Design method for double-iteration Turbo equalizer of OTFS-MIMO-ISAC system

Abstract

The invention discloses a method for designing a double-iteration Turbo equalizer of an OTFS-MIMO-ISAC system, and the method comprises the steps: carrying out the sparse processing of covariance matrix inversion, and reducing the matrix inversion cost of conventional MMSE equalization through linear complexity derivation; the sparse linear system is solved through the TFQMR algorithm to reduce the complexity of the first iteration, and a sparse approximate inverse matrix is generated in the subsequent iteration through the FSPAI algorithm to further realize the balance between the calculation efficiency and the performance; and dynamically optimizing channel information by adopting an iterative channel estimation method and utilizing MMSE (Minimum Mean Square Error) equilibrium feedback. Through outer iteration and inner iteration, the symbol estimation precision is gradually improved, meanwhile, the calculation complexity is reduced, and a low-complexity and high-precision equalization and channel estimation solution is provided for a high-mobility wireless communication system.

Description

Technical Field

[0001] The present invention relates to wireless communication technologies, and particularly 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 high-speed mobile environments 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), resulting in a significant degradation in its performance and 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 expands each symbol to 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-variation, 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 critical in high-dimensional MIMO environments.

[0004] In current research, although various equalization and detection methods have been proposed, such as Turbo detectors based on the Message Passing Algorithm (MPA) and the Sum Product Algorithm (SPA), 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 in 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 dual-iteration Turbo equalizer design method for 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 solution adopted by the present invention is as follows: A dual-iteration Turbo equalizer design method for an OTFS-MIMO-ISAC system, the method comprising the following steps: 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; S2, converting the symbol sequence to the time-frequency domain through an inverse symplectic finite Fourier transform to obtain a time-frequency domain matrix; S3, applying a Heisenberg transform to convert the time-frequency domain matrix to a time-domain signal; 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 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 delay-Doppler two-dimensional coupling characteristics; S5, remapping the received signal back to the DD domain through a Wigner transform and a symplectic finite Fourier transform; 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; Specifically, when performing the outer iteration, the MMSE estimator estimates the transmit symbols from the received sequence based on the prior information of the current outer 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 extrinsic 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 outer iteration; When performing internal iteration, the TFQMR algorithm is adopted 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 the approximation of . ; S7. Based on the estimated transmitted symbols, perform channel estimation to obtain channel information.

[0007] Step S1 further includes: Multiple user equipments are distributed around the base station. Each user equipment has its own antenna array. The downlink transmission in the MIMO-OTFS system is adopted between the base station and each user equipment. The base station is deployed with antennas to serve user equipments. The base station is equipped with a uniform linear array, and the antenna spacing is set to half wavelength. The input bit stream of each user equipment is encoded by an encoder to obtain a coded vector , where is the generator matrix of the encoder, is the length of the input bit stream of user equipment , and its relationship with the length of the coded bit stream and the code rate is: ; The coded vector is interleaved to obtain , and then mapped to a symbol sequence , where is the length of the symbol sequence of user equipment , and is the symbol set of user equipment .

[0008] Furthermore, in step S2, for user equipment , perform a fast Fourier transform of points in the delay dimension and an inverse fast Fourier transform of points in the Doppler dimension, and allocate 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: ; where is the DFT matrix of points, which is used to perform a fast Fourier transform in the delay dimension; is the IDFT matrix of points, which is used to perform an inverse fast Fourier transform in the Doppler dimension.​ Denote the time-frequency domain matrix; Denote the symbol matrix, which is a symbol sequence The matrix obtained after mapping in the delay-Doppler domain.

[0009] Step S3 further includes: 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 : ; In the formula, is the DFT matrix of points, denotes the conjugate transpose of the matrix; Vectorize the time-domain transmission matrix into a time-domain signal : ; Among them, , denotes the Kronecker product, denotes the vectorization function, is the IDFT matrix of points, denotes the symbol matrix.

