Coefficient Matrix Research Articles

On the Quasi-diagonalization and Uncoupling of Gyroscopic Circulatory Multi-degree-of-freedom Systems

Abstract A new central result that gives the necessary and sufficient conditions for two n by n skew-symmetric matrices and one symmetric matrix to be simultaneously quasi-diagonalized by a real orthogonal congruence is proved. Based on this result, the decomposition of linear multi-degree-of-freedom dynamical systems with gyroscopic, circulatory, and potential forces is investigated through a real linear coordinate transformation generated by an orthogonal matrix. Several sets of conditions, applicable to real-life structural and mechanical systems arising in aerospace, civil, and mechanical engineering, under which such a coordinate transformation exists are found, thereby allowing these systems to be decomposed into independent, uncoupled subsystems, each with a maximum of two degrees of freedom. The conditions are expressed in terms of the coefficient matrices of the system. A specific form for the circulatory (gyroscopic) matrix is posited, and when the gyroscopic (circulatory) matrix is simple—a situation that commonly appears in real-life applications—it is shown that just a single necessary and sufficient condition is required for the decomposition of the multi-degree-of-freedom system. Numerical examples are provided throughout to demonstrate the analytical results.

Analysis for sparse channel representation based on dictionary learning in massive MIMO systems

AbstractThe accuracy analysis of dictionary sparse representation for channels in massive MIMO systems is a relatively unexplored field. Existing research has primarily focused on investigating the accuracy of dictionary sparse representation using simulation in massive MIMO systems, but has not provided quantitative accuracy analysis. To address this gap, the correlation numerical proportional factor is proposed to represent the accuracy performance of non‐zero elements in the coefficient matrix. Additionally, a qualitative analytical formula for dictionary sparse representation accuracy is provided and an optimal upper bound for the correlation numerical proportional factor is established. Furthermore, the innovation indicates that the accuracy of dictionary sparse representation is mainly influenced by the cross‐correlation between the pilots matrix and the dictionary matrix, as well as sparsity. The author has also developed a method for minimizing the correlation numerical proportional factor. In order to obtain an optimal sparse representation coefficient matrix, a cross‐correlation matrix is constructed and an analytical expression is derived for it as well as its use as an optimal hard decision threshold is determined. Finally, a sparse representation coefficient optimization algorithm is proposed using this optimal threshold. Simulation results demonstrate that this algorithm can significantly improve channel sparse dictionary representation accuracy.

Photometric stereo light calibration optimization and high-reflective distance fitness compensation method to improve reconstruction

This paper proposes an interpolated light calibration optimization and high-reflective area compensation method to solve the accuracy loss caused by idealization of the photometric stereo (PS) model and high-reflective area. The spatial distribution model of light intensity is defined as a cubic interpolation function, which is used to obtain an intensity coefficient matrix to optimize the PS model. A light source adaptation model is adopted to select the appropriate position of the light source based on the characteristics of the object. Finally, a distance fitness compensation model is established to repair the high-reflective area, which selects the optimal pixel values of the non-high-reflective area. The experiment result shows that the proposed method can obtain the light source spatial characteristics, and high-reflective areas could be repaired.

High thermolectric properties of disordered Ge0.8-ySn0.2MnyTe semiconductor

For layered GeTe-based materials, the typical strategy to reduce the thermal conductivity of the crystal lattice is to introduce atomic disorder, increase the lattice non-harmonicity, and maximize the fluctuations of the quality/stress field in the sample. Since Sn/Mn can easily form solid solutions in the GeTe matrix, the degree of disorder in the sample is much higher than the level reported in IV-VI alloys in the past, which leads to a strong phonon scattering and a sharp decrease in the lattice thermal conductivity over the entire temperature range. In addition, Mn plays a significant role in changing the valence band structure of GeTe, i.e., Mn doping enhances the convergence of the light valence band and the heavy valence band and the change of the bandgap, significantly increasing the Seebeck coefficient of the GeTe matrix, resulting in a significant improvement in the thermoelectric performance of the Sn/Mn co-doped synthesized GeTe-based materials. Finally, we found that treating the sample with layered disorder is an effective method to improve the thermoelectric performance, and this study has certain reference value for the development of novel toxic-free, high-performance thermoelectric materials.

