Anti-symmetric filter design method based on anti-auto-inverse matrix theory

Abstract

The invention discloses an anti-symmetric filter design method based on an anti-auto-inverse matrix theory, and belongs to the technical field of digital signal processing. According to the method, a design problem of an antisymmetric filter is modeled into an optimization problem of an impulse response vector h, and an inverse self-inverse constraint h =-Jh (J is an inverse unit matrix) is applied, so that an optimization target minize | A -d | is constructed. An iterative solution technology of a generalized coupling Sylster transpose matrix equation is adopted, and through parameter initialization, residual matrix updating, step length dynamic adjustment and anti-auto-back projection operation, it is ensured that a filter coefficient strictly meets antisymmetry. Furthermore, the calculation complexity is reduced by using block calculation, fast multiplication and a precondition technology, and the frequency domain performance is optimized in combination with a weighted least square method and a Lagrange multiplier method. According to the filter designed by the invention, the strict linear phase characteristic is maintained, meanwhile, the stop-band attenuation can reach more than 50dB, the gradient of a transition band is improved by 30%, and the group delay fluctuation is controlled within 0.1 sampling point, so that the filter is suitable for medical image and radar signal processing and high-fidelity audio systems, and has the characteristics of high calculation efficiency and excellent performance.

Description

Technical Field

[0001] This invention belongs to the field of digital signal processing technology, specifically relating to an antisymmetric filter design method based on the theory of inverse self-reflexive matrices. Background Technology

[0002] Filter design in digital signal processing systems has always been a core topic in the field. Antisymmetric filters, due to their inherent linear phase characteristics, possess unique advantages in phase-sensitive applications such as medical image reconstruction and stereo audio processing. However, existing antisymmetric filter design methods still face significant technical bottlenecks in mathematical modeling, computational efficiency, and performance optimization. Meanwhile, inversely inverse matrices, as a matrix type with a special algebraic structure, have mathematical properties that highly align with the design requirements of antisymmetric filters, but this theoretical advantage has not yet been fully explored in the field of digital signal processing. This paper systematically reviews currently disclosed patent technologies, analyzes the current state of antisymmetric filter design technology, and explores the application potential of inversely inverse matrices in this field, providing a theoretical foundation and technical reference for subsequent technological innovation.

[0003] Filters, as the most fundamental and critical component in digital signal processing systems, directly impact the signal processing quality of the entire system. Traditional filter design methods often focus on optimizing amplitude-frequency characteristics while relatively neglecting the precise control of phase-frequency characteristics. This design philosophy has shown significant limitations in many modern applications. In medical image reconstruction, phase distortion leads to image artifacts and loss of detail; in stereo audio processing, phase distortion disrupts sound field localization and spatial perception; and in specialized fields such as radar signal processing and seismic exploration, phase fidelity is directly related to the accuracy of target recognition and localization. These application scenarios place stringent requirements on the phase characteristics of filters, making antisymmetric filters with linear phase characteristics an irreplaceable technical choice.

[0004] The mathematical essence of an antisymmetric filter lies in the symmetry of its impulse response sequence, h(n) = -h(N-1-n). This special structure endows the filter with inherent linear phase characteristics, ensuring that the signal does not suffer phase distortion after passing through the filter. However, existing antisymmetric filter design methods face three major technical bottlenecks: In terms of mathematical modeling, traditional optimization algorithms struggle to strictly maintain the antisymmetric characteristics of the filter during iteration, often resulting in a gradual degradation of antisymmetry during the design process, leading to the final filter failing to achieve the theoretically expected phase linearity; in terms of computational efficiency, existing design algorithms generally have high complexity, especially when designing high-order filters, where computational resource consumption increases exponentially, making the design process too time-consuming and unable to meet real-time processing requirements; and in terms of performance optimization, antisymmetric filters designed using traditional methods often perform poorly in key indicators such as stopband attenuation and transition band steepness, failing to meet the requirements of high-precision applications.

[0005] Persymmetric matrices, as a type of matrix with a special algebraic structure, possess mathematical properties that are highly compatible with the design requirements of antisymmetric filters. A persymmetric matrix is ​​a matrix A that satisfies JAJ = A^T, where J is the inverse identity matrix (with 1s on the secondary diagonal and 0s elsewhere). These matrices exhibit structural symmetry along the secondary diagonal, which is highly consistent with the mathematical description of antisymmetric filters. The theoretical advantages of persymmetric matrices include: accurately describing and preserving antisymmetric properties, simplifying constraints in optimization problems; their special structure can be used to reduce computational complexity and achieve fast algorithms; and they also maintain good numerical stability. However, in the field of digital signal processing, these theoretical advantages of persymmetric matrices have not yet been fully explored and applied, especially in the specific problem of filter design, where related research is still in its early stages.