[0010] Step S4 further includes: Add a cyclic prefix with a length of to each OTFS frame of the time-domain signal , and then convert it into an analog signal to obtain the transmitted signal; among them, the length of the cyclic prefix satisfies the following condition: , where is the maximum delay of the user equipment , denotes the bandwidth, is the subcarrier spacing; The received signal of the user equipment is expressed as: ; Among them is the additive white Gaussian noise in the time domain; denotes the channel impulse response of the user equipment , where and respectively denote the time delay and Doppler frequency shift of the signal, and t is the time; After removing the cyclic prefix, the received signal at intervals Sampling is performed, and the discrete received signal is expressed as: ; wherein , denotes 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: ; wherein is the 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 the permutation matrix.

[0011] Step S5 further includes: 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 , wherein is the identity matrix, 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 reception sequence , the symbol sequence transmitted and the reception sequence received The relationship between them is expressed as follows: ; In the formula, is the time-domain effective channel matrix of the user equipment , is the additive white Gaussian noise vector; Simplify the relational expression to , where is the DD-domain noise vector, represents the effective DD-domain channel matrix of the user equipment .

[0012] Furthermore, in step S6, the external iteration includes the following steps: S61. For the user equipment , use the MMSE estimator to estimate the transmitted symbol from the reception sequence . The input of the MMSE estimator includes the mean value and variance of each symbol , ; The transmitted symbol estimated by the MMSE estimator is: ; In the formula, is the covariance formula, represents the discrete-valued signal after sampling of the time-domain signal ; Let , , and the covariance matrix of the reception sequence , then the transmitted symbol is expressed as: ; where represents the mean vector, represents the th column of the effective DD-domain channel matrix of the user equipment , ; Modify the above formula to: ; where the initial values of the mean and the variance are taken as 0 and 1 respectively; Using the Woodbury matrix identity, the estimated transmitted symbol is expressed 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: ; ; where is obtained from being a Hermitian matrix, 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; By substituting the mean and variance into the calculation formula of the corresponding log-likelihood ratio, the extrinsic LLR of is derived as: ; ; where, and represent the real and imaginary parts of the corresponding symbol, and is deinterleaved into and used as the prior information of the decoder; S63, the extrinsic information output by the decoder is obtained through the interleaver as , and fed back to the MMSE estimator, and it is derived that: ; where , from the symbol set , the new mean and variance are expressed as:

[0013] ; ​S64, sparsify the covariance matrix using the following formula : ; where, 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 .

[0014] Furthermore, in step S6, the internal iteration includes the following steps: Convert the problem of calculating the original into solving the equivalent sparse linear system , and use the TFQMR algorithm to solve the equation. The solving process includes: 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: ; ; 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 is then generated by the conjugate transpose of the matrix ; and are scalar coefficients calculated at each iteration, is the number of iterations; After each iteration, according to the current residual and the updated solution Update the approximate solution, and the solution update formula is: ; where is the increment obtained from the Krylov subspace ; 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 ; 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, and 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 : ; where tr(·) and det(·) represent the trace and determinant of the matrix respectively, ; represents the noise power, represents the identity matrix; 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 : ; where is the -th column of the identity matrix, Indicates the index The numerical value of the k-th column of the corresponding sparse approximate Cholesky factor; selected from the index , where represents the th row of the matrix and the operation with the set , adding 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: ; Continue to iterate until or the maximum value of is less than the preset tolerance .

[0015] Furthermore, in step S7, the new estimate of the channel gain is expressed as: ; where represents the conjugate complex number of, represents the effective DD domain channel matrix of the user equipment , represents the estimated transmitted symbol, represents the received sequence, represents the expected value function.

[0016] Compared with the prior art, the beneficial effects of the present invention are as follows: 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. By combining the TFQMR and FSPAI algorithms, the computational complexity of symbol estimation and channel estimation is significantly reduced in the multi-user equipment scenario.

[0017] Second, 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 through the dual-iteration structure of outer iteration and inner iteration, approaching the performance of the optimal detector.