Sparse representation-based noise reduction for BOTDR monitoring signals in foundation pit anchor cables

Abstract In foundation pit monitoring, the Brillouin optical time-domain reflectometer (BOTDR) system typically has a low signal-to-noise ratio (SNR). To solve this problem, a BOTDR signal noise-suppression method based on sparse representation is proposed in this paper. The initial dictionary is constructed from the eigenvectors of the normalized graph Laplace matrix, and K-SVD and OMP algorithms are combined to update the dictionary and coefficient matrix to sparsely represent the essential characteristics of the BOTDR signal, filtering out random noise lacking sufficient similarity data and improving the quality of the reconstructed signal. To verify the effectiveness of the proposed algorithm, a BOTDR temperature-sensing experimental system was designed and used to test the denoising performance of the sparse representation algorithm quantitatively. The experimental results showed that the average SNR of the algorithm on six temperature levels was 27.4%, 15.4%, 13.1%, and 17.9% higher than that of WDD, EMT, EMD, and the raw signals, respectively. The average sample entropy (SE) was 24.4%, 16.0%, 15.4%, and 47.9% lower than that of WDD, EMT, EMD, and the raw signals, respectively. Additionally, the proposed Laplace-based dictionary outperformed the DCT dictionary. Finally, the monitoring project of the Beijing Stomatological Hospital was used as a test site. The sparse representation algorithm was used to conduct noise reduction experiments on the on-site anchor cable fiber optic monitoring signals. The average signal SE decrease of monitoring signals at the site reached 45.2%. This study provides an effective signal denoising method for the application of BOTDR technology in foundation pit monitoring.

APPLICATION OF OPTIMIZATION METHOD IN MULTIDIMENSIONAL SPACE TO THE PROBLEM OF DETERMINING FULLERENE CONCENTRATION IN FULLERENE-CONTAINING MIXTURE

The work proposes a method for the quantitative processing of experimental data from ultraviolet spectroscopy of substituted fullerene mixtures, taking into account errors in the experimental data. This task can be written as a system of Vierordt equations (a system of linear equations). The coefficients of the system are the extinction coefficients of individual substituted fullerenes, which are difficult to measure accurately. Therefore, the mathematical model of the problem contains significant errors that affect the reliability and stability of the solution. To minimize the influence of errors, the work [1] proposes a method, which involves reducing the considered system of linear equations to a linear programming problem. This study generalizes the work [1] and proposes to use linear programming problems to find interval values of correction factors for parameters containing errors. Since decreasing the discrepancy reduces the solution error only for well-conditioned coefficient matrices, the values of correction coefficients that minimize the condition number of the coefficient matrix are found in the obtained intervals. The found correction values allowed to significantly improve the conditionality of the system coefficient matrix and obtain a more accurate solution. By using the refined extinction coefficients, the distribution of molar fractions of fullerene fragments and fullerene-containing compounds in the mixture of products formed by the attachment of cyanoisopropyl radicals was obtained, as well as the distribution of molar fractions of fullerene fragments of different degrees of substitution for samples of fullerene-containing polymethyl methacrylate and the distribution of molar fractions of fullerene fragments of different degrees of substitution for samples of fullerene-containing polystyrene. The results are correct and consistent with the theory, both for low-molecular-weight mixtures and for fullerene-containing polymers.

Detection of Traffic Anomalies Based on Their Frame Wavelet Transformations Processing

Relevance. The active transition to a massive digital infrastructure based on Internet of Things (IoT) technology has brought telecommunications networks to the level of dominant information resources. The one-time increase in the number of existing Internet services is inextricably linked to the growing variety of network anomalies on telecommunications equipment. In turn, existing methods of detecting network threats do not allow timely assessment of network traffic, which is characterized by a large number of parameters, and the detected anomalies from external interference do not have pronounced patterns. The purpose of the study is to increase the efficiency of detecting traffic anomalies based on the results of processing its frame wavelet transform. The scientific task is to develop scientific and methodological approaches that allow effective analysis and timely detection of anomalies in network traffic. A comparative review of search methods for detecting network traffic anomalies, algorithms for detecting uncontrolled anomalies, traffic analysis methods based on local emission factor, binary trees, optical emission spectroscopy. Decision. The results of the study of the possibility of detecting anomalies in the bitstream traffic based on the results of its multiple-variable transformation in the Haar wavelet basis are considered. The choice for further processing of the coefficients of the traffic decomposition matrix along the time shift variable is justified. It is proved that multiple-scale transformations not only increase the structural differences in traffic, but also open up the possibility of localization of anomalies that caused these differences. The scientific novelty of the work is determined by the author's approach to detecting network traffic anomalies during the transition from the direct representation of a signal in the form of its discrete samples to coefficients formed from the matrices of its wavelet transformations, and, as a result, increasing its contrast with other signals with a similar structure. Theoretical significance. The necessity and sufficiency of using wavelet coefficients instead of time samples of signals in the basis of the parent wavelet from the matrix of the generated frame is proved. The relationship between the Hurst indicators and the coefficients of the cross-correlation functions has been established. Practical significance. The results obtained in the work, in the future, can be used in the construction of models for evaluating network traffic in conditions of deliberate, as well as methods for searching and synthesizing effective methods of protection against them.