[0006] In recent years, patent technologies in the field of filter design have shown a diversified development trend, with various innovative design methods and structures constantly emerging. Through a systematic analysis of published patents, we can broadly categorize these technologies into three main areas: structural innovation, algorithm optimization, and materials and processes. Each category provides valuable references for antisymmetric filter design at different levels.

[0007] In terms of structural innovation, multilayer filter design has become an important trend. The patent "A Multilayer Filter" (CN113679454B) obtained by Wuhan Fangu Electronic Technology Co., Ltd. significantly improves the selectivity and stability of filters through innovative multilayer structure design. This technology exhibits superior anti-interference capabilities in different operating frequency bands, its core being the optimization of overall performance through the synergistic effect of multiple functional layers. The multilayer structure design concept provides structural possibilities for antisymmetric filters to achieve steeper transition bands and higher stopband attenuation.8 Similarly, the patent "Single-Channel Filter Realizing Multi-Channel Filter" (CN113889724B) from Suzhou Nuotai Communication Co., Ltd. achieves single-channel processing of multi-channel signals through innovative design. This structural simplification method reduces hardware complexity and cost, and its design concept is inspiring for the compact implementation of antisymmetric filters.5

[0008] Patents related to algorithm optimization directly address the computational efficiency issues in filter design. The patent "Parameter Optimization Method for W-Band Coupled Multi-Band Filter" (CN119494312A) applied for by the Aerospace Science and Industry Communication Technology Research Institute proposes a systematic design process, with clear optimization objectives at each step, from determining the ideal characteristic function to adjusting the final performance indicators. This technology utilizes coupling matrix synthesis software to automatically generate accurate coupling matrices, significantly improving design efficiency and ensuring seamless integration from design to manufacturing. The patent "Symmetrical S-Band Filter Based on GaAs IPD" (CN119496482A) applied for by Anhui Silicon Microelectronics Technology Co., Ltd. significantly improves out-of-band suppression through innovative symmetrical structure design, including optimized configuration of multi-order resonant circuits and coupling capacitors. This optimization approach for symmetrical structures shares many similarities with the design requirements of antisymmetric filters, and its technical path is worth learning from.

[0009] Patent technology trend analysis indicates that filter design is moving towards high performance, low cost, and miniaturization. VIAVI Technology Co., Ltd.'s patent for a "multi-transmission optical filter" (CN114556169B) demonstrates the potential for cross-domain technology integration. This technology, through optimized materials and structure, improves transmittance while reducing optical loss, and its design concept can be transferred to the field of electronic filters. It is worth noting that existing patents directly addressing specialized technologies for digital antisymmetric filters are relatively scarce; most related innovations are scattered across general methods of various filter designs, leaving ample room for the application of specialized technologies such as inverse self-reflex matrices.

[0010] Key technical bottlenecks in the design of antisymmetric filters severely restrict their performance in high-precision applications. In-depth analysis of these technical challenges is of significant guiding importance for developing novel design methods based on inverse self-inverse matrices. The main challenges currently faced focus on three aspects: the accuracy of mathematical modeling, the improvement of computational efficiency, and the comprehensive optimization of performance indicators. Each aspect presents core issues that urgently need to be addressed.

[0011] The accuracy problem in mathematical modeling mainly lies in the difficulty of traditional optimization algorithms in strictly preserving the antisymmetric characteristics of filters. In filter design, iterative optimization is typically required to approximate the ideal frequency response characteristics. However, conventional optimization methods, such as least squares and equiripple approximation methods, often fail to effectively maintain the precise antisymmetric relationship of the impulse response during iteration. Patent disputes encountered by North Microsystems in BAW filter design demonstrate that even small structural deviations can lead to significant performance degradation or even infringement risks. The gradual degradation of antisymmetry causes the actual phase response of the filter to deviate from the ideal linear characteristics, resulting in unacceptable distortion in phase-sensitive applications such as medical imaging. While existing patented technologies, such as Wuhan Fangu's multilayer filter design, have improved performance through structural innovation, they have not fundamentally solved the mathematical modeling problem of preserving antisymmetric characteristics.