[0018] Third, in 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

[0019] Figure 1 is the model diagram of the OTFS-MIMO-ISAC system of the present invention; Figure 2 is the flow chart of the dual-iteration Turbo equalizer design method of the OTFS-MIMO-ISAC system of the present invention; Figure 3 is the schematic diagram of the comparison of bit error rates under different algorithms; Figure 4 is the schematic diagram of the comparison of channel estimation MSE under different algorithms; Figure 5 is the schematic diagram of the comparison of detection probabilities under different numbers of user equipment; Figure 6 For the number of outer iterations is the schematic diagram of the influence relationship of sparsity. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0020] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0021] The present invention discloses a dual-iteration Turbo equalizer design method for an OTFS-MIMO-ISAC system, and the method includes the following steps: 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; S2, converting the symbol sequence to the time-frequency domain through the inverse symplectic finite Fourier transform to obtain a time-frequency domain matrix; S3, applying the Heisenberg transform to convert the time-frequency domain matrix into a time-domain signal; S4, distributing the time-domain signal to multiple transmit antennas through the MIMO architecture for multi-antenna transmission, and adding 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, remapping the received signal back to the DD domain through the Wigner transform and the symplectic finite Fourier transform; S6. Input the received signal mapped back to the DD domain 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, 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 transmitted symbols from the received sequence based on the prior information of this 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 to obtain new mean and variance and sparsify the covariance matrix variance as the prior information for starting the next external iteration; When performing internal iterations, the TFQMR algorithm is used to sparsify the calculation of the inverse matrix of the covariance matrix of the problem into solving an equivalent sparse linear system, and then the FSPAI algorithm is used to deduce an approximation of; S7. Based on the estimated transmitted symbols, perform channel estimation to obtain channel information.

[0022] 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, and the downlink transmission in the MIMO-OTFS system is considered. The BS deploys antennas to provide services for user equipments. The centralized massive MIMO is equipped with a uniform linear array (ULA) at the base station, and the antenna spacing is set to half wavelength. Without loss of generality, for simplicity, it is assumed that the influence of other user equipments has been effectively isolated or eliminated, and only one user equipment is concerned, ignoring the dependence of the channel on the user equipment index.

[0023] The equalization processing flow of the channel in the present invention is as Figure 2As shown, after the transmitted signal is modulated by OTFS, it is first mapped from the DD-domain symbols to the time-frequency domain through the 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 a 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 two-dimensional coupling characteristics of delay and Doppler. 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 dual-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 dual-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.

[0024] The experimental procedure of this embodiment is as follows: Step 1, bit stream coding and mapping: First, for each user equipment input bit stream is encoded by the encoder to obtain the encoded vector , is the generator matrix of the encoder, is the length of the input bit stream of user equipment , and its relationship with the length of the encoded bit stream and the code rate is: ; The encoded vector is interleaved to obtain , and mapped to the symbol sequence , where is the length of the symbol sequence of user equipment , is user equipment Symbol set.

[0025] 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 a user equipment , perform -point fast Fourier transform (FFT) in the delay dimension and perform -point inverse fast Fourier transform (IFFT) in the Doppler dimension. Each user equipment is assigned a specific DD resource blocks to ensure that signals between user equipments do not interfere with each other. The mathematical expression is: (1); where is the -point DFT matrix used to perform FFT in the delay dimension; is the -point IDFT matrix used to perform IFFT 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.

[0026] 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 -point inverse fast Fourier transform (IFFT) on each column of the time-frequency domain matrix and applies 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: (2); where is the pulse shaping matrix of user equipment , whose diagonal elements are samples of . The duration of the pulse shaping waveform is , and the sampling interval is , that is: (3).

[0027] For simplicity of 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 window) can also be used to optimize the signal transmission performance. The time-domain transmission matrix can be vectorized into the time-domain signal : (4); Among them, , represents the Kronecker product, and

[0028] 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.

[0029] 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: (5); Among them, 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: (6); Among them, is the delay index of the -th path of user equipment , is the Doppler frequency shift index of the -th path of user equipment , is the fractional Doppler frequency shift of the -th path of user equipment . For traditional broadband systems, the sampling time is small enough, so 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 practical systems. This embodiment assumes that is large enough to ignore the fractional Doppler, which helps to understand the sparse characteristics of the channel matrix.

[0030] 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: (7); where is the additive white Gaussian noise (AWGN) in the time domain, and its distribution is .

[0031] After removing the cyclic prefix, the received signal is sampled at intervals . By substituting Equation (5), the discrete received signal can be expressed as: (8); where , represents taking the modulo operation.

[0032] 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: (9); is the diagonal matrix, representing the Doppler effect and additional frequency offset: , where , so the received signal vector of the user equipment can be expressed as: (10); where is the AWGN noise vector.

