and must be carefully selected from a large number of simulations to achieve a balance between the complexity of calculating and the performance degradation.

[0199] Apply the above - defined to , and use the sparsification criterion to reduce the complexity of the equalizer. It can be seen from the previous derivation that is the sum of , where contains only one non - zero element in each row and each column, and its sparse pattern is given by:

[0200] (31);

[0201] where sp(·) represents the sparse pattern of the matrix, and represent the permutation matrices in the Doppler dimension and the delay dimension. The sparse pattern of the covariance matrix can be derived by the following formula:

[0202] (32);

[0203] where , , is the set of path pairs of the user equipment , satisfying that if , then, which indicates that is symmetric.

[0204] For the user equipment , after applying the sparsification criterion, the covariance matrix can be calculated with lower complexity. When the signal - to - noise ratio is low, is mainly dominated by the second term , so the diagonal elements of have a larger magnitude than the non - diagonal elements. According to the sparsification criterion for After sparsification, most of the non - diagonal elements can be ignored. However, at high signal - to - noise ratios, the non - diagonal elements may be of the same order of magnitude as the diagonal elements. In this case, direct calculation should be avoided . However, with the help of the decoder, the prior information is highly reliable in the second iteration, so the updated variance is close to zero, which weakens the first term on . Therefore, the second term dominates again, and the sparsification criterion can be applied again.

[0205] In the outer iteration of the Turbo equalizer, first, MMSE estimation based on prior information is to be performed. In the initial iteration, there is , due to the lack of prior information. In this case, the non - diagonal elements of are of comparable magnitude to the diagonal elements. Therefore, the sparsity of can only be slightly enhanced. Thus, the direct calculation of is too complex. To reduce the computational complexity of MMSE estimation, two low - complexity algorithms for calculating are designed, one of which is used in the initial outer iteration and the other is applied in subsequent iterations. The former uses the TFQMR algorithm to transform the problem of calculating into solving an equivalent sparse linear system, while the latter uses the FSPAI algorithm to derive an approximation of after applying the proposed sparsification criterion. According to the previous derivation, and where

[0206] Regarding the former, we observe that is the common part of all estimated symbols , where in the initial iteration , because is a all - zero vector. Therefore, the problem can be solved for all by solving an equivalent sparse linear system , thus avoiding the direct calculation of , where is the unknown vector. For the calculation of , assuming that all (for and user equipment ) are almost the same under a given , the calculation of can be converted to solving another sparse linear system , where is the unknown vector. By Obtained has a very small computational amount because it is sparse and has only non-zero elements. In this way, we can solve two equivalent sparse linear systems, namely and , instead of directly calculating , thereby reducing the computational amount in the initial iteration.

[0207] TFQMR algorithm: Based on the above sparse linear system, an iterative algorithm called TFQMR is used to solve the sparse linear system because TFQMR has no requirements for the coefficient matrix of the linear system . In addition, TFQMR has a smaller computational amount than some other candidate iterative algorithms (such as the Generalized Conjugate Residual (GCR) algorithm), and the global convergence of TFQMR applied in the system can be guaranteed through a finite number of iterations. If the MMSE estimation is only performed once without subsequent external iterations, the iterative equalizer is simplified to a traditional LMMSE equalizer. Therefore, TFQMR can be easily extended to any other system with an LMMSE equalizer.

[0208] For the linear system , TFQMR starts from an initial guess and constructs the Krylov subspace , where is the initial residual. Through multiple iterations, the residual is gradually reduced. At each iteration, TFQMR generates two orthogonal sequences: one is the sequence composed of , and the other is the sequence composed of . The two respectively satisfy the following relationships:

[0209] (33);

[0210] (34);

[0211] where and are orthogonal sequences generated by the bi-Lanczos process, calculating is the new vector generated after the matrix acts on the current vector , and calculating is generated by the conjugate transpose of the matrix . and is a scalar coefficient calculated in each iteration. With these orthogonal sequences, we can effectively solve the linear system without explicitly computing the transpose of the matrix. After each iteration, TFQMR updates the approximate solution based on the current residual and the updated solution to update the approximate solution and compute the new residual. Specifically, the update formula for the solution is as follows:

[0212] , (35)

[0213] where is the increment obtained from the Krylov subspace . In this way, TFQMR gradually improves the current solution. After each iteration, TFQMR calculates the current residual and checks whether the stopping condition is satisfied. If the residual norm is less than the preset tolerance , the algorithm stops and outputs ; if convergence is not achieved, TFQMR will continue to iterate and further optimize the solution by updating and .

