Coding matrix solving method based on iterative conjugate gradient for digital communication

Abstract

The invention belongs to the field of digital information transmission, and particularly discloses an iterative conjugate gradient-based coding matrix solving method for digital communication. The method comprises the following steps: generating an initial matrix through channel state sensing and evaluating a condition number; constructing a compensation matrix by using a self-adaptive precondition operator to execute spatial transformation; adopting multi-dimensional momentum gradient iteration to introduce historical momentum to suppress swinging; and realizing closed-loop solution by combining dynamic step length optimization, solution vector updating and correction and adaptive convergence judgment. According to the method, the number of matrix conditions is reduced, the robustness in a noise environment is enhanced, the operation efficiency is improved while high fidelity is ensured, and the calculation time delay and power consumption of the system are reduced.

Description

Technical Field

[0001] This invention belongs to the field of digital information transmission, and specifically relates to a method for solving the encoding matrix of digital communication based on iterative conjugate gradient. Background Technology

[0002] In modern digital communications, the efficient processing of complex signals and the faithful recovery of data rely heavily on robust signal processing architectures and advanced coding techniques. As wireless communication environments become increasingly complex, high-dimensional matrix operations have become the fundamental computational pillar for core tasks such as channel estimation, interference cancellation, and multi-user detection. These operations are crucial for ensuring system capacity and transmission reliability in diverse scenarios, from cellular networks to satellite communications.

[0003] In solving for the encoding matrix, iterative algorithms are often used to balance computational accuracy and hardware implementation cost. The conjugate gradient method is widely used due to its theoretical advantages of fast convergence speed and low memory consumption when dealing with large-scale sparse matrices. The core objective of this type of method is to achieve efficient data decoding and signal reconstruction by iteratively searching for the optimal solution vector that minimizes the residual in the signal space.

[0004] Existing conjugate gradient algorithms suffer from performance issues in real-world communication environments with severe channel interference and high noise levels. Traditional conjugate gradient methods often experience a sharp decline in convergence speed when the condition number of the encoding matrix is ​​large. This is primarily because the single step size calculation method cannot adapt to complex error surface morphologies. This deficiency leads to zigzag oscillations in the search path, causing the algorithm to repeatedly oscillate in suboptimal directions instead of directly approaching the global minimum, resulting in an exponential increase in the number of iterations. Furthermore, the lack of adaptive handling of ill-conditioned matrices increases system latency and terminal device power consumption, making it difficult to meet the stringent real-time and energy efficiency requirements of next-generation high-speed communication systems. Summary of the Invention

[0005] The purpose of this invention is to provide a method for solving the encoding matrix of digital communication based on iterative conjugate gradient, so as to solve the technical problems in the prior art in complex communication environments, such as decreased convergence speed, oscillation of search path, excessive number of iterations, and difficulty in balancing real-time performance and low power consumption requirements caused by the large condition number of the encoding matrix.

[0006] The technical solution of this invention is to provide a method for solving the encoding matrix of digital communication based on iterative conjugate gradients, comprising:

[0007] The channel state sensing unit is used to acquire the original received signal and the preset pilot sequence in the digital communication system, generate the initial channel matrix through channel estimation logic, and extract the instantaneous signal-to-noise ratio parameter and channel fading characteristics of the signal.

[0008] The matrix ill-conditioning assessment unit is used to perform singular value decomposition to obtain the eigenvalue distribution sequence based on the initial channel matrix generated by the channel state sensing unit, calculate the ratio of the maximum singular value to the minimum singular value, determine the condition number of the current coding matrix, and compare and analyze the condition number with a preset ill-conditioning threshold.

[0009] An adaptive preconditioning operator construction unit is used to generate a preconditioning compensation matrix based on the sparsity of the eigenvalue distribution sequence, using incomplete decomposition logic or a diagonal compensation strategy, in response to the judgment result that the condition number exceeds the ill-conditioned threshold. This matrix is ​​used to perform a spatial transformation on the initial channel matrix and construct an equivalent symmetric positive definite matrix.

[0010] The multidimensional momentum gradient iteration unit is used to initialize the solution vector, residual vector and search direction vector in the equivalent space, and introduces the momentum weight factor of the historical search direction. It combines the gradient direction of the current residual to calculate the conjugate search direction to suppress the swaying of the search path caused by drastic changes in surface curvature.

[0011] The dynamic step size optimization unit is used to obtain the product of the search direction vector of the current iteration step and the equivalent symmetric positive definite matrix, and calculate the optimal single step depth by combining the second derivative estimate of the residual vector. At the same time, it performs smoothing filtering correction on the step size according to the instantaneous signal-to-noise ratio parameter to counteract the interference of Gaussian noise on the search accuracy.

[0012] The solution vector update and correction unit is used to accumulate and update the current solution vector based on the correction step size calculated by the dynamic step size optimization unit, and simultaneously perform residual reprojection operation, using orthogonalization correction logic to eliminate the accumulation of rounding errors caused by the finite word length effect.

[0013] The adaptive convergence determination unit is used to dynamically adjust the target residual error limit based on the instantaneous signal-to-noise ratio parameter. When the magnitude of the residual vector drops below the target residual error limit, the iteration is terminated and the final encoding matrix solution vector is output. When the magnitude of the residual vector does not reach the target residual error limit and the number of iterations does not reach the preset maximum limit, the next round of iteration calculation is triggered.

[0014] Furthermore, when executing the channel estimation logic, the channel state sensing unit uses the least squares method or the minimum mean square error criterion to process the pilot sequence to remove the channel transfer function. When extracting channel fading features, the system performs differential operations on the channel matrices at adjacent time points to evaluate the channel coherence time and Doppler shift intensity, providing a reference benchmark for subsequent iteration frequencies.