[0033] Step 5, for the user equipment , the received signal vector is rearranged into a matrix , arranged by columns. Considering using a rectangular received pulse shaping waveform, apply the to each column of the matrix point fast Fourier transform (FFT) to obtain the time-frequency domain received signal Specific formulas are as follows: (11).

[0034] After FFT processing, the time-frequency domain received signal can be expressed as , where is 's 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.

[0035] 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: (12); Simplify equation (12) to , where is the DD domain noise vector, so the effective DD domain channel matrix of the user equipment can be expressed as: (13); where is the component caused by the th path. According to previous studies, it can be deduced that: (14); where, are respectively the Doppler index and delay index of the th element of the user equipment , and respectively represent the Doppler frequency shift index and delay index of the th path of the user equipment , is the phase factor used to represent the Doppler effect.

[0036] Obviously, contains only one non-zero element in each row and each column, so is a sparse matrix, with non-zero elements in each row and each column. In subsequent processing, the received sequence is fed into a Turbo equalizer to recover information bits. In practical applications, the size of an OTFS frame (i.e., ) is usually very large. The design of this frame structure is to adapt to the complex characteristics of high-speed mobile environments and multipath channels, thereby improving the robustness of the system and the data transmission efficiency. 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.

[0037] Step 6, the dual-iteration sparse MMSE Turbo equalizer designed in the present invention aims to provide a lower-complexity equalizer for the channel when the OTFS frame is large. The received signal is passed 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 a priori 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 -th constellation, which corresponds to two bits, namely [0,0], [0,1], [1,0], and [1,1].

[0038] For a user equipment , the first step is to estimate the transmitted symbols from the received sequence using an MMSE estimator. The input of the MMSE estimator includes the mean and variance of each symbol , where , represents the discrete-valued signal after sampling the time-domain signal . To minimize the mean square error (MMSE) , the MMSE estimator derives the estimated transmitted symbol from Equation (12) as follows: (15).

[0039] For the convenience of symbol representation, the covariance matrix of the received sequence is denoted as , and calculating its inverse matrix largely determines the complexity of the equalizer.

[0040] The estimated transmitted symbol is mapped to the corresponding log-likelihood ratio (LLR) of the decoder. To obtain potential full-channel diversity gain, the posterior probability of each bit with respect to the entire received sequence is ideal, 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 of the estimated transmitted symbol is introduced to approximate . The definition of LLR is as follows: (16); 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.

[0041] Finally, the MMSE estimator outputs the extrinsic information as the prior information of the decoder. Thereafter, the decoder further reduces interference and feeds the extrinsic information back 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.

[0042] 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: (17); (18); (19); 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 also be derived that , where represents the th column of . By substituting Equation (18) and into Equation (15), the estimated symbol It can be expressed as: (20); where represents the mean vector. Since the derivation of depends on prior knowledge the result derived from is independent of (21); where initially and thus excluding the prior knowledge of itself. To avoid repeatedly calculating the inverse matrix in the above formula, the Woodbury matrix identity is used: (22); where , by substituting into Equation (22), the estimated symbol can be expressed as: (23); It can be seen from the above formula that is used twice, that is, and , which largely determines the complexity of the MMSE estimation.

[0043] Step 2, calculate the extrinsic information of the decoder: To calculate , assume that follows a Gaussian distribution, that is, , where and , according to the estimated transmitted symbol the mean and variance of the estimated symbol can be calculated as follows: (24); (25); where is obtained from being 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 derived as (26); (27); Then, is deinterleaved into , and used as the prior information of the decoder.

[0044] 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, is obtained and fed back to the MMSE estimator. According to the definition of LLR in Equation (16), it can be deduced that: (28); where , from the symbol set , the new mean and variance can be expressed as: (29); (30).

[0045] In the last step of the outer iteration, the covariance matrix needs to be sparsified for facilitating the calculation of its inverse matrix . For this purpose, some sparsification criteria are proposed with the help of graph theory, and the specific criteria are as follows: 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 sparsification criteria are proposed using graph theory to sparsify.

[0046] Criterion 1: (Sparsify from the edge constraint): 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. In the case of a significant performance degradation, try to make .

[0047] Criterion 2 (Sparsify from the node constraint): 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 imprecision, the threshold should be much smaller than the maximum degree.