[0214] FSPAI algorithm: After the first outer iteration, new means and variances are obtained as prior information for starting the next outer iteration. However, since the variance matrix is no longer the identity matrix, has completely different values at . Therefore, TFQMR needs to use times in each outer iteration to derive each , which is computationally expensive for subsequent outer iterations. With the help of the decoder, most of the prior information of the MMSE estimator is reliable enough, so the corresponding new variance of the estimated symbol is close to zero. Therefore, the second term of dominates, and the magnitude of the diagonal elements is much larger than that of other elements. By sparsifying according to the criterion, the sparsity of can also be greatly improved. Therefore, we can use the FSPAI algorithm to generate a sparse approximation of , which is inherently parallelizable and can further save computational time.

[0215] Select a predefined sparse pattern , for example , compute the matrix , initialize the sparse approximate Cholesky factor such that , where is a lower triangular matrix and can be obtained by minimizing the Frobenius norm with respect to the prescribed sparse pattern and its initial value is based on . By minimizing the Kaporin condition number the sparse pattern of is dynamically captured, that is where tr(·) and det(·) denote the trace and determinant of a matrix respectively

[0216] (36)

[0217] .

[0218] For each column of , the index set of its non - zero elements and the corresponding values are calculated independently. By selecting a new index and adding it to , the approximation is iteratively improved. For each new index , is updated by minimizing the new Kaporin condition number :

[0219] , (37)

[0220] where is the -th column of the identity matrix. Selecting an index from , the index with the largest is added to to form a new index set and , where measures the reduction in the condition number by adding the new index . By adding the new index , the minimum value of the Kaporin condition number is:

[0221] , (38)

[0222] where measures the reduction in the condition number by adding the new index . The complexity of FSPAI is mainly determined by calculating . Therefore, the cardinality of should not exceed a certain value or the maximum value of​ Since the matrix factorization and iterative optimization processes for each user equipment are independent, the calculations for all user equipments can be processed in parallel, thereby further improving the calculation efficiency.

[0223] Step 7, after using the above Turbo equalizer, relatively accurate restored mapped symbols of all user equipments can be obtained. Then, based on this, the transmission signal of the specified user equipment can be reconstructed, and interference can be subtracted from the received signal. Matching estimation is performed to obtain the channel estimation of the expected user equipment, that is, this channel estimator is a matched channel estimator after partial interference cancellation. Using the designed sparse double iterative equalization, the restored signals of all user equipments can be determined at the BS. Reconstruct and eliminate the intra-frame interference, and then perform conjugate matching calculations based on the restored expected user equipment signals. The new estimation of the channel gain can be expressed as:

[0224] (39);

[0225] where the arithmetic mean of the data block is used as the mathematical expectation. This channel estimator does not require much calculation.

[0226] The simulation experiment mainly simulates and discusses the BER performance of different detectors, and compares the mean square error MSE of different channel estimators, where MSE is defined as: , and the detection probabilities for different estimation algorithms are also compared. To evaluate the computational complexity of FSPAI, the sparsity of the approximate Cholesky factor is studied. The simulation settings are M = 64, N = 32, and the length of CP is 16. It is assumed that each path gain follows a Rayleigh distribution with a uniform power delay profile, that is , the delay and Doppler exponents are uniformly selected such that and , where and are equal to 6. Compared with the double iterative sparse Turbo equalizer, the non-Turbo equalizer is used for the uncoded OTFS system, where the MMSE estimator regards the extrinsic information generated previously as the prior information in the next extrinsic iteration.