Lippmann-Schwinger equation representation of Green's function and its preconditioned generalized over-relaxation iterative solution in wavelet domain

The calculation of Green's function is the core of seismic forward and inverse methods based on integral operators. When the Lippmann-Schwinger (L-S) equation is used to calculate Green's function in strongly scattering media, both the Born scattering series and the numerical iterative method encounter issues of slow convergence or divergence. Although the renormalization method derived from quantum mechanics can effectively address the convergence problem of Born scattering series in strong scattering problems, it is acknowledgeed that the convergence conditions and rates of convergence of different reformulation series may vary, and no universal convergence reformulation scattering series exists. Numerical methods for solving integral equations tend to be more general and mathematically robust. In this work, we focus on the numerical solution method of L-S equations. By using a wavelet-domain preconditioner to a reformulated or equivalent Lippmann-Schwinger (L-S) equation, we present an iterative method for numerically solving the equivalent L-S equation aimed at improving the rate of convergence and iteration efficiency in strongly inhomogeneous media. Following Jakobsen et al. (2020), we first introduce a small imaginary component into the background wave number，then rewrite the L-S equation to derive the equivalent complex wave number L-S equation. This reformulation ensures that the coefficient matrix exhibits a banded structure after numerical discretization, allowing the wavelet coefficient matrix to maintain good sparsity. We employ a multi-level fill-in incomplete LU (ILU) factorization method along with a block ILU-based algebraic recursive multilevel solve (ARMS) method in the wavelet domain to generate sparse approximate inverses as preconditioning operators, thereby accelerating the convergence of the generalized successive over-relaxation (GSOR) iterative method. This method is applied to compute numerical Green's functions in strongly inhomogeneous media. Numerical results demonstrate that our method yields simulation outcomes consistent with those obtained from the direct method for solving the original real wave number L-S equation. By testing various preconditioners, we find that the ARMS preconditioner offers significant advantages in operator generation efficiency and non-zero element filling ratio, effectively accelerating the convergence of the GSOR iterative method while achieving higher computational efficiency.

Acoustic and soliton propagation using fully-discrete energy preserving partially implicit scheme in homogeneous and heterogeneous mediums

This study presents an energy preserving partially implicit scheme for the simulation of wave propagation in homogeneous and heterogeneous mediums. Despite its implicit nature, the developed scheme does not require any explicit numerical or analytical inversion of the coefficient matrix. Theoretical analysis and numerical experiments are performed to validate the energy preserving properties of the fully-discrete scheme. Convergence analysis is also performed to assess the rate of convergence of the developed scheme. The efficiency and accuracy of the developed scheme are validated by numerical solutions of wave propagation in layered heterogeneous mediums. Furthermore, simulations of soliton propagation following nonlinear sine-Gordon and Klein-Gordon equations in homogeneous and heterogeneous mediums are discussed. Numerical solutions are also compared with the results available in the literature. The present method accurately resolves the physical characteristics of the chosen problems, competing well with existing multi-stage time-integration methods. Moreover, it has significantly lower computational complexity than the four-stage, fourth-order Runge-Kutta-Nyström method.