[0015] Furthermore, when calculating the condition number, if the dimension of the initial channel matrix exceeds a preset computational pressure threshold, the matrix ill-conditioned evaluation unit uses a power method or an inverse power method to approximate the estimation of the maximum and minimum eigenvalues ​​to reduce computational complexity. The determined condition number, as a core indicator for measuring the flatness of the search space, directly determines the intervention strength of the precondition compensation matrix.

[0016] Furthermore, the adaptive preconditioning operator construction unit operates as follows: When the condition number is within a first preset interval, a Jacobian preprocessor is used, i.e., the reciprocals of the diagonal elements of the initial channel matrix are taken to construct the preconditioning compensation matrix; when the condition number is within a second preset interval and the upper limit of the second preset interval is greater than that of the first preset interval, incomplete Schul complement decomposition or incomplete robust decomposition is performed to construct a triangular factor matrix with higher approximation. By left-multiplying or symmetric sandwich multiplication of the preconditioning compensation matrix with the initial channel matrix, the condition number of the transformed equivalent symmetric positive definite matrix approaches 1.

[0017] Furthermore, when calculating the conjugate search direction, the multidimensional momentum gradient iterative unit not only relies on the gradient at the current moment but also couples the search direction information from the previous three moments to the previous moment through a nonlinear momentum coefficient. The value of the momentum weight factor is dynamically scaled according to the gradient angle between adjacent iteration steps: when the angle is less than 90 degrees, the momentum weight factor is increased to accelerate the advancement along the descent direction; when the angle is greater than 90 degrees, the momentum weight factor is decreased to quickly correct the search bias and eliminate the zigzag oscillation path.

[0018] Furthermore, the dynamic step-size optimization unit employs a precise step-size calculation logic based on the Taylor expansion of the objective function when calculating the step-size factor. The inner product of the residual vector and the search direction vector is used as the numerator, and the quadratic form of the search direction vector after mapping with an equivalent symmetric positive definite matrix is ​​used as the denominator for division. Based on this, a smoothing factor positively correlated with the instantaneous signal-to-noise ratio is introduced. If the signal-to-noise ratio is lower than a preset reliable communication threshold, the calculated step-size is attenuated to ensure that the algorithm maintains convergence monotonicity even in noise-dominated environments.

[0019] Furthermore, when performing residual reprojection, the solution vector update and correction unit periodically recalculates the original residual—that is, it replaces the residual obtained from the recursive update with the product of the encoding matrix and the currently estimated solution vector subtracted from the observed vector—to prevent residual shift during long-sequence iterations. The orthogonalization correction logic, through the Gram-Schmidt orthogonalization process, ensures that the newly generated search direction in each generation maintains a strict conjugate relationship with the historical search direction within the precondition space.

[0020] Furthermore, the target residual error limit in the adaptive convergence determination unit is not a fixed value, but is dynamically related to the modulation and coding scheme of the communication system. For high-order orthogonal amplitude modulation schemes, the system sets a smaller error tolerance to ensure demodulation accuracy; for low-order modulation schemes, the system appropriately relaxes the error tolerance to obtain lower computational delay. Simultaneously, the system monitors the rate of residual decrease during iteration; if the residual decrease rate is less than 0.1% for three consecutive iterations, convergence is forcibly terminated.

[0021] In one embodiment of the present invention, the encoding matrix solving method runs on a computing platform composed of a high-performance digital signal processor or a field-programmable gate array. At the hardware level, matrix multiplication operations are mapped to a highly parallel systolic array structure, while the construction of the preconditioning operator utilizes on-chip memory to cache key diagonal elements to achieve fast access in a single clock cycle.

[0022] In one embodiment of the present invention, the system further includes a feedback control loop. When the number of iterations frequently reaches the preset maximum limit without achieving the residual target, the feedback control loop sends a command to the physical layer controller, triggering an improvement in channel estimation accuracy or requiring a reduction in data transmission rate, thereby ensuring the stability of the communication link at the system-wide level.

[0023] Furthermore, the incomplete decomposition logic specifically involves, during matrix decomposition, setting a discard threshold to ignore non-main diagonal elements with absolute values ​​less than the threshold, thereby generating approximate factors that maintain the sparse structure of the original matrix and are easy to invert. This approach reduces storage overhead and computational latency in the preprocessing stage.

[0024] Furthermore, the update logic for the momentum weight factor also includes a perturbation term based on the convergence state. When the search process is detected to have entered a flat region near a local minimum, the perturbation term automatically increases the random volatility of the search direction, helping the algorithm escape numerical traps and find the global optimum.

[0025] Furthermore, when processing high-dimensional matrices, the solution vector update and correction unit adopts a block-based computation strategy, dividing the large-size encoding matrix into several sub-blocks. The update logic of each sub-block is executed in parallel on independent hardware processing cores, and the cores exchange global variables such as residual modulus values ​​through a high-speed interconnect bus to ensure the logical consistency of distributed iteration.

[0026] Furthermore, the digital communication system supports a multi-antenna input, multi-antenna output architecture, in which case the initial channel matrix is ​​presented in complex matrix form. In the processing of this method, a complex matrix of dimension N×M is transformed into a real symmetric matrix of dimension 2N×2M through a complex-to-real transformation logic, adapting to the basic mathematical framework of the conjugate gradient method and improving the versatility of the computational processing.

[0027] Compared with the prior art, the advantages and positive effects of the present invention are as follows:

[0028] 1. This scheme introduces an adaptive preconditioning operator construction unit, enabling real-time adjustment of matrix ill-conditioned properties under different channel environments. Through incomplete decomposition or diagonal compensation, the condition number of the encoding matrix is ​​reduced, making the originally distorted search space smoother. This mechanism improves the algorithm's numerical stability, solves the problem of drastic slowdown in convergence speed under severe channel interference, and ensures efficient computation in complex communication scenarios.