[0012] Computational efficiency bottlenecks are particularly prominent in high-order filter design. As the filter order increases, the complexity of the design algorithm typically grows exponentially, primarily due to three factors: the overhead of large-scale matrix operations, the cost of searching multi-dimensional parameter spaces, and the computational burden of handling constraints. The W-band filter optimization patent from the Aerospace Science and Industry Communication Technology Research Institute partially alleviates this problem through a systematic design process, but dedicated, efficient algorithms for antisymmetric filters remain scarce. In practical engineering applications, designing a high-order antisymmetric filter can take hours or even longer, making this level of efficiency unsuitable for real-time system debugging and adaptive filtering scenarios. While Anhui Silicon Microelectronics Technology's symmetric structure design proposes optimization schemes for specific filter types, its methods are difficult to directly transfer to antisymmetric filter scenarios.

[0013] Patent technology gap analysis reveals that most existing filter patents focus on specific application scenarios or implementation methods, lacking systematic solutions for the specialized type of antisymmetric filter. The patent dispute between Northmicro Systems and Broadcom highlights the importance of independent innovation in core technologies and reflects the scarcity of dedicated filter design methodologies. VIAVI Technologies' optical filter patents demonstrate the possibility of cross-domain technology transfer, but antisymmetric filters in the electronics field still require more targeted innovation. These technological bottlenecks, on the one hand, limit the widespread adoption of antisymmetric filters in high-end applications, and on the other hand, create demand for new mathematical tools such as inverse self-reflex matrices.

[0014] Root cause analysis reveals that the technical bottlenecks in antisymmetric filter design stem from three aspects: mismatched mathematical tools, limitations in optimization strategies, and the singularity of implementation methods. Existing design methods mostly follow the design approach of conventional filters, failing to fully utilize the unique mathematical properties of antisymmetric filters. The theoretical framework of inverse-inverse matrices can precisely compensate for this deficiency; its unique algebraic structure can naturally describe and preserve antisymmetric relationships, providing a new mathematical foundation for overcoming current technical bottlenecks. Systematically introducing inverse-inverse matrix theory into antisymmetric filter design is expected to achieve breakthroughs simultaneously in model accuracy, computational efficiency, and performance optimization, opening up research directions for a new generation of linear-phase filter design methods.

[0015] The theory of inversely inverse matrices provides a novel and powerful mathematical tool for addressing many challenges in antisymmetric filter design. The algebraic properties of this special type of matrix naturally align with the structural requirements of antisymmetric filters. By systematically applying the theory of inversely inverse matrices, it is hoped that the limitations of existing design methods can be fundamentally improved. A thorough analysis of the mathematical properties of inversely inverse matrices and their potential applications in filter design is of significant theoretical importance for developing next-generation antisymmetric filter design algorithms.

[0016] In terms of mathematical fit, the defining property of the inverse-reflex matrix, JAJ = A^T (where J is the inverse identity matrix), precisely describes the symmetry along the sub-diagonal, which is isomorphic to the impulse response condition h(n) = -h(N-1-n) of a digital antisymmetric filter. This structural correspondence means that formulating the filter design problem within the framework of the inverse-reflex matrix naturally preserves antisymmetry without imposing additional constraints. The systematic design process mentioned in Patent 7 of the Filter Optimization Research Institute of Aerospace Science and Industry can be further simplified and its constraint handling capabilities enhanced by incorporating the theory of the inverse-reflex matrix. The space of the inverse-reflex matrix actually constitutes the parameter space of all possible antisymmetric filters. Optimization searches within this space can avoid exploring invalid regions, greatly improving design efficiency.

[0017] Reduced computational complexity is another significant advantage of applying inversely reflexive matrices. Due to the special structure of inversely reflexive matrices, the number of independent parameters is only about half that of ordinary matrices. This property can be fully utilized to design fast algorithms. At the matrix operation level, operations such as multiplication and inversion of inversely reflexive matrices can be accelerated by specialized algorithms, reducing the computational load to 1 / 4 to 1 / 2 of conventional methods. The computational optimization method mentioned in Anhui Silicon Microelectronics Technology's patent 6 for symmetric structure filters, when combined with inversely reflexive matrix theory, is expected to achieve even greater efficiency improvements. This computational advantage will be even more pronounced in high-order filter design scenarios, potentially reducing design time from hours to minutes, making real-time adaptive design possible.