Temporal asynchrony analysis for dynamic operation of hydraulic-thermal-electricity multiple energy networks based on holomorphic embedding method

Analyzing the operational states of multiple energy networks (MEN) in multi-energy systems is crucial for ensuring system stability. The dynamic operational characteristics of different energy flows pose challenges for computational analysis. Traditional steady-state methods are inadequate for addressing the dynamics of MEN, especially when dealing with temporal discrepancies between hydraulic and thermal flows in thermal networks (TN) and the heterogeneity between TN and electrical networks. Therefore, this paper proposes a novel holomorphic embedding method (HEM) based on multi-stage decomposition method. The developed HEM constructs a time coefficient matrix and utilize inner-outer loop recursion to handle the time lag between thermal flow and hydraulic flow in the TN. Additionally, we reconstruct a holomorphic matrix, integrating hydraulic flow to bridge thermal and electric power flows, thereby improving the operational heterogeneity among different networks. Real-case simulations show that when the Taylor expansion order in HEM is equal to 4, the proposed method achieves a mere 1% discrepancy from actual operational data, enhancing computational efficiency by 60% compared to the Newton-Raphson method. Moreover, in this real-case, the TN exhibits a maximum delay response time of 180s compared to electrical networks. Exploiting this delay time effectively increases renewable energy generation within multi-energy systems by 961.58kW per day.

Improving the performance of self-organizing map using reweighted zero-attracting method

In this paper, we introduce a novel approach to enhance the accuracy and convergence behavior of Self-Organizing Maps (SOM) by incorporating a reweighted zero-attracting term into the loss function. We evaluated two SOM versions: conventional SOM and robust adaptive SOM (RASOM). The enhanced versions, reweighted zero-attracting SOM (RZA-SOM) and reweighted zero-attracting RASOM (RZA-RASOM), include an l1 norm in the error function to add a zero-attractor term, which improves weight coefficient adjustments while preserving topology. The models were assessed for convergence speed and misadjustment under sparsity assumptions of the true coefficient matrix, and their robustness was tested under conditions of increased non-zero taps. Using six different datasets, we compared the performance of RZA-SOM and RZA-RASOM against conventional SOM and RA-SOM in terms of accuracy, quantization error, and topology preservation. Experimental results consistently demonstrated that RZA-SOM and RZA-RASOM surpassed the performance of conventional SOM and RA-SOM.

Construction and non-vanishing of a family of vector-valued Siegel Poincaré series

Using Poincaré series of K-finite matrix coefficients of integrable antiholomorphic discrete series representations of Sp2n(R), we construct a spanning set for the space Sρ(Γ) of Siegel cusp forms of weight ρ for Γ, where ρ is an irreducible polynomial representation of GLn(C) of highest weight ω∈Zn with ω1≥…≥ωn>2n, and Γ is a discrete subgroup of Sp2n(R) commensurable with Sp2n(Z). Moreover, using a variant of Muić's integral non-vanishing criterion for Poincaré series on unimodular locally compact Hausdorff groups, we prove a result on the non-vanishing of constructed Siegel Poincaré series.

Aeroelastic Structural Analysis to Calculate Symmetrical Divergence Modes of an Aircraft Wing Using Aerodynamic Lifting Line Theory

This work explores the use of numerical matrix iteration to determine an aircraft wing's divergence speed using the aerodynamic span effects approach. The present work begins with the construction of equilibrium equations in differential and integral forms. In this paper, integral formulas are used because it serves as a convenient basis for numerical solutions of complex practical problems. Second, the straight-tapered wing is divided into a number of Multhopp stations. Subsequently, the stations' torsional influence coefficient matrix has been calculated. Third, lifting line theory was used as a suitable choice of aerodynamic theory, and the governing equations were represented in matrix form. Finally, aerodynamic span effects are taken into consideration with induction effects according to Prandtl’s lifting line theory to calculate the symmetrical divergence speed from the lowest eigenvalue of the homogenous governing integral equation. To get the solution to converge, a matrix has been iterated using the MATLAB environment. The obtained results will aid modern aircraft designers in their understanding of wing instability in steady motion.

