Two-dimensional recursive filter design method based on P system and collaborative optimization strategy

Abstract

The invention belongs to the field of circuits and systems, and particularly relates to a two-dimensional recursive filter design method based on a P system and a collaborative optimization strategy, which comprises the following steps of: 1) establishing a model by performing mathematical analysis on a transfer function of a two-dimensional recursive filter; 2) combining a crossover mutation operator which is strong in global search capability but is easy to fall into local optimum, a cosine enhancement operator which is strong in local search capability but is slightly insufficient in population diversity mining, and a mixed membrane structure of a P system into a novel mixed operator capable of balancing convergence and diversity; 3) embedding an object updating rule represented by the operator into a P system with a mixed membrane structure to form a novel P system optimization algorithm (PHOA); 4) taking a numerical value combination of objects in the object set of the novel P system optimization algorithm as a model parameter group of a two-dimensional recursive filter, substituting the model parameter group into the two-dimensional recursive filter model, and calculating an error function between a two-dimensional recursive filter model parameter amplitude response M (omega1, omega2) and an ideal two-dimensional filter amplitude response Md (omega1, omega2); and 5) by minimizing the objective function, obtaining a parameter set and combining the parameter set with a filter model structure to form an optimal two-dimensional recursive digital filter meeting conditions. According to the scheme, the stability of the filter can be ensured, and compared with the previous scheme, the amplitude response of the two-dimensional recursive digital filter is closer to that of an ideal two-dimensional recursive digital filter.

Description

Technical Field

[0001] The present invention belongs to the field of circuits and systems, and particularly relates to a two-dimensional recursive filter design method based on a P system and a collaborative optimization strategy. Background Art

[0002] Digital signal processing (DSP) has attracted widespread attention due to its high reliability, small physical size, accuracy, flexibility, and low cost. Digital filters are a key component of DSP. Digital filters can be categorized into two types: infinite impulse response (IIR) and finite impulse response (FIR). IIR filters achieve excellent frequency selectivity and amplitude-frequency response at a relatively low order. Therefore, they are widely used in fields such as communications, control, speech, image processing, medicine, and radar. They generally offer better performance and lower computational cost than FIR filters.

[0003] There are two main approaches to IIR filter design: transformation techniques and optimization techniques. Transformation techniques convert digital filters into analog filters, but they require a significant amount of prior knowledge and are inefficient. Optimization techniques directly solve for optimal coefficients under a defined optimization criterion, offering high design efficiency and effectively addressing HR filter design issues. Traditional gradient-based design methods are prone to falling into local optimal solutions when dealing with the nonlinearities and multimodal error surfaces of IIR filters, making it difficult to find a global optimal solution.

[0004] To address this problem, researchers have employed a variety of heuristic and evolutionary optimization algorithms, such as genetic algorithms (GAs), particle swarm optimization (PSOs), and differential evolution (DEs). However, these methods still face the problem of local optimality. Therefore, this paper proposes an algorithm based on P-system and collaborative optimization strategy (PHOA), which excels in global search. Therefore, it is applied to the design of complex two-dimensional IIR filters and compared with other methods to demonstrate its advantages. Summary of the Invention

[0005] In order to solve the problems existing in the prior art, the present invention proposes an improved global optimization algorithm, which adopts a P system collaborative optimization algorithm (PHOA) that combines a hybrid membrane structure, a global search cross-mutation operator and a cosine increase operator to promote the overall anti-cheating performance of the algorithm. It is applied to the design of a two-dimensional recursive filter to obtain the optimal parameter set of the filter, so that the amplitude response M(ω1, ω2) of the two-dimensional recursive filter designed by the present invention is closer to the amplitude response M(ω1, ω2) of the ideal two-dimensional filter. d (ω1, ω2).