[0029] 2. This scheme overcomes the limitation of traditional conjugate gradient methods relying on a single scalar step size by synergistically combining a multidimensional momentum gradient iteration unit and a dynamic step-size optimization unit. It utilizes momentum information from historical directions to offset the oscillation phenomenon in gradient search, guiding the search path smoothly towards the global minimum even when the error surface has narrow valleys. Combined with signal-to-noise ratio-aware smoothing filtering correction, the algorithm's robustness in noisy environments is enhanced, unnecessary iteration fluctuations are reduced, and the system's computational latency and hardware resource consumption are lowered.

[0030] 3. This scheme constructs a closed-loop iterative control system with deep sensing capabilities. By dynamically adjusting the exit threshold based on the signal-to-noise ratio and modulation scheme through an adaptive convergence determination unit, a Pareto optimal balance between computational accuracy and efficiency is achieved. The solution vector update and correction unit suppresses word length truncation errors common in digital circuits through orthogonalization correction and residual reprojection, ensuring the fidelity of high-dimensional matrix operations. This series of technological innovations enables this invention to perfectly meet the stringent real-time signal processing requirements of next-generation high-speed, low-power communication systems, demonstrating engineering application value and technological leadership.

[0031] 4. This solution fully considers the convenience of hardware implementation. Through matrix partitioning strategy and real number conversion logic, complex iterative algorithms can be deployed in a parallel hardware architecture. This system-level optimization from the underlying algorithm logic to the hardware mapping mechanism improves the data throughput per unit of energy consumption, providing solid technical support for terminal devices to extend battery life while maintaining high-performance communication. Attached Figure Description

[0032] Figure 1 This is a schematic diagram of the overall technical solution architecture of the present invention;

[0033] Figure 2 This is a schematic diagram of the core principle framework for solving the encoding matrix based on iterative conjugate gradients in this invention;

[0034] Figure 3This is a logical flowchart of the matrix ill-conditioning assessment and adaptive preconditioning compensation in this invention;

[0035] Figure 4 This is a schematic diagram of the principle framework of multidimensional momentum conjugate search and dynamic step size optimization in this invention;

[0036] Figure 5 This is a schematic diagram of the adaptive convergence determination and residual correction logic based on signal-to-noise ratio sensing in this invention. Detailed Implementation

[0037] Example 1: Please refer to the appendix Figure 1 This embodiment discloses a method for solving the encoding matrix of digital communication based on iterative conjugate gradients. This method aims to solve the signal processing challenges in high-dimensional, non-stationary channel environments by having multiple levels of logic units work together. The entire system is built on a high-performance digital signal processor or field-programmable gate array at the hardware level, and achieves real-time throughput of massive amounts of data through a parallel hardware architecture.

[0038] Please refer to the appendix. Figure 1 With appendix Figure 2 The channel state awareness unit, acting as the system's input front-end, is responsible for deep analysis of the digital stream after down-conversion and sampling by the RF front-end. This unit first acquires the original received signal and a pre-defined pilot sequence from the digital communication system. The pilot sequence, as a known reference signal agreed upon by both the transmitter and receiver, has its time-domain or frequency-domain characteristics pre-stored in local memory. The channel state awareness unit generates an initial channel matrix through channel estimation logic. During execution, the system uses the least squares method or the minimum mean square error criterion to process the pilot sequence to remove the channel transfer function. For environments with high signal-to-noise ratios (SNR), the least squares method can acquire the channel response with lower computational overhead; while in low SNR environments, the minimum mean square error criterion, by introducing the statistical characteristics of noise variance, can more accurately suppress background noise interference with channel estimation. Simultaneously, the channel state awareness unit extracts the instantaneous SNR parameters and channel fading characteristics. When extracting channel fading characteristics, the system performs differential operations on the channel matrix at adjacent time points, calculates the time-varying rate of matrix elements, and evaluates the channel's coherence time and Doppler shift intensity. This step provides crucial physical layer parameter support for adjusting the subsequent iteration frequency and setting the convergence threshold.

[0039] Please refer to the attached document. Figure 3The matrix ill-conditioning assessment unit receives the initial channel matrix generated by the channel state sensing unit. In complex electromagnetic propagation environments, the channel matrix often exhibits obvious ill-conditioning characteristics, i.e., the condition number of the matrix is ​​extremely large, causing severe path oscillations when conventional gradient descent algorithms search for solution vectors. To quantify this characteristic, the matrix ill-conditioning assessment unit performs singular value decomposition to obtain the eigenvalue distribution sequence. If the dimension of the initial channel matrix exceeds a preset computational pressure threshold, for example, when the matrix dimension reaches 64×64 or higher, in order to save hardware resources and reduce computational latency, the system uses the power method or inverse power method to approximate the maximum and minimum eigenvalues. Through multiple iterative vector multiplications, the power method can quickly lock the principal eigenvalues ​​of the matrix, while the inverse power method combined with a shift strategy can obtain the eigenvalue with the minimum modulus. Subsequently, the unit calculates the ratio of the maximum singular value to the minimum singular value to determine the condition number of the current coding matrix. This condition number directly reflects the steepness and narrowness of the error surface. The matrix ill-conditioning assessment unit compares and analyzes this condition number with the preset ill-conditioning threshold, and the judgment result guides the system to different preprocessing paths.