Implementation of spectral methods on Ising machines: toward flow simulations on quantum annealers

Abstract We investigate the possibility and current limitations of flow computations using quantum annealers by solving a fundamental flow problem on Ising machines. As a fundamental problem, we consider the one-dimensional advection-diffusion equation. We formulate it in a form suited to Ising machines (i.e., both classical and quantum annealers), perform extensive numerical tests on a classical annealer, and finally test it on an actual quantum annealer. To make it possible to process with an Ising machine, the problem is formulated as a minimization problem of the residual of the governing equation discretized using either the spectral method or the finite difference method. The resulting system equation is then converted to the Quadratic Unconstrained Binary Optimization (QUBO) form through the quantization of variables. It is found in numerical tests using a classical annealer that the spectral method requiring a smaller number of variables has a particular merit over
the finite difference method because the accuracy deteriorates with the increase of the number of variables. We also found that the computational error varies depending on the condition number of the coefficient matrix. In addition, we extended it to a two-dimensional problem and confirmed its fundamental applicability. From the numerical test using a quantum annealer, however, it turns out that the computation using a quantum annealer is still challenging due largely to the structural difference from the classical annealer, which leaves a number of issues toward its practical use.

Phylogenetic Study of Several Parasitic Plant Species Based on The atp-1 Gene Sequence

The distinction between parasitic and non-parasitic plants can be determined by analyzing the atp-1 gene, which plays a vital role in respiration and is known for its high mutation rate. This study analyzed the kinship of parasitic plant subclass species through the construction of a phylogenetic tree based on atp-1 gene sequences. The atp-1 gene sequences of parasitic and non-parasitic plants with a total of 29 species were obtained from NCBI. The sequences were then aligned with ClustalW and analyzed for mutation patterns. Sequences that have been aligned, phylogenetic trees were made with MEGA11 software with the Maximum Likelihood method and analyzed using the iTOL website. The sequences were analyzed for similarity and kinship with Matrix Coefficient and Haplotype Construction. The atp-1 gene proved that parasitic plants (hemiparasites) are furthermore related to non-parasitic plants compared to holoparasite parasitic plants. Besides that, the kinship of parasitic plants can be analyzed by several methods, namely matrix coefficient to measure similarity, DnaSP to analyzing haplotype, and haplotype network to find out detailed information on mutations that occur. Matrix coefficients can also be used to measure specific similarities between species. It was found that the same subclass had high similarities, for example the species Santalum album and Heisteria parvifolia with a genetic distance value of 0.00574. Meanwhile, different subclasses have low similarity, such as Cassytha filiformis and Ombrophytum with a genetic distance value of 0.07871. This study shows that the atp-1 gene is effective in analyzing the kinship between parasitic and non-parasitic plants. Hemiparasites are genetically closer to non-parasitic plants than holoparasites, with higher genetic similarity within the same subclass.

Research on power battery anomaly detection method based on improved TimesNet

Health monitoring and abnormality detection of power batteries for new energy vehicles has been one of the hot topics in recent years. Accurate and efficient power battery anomaly detection is crucial to ensure stable operation of the battery system and energy saving. However, power battery data are often non-linear and unstable due to external factors, such as temperature conditions, which pose challenges for anomaly detection. Existing methods for collecting battery data’s temporal and spatial features are available, but many of them yield suboptimal detection accuracy and feature extraction efficiency due to their disregard for the multi-periodicity of the data. This paper proposes a novel network structure for power battery anomaly detection based on an improved TimesNet. Firstly, the original battery data undergo preprocessing, and the feature correlation coefficient matrix is established using the MIC algorithm. Secondly, the improved TimesNet network is employed to convert the one-dimensional time feature sequence into a two-dimensional tensor, enabling the extraction of temporal features and dependencies at various cycle scales. Finally, the two-dimensional tensor is transformed into a one-dimensional time series, which undergoes information-weighted aggregation to obtain the final anomaly detection results. To assess the effectiveness and generalization of the proposed model, experiments are conducted using Battery and four public datasets for anomaly detection. The experimental results demonstrate that the proposed model outperforms the transformer, autoformer, TimesNet, and Dlinear models, achieving an improvement of 1%–19% in the F1 value and 1%–3% in the ACC compared to the other models.

Extended influence coefficient method for meshing stiffness evaluation of spur gears considering rotor flexibility

Rotor-supported gears are applied in transmission systems, and the rotor flexiblility leads to the misalignment of the gear pair with multiple degrees of freedom, which ultimately leads to the decrease of meshing stiffness. To evaluate the meshing stiffness of rotor-supported gears, the extended influence coefficient method (EICM) is proposed in this paper. Firstly, the gear contact position adjustment under misalignment condition is carried out. Then, the extended influence coefficients of bending, shearing, axial compression, Hertzian contact and fillet foundation deformation of gears are derived to obtain the extended influence coefficient matrix. Afterward, the load-deformation equations under different contact states are obtained, and the meshing stiffness is calculated. Subsequently, a cantilever rotor-gear coupling dynamic model is established by the rigid body element-extended influence coefficient method (RBE-EICM), and the dynamic meshing stiffness considering rotor flexibility is obtained. The accuracy of EICM and RBE-EICM is verified by comparison with finite element method, literature methods and experimental data, respectively. Finally, the effects of supporting layout, input torque and rotor diameter on the dynamic characteristics of cantilever rotor-supported gears are discussed.