[0006] The two-dimensional recursive filter design based on the P system and x collaborative optimization strategy described in the present invention includes the following steps:

[0007] 1) By deriving the transfer function of the two-dimensional recursive filter, a corresponding mathematical model is established, and the design problem of the two-dimensional recursive filter is transformed into a global optimal parameter estimation problem under constraint conditions;

[0008] 2) The amplitude response diagram of the filter is divided into (2×N1 + 1)×(2×N2 + 1) grids;

[0009] 3) A P-system collaborative optimization algorithm based on a hybrid membrane structure is proposed, which integrates collaborative operators, namely: a crossover and mutation update operator with strong global search and a cosine enhancement operator to enhance the exploration ability of the algorithm;

[0010] 4) Set the initial environment and parameters of the newly formed P-system optimization algorithm: the initial solution set n (the number of solution sets in the first-level membrane n1, the number of solution sets in the second-level membrane n2, n1 + n2 = n), the number of membrane layers m, the mutation probability P i , the local exploitation probability P s , the maximum number of iterations T max ;

[0011] 5) Take the sum of squared errors (SSE) between the amplitude response M(ω1, ω2) of the two-dimensional recursive filter designed by this algorithm and the amplitude response M d (ω1, ω2) of the ideal two-dimensional filter as the objective function, as shown below:

[0012]

[0013] In the formula, J is the objective function, N1 and N2 represent the number of grids, indicating that the actual amplitude response and the expected amplitude response of the filter are composed of (2×N1 + 1)×(2×N2 + 1) grids. {a ij , b k , c k , d k , H0} represents the coefficients of the filter, that is, the parameters that the algorithm needs to obtain.

[0014] The membrane structure of the method for designing a two-dimensional recursive filter based on a P-system and a collaborative optimization strategy is as Figure 1 shown. It is a hybrid membrane structure with a three-layer membrane structure. The outermost layer is the first-level membrane (surface membrane) for global search to enhance the diversity of the solution set. The second-level membrane and the third-level membrane focus on local exploitation and the solutions are sent to the first-level membrane to balance the diversity and convergence of the solution set of the overall algorithm.

[0015] For the method for designing a two-dimensional recursive filter based on a P-system and a collaborative optimization strategy, by minimizing the objective function, the designed filter approaches the ideal filter. For design purposes, the transfer function of the designed two-dimensional recursive filter is:

[0016]

[0017] where N represents the dimension of the filter; {a ij , b k , c k , d k , H0} represent the coefficients of the filter, z1 = e -jω1 , z2 = e -jω2 , ω1, ω2 ∈ (-π, π).

[0018] The design task of a two-dimensional recursive filter is equivalent to finding a transfer function H(z1, z2) such that M(ω1, ω2) approximates the amplitude response M d (ω1, ω2) of an ideal two-dimensional recursive filter. The above mathematical process can be transformed into finding the minimum value of a function:

[0019]

[0020] where ω1 = πk1 / N1, ω2 = πk2 / N2, p is a positive integer (usually p = 2). N1, N2 represent the number of grids, indicating that the actual amplitude response and the desired amplitude response of the filter consist of (2 × N1 + 1) × (2 × N2 + 1) grids. Since we are dealing with first-order factors in the denominator, the stability condition is known:

[0021] |b k + c k | - 1 < d k < 1 - |b k - c k |, k = 1, 2,... N (4) Therefore, the design problem of a two-dimensional recursive filter can be simplified to a constrained minimization problem:

[0022]

[0023] For illustrative purposes, without loss of generality, the case we consider is as follows: N = 2. Therefore, H(z1, z2) can be written in the form of Equation (6);

[0024]

[0025] where z1 = e -jω1 , z2 = e -jω2 . So the amplitude response of the two-dimensional recursive filter can be written in the following form:

[0026]

[0027] where:

[0028]

[0029]

[0030] Therefore, the two-dimensional recursive filter is transformed into {a 01 , a 02 , a 10 , a 11 , a 12 , a 20 , a 21 , a 22 , b1, b2, c1, c2, d1, d2, H0} these 15 parameters;

[0031] Desired amplitude response of a two-dimensional circularly symmetric low-pass recursive digital filter:

[0032]

[0033] It is a filter with a narrow pass range that can effectively filter out high-frequency noise.