[0048] The first criterion eliminates edges whose magnitudes are negligible compared to the corresponding diagonal elements because these edge pairs have little impact on matrix operations. Jacobi scaling is used to normalize the diagonal elements for ease of selection . When the magnitude of the diagonal elements of 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 strike a balance between the complexity of calculating

[0049] 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 , contains only one non - zero element in each row and each column, and its sparse pattern is given by: (31); 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: (32); where, , , is the set of path pairs of the user equipment , satisfying that if , then, which indicates that is symmetric.

[0050] 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 composed of the second term dominant, so the diagonal elements have a larger magnitude than the non - diagonal elements. After sparsifying according to the sparsification criterion, 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 of should be avoided. But 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 . Thus, the second - term dominates again, and the sparsification criterion can be applied again.

[0051] 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, so the sparsity of can only be slightly enhanced. Thus, directly calculating has too high a complexity. 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, after applying the proposed sparsification criterion, uses the FSPAI algorithm to derive an approximation of and , which will be discussed separately in the following sections.

[0052] 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. Thus, the problem for all can be solved by solving an equivalent sparse linear system , thus avoiding direct calculation of , where is the unknown vector. For calculating , assume that all (for and user equipment ) at a given is almost the same below, calculate can be transformed into solving another sparse linear system , where is the unknown vector. By obtain has a small computational cost because it is sparse, with only non-zero elements. In this way, we can reduce the computational cost in the initial iteration by solving two equivalent sparse linear systems, namely and , instead of directly calculating .

[0053] 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 cost 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 by a finite number of iterations. If the MMSE estimation is only performed once without subsequent external iterations, the iterative equalizer is reduced to a traditional LMMSE equalizer. Therefore, TFQMR can be easily extended to any other system with an LMMSE equalizer.

[0054] 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: (33); (34); 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 computed in each iteration. With these orthogonal sequences, we can efficiently 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 and calculates the new residual. Specifically, the update formula for the solution is as follows: , (35) 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 .

[0055] 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 results in a large computational cost 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 symbols 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, can also be greatly sparsified. Therefore, we can use the FSPAI algorithm to generate a sparse approximation, which is inherently parallelizable and can further save computational time.

[0056] Select a predefined sparse pattern , for example , calculate the matrix , initialize the sparse approximate Cholesky factor such that , where is a lower triangular matrix that can be obtained by relative to The specified sparse pattern minimizes the Frobenius norm to obtain, and its initial value is based on obtained. By minimizing the Kaporin condition number to dynamically capture the sparse pattern, that is (36) where tr(·) and det(·) represent the trace and determinant of the matrix, respectively .

[0057] For each column of , independently calculate its non-zero element index set and the corresponding values. By selecting a new index and adding it to , iteratively improve the approximation. For each new index , by minimizing the new Kaporin condition number to update : , (37) where is the th column of the identity matrix. Select an index from , add the index with the largest to to form a new index set and , where measures the reduction in the condition number due to adding the new index . By adding the new index , the minimum value of the Kaporin condition number is: , (38) where measures the reduction in the condition number due to 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 is less than a preset tolerance

[0058] ​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 the interference can be subtracted from the received signal. Perform a matching estimation to obtain the channel estimation of the expected user equipment, that is, this channel estimator is a matching channel estimator after partial interference cancellation. Using the designed sparse double iterative equalization, all user equipments can be determined at the BS 's restored signal. Reconstruct and eliminate the intra-frame interference, and then perform a conjugate matching calculation based on the restored signal of the expected user equipment. The channel gain 's new estimation can be expressed as: (39); where the arithmetic mean of the data block is used as the mathematical expectation. This channel estimator does not require much calculation.

[0059] 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 of different estimation algorithms are also compared. To evaluate the computational complexity of FSPAI, the sparsity of the approximate Cholesky factor is studied. The simulation sets 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 before as the prior information in the next extrinsic iteration.

[0060] In 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, through the collaborative optimization of outer-iteration soft information exchange and inner-iteration low-complexity algorithms (TFQMR / FSPAI), the receiver significantly improves the detection accuracy. At the first outer iteration, the equalizer performance 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.

[0061] Figure 4 The mean square error (MSE) performance comparison 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.