[0227] At Figure 3The bit error rate performance of the dual-iterative sparse Turbo equalizer designed by the present invention in the OTFS system is shown. Compared with the traditional LMMSE, MPA and near-optimal Coded SPA detectors, the receiver significantly improves the detection accuracy through the collaborative optimization of outer-iteration soft information exchange and inner-iteration low-complexity algorithms (TFQMR / FSPAI). At the first outer iteration, the performance of the equalizer is already comparable to that of the traditional LMMSE. After 5 outer iterations, its bit error rate achieves a gain of approximately 4 dB compared to LMMSE, a 1.3 dB improvement compared to MPA, and approaches the theoretical limit performance of Coded SPA. The complexity of Coded SPA grows exponentially with the number of paths P (P = 4 is set in the simulation), while the proposed scheme in the present invention reduces the computational complexity to a linear order through sparse matrix approximation and iterative truncation strategies, without relying on pilot overhead. This result verifies the unique advantage of the dual-iterative architecture in balancing performance and complexity in high-mobility scenarios.

[0228] Figure 4 The comparison of the mean square error (MSE) performance of different channel estimation methods in a dynamic environment is shown. Generally speaking, the classical LS algorithm does not consider the channel statistical characteristics, and its MSE is relatively high at low signal-to-noise ratios, and the improvement is limited as the signal-to-noise ratio increases, indicating its insufficient robustness to noise and interference. In contrast, the MMSE algorithm shows a stable downward trend by utilizing the channel prior information, verifying its reliability in complex environments. The iterative optimization method designed by the present invention exhibits unique performance advantages: in the low signal-to-noise ratio region, the estimation accuracy of the first iteration is already much lower than that of the two algorithms, and as the signal-to-noise ratio increases, the gap between the two estimation methods and the dual-iterative algorithm becomes more obvious, confirming the correction ability of the iterative mechanism for residual errors. This characteristic enables it to have both noise resistance and convergence efficiency under dynamic channel conditions, providing a new idea for low-complexity and high-precision estimation.

[0229] In Figure 5 it shows the influence of the number of user equipments on the target detection probability. As the number of UEs increases, the detection probability It will increase. This is because as the number of user devices increases, the chance for more user devices to approach the target also increases. The LS, a classical estimator based on the least squares criterion, ignores multi-path interference (MPI) and pilot contamination, and its detection probability (black line) is always low, exposing its sensitivity to low signal-to-noise ratio. The MMSE optimizes the estimation accuracy through the channel covariance matrix, but its static model shows spatial correlation degradation in the multi-user device scenario, and its performance drops rapidly when the number of user devices increases. Through an iterative optimization mechanism, for the proposed equalizer, the TFQMR is adopted for sparse channel matrix approximation in the first outer iteration to initially suppress multi-user device interference (MUI), and the detection probability is improved significantly compared with the MMSE. Through five outer iteration soft information exchanges and dynamic threshold sparsification, when the number of user devices increases, the detection probability slightly decreases, verifying the effectiveness of iterative interference reconstruction.

[0230] Figure 6 The evolution characteristics of the channel sparsity with the number of iterations in different signal-to-noise ratio scenarios for Turbo and non-Turbo equalizers are compared. The sparsity difference stems from the differences in the handling of residual interference between the two methods: the non-Turbo scheme relies on a single global approximation and is vulnerable to noise-dominated path interference at low signal-to-noise ratios; the Turbo scheme gradually enhances the weight coefficients of the true multi-path components through iterative path strength reweighting, thereby achieving a more accurate sparse representation. Although the non-Turbo equalizer shows a rapid increase in sparsity at high signal-to-noise ratios, indicating that a single matrix approximation can effectively capture the dominant paths under good channel conditions, its sparsity growth at low signal-to-noise ratios lags significantly, and it cannot reach a high sparsity even after multiple iterations, exposing the misjudgment problem of traditional methods for weak paths under noise interference. The dual-iteration mechanism of the Turbo equalizer realizes stable sparsity growth in both 6dB and 10dB scenarios through outer iteration soft information feedback and inner iteration dynamic threshold adjustment. Especially at 10dB, the Turbo scheme only needs 4 iterations to exceed the final performance of the non-Turbo scheme, verifying the progressive optimization ability of the outer loop for channel prior information.