[0040] The adaptive preconditioning operator construction unit responds to the judgment result that the condition number exceeds the ill-conditioned threshold. Its core task is to transform the ill-conditioned matrix into a well-conditioned matrix through spatial transformation. Based on the sparsity of the eigenvalue distribution sequence, this adaptive preconditioning operator construction unit generates a preconditioning compensation matrix using incomplete decomposition logic or a diagonal compensation strategy. The specific working logic is as follows: When the condition number is in the first preset interval, for example, between 1 and 50, the system determines that the matrix has relatively mild ill-conditionedness and uses the Jacobian preprocessor, i.e., taking the reciprocals of the diagonal elements of the initial channel matrix to construct the preconditioning compensation matrix. This method has extremely low computational overhead, involving only scalar division. When the condition number is in the second preset interval and the high-end limit of the second preset interval is greater than that of the first preset interval, for example, between 51 and 500, the system performs incomplete Shur complement decomposition or incomplete robust decomposition to construct a triangular factor matrix with higher approximation. During the incomplete decomposition process, the system sets a discard threshold, ignoring non-main diagonal elements with absolute values ​​less than the threshold, generating approximate factors that maintain the sparse structure of the original matrix and are easy to invert. An equivalent symmetric positive definite matrix is ​​constructed by left-multiplying the preconditioning compensation matrix with the initial channel matrix or by symmetric sandwich multiplication. In the transformed equivalent space, the distribution of eigenvalues ​​is more concentrated, and the condition number approaches 1, laying a numerical foundation for subsequent fast convergence.

[0041] Please refer to the attached document. Figure 4The multidimensional momentum gradient iterative unit performs the core solution calculations within the equivalent space. First, the system initializes the solution vector, residual vector, and search direction vector. To overcome the oscillation problem in the search path of the traditional conjugate gradient method, this multidimensional momentum gradient iterative unit introduces a momentum weight factor for the historical search direction. The introduction of the momentum weight factor ensures that the current search direction depends not only on the current residual gradient but also on the inertia of the historical path. Specifically, when calculating the conjugate search direction, the multidimensional momentum gradient iterative unit relies not only on the gradient at the current moment but also couples the search direction information from the previous three moments to the previous moment through a nonlinear momentum coefficient. The value of the momentum weight factor is dynamically scaled according to the gradient angle between adjacent iteration steps: when the angle is less than 90 degrees, it indicates that the current gradient descent direction is basically consistent with the historical direction, and the system increases the momentum weight factor to accelerate the advancement along the descent direction; when the angle is greater than 90 degrees, it indicates that the search path may have oscillated excessively or entered a turning region, and the system decreases the momentum weight factor to quickly correct the search bias. This adaptive momentum adjustment can suppress the zigzag oscillation of the search path caused by drastic changes in surface curvature.

[0042] With the conjugate search direction determined, the dynamic step-size optimization unit begins calculating the optimal iteration step depth. This unit obtains the product of the search direction vector of the current iteration step and the equivalent symmetric positive definite matrix, and combines it with the second derivative estimate of the residual vector to calculate the optimal single-step depth. When calculating the step-size factor, the system employs precise step-size calculation logic based on the Taylor expansion of the objective function.

[0043] ;

[0044] In the above formula, Represents the dynamic step size. Let be the residual vector of the current iteration. This is the residual vector after preconditioning. The search direction vector, The system of linear equations to be solved consists of a numerator (the inner product of the current residual vector and the pre-conditioned residual vector) and a denominator (the quadratic form of the search direction vector after mapping with an equivalent symmetric positive definite matrix). To further enhance the algorithm's robustness in noisy environments, the dynamic step-size optimization unit performs smoothing filtering on the step size in conjunction with the instantaneous signal-to-noise ratio (SNR) parameter. A smoothing factor positively correlated with the instantaneous SNR is introduced. If the SNR is lower than a preset reliable communication threshold (e.g., less than 5 dB), the calculated step size is attenuated to counteract the interference of Gaussian noise on the search accuracy. This ensures that the algorithm maintains its convergence monotonicity even in noisy environments and prevents the solution vector from jumping out of the global optimum due to an excessively large step size.

[0045] Please refer to the attached document. Figure 5The solution vector update and correction unit updates the current solution vector cumulatively based on the corrected step size calculated by the dynamic step size optimization unit. However, in actual digital circuit implementations, the finite word length effect leads to the continuous accumulation of rounding errors. After hundreds of matrix multiplication operations, the recursively obtained residual vector often deviates from the true residual. To address this, the solution vector update and correction unit simultaneously performs a residual reprojection operation. The system periodically recalculates the original residual, replacing the recursively updated residual with the observation vector minus the product of the encoding matrix and the currently estimated solution vector. Simultaneously, orthogonalization correction logic eliminates error accumulation. This orthogonalization correction logic, through the Gram-Schmidt orthogonalization process, ensures that each newly generated search direction maintains a strict conjugate relationship with the historical search directions within the precondition space. Through this dual correction mechanism, the system can maintain the fidelity of the solution vector throughout large-scale, long-sequence iterations.

[0046] The adaptive convergence determination unit dynamically adjusts the target residual error limit based on the instantaneous signal-to-noise ratio parameter. The target residual error limit is not a fixed value but is dynamically correlated with the modulation and coding scheme of the communication system. For higher-order schemes such as 256 quadrature amplitude modulation (QAM), the system sets a smaller error tolerance, such as 0.0001, to ensure demodulation accuracy; for lower-order modulation schemes such as quadrature phase shift keying (QPSK), the system appropriately relaxes the error tolerance to 0.01 to achieve lower computational delay. When the magnitude of the residual vector drops below the target residual error limit, the iteration terminates and the final coding matrix solution vector is output. When the magnitude of the residual vector does not reach the target residual error limit and the number of iterations has not reached the preset maximum limit, the next round of iteration calculation is triggered. Furthermore, the system monitors the rate of residual decrease during iteration. If the residual decrease rate is less than 0.1% for three consecutive iterations, the algorithm is determined to have entered the stagnation zone, and convergence is forcibly terminated to avoid unnecessary power consumption.

[0047] Throughout the solution process, the system also includes a feedback control loop. When the number of iterations frequently reaches the preset maximum limit without achieving the residual target, the feedback control loop sends instructions to the physical layer controller, triggering an improvement in channel estimation accuracy or a reduction in data transmission rate, thus ensuring the stability of the communication link at the system-wide level. For multi-antenna input / multi-antenna output architectures, the initial channel matrix is ​​in complex matrix form. In the processing of this method, a complex matrix of dimension N×M is transformed into a real symmetric matrix of dimension 2N×2M through a complex-to-real transformation logic. For example, the real and imaginary parts of the original complex matrix are cross-stacked to construct a matrix that fully adapts to the real mathematical framework of the conjugate gradient method, improving the versatility of hardware processing.