[0062] 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 of more user devices approaching the target also increases. The LS, a classic estimator based on the least squares criterion, ignores multipath interference (MPI) and pilot contamination, and its detection probability (black line) is always low, revealing 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, TFQMR is used 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 iterations of soft information exchange and dynamic threshold sparsification, when the number of user devices increases, the detection probability drops slightly, verifying the effectiveness of iterative interference reconstruction.

[0063] Figure 6 The evolution characteristics of the channel sparsity with the number of iterations in different signal-to-noise ratio scenarios are compared between the Turbo and non-Turbo equalizers. 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 multipath 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 when the channel conditions are good, its sparsity growth at low signal-to-noise ratios lags significantly, and it cannot reach a high sparsity even after multiple iterations, revealing the misjudgment problem of traditional methods for weak paths under noise interference. The double-iteration mechanism of the Turbo equalizer achieves stable sparsity growth in both the 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.

[0064] 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.

[0065] 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 for a double-iterative Turbo equalizer in 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 the 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 delay-Doppler two-dimensional coupling characteristics; S5. Remap the received signal back to the DD domain through the Wigner transform and the 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 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; Specifically, when performing the outer iteration, the MMSE estimator estimates the transmit symbols from the received sequence based on the prior information of the current outer iteration, and then maps the estimated transmit symbols to the corresponding log-likelihood ratio of the soft input soft output decoder as the prior information of the soft input soft output decoder. The decoder feeds back the calculated extrinsic information to the MMSE estimator, obtains the new mean and variance and sparsifies the covariance matrix variance therefrom as the prior information for starting the next outer iteration; When performing internal iterations, 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 ; S7. Based on the estimated transmit symbols, perform channel estimation to obtain channel information.

2. The dual-iteration Turbo equalizer design method for the OTFS-MIMO-ISAC system according to claim 1, characterized in that 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 serve 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 device to obtain a coded vector through an encoder, , where is the generator matrix of the encoder, is the length of the input bitstream of the user device 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 device and is the symbol set of the user device 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 fast Fourier transform of points on the delay dimension, and perform inverse fast Fourier transform of points on the Doppler dimension. Allocate each user equipment to the specified DD resource blocks to ensure that 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 dual-iteration Turbo equalizer design method for the OTFS-MIMO-ISAC system according to claim 1, characterized in that, 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; where 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 channel impulse response of the user equipment 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 represented 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-iteration Turbo equalizer design method for the OTFS-MIMO-ISAC system according to claim 1, characterized in that Step S5 further includes: S51, for a 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 columns and a fast Fourier transform of points on rows; specifically, use a rectangular received pulse shaping waveform , apply a fast Fourier transform of points to each column of the matrix to obtain a 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 rows to obtain a DD domain received matrix ; in the formula, is the DFT matrix of points, and is the IDFT matrix of points; S52, vectorize the DD domain reception matrix to obtain a reception sequence , the symbol sequence transmitted and the received reception sequence are related as follows: ； In the formula, is the time-domain effective channel matrix of the user equipment , and is the 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 outer iteration includes the following steps: S61, for a user equipment , estimate transmitted symbols from a received sequence using a MMSE estimator, wherein the input to 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 the 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 initial values of the mean and the variance are 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: ； ； Among them is derived 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 and the variance into the calculation formula of the corresponding log-likelihood ratio, the external LLR is obtained 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, obtaining the extrinsic information output by the decoder through the interleaver , and feeding it back to the MMSE estimator, it is deduced 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-iteration Turbo equalizer design method for the OTFS-MIMO-ISAC system according to claim 1, characterized in that In step S6, the inner 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 , a Krylov subspace is constructed, where is the initial residual; through multiple iterations, the residual is gradually reduced; at each iteration, two orthogonal sequences are generated, one is the sequence composed of , and the other is the 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. 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 , denotes 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 sparsity pattern; dynamically capture the sparsity pattern of by minimizing the Kaporin condition number : ​ ； where tr(·) and det(·) represent 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 : ； Among them 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 , add the index with the largest to to form a new index set and , where is used to measure the reduction of the condition number by adding the new index ; by adding the new index , the minimum value of the Kaporin condition number is: ； Keep iterating until or the maximum value of is less than a preset tolerance 9. The design method of the dual-iteration Turbo equalizer 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 for 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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