[0231] Although the preferred embodiments of the present application have been described, those skilled in the art can make additional changes and modifications once they learn the basic creative concept. Therefore, the appended claims are intended to be construed to include the preferred embodiments as well as all changes and modifications falling within the scope of the present application.

[0232] Obviously, those skilled in the art can make various changes and modifications to the present application without departing from the spirit and scope of the present application. Thus, if these modifications and variations of the present application fall within the scope of the claims of the present application and their equivalent technologies, the present application also intends to include these changes and modifications.

Claims

1. A design method of a double-iteration Turbo equalizer for an OTFS-MIMO-ISAC system, characterized in that, The method includes the following steps: S1. Encode the input bit stream of each user equipment through an encoder, and map the obtained encoded vector to a symbol sequence after interleaving; S2. Convert the symbol sequence to the time-frequency domain through an inverse symplectic finite Fourier transform to obtain a time-frequency domain matrix; S3. Apply a Heisenberg transform to convert the time-frequency domain matrix to a time-domain signal; S4. Allocate the time-domain signal to multiple transmit antennas through a MIMO architecture for multi-antenna transmission, and add a cyclic prefix to each transmit signal to combat multipath delay; when the transmit signal is transmitted through a time-varying channel, it suffers from inter-symbol interference caused by multipath effects, inter-carrier interference caused by Doppler frequency shift, and the influence of additive white Gaussian noise, forming a received signal with two-dimensional coupling characteristics of delay and Doppler; S5. Remap the received signal back to the DD domain through a Wigner transform and a symplectic finite Fourier transform; S6. Input the received signal remapped to the DD domain into the constructed double-iterative sparse MMSE Turbo equalizer to estimate the transmit symbols of all user equipments; the double-iterative sparse MMSE Turbo equalizer includes an MMSE estimator and a soft input soft output decoder, and this double-iterative sparse MMSE Turbo equalizer performs external iterations by iteratively passing soft information between the MMSE estimator and the soft input soft output decoder, and performs internal iterations within the MMSE estimator; Specifically, when performing external iterations, the MMSE estimator estimates the transmit symbols from the received sequence based on the prior information of the current external iteration, and then maps the estimated transmit symbols to the corresponding log-likelihood ratios of the soft input soft output decoder as the prior information of the soft input soft output decoder. The decoder feeds back the calculated external information to the MMSE estimator, obtains new mean and variance and sparsifies the covariance matrix variance therefrom as the prior information for starting the next external iteration; When performing internal iteration, the TFQMR algorithm is used to sparsify the calculation of the inverse matrix of the covariance matrix, transforming the problem into solving an equivalent sparse linear system, and then the FSPAI algorithm is used to derive an approximation value; S7. Based on the estimated transmit symbols, perform channel estimation to obtain channel information.

2. The design method of the double-iteration Turbo equalizer for the OTFS-MIMO-ISAC system according to claim 1, wherein Step S1 further includes: Multiple user devices are distributed around the base station. Each user device has its own antenna array, and the downlink transmission in the MIMO-OTFS system is adopted between the base station and each user device; The base station is deployed with antennas to provide services for user devices; The base station is equipped with a uniform linear array, and the antenna spacing is set to half wavelength; Encode the input bitstream of each user equipment through an encoder to obtain a coded vector , where is the generator matrix of the encoder, is the length of the input bitstream of the user equipment and the relationship between it and the length of the coded bitstream and the code rate is: ; Interleave the coded vector to obtain , and map it to a symbol sequence , where is the length of the symbol sequence of the user equipment and is the symbol set of the user equipment.