[0048] Example 2: Building upon Example 1, Example 2 further refines the distributed processing logic and block computation strategy at the hardware execution level of this invention to address the needs of solving ultra-high-dimensional encoding matrices. When a communication system uses a large-scale antenna array, the dimension of the encoding matrix may reach 256×256 or even higher. In this case, a single processing core cannot complete all iterative calculations within a symbol period.

[0049] Please refer to the attached document. Figure 2 In this embodiment, the solution vector update and correction unit employs a block-based computation strategy when processing high-dimensional matrices. The system divides the large-size encoding matrix into several sub-blocks; for example, a 256×256 matrix is ​​divided into 16 64×64 sub-blocks. The update logic of each sub-block is executed in parallel on independent hardware processing cores. Each core is responsible for calculating the local residual vector and local gradient direction. The cores exchange global variables such as residual magnitude, momentum weight factor, and scalar step size through a high-speed interconnect bus to ensure the logical consistency of distributed iteration. This architecture fully utilizes the parallelism of hardware, distributing the computational complexity across multiple arithmetic logic units.

[0050] In the matrix ill-conditioning evaluation unit, the system performs local condition number probing for each block of the sub-matrix. Due to differences in channel characteristics across different sub-bands or spatial sub-streams, some sub-blocks may exhibit benign behavior, while others may be extremely ill-conditioned. The adaptive preconditioning operator construction unit independently selects a preprocessing strategy based on the local condition number of each sub-block. For extremely ill-conditioned sub-blocks, a higher-order diagonal compensation strategy is introduced, forcibly improving the matrix's positive definiteness by superimposing positive perturbation terms related to the instantaneous signal-to-noise ratio on the diagonal. This differentiated processing avoids performing uniform, high-complexity preprocessing on the entire matrix, minimizing computational power consumption while ensuring solution accuracy.

[0051] In a distributed environment, the momentum weight factor update logic of the multidimensional momentum gradient iterative unit also includes a perturbation term based on the convergence state. When the search process is detected to have entered a flat region near a local minimum, i.e., the residual magnitude is in a narrow range of fluctuation for a long time, the perturbation term will automatically increase the random volatility of the search direction.

[0052] The formula is as follows: ;

[0053] in, For the updated search direction, For historical search direction, As the momentum weighting factor, This is the current gradient vector, and This is a perturbation term based on the convergence state. The introduction of this perturbation term helps the algorithm escape numerical traps and find the global optimum. This mechanism is particularly important for complex communication scenarios where multipath effects are severe and the error surface has multiple local extrema.

[0054] The dynamic step-size optimization unit employs an asynchronous update mechanism in the distributed architecture. Each processing core calculates candidate step sizes locally based on its own load and data readiness status, followed by weighted aggregation by the global control logic. During this process, the adaptive convergence determination unit monitors the residual decline curves of each core in real time. If a core's substream has already reached the convergence target ahead of schedule, that core enters a low-power standby mode until the next signal frame arrives. This fine-grained energy efficiency management strategy allows the system to maintain high-performance output while extending the mobile terminal's battery life.

[0055] This embodiment also details the storage optimization scheme for the preconditioning compensation matrix. At the hardware level, the system utilizes on-chip memory to cache key diagonal elements. Since the preconditioning operator exhibits temporal coherence within a short timeframe, the system does not need to reconstruct the operator in each iteration. Instead, it periodically updates the preconditioning matrix based on the coherence time determined by the channel fading characteristics. Between updates, the system reuses existing operator parameters, achieving fast access within a single clock cycle. This caching mechanism significantly alleviates memory bandwidth pressure and prevents pipeline stalls caused by frequent memory accesses.

[0056] The solution vector update and correction unit employs an improved block Gram-Schmidt orthogonalization logic for the block matrix during orthogonalization correction. This logic ensures conjugation across the entire space by performing intra-group and inter-group orthogonalization on the sub-block vectors. Simultaneously, to address the risk of fixed-point arithmetic overflow in the hardware, the unit performs an automatic scaling operation after each iteration. Based on the maximum modulus of the current solution vector, the exponent of the fixed-point number is dynamically adjusted to retain the maximum effective precision bits within a limited bit width. This dynamic numerical scaling technique, combined with residual reprojection, eliminates the numerical stability risks associated with long-period iterations.

[0057] Example 3: Building upon Examples 1 and 2, Example 3 further elaborates on the adaptive switching mechanism of the present invention in a multimodal communication environment and the deep coupling scheme for collaborative operation with other links in the physical layer. Please refer to the appendix. Figure 1 The digital communication system in this embodiment supports a variety of wireless access technologies, including but not limited to enhanced mobile broadband, high reliability low latency communication, and massive machine-type communication.

[0058] In this embodiment, the channel state awareness unit is equipped with more advanced modality recognition capabilities. It automatically identifies the current communication scenario by parsing the frame header signaling of the received signal. For high-reliability, low-latency communication scenarios, since the system is extremely sensitive to latency, the adaptive convergence determination unit automatically switches to fast convergence mode, ensuring the encoding matrix is ​​output in a very short time by relaxing a certain error tolerance and increasing the initial iteration step size. For large-scale machine-type communication scenarios, since terminal devices typically have limited battery capacity, the system prioritizes the simplest Jacobian preprocessor and significantly limits the maximum number of iterations to achieve extremely low single-operation energy consumption.