[0034] The two-dimensional recursive filter design method based on P system and coordinated optimization strategy is described. In the initial stage of the algorithm, the initial solution set is distributed to the primary membrane and the secondary membrane in a certain proportion for data mining. The primary membrane performs crossover mutation operator update on the candidate solution set it distributes, and the secondary membrane performs cosine spiral update on the candidate solution set it distributes. There is a certain probability P in the solution set update of each secondary membrane. s The partial solution set is sent to the third-level membrane to execute the local update rule and return; m ), each secondary membrane sends the better candidate solutions in the solution set to the primary membrane to adjust its search direction, thereby enhancing the anti-deception and convergence of the overall algorithm;

[0035] 1) Crossover operator

[0036] In the current iteration of the algebraic first-level membrane candidate solution, select X pairs of solutions (X≤n1 / 2), and perform the crossover operator on the X pairs of solutions to update them.

[0037]

[0038] In the formula, x1 and x2 are parent individuals, c is the crossover point, and x child1 and x child2 are the generated offspring;

[0039] 2) Mutation operator

[0040] The n1 candidate solution set is updated using the mutation operator. The update formula is as follows:

[0041]

[0042] Where r is the weight value, with a value range before (0, 1), and x i,j is the j-th dimensional variable of the current individual i, and x g,j is the value of the j-th dimensional variable of the global optimal solution, and x w,j is the value of the j-th dimensional variable of the local worst evaluation value. P i is the mutation probability. If the mutation probability is satisfied, a mutation operation based on the global optimal variable is performed on the variable;

[0043] 3) Update of the position of the secondary membrane helix

[0044]

[0045] Where α1 = a + (1 - a)·cos(ε·π·t / T max ), α2 = (1 - a)·(1 - cos(ε·π·t / T max ), both a and ε are coefficient constants (the best effect is achieved when a = 0.7 and ε = 1), t is the current iteration number, and T max is the maximum number of iterations. β = exp{bl}·cos(2·π·b), b is a random number within [0, 1], and l = exp{3·cos(π·(T max - t + 1) / T max )};

[0046] 4) Local update of the tertiary membrane

[0047]

[0048] Where TF = ±1, P = (1 - t / T max ) t / Tmax ;

[0049] The present invention incorporates a crossover mutation operator and a cosine enhancement strategy into the standard P-system optimization algorithm to form a PHOA optimization algorithm. The crossover mutation operator of the primary membrane and the cosine enhancement operator in the secondary membrane cooperate with each other, not only enhancing the global exploration ability in the optimization process in the early stage of the search, but also enabling the neighborhood search mechanism to enhance the local mining ability of the algorithm in the later stage of the search.

[0050] Compared with previous optimization algorithms, the filter designed by this solution has the following two advantages:

[0051] 1. The amplitude response of the two-dimensional recursive filter designed by this solution is closer to the amplitude response of the ideal two-dimensional recursive filter;

[0052] 2. The stability of the filter can be guaranteed during the design process of this solution. Description of the Drawings

[0053] Figure 1 Schematic diagram of the nested membrane structure;

[0054] Figure 2 Ideal amplitude response image;

[0055] Figure 3 Amplitude response image of GENETICA;

[0056] Figure 4 Amplitude response image of NN;

[0057] Figure 5 Amplitude response image of PSO;

[0058] Figure 6 Amplitude response image of GA;

[0059] Figure 7 Amplitude response image of PHOA;

[0060] Figure 8 Overall flowchart of PHOA;

[0061] Figure 9 : Implementation process of the design method of a two-dimensional recursive filter based on a P system and a cooperative optimization strategy;

[0062] Figure 10 : Parameter sets of various optimization algorithms and the value of the objective function J. Specific implementation manners

[0063] [[ID=4__0]]The present invention will be further described in detail below with reference to the accompanying drawings and specific implementation manners. Using the design of a two-dimensional recursive filter based on the P system optimization algorithm, the implementation scheme is as follows:

[0064] 1) Establish a model by performing mathematical analysis on the transfer function of the two-dimensional recursive filter;