3. The dual-iteration Turbo equalizer design method for the OTFS-MIMO-ISAC system according to claim 1, characterized in that, In step S2, for the user equipment , perform the fast Fourier transform of points on the delay dimension, and perform the inverse fast Fourier transform of points on the Doppler dimension. Assign each user equipment to the specified DD resource blocks to ensure that the signals between user equipments do not interfere with each other. The mathematical expression is: ； Among them, is the DFT matrix of points, used to perform fast Fourier transform in the delay dimension; is the IDFT matrix of points, used to perform inverse fast Fourier transform in the Doppler dimension; represents the time-frequency domain matrix; represents the symbol matrix, which is the matrix obtained after the symbol sequence is mapped in the delay-Doppler domain.

4. The dual-iteration Turbo equalizer design method for the OTFS-MIMO-ISAC system according to claim 1, characterized in that, Step S3 further includes: Perform the inverse fast Fourier transform of points on each column of the time-frequency domain matrix and apply the pulse shaping matrix to obtain the time-domain transmission matrix : : ； In the formula, is the DFT matrix of the points, denotes the conjugate transpose of the matrix; Vectorize the time-domain transfer matrix into a time-domain signal : ； Among them, , denotes the Kronecker product, denotes the vectorization function, is the IDFT matrix of the points, denotes the sign matrix.

5. The design method of the double-iteration Turbo equalizer for the OTFS-MIMO-ISAC system according to claim 1, wherein Step S4 further includes: In the time-domain signal add a cyclic prefix with a length of to each OTFS frame, and then convert it into an analog signal to obtain the transmitted signal; wherein, the length of the cyclic prefix satisfies the following conditions: , where is the maximum delay of the user equipment , represents the bandwidth, is the subcarrier spacing; User equipment The received signal Is expressed as: ； wherein is additive white Gaussian noise in the time domain; represents the user equipment channel impulse response, wherein and respectively represent the time delay and Doppler frequency shift of the signal, and t is time; After removing the cyclic prefix, the received signal at intervals of is sampled, and the discrete received signal is expressed as: ； Among them , represents taking the modulo operation, is the delay index of the th path of the user equipment, is the Doppler shift index of the th path of the user equipment, is the number of paths of the user equipment , is the separated time point; Starting from the discrete received signal, the received signal vector of the user equipment is expressed as: is expressed as: ； wherein is an additive white Gaussian noise vector, is the time-domain effective channel matrix of the user equipment , , is the channel gain of the th path of the user equipment , is a permutation matrix.