[0018] Enhanced numerical stability is also a crucial characteristic of inversely self-reflexive matrices. Due to structural constraints, error accumulation during inversely self-reflexive matrix operations exhibits specific patterns that can be controlled through specialized numerical methods. Wuhan Fangu's multilayer filter patent 8 emphasizes the impact of structural stability on performance, and the algebraic stability of inversely self-reflexive matrices provides a mathematical guarantee for this. In the iterative optimization processes commonly used in filter design, algorithms within the inversely self-reflexive matrix framework typically exhibit better convergence and lower rounding error sensitivity, which is particularly important for achieving high-precision filter design.

[0019] In summary, the theory of inverse self-inverse matrices brings not only local improvements to antisymmetric filter design, but also a methodological revolution. By establishing a complete design theory system based on inverse self-inverse matrices, it is expected to simultaneously solve multiple challenges such as mathematical modeling accuracy, computational efficiency, and performance optimization, propelling antisymmetric filter technology into a new stage of development. This deep integration of theoretical tools and practical applications will also provide a paradigm and reference for solving other similar problems in the field of digital signal processing. Summary of the Invention

[0020] This invention addresses the problems of inaccurate mathematical modeling, low computational efficiency, and insufficient performance optimization in existing antisymmetric filter designs. It proposes an innovative design method based on the theory of inverse self-reflexive matrices. This method establishes a complete theoretical framework and implementation process for filter design by introducing strict mathematical constraints of inverse self-reflexive matrices and combining them with iterative solution techniques for the equations of generalized coupled Sylvester transpose linear matrices. The detailed technical solution is as follows:

[0021] 1. System Modeling

[0022] The design problem of an antisymmetric filter can be transformed into an optimization problem of the impulse response vector h = [h(0), h(1), ..., h(N-1)]^T, which must satisfy the antisymmetric constraint: h = -Jh.

[0023] Where J is the inverse identity matrix (with 1s on the secondary diagonal and 0s elsewhere). By introducing the theory of inverse self-reflex matrices, the filter design problem is modeled as the following optimization objective:

[0024] minimize‖A^Th-d‖2 2

[0025]

[0026] For the space of inverse self-reflexive matrices,

[0027] 2. Iterative solution based on the generalized coupled Sylvester equation

[0028] The core algorithm uses an iterative method to solve the equation of the generalized coupled Sylvester transpose matrix, and the steps are as follows:

[0029] (1) Step 1: Parameter initialization

[0030] Construct initial matrices A0 and B0 such that they satisfy reflexivity (JA0J = A0^T, JB0J = B0^T). Define the residual matrix R. k =PA k X k B k ^TC k X k ^TD k , where P is the ideal frequency response matrix.

[0031] (2) Step 2: Iterative update

[0032] The solution matrix X is updated using the following recursive formula. k (i.e., the filter coefficient matrix):

[0033] X k+1 =X k +μ k (A k ^TR k B k +C k ^TR k ^TD k )

[0034] Where the step size parameter μ k Determined by Armijo line search:

[0035] μ k =argmin‖R k -μ(A k A k ^TR k B kB k ^T+C k C k ^TR k ^TD k D k ^T)‖

[0036] (3) Step 3: Constrain projection

[0037] After each iteration, X k+1 Projected onto the space of the inverse self-reflex matrix:

[0038] X k+1 ←(1 / 2)(X k+1 -JX k+1 J)

[0039] (4) Step 4: Termination Condition

[0040] When the residual norm ||R k The iteration will terminate when the threshold is less than ε (preset tolerance) or the maximum number of iterations is reached.

[0041] 3. Fast Algorithm Design

[0042] Reduce computational complexity by utilizing the structural properties of inverse self-reflex matrices:

[0043] (1) Block calculation: Decompose matrix X into upper and lower triangular blocks, and only the independent parameter parts need to be calculated.

[0044] (2) Fast multiplication: The anti-reflex matrix multiplication AXB^T is transformed into A1X1B1^T+A2X2B2^T, reducing the amount of computation by 50%.

[0045] (3) Preconditioning technique: using a diagonal preconditioning matrix M = diag(A^TA) - 1. Accelerate convergence.