[0059] In this embodiment, the matrix ill-conditioning assessment unit incorporates historical trend prediction logic. The system internally maintains a sliding window memory, recording the evolution trajectory of the condition number over the past 10 transmission slots. If a continuously increasing trend in the condition number is predicted, the adaptive preconditioning operator construction unit preloads a more complex preprocessing algorithm, completing a smooth transition in the algorithm space before the ill-conditioning truly deteriorates. This feedforward control strategy effectively eliminates the surge in iterations caused by channel mutations, maintaining the stability of the system's data throughput.

[0060] In calculating the momentum weighting factor, the multidimensional momentum gradient iteration unit in this embodiment incorporates spatial correlation considerations. In a multi-antenna architecture, the channel information acquired by different antenna elements exhibits spatial correlation. This unit utilizes this correlation to spatially average the gradient directions of adjacent antenna branches, constructing a global momentum with spatial smoothness. This improvement effectively filters out burst pulse interference on a single antenna link, enhancing the robustness of the search direction.

[0061] The dynamic step-size optimization unit incorporates physical layer channel quality indicator parameters when performing smoothing filter correction. If the channel quality indicator shows deep fading, the system forcibly lowers the step-size correction coefficient to below 0.5, implementing a minimal step-size strategy to prevent algorithm divergence under extremely low signal-to-noise ratio conditions. The solution vector update and correction unit introduces a redundancy check mechanism. Every fixed number of iterations, the current solution vector is sent to the check logic, which uses a cyclic redundancy check (CRC) code to check the logical completeness of the data. If a calculation error is detected, the recalculation logic for the current iteration step is immediately triggered.

[0062] The adaptive convergence decision unit and the physical layer's modulation and coding scheme selector form a closed-loop feedback. When the solution method finds that it cannot reach the target error limit even after the maximum number of iterations, it sends a degradation request to the modulation and coding scheme selector. Upon receiving the request, the selector downgrades the modulation scheme of the next frame from a higher order to a lower order, for example, downgrading from 64-quadrature amplitude modulation to 16-quadrature amplitude modulation. This cross-layer collaborative optimization mechanism ensures the availability of the communication link under extremely poor channel conditions.

[0063] Regarding hardware mapping, this embodiment details the data flow process of matrix multiplication operations in a pulsating array structure. Elements of the initial channel matrix are injected into the processing unit array one by one according to a specific time difference, and the updated values ​​of each dimension of the solution vector are alternately passed within the array. This pulsating computation flow enables extremely high clock frequencies and minimizes the wiring length between registers, reducing dynamic power consumption at high frequencies. Simultaneously, the trigonometric factors of the preconditioning compensation matrix are mapped to the edge processing units of the array, where their specially designed reciprocal calculation units quickly complete the preprocessing transformation.

[0064] Finally, the method in this embodiment also possesses self-learning characteristics. The system background continuously collects convergence speed data under different channel characteristics and dynamically corrects the mapping table between the ill-conditioned threshold and the momentum weight factor using a simple linear regression model. Through this long-term online evolution, the coding matrix solution method can gradually adapt to multipath propagation patterns in specific geographical environments, achieving continuous performance improvement. This systematic design, integrating sensing, evaluation, preprocessing, iteration, correction, and global feedback, makes this invention highly resilient in modern complex communication systems.

[0065] In summary, this embodiment constructs a digital communication coding matrix solution that balances accuracy, speed, and power consumption through multi-dimensional technological innovation. From the underlying hardware array optimization to the mid-layer distributed block strategy and the upper-layer cross-layer protocol collaboration, each layer is specifically enhanced around the numerical characteristics of the conjugate gradient method. It solves the failure problem of existing technologies under ill-conditioned channels and provides solid algorithmic support for the physical layer implementation of next-generation ultra-high-speed mobile communication systems. Through the organic combination of adaptive preconditioning, momentum gradient correction, and dynamic step size optimization, this invention successfully achieves efficient signal recovery in non-ideal channel environments, demonstrating its engineering practicality and technological leadership.

[0066] In practical implementation, the interfaces between units adhere to strict timing constraints. After estimating a frame of signal, the channel state awareness unit notifies the matrix ill-conditioning evaluation unit to initiate calculations via an interrupt signal. The register configurations within each unit are initialized via a dedicated configuration bus, allowing the system to flexibly adjust parameters according to different communication protocol standards. This modular and parameterized design approach ensures that this invention can be widely applied in various digital communication fields such as 5G and its evolution technologies, satellite communications, and high-bandwidth microwave links.

[0067] All thresholds, step sizes, weighting factors, and error limits mentioned in the embodiments can be fine-tuned according to the specific hardware platform's bit width and computing resources. For example, on a fixed-point 16-bit computing platform, the frequency of residual reprojection can be appropriately increased to compensate for more severe truncation noise; while on a floating-point 32-bit platform, the reprojection cycle can be appropriately extended to achieve higher processing efficiency. This flexibility makes this method highly adaptable to different platforms.

[0068] Finally, it should be noted that the various units, modules, and their connections described in this embodiment are based on functional logic. In actual chip design or software development, these functions can be merged, split, or recombined as needed, as long as the core iterative conjugate gradient solution logic remains unchanged, all of which fall within the protection scope of this invention. Through this comprehensive technical layout, this invention provides a standardized, high-performance solution paradigm for the core computing units of digital communication systems.

[0069] Due to space limitations, this embodiment does not elaborate on certain common matrix operation details. However, in actual engineering implementations, these details should all follow standard numerical calculation specifications. The implementation of the entire encoding matrix solution method relies not only on the ingenious design of the algorithm but also on a deep understanding of the physical characteristics of communication and a precise grasp of the characteristics of hardware resources. It is based on this system-level thinking that this invention achieves fast and accurate solution of the encoding matrix in complex communication environments.

[0070] In future technological evolution, as antenna scale further expands, the segmentation strategy and adaptive preprocessing mechanism proposed in this embodiment will demonstrate even greater technical value. By transforming complex matrix operations into a scalable combination of hardware modules, this invention can easily handle terabyte-level data transmission demands. Simultaneously, its unique noise immunity and error accumulation immunity characteristics will provide reliable technical support for more challenging application scenarios such as high-frequency communication.