6. The dual-iterative Turbo equalizer design method for the OTFS-MIMO-ISAC system according to claim 1, wherein Step S5 further includes: S51, for the user equipment , rearrange the received signal vector into a matrix , perform a sparse fast Fourier transform, including performing an inverse fast Fourier transform of points on the columns and a fast Fourier transform of points on the rows; specifically, use a rectangular received pulse shaping waveform , apply a fast Fourier transform of points to each column of the matrix to obtain the received signal in the time-frequency domain , ; then represent the received signal in the time-frequency domain as , where is the identity matrix of , perform a fast Fourier transform of points on the rows to obtain the DD-domain received matrix ; in the formula, is the DFT matrix of points, is the IDFT matrix of points; S52, vectorize the DD domain reception matrix to obtain the received sequence , the symbol sequence transmitted and the received sequence received The relationship between them is expressed as follows: ； wherein, is the time-domain effective channel matrix of the user equipment , and is an additive white Gaussian noise vector; Simplify the relationship to , where is the DD domain noise vector,[[]] represents the user equipment 's effective DD domain channel matrix.[[]] 7. The design method of the double-iterative Turbo equalizer for the OTFS-MIMO-ISAC system according to claim 1, characterized in that In step S6, the external iteration includes the following steps: S61, for a user equipment , estimate transmitted symbols from a received sequence using a MMSE estimator, wherein the input of the MMSE estimator includes the mean and variance of each symbol , ; the transmitted symbols estimated by the MMSE estimator are: ； In the formula, is the covariance formula, represents the time-domain signal which is the signal of discrete values after sampling; Let , , and the covariance matrix of the received sequence is , then the transmitted symbol is expressed as: ； wherein represents the mean vector represents the user equipment of the effective DD domain channel matrix the th column ; Modify the above formula to: ； where the mean value and the variance are initially set to 0 and 1 respectively; Using the Woodbury matrix identity, represent the estimated transmit symbols as: ； S62, map the estimated transmitted symbol to the corresponding log-likelihood ratio of the soft-input soft-output decoder, and calculate the mean and variance of the estimated symbol: ； ； wherein is obtained from is a Hermitian matrix, and the symbol set is Each symbol represents the th constellation, which corresponds to two bits, namely [0, 0], [0, 1], [1, 0], and [1, 1]; is a real number, represents the complex conjugate of By substituting the mean value and the variance into the calculation formula of the corresponding log-likelihood ratio, the external LLR is derived as follows: ； ； wherein, and represent the real and imaginary parts of the corresponding symbol, and is deinterleaved into and used as the prior information for the decoder; S63, the extrinsic information output by the decoder is obtained through the interleaver , and fed back to the MMSE estimator, and it is derived that: ； Among them , from the symbol set , the new mean and variance are expressed as: ； ； S64, sparsify the covariance matrix using the following formula :[[]] ； Among them, represents the sparse pattern of the matrix, represents the user equipment 's symbol covariance matrix, represents the noise power, represents the identity matrix, represents the path and the path 's difference in Doppler shift indices, represents the path and the path 's difference in delay indices, is the set of path pairs of the user equipment ​ 8. The dual-iterative Turbo equalizer design method for the OTFS-MIMO-ISAC system according to claim 1, wherein In step S6, the internal iteration includes the following steps: Convert the problem of calculating the original into the problem of solving an equivalent sparse linear system , and use the TFQMR algorithm to solve the equation. The solution process includes: For a linear system , starting from an initial guess , construct the Krylov subspace , where is the initial residual; through multiple iterations, gradually reduce the residual ; at each iteration, generate two orthogonal sequences, one sequence composed of , and the other sequence composed of , expressed as: ； ； Among them, and are orthogonal sequences generated by the double Lanczos process, and the calculation of is the new vector generated after the matrix acts on the current vector , and the calculation of is generated by the conjugate transpose of the matrix ; and are scalar coefficients calculated in each iteration, is the number of iterations; After each iteration, based on the current residual and the updated solution to update the approximate solution, and the update formula for the solution is: ； Among them, is the increment obtained from the Krylov subspace ; Calculate the new residual , check whether the stopping condition is satisfied; if the residual norm is less than the preset tolerance , then the algorithm stops and outputs ; if convergence is not achieved, continue the iteration and further optimize the solution by updating and . Select a predefined sparse pattern , calculate the matrix , represents the symbol covariance matrix of the user equipment ; The calculation process includes: Based on , initialize the sparse approximate Cholesky factor , such that , where is a lower triangular matrix obtained by minimizing the Frobenius norm with respect to the prescribed sparse pattern of ; dynamically capture the sparse pattern of by minimizing the Kaporin condition number : ​ ； where tr(·) and det(·) denote the trace and determinant of a matrix, respectively, ; denotes the noise power, denotes the identity matrix; For each column of , independently compute its non - zero element index set and corresponding values, and iteratively improve the approximation by selecting new indices and adding them to ; specifically, for each new index , update by minimizing the new Kaporin condition number : ； wherein is the th column of the identity matrix, denotes the index corresponding to the numerical value of the k-th column of the sparse approximate Cholesky factor; select the index from where represents the operation of the th row of the matrix with the set and add the index with the largest to to form a new index set and where is used to measure the reduction in the condition number by adding the new index ; by adding the new index , the minimum value of the Kaporin condition number is: ； Iterate continuously until or the maximum value of .

9. The dual-iteration Turbo equalizer design method for the OTFS-MIMO-ISAC system according to claim 1, characterized in that In step S7, the new estimate of the channel gain is expressed as: ； wherein denotes the conjugate complex number of denotes the user equipment the effective DD domain channel matrix of denotes the estimated transmitted symbol denotes the received sequence denotes the expected value function

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