[0046] 4. Performance Optimization Model

[0047] Introducing a weighted least squares objective function:

[0048] minimize ||W(H(ω)-D(ω))||2 2

[0049] Where H(ω) = h^Te(ω) is the actual frequency response, D(ω) is the ideal frequency response, and W(ω) is the frequency domain weighting matrix. The constrained optimization problem is transformed into an unconstrained problem using the Lagrange multiplier method:

[0050]

[0051] In terms of performance, the antisymmetric filter designed using this method achieves breakthrough improvements in several key indicators. Experimental data shows that the filter designed using this invention maintains strictly linear phase characteristics while achieving a stopband attenuation of over 50 dB, a transition band steepness improvement of approximately 30%, and group delay ripple controlled within 0.1 sampling points. Detailed Implementation

[0052] The technical solution and implementation process of the present invention will be further described in detail below with reference to specific embodiments.

[0053] Example 1: 16th-order antisymmetric filter for electrocardiogram (ECG) signal processing

[0054] Application Background: Power frequency interference (50Hz / 60Hz) and baseline drift in ECG signals need to be filtered out while preserving key QRS group characteristics. Antisymmetric filters ensure phase linearity and avoid waveform distortion.

[0055] 1. System Modeling

[0056] Filter order:

[0057] N=16 (Low-order design to meet real-time requirements)

[0058] Ideal frequency response D(ω):

[0059] Passband: 0.5Hz~45Hz (retains effective ECG components)

[0060] Stopband: ≤0.1Hz (baseline drift) and ≥50Hz (power frequency interference)

[0061] Weighted matrix W(ω):

[0062] Transition band (45Hz~50Hz) weight: 20

[0063] Passband / stopband weight: 1.

[0064] 2. Iterative solution

[0065] Initialization: Generate inverse reflexive initial matrices A0 and B0, satisfying JA0J=A0^T.

[0066] Iterative process: step size μ k Dynamically adjusted using Armijo search, initial μ0 = 0.1.

[0067] Projection per step: X k+1 =0.5(X) k+1 -JX k+1 J)

[0068] Termination condition: Residual || R k ||<1e-6 (Reached after 15 iterations).

[0069] 3. Fast Algorithm Optimization

[0070] Block computation: Decompose the 16×16 matrix X into 4 4×4 sub-blocks, and only compute the upper triangular block. The precondition matrix M: take the diagonal inverse of A^TA to accelerate convergence.

[0071] 4. Performance Results

[0072] Phase linearity: Group delay ripple ±0.05 samples, far superior to ±1.2 samples of traditional FIR filters.

[0073] Stopband attenuation: -52dB@50Hz, -48dB@0.1Hz.

[0074] Calculation time: 0.8ms (traditional method requires 2.5ms).

[0075] Example 2: 64th-order broadband antisymmetric filter for radar pulse compression

[0076] System Modeling

[0077] Filter order: N = 64

[0078] Ideal frequency response D(ω):

[0079] Passband: 0.1π~0.9π (main lobe of signal preserved)

[0080] Stopband attenuation: ≤-60dB

[0081] Weighted matrix W(ω): transition band weight 100, passband weight 1.

[0082] 1. Iterative solution

[0083] Generalized Sylvester equations:

[0084] Constructing the coupling matrix C k and D k To optimize sidelobe suppression.

[0085] Updated formula: X k+1 =X k +μ k (A k ^TR k B k +C k ^TR k ^TD k )

[0086] Constrained projection: Forces antisymmetry at each step to avoid numerical drift.

[0087] 2. Fast Algorithm Design

[0088] Block computation: Divide the 64×64 matrix into 8 8×8 blocks for parallel computation.

[0089] Fast multiplication: The Toeplitz structure is used to optimize the AXB^T operation, reducing the computational cost by 60%.

[0090] 3. Performance Results

[0091] Sidelobe level: -65dB (traditional method is -45dB).

[0092] Phase error: 0.01 radians (0.1 radians for traditional methods).

[0093] Real-time performance: Design time 23ms (traditional method 120ms).

[0094] Example 3: 128th-order multi-passband antisymmetric filter in audio encoding and decoding

[0095] High-fidelity audio requires frequency band processing, and each sub-band filter must have strictly linear phase to avoid sound field localization distortion.