[0071] In summary, the proposed method for solving the encoding matrix of digital communication based on iterative conjugate gradients systematically overcomes the shortcomings of traditional algorithms, such as slow convergence and low accuracy, in ill-conditioned matrices and noisy environments, by constructing a complete logical system encompassing sensing, evaluation, preconditioning, momentum iteration, step size optimization, correction, and decision-making. The improvements in the underlying algorithm logic and the optimizations in hardware implementation complement each other, jointly promoting the development of digital signal processing technology towards greater efficiency and robustness.

[0072] The collaborative operation of each unit is not limited to a single solution task. For example, the fading features extracted by the channel state awareness unit are not only used to assist iteration but can also be fed back to the system's power control module, enabling cross-functional resource reuse. The condition number output by the matrix ill-conditioning evaluation unit can also serve as an important indicator of the current communication link quality, providing a basis for routing and switching decisions by the upper-layer protocol stack. This multi-dimensional application potential further enhances the overall technical value of this invention.

[0073] In the specific algorithm execution steps, each mathematical operation unit has undergone meticulous time-delay alignment design. The calculation of momentum gradient and the estimation of step size are interleaved in the hardware pipeline, maximizing the utilization of each functional module. Through this extreme engineering optimization, this invention achieves high-precision coding matrix solving while successfully reducing the average number of clock cycles per iteration to an industry-leading level.

[0074] Finally, the examples and parameters provided in this specific embodiment are intended to demonstrate the implementation process of the invention through specific numerical and logical flows, and should not be construed as limiting the scope of the invention. Without departing from the core concept of this invention, any improved scheme based on the conjugate gradient framework and incorporating momentum correction, preconditioning transformation, or adaptive correction falls within the scope of this invention's technical contribution. Through this rigorous and detailed description, this invention provides those skilled in the art with a clear and engineering-featured implementation guide, ensuring that the invention's results can be transformed into practical communication productivity.

[0075] In the context of rapid advancements in digital communication technology, efficient and stable matrix operations remain the cornerstone of physical layer algorithms. This invention, through deep optimization of the iterative conjugate gradient method, not only improves the performance of individual algorithms but also provides strong support for the robust operation of the entire communication system through a systematic architectural design. Whether facing fast-fading channels due to high mobility or severe interference from ultra-dense networking, the method described in this invention maintains excellent solution efficiency and demonstrates strong engineering robustness.

[0076] This comprehensive innovation, from theoretical modeling to hardware implementation, and from local optimization to global collaboration, is the key to this invention's ability to surpass existing technologies. Through the progressive explanations of Embodiments 1, 2, and 3, the technical details of this invention are fully demonstrated. These details collectively constitute a complete, rigorous, and efficient digital communication coding matrix solving technology system, making a substantial contribution to promoting technological progress in the communications industry.

[0077] The linear scalability exhibited by the method of this invention when processing high-dimensional matrices makes it an ideal choice for cutting-edge fields such as future ultra-large-scale antenna systems and terahertz communication. With the continuous improvement of computing device performance, the algorithm logic described in this invention will be able to support higher iteration frequencies and more complex preprocessing strategies, further unlocking the transmission potential of communication links. This forward-looking technological foresight ensures that this invention maintains a leading position in long-term technological competition.

[0078] In conclusion, it is reiterated that the specific embodiments described in this invention are based on optimal practices of current communication technologies. With the emergence of new materials, processes, and protocols, the related parameter configurations and hardware implementations may evolve, but the core solution concept based on iterative conjugate gradients and adaptive preprocessing will continue to play a vital role. This invention, through standardized interface definitions and modular logic design, leaves ample room for such future evolution, demonstrating exceptional design wisdom and technological depth.

[0079] By real-time sensing and ill-conditioning of the initial channel matrix, this method achieves precise allocation of computational resources; by constructing an adaptive preconditioning operator, it eliminates geometric distortions in the search space; by using multidimensional momentum gradient iteration and dynamic step size optimization, it achieves rapid traversal on complex surfaces; and finally, by correcting the solution vector and using adaptive convergence determination, it ensures the absolute reliability of the output results. Each step of this closed-loop process reflects a profound insight into the essential laws of digital communication.

[0080] The method of this invention is not only applicable to solving the coding matrix, but its core logic can also be extended to multiple communication signal processing stages such as equalizer design, precoding vector calculation, and multi-user detection. This cross-domain applicability greatly enhances the technical influence and commercial application prospects of this invention. As a fundamental algorithmic innovation, it will provide a continuous driving force for the performance optimization of modern digital communication systems.

[0081] This detailed description of the embodiments has now covered all the core points and technical details of the invention. Through detailed logical derivation, accurate mathematical description, and close integration with accompanying drawings, the method of this invention has achieved complete implementability. Those skilled in the art can reproduce this high-performance encoding matrix solving system on different hardware platforms based on the guidance of this specification, and share the technical benefits brought by this invention.

[0082] In the future development of wireless communication standards, the adaptive iteration and cross-layer collaboration concepts advocated in this invention are expected to become key technological paths for improving system spectral efficiency and energy efficiency ratio. We anticipate that the application of this invention will make a greater contribution to the prosperity of the global communications industry and usher in a new chapter of high-speed, low-power communication. This concludes the detailed implementation of this invention.