Cultivated Land Information Extraction and Spatial Pattern Analysis in Beijing Based on Deep Learning

Abstract. Cultivated land is the basic resource and material condition for human survival, providing the necessary material basis for agricultural development and agricultural modernization (Tian and Shi, 2024). For this reason, this paper takes Beijing as an experimental area to study the cultivated land extraction method and analyze the distribution pattern of cultivated land and the degree of fragmentation. The results show that (1) the extraction results are evaluated by using confusion matrix and Kappa coefficient, and the Kappa coefficient is obtained to be 0.8358.(2) The cultivated land in Beijing is mainly distributed in the southeast as well as the northwest of the local area of gentle terrain, and the The range of the extremely high value of cultivated land kernel density in 2017–2022 is significantly reduced, the slope of the area with the smallest reduction in the proportion of cultivated land is less than 5°, the reduction in the area of cultivated land in water resource-rich areas is small, and the reduction in the proportion of cultivated land in road-intensive areas is large. (3) Due to various natural factors and human activities, the degree of cultivated land fragmentation in Beijing is increasing, the boundary shape of cultivated land patches tends to be irregular.

Joint passive beamforming and deployment design for active intelligent reflecting surface-assisted energy harvesting-cognitive radio sensor networks.

To prolong the network lifetime of energy harvesting-cognitive radio sensor networks (EH-CRSNs), this paper integrates active intelligent reflecting surface (IRS) to support both downlink EH and uplink data transmissions and formulates a constrained non-convex optimization problem that maximizes the net energy gain. To solve this problem, a joint passive beamforming and IRS deployment mechanism is proposed to determine the optimal deployment location of the active IRS and optimally configure the IRS reflection coefficient matrices. Specifically, the net energy gain maximization problem at a specified location is divided into two sub-problems according to its characteristics: maximizing the cumulative energy harvested by all CRSNs nodes in the downlink and minimizing the energy consumption of cluster heads in uplink data transmissions, with each sub-problem being solved independently. Furthermore, this paper explores how factors such as the number of active reflecting elements and the amplification power budget influence the optimal deployment location of the active IRS. The findings from this investigation can guide the design of active IRS-assisted EH-CRSNs under specified parameters. Simulation results confirm the aforementioned conclusions and highlight the benefits of the proposed joint mechanism in enhancing net energy gain over various benchmark mechanisms.

An Improved Nonnegative Matrix Factorization Algorithm Combined with K-Means for Audio Noise Reduction

Clustering algorithms have the characteristics of being simple and efficient and can complete calculations without a large number of datasets, making them suitable for application in noise reduction processing for audio module mass production testing. In order to solve the problems of the NMF algorithm easily getting stuck in local optimal solutions and difficult feature signal extraction, an improved NMF audio denoising algorithm combined with K-means initialization was designed. Firstly, the Euclidean distance formula of K-means has been improved to extract audio signal features from multiple dimensions. Combined with the initialization strategy of K-means decomposition, the initialization dictionary matrix of the NMF algorithm has been optimized to avoid getting stuck in local optimal solutions and effectively improve the robustness of the algorithm. Secondly, in the sparse coding part of the NMF algorithm, feature extraction expressions are added to solve the problem of noise residue and partial spectral signal loss in audio signals during the operation process. At the same time, the size of the coefficient matrix is limited to reduce operation time and improve the accuracy of feature extraction in high-precision audio signals. Then, comparative experiments were conducted using the NOIZEUS and NOISEX-92 datasets, as well as random noise audio signals. This algorithm improved the signal-to-noise ratio by 10–20 dB and reduced harmonic distortion by approximately −10 dB. Finally, a high-precision audio acquisition unit based on FPGA was designed, and practical applications have shown that it can effectively improve the signal-to-noise ratio of audio signals and reduce harmonic distortion.