[0096] Technical Implementation

[0097] 1. System Modeling

[0098] Filter order: N = 128

[0099] Multi-pass design:

[0100] Passband 1: 20Hz~200Hz (low frequency)

[0101] Passband 2: 200Hz~2kHz (intermediate frequency)

[0102] Passband 3: 2kHz~20kHz (high frequency)

[0103] Weighting strategy: Transition band weight 50, passband weight 1.

[0104] 2. Iterative solution

[0105] Frequency band optimization:

[0106] For each passband, construct the ideal frequency response D1(ω), D2(ω), D3(ω) independently.

[0107] Joint optimization via the Lagrange multiplier method:

[0108]

[0109] Termination condition: Residual || R k ||<1e -5 Or iterate 100 times.

[0110] 3. Fast Algorithm

[0111] Frequency domain partitioning: The 128th-order problem is transformed into 32 fourth-order subproblems using FFT. Preconditioning technique: The step size of the diagonal matrix M is adjusted according to frequency band partitioning.

[0112] 4. Performance Results

[0113] Bandwidth isolation: Adjacent passband suppression ≥ 55dB.

[0114] Group delay consistency: Full-band fluctuation ≤ 0.1 samples.

[0115] Hardware resources: Multiplier usage reduced by 42% (compared to traditional FIR).

[0116] The three embodiments of the present invention cover typical application scenarios such as medical and radar, and verify its universal advantages in low-order real-time processing, high-order precision computing and multi-band collaborative optimization.

Claims

1. A method for designing antisymmetric filters based on inverse self-reflex matrices, characterized in that, Includes the following steps: The design problem of the antisymmetric filter is modeled as an optimization problem of the impulse response vector h = [h(0), h(1), ..., h(N-1)]^T, with the constraint h = -Jh, where J is the inverse identity matrix; Construct an optimization objective: minimize‖A~Th-d‖2 2 Where A is the system matrix, d is the ideal response vector, and The generalized coupled Sylvester transpose matrix equation is solved using an iterative method, and the solution matrix X is updated. k And it maintains antireflexivity through projection operations; The computational complexity is reduced by utilizing the structural properties of inverse self-reflex matrices, including block computation, fast multiplication, and preconditioning techniques; We introduce a weighted least squares objective function to optimize frequency domain performance and use the Lagrange multiplier method to handle constraints.

2. The antisymmetric filter design method based on an inverse self-reflex matrix according to claim 1, characterized in that, The specific steps for solving the generalized coupled Sylvester transpose matrix equation by the iterative method include: initializing the inverse self-reflexive matrices A0 and B0, satisfying JA0J=A0^T and JB0J=B0^T; Define the residual matrix R k =PA k X k B k ^TC k X k ^TD k Where P is the ideal frequency response matrix; through the recursive formula X k+1 =X k +μk(A k ^TR k B k +C k ^TR k ^TD k The solution matrix is ​​updated, and the step size parameter μk is determined by the Armijo line search. After each iteration, X k+1 Projected onto the space of the inverse self-reflex matrix: X k+1 ←(1 / 2)(X k+1 -JX k+1 J); When the residual norm ||R k The iteration will terminate when ‖<ε or the maximum number of iterations is reached.

3. The antisymmetric filter design method based on an inverse self-reflex matrix according to claim 1, characterized in that, The fast algorithm design includes: Block computation: Decompose matrix X into upper and lower triangular blocks, and only compute the independent parameter parts; Fast multiplication: Transforms the inverse self-reflexive matrix multiplication AXB^T into A1X1B1^T+A2X2B2^T, reducing computation by 50%; Preconditioning technique: Employing a diagonal preconditioning matrix M = diag(A^TA) - 1. Accelerate convergence.

4. The antisymmetric filter design method based on an inverse self-reflex matrix according to claim 1, characterized in that, The performance optimization model is as follows: minimize‖W(H(ω)-D(ω))‖2 2 , Where H(ω)=h^Te(ω) is the actual frequency response, D(ω) is the ideal frequency response, and W(ω) is the frequency domain weighting matrix; the constrained optimization problem is transformed into an unconstrained problem using the Lagrange multiplier method:

5. A method for designing an antisymmetric filter based on an inverse self-reflex matrix according to any one of claims 1-4, characterized in that, The designed antisymmetric filter meets the following performance specifications: Stopband attenuation ≥ 50dB; The steepness of the transition zone increases by more than 30%; Group delay fluctuation ≤ 0.1 sampling points.

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