Claims

1. A coding matrix solving system for digital communication based on iterative conjugate gradients, characterized in that, include: The channel state sensing unit is used to acquire the original received signal and the preset pilot sequence in the digital communication system, generate the initial channel matrix through channel estimation logic, and extract the instantaneous signal-to-noise ratio parameter and channel fading characteristics of the signal. The matrix ill-conditioning assessment unit is used to perform singular value decomposition based on the initial channel matrix to obtain the eigenvalue distribution sequence, calculate the ratio of the maximum singular value to the minimum singular value, determine the condition number, and compare and analyze it with a preset ill-conditioning threshold. An adaptive preconditioning operator construction unit is used to generate a preconditioning compensation matrix based on the sparsity of the eigenvalue distribution sequence, using incomplete decomposition logic or a diagonal compensation strategy, in response to the condition number exceeding the ill-conditioned threshold, so as to perform a spatial transformation on the initial channel matrix and construct an equivalent symmetric positive definite matrix. The multidimensional momentum gradient iteration unit is used to initialize the solution vector, residual vector and search direction vector in the equivalent space, and introduces momentum weight factor to calculate the conjugate search direction in combination with the gradient direction of the current residual, so as to suppress the swaying of the search path caused by drastic changes in surface curvature. The dynamic step size optimization unit is used to obtain the product of the search direction vector and the equivalent symmetric positive definite matrix, and calculate the optimal step size by combining the second derivative estimate of the residual vector. At the same time, it performs smoothing filtering correction on the step size according to the instantaneous signal-to-noise ratio parameter to counteract the interference of Gaussian noise on the search accuracy. The solution vector update and correction unit is used to update the solution vector based on the correction step size and simultaneously perform residual reprojection operation, using orthogonalization correction logic to eliminate the accumulation of rounding errors caused by the finite word length effect. An adaptive convergence determination unit is used to dynamically adjust the target residual error limit according to the instantaneous signal-to-noise ratio parameter, and determine whether to terminate the iteration or trigger the next round of iteration calculation based on the comparison result between the magnitude of the residual vector and the target residual error limit.

2. The coding matrix solving system for digital communication based on iterative conjugate gradients according to claim 1, characterized in that, When executing channel estimation logic, the channel state sensing unit uses the least squares method or the minimum mean square error criterion to process the pilot sequence to remove the channel transfer function; when extracting channel fading features, it performs differential operations on the channel matrix at adjacent time points to evaluate the channel coherence time and Doppler shift intensity.

3. The coding matrix solving system for digital communication based on iterative conjugate gradients according to claim 1, characterized in that, When calculating the condition number, if the dimension of the initial channel matrix exceeds a preset computational pressure threshold, the matrix ill-conditioning assessment unit uses the power method or inverse power method to approximate the estimation of the maximum and minimum eigenvalues. The determined condition number serves as the core indicator for measuring the flatness of the search space and determines the intervention strength of the preconditioning compensation matrix.

4. The coding matrix solving system for digital communication based on iterative conjugate gradients according to claim 1, characterized in that, The working logic of the adaptive preconditioning operator construction unit is as follows: when the condition number is in the first preset interval, the Jacobi preprocessor is used to take the reciprocal of the diagonal elements of the initial channel matrix to form the preconditioning compensation matrix; when the condition number is in the second preset interval and the high-end limit of the second preset interval is greater than the first preset interval, incomplete Shur complement decomposition or incomplete robust decomposition is performed to construct the preconditioning compensation matrix with triangular factors.

5. A coding matrix solving system for digital communication based on iterative conjugate gradients according to claim 4, characterized in that, The specific execution process of the incomplete decomposition logic is as follows: During the matrix decomposition process, by setting a discard threshold, non-main diagonal elements with an absolute value less than the discard threshold are ignored, and an approximate factor that maintains the sparse structure of the original matrix and is invertible is generated, so as to reduce the storage overhead and computational latency of the preprocessing stage.

6. The coding matrix solving system for digital communication based on iterative conjugate gradients according to claim 1, characterized in that, When calculating the conjugate search direction, the multidimensional momentum gradient iteration unit couples the search direction information from the previous three time steps to the previous time step through a nonlinear momentum coefficient; the value of the momentum weight factor is dynamically scaled according to the gradient angle between adjacent iteration steps: when the angle is less than 90 degrees, the momentum weight factor is increased; when the angle is greater than 90 degrees, the momentum weight factor is decreased.

7. A coding matrix solving system for digital communication based on iterative conjugate gradients according to claim 1, characterized in that, When calculating the step size, the dynamic step size optimization unit uses the inner product of the residual vector and the search direction vector as the numerator and the quadratic form of the search direction vector after mapping through an equivalent symmetric positive definite matrix as the denominator for division; and when the instantaneous signal-to-noise ratio is lower than the preset reliable communication threshold, the calculated step size is attenuated.

8. A coding matrix solving system for digital communication based on iterative conjugate gradients according to claim 1, characterized in that, When performing residual reprojection, the solution vector update and correction unit periodically recalculates the original residual and replaces the residual obtained by recursive update by subtracting the product of the encoding matrix and the current estimated solution vector from the observation vector. The orthogonalization correction logic ensures that the newly generated search direction and the historical search direction maintain a conjugate relationship in the precondition space through the Gram-Schmidt orthogonalization process.

9. A coding matrix solving system for digital communication based on iterative conjugate gradients according to claim 1, characterized in that, The target residual error limit in the adaptive convergence determination unit is dynamically associated with the modulation and coding scheme of the communication system: for high-order orthogonal amplitude modulation schemes, a first error tolerance is set; for low-order modulation schemes, a second error tolerance greater than the first error tolerance is set; and if the residual decrease ratio of three consecutive iterations is less than 0.1%, then convergence exit is forcibly executed.

10. A coding matrix solving system for digital communication based on iterative conjugate gradients according to claim 1, characterized in that, The system further includes: a block calculation module, used to divide a large-size coding matrix into several sub-blocks and execute the update logic of each sub-block in parallel on independent hardware processing cores; a real number conversion module, used to convert a complex initial channel matrix of dimension N×M into a real symmetric matrix of dimension 2N×2M through complex-to-real number transformation logic; and a feedback control loop, used to trigger an improvement in channel estimation accuracy or a reduction in data transmission rate when the number of iterations reaches a preset maximum limit and the residual target is not achieved.

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