[0065] 2) Set the initial environment and parameters of the PHOA optimization algorithm: topological structure, number of cycles T of P system operation max , initial population n, number of membrane layers m, mutation probability P i , local exploitation probability P s , maximum number of cycles T max ;

[0066] 3) Use the sum of squared errors (SSE) J as the objective function to find an optimal combination in a 15-dimensional search space to minimize J;

[0067] 4) Use the matlab software to obtain the optimal parameter set of the two-dimensional recursive filter using the PHOA optimization algorithm;

[0068] 5) Compare the results of the present invention's scheme with other optimization algorithms, and the results are shown in Figure 10 , Figures 2 to 7For their amplitude response images, Figure 7 For the amplitude response image of the PHOA optimization algorithm, it can be seen from the figure that the amplitude response of the two-dimensional recursive filter obtained by this scheme is closest to the ideal two-dimensional filter amplitude response, and its stopband ripple is the smallest. Figure 10 The values in it can also illustrate this scheme.

Claims

1. A design method of a two-dimensional recursive filter based on P systems and cooperative optimization strategies, characterized by including The following steps: 1) Analyze the low-pass, high-pass, band-pass, and band-stop frequency range response characteristics of a two-dimensional recursive filter. Divide the two-dimensional frequency ranges [-ω 1max , ω 1max and [-ω 2max , ω 2max into 2N2 and N2 intervals respectively, that is, a total of (2×N1 + 1)×(2×N2 + 1) grid points are formed in the two-dimensional frequency plane; 2) Combine a crossover and mutation operator with strong global search ability but prone to local optima and a cosine enhancement operator with strong local search ability but slightly insufficient exploitation of population diversity into a new hybrid operator that can balance convergence and diversity; 3) Embed the object update rule represented by this operator into a P system with a hybrid membrane structure, and combine the communication rule, selection rule, rewriting rule, and termination rule of the standard P system optimization algorithm to form a new P system optimization algorithm; Set the initial environment and parameters of the newly formed P-system optimization algorithm: the initial object set n (the number of solution sets of the first-level membrane n1, the number of solution sets of the second-level membrane n2, n1 + n2 = n), the number of membrane layers m, the mutation probability P i , the local exploitation probability P s , the maximum number of iterations T max ; 4) Use the numerical combination of objects in the new P-system optimization algorithm object set as the model parameter set of the two-dimensional recursive filter, substitute it into the two-dimensional recursive filter model, and calculate the error function between the amplitude response M(ω1, ω2) of the two-dimensional recursive filter model parameters and the amplitude response M d (ω1, ω2) of the ideal two-dimensional filter; 5) Through the search, update, and multiple selections of the new P system optimization algorithm, identify the parameter combination that can minimize the error function; according to the model structure and parameter selection of the two-dimensional recursive filter, select the corresponding values of electronic components to fabricate the circuit of the filter.

2. The design method of a two-dimensional recursive filter based on the P system and the collaborative optimization strategy according to claim 1, characterized in that The transfer function of the two-dimensional recursive filter is as shown in Equation (1), and the error function in step 4), also known as the objective function, is as shown in Equation (2): where N is the dimension of the filter, b k , c k , d k are the coefficients of the filter, z1 and z2 can be transformed into indeterminates through Laplace transform, and ω1, ω2 ∈ [-π, π], z1 = e -jw1 , z2 = e -jw2 ; thus, the optimal design problem of the two-dimensional recursive filter is transformed into a parameter estimation problem for a complex model; by finding a set of parameters in the search space, this set of parameters can be expressed in the form of M(ω1, ω2) = H(e -jw1 , e -jw2 ), and making the frequency amplitude response of the two-dimensional recursive filter as close as possible to the frequency amplitude response M d (ω1, ω2); where p is a positive integer, and k1, k2, N1, N2 are integers, and (a ij , b k , c k , d k , H0) are the parameters to be identified.

3. A method for designing a two-dimensional recursive filter based on a P system and a cooperative optimization strategy according to claim 1, characterized in that The hybrid membrane structure in step 3), which is a surface membrane (primary membrane) with M s_level number of secondary membranes, and each secondary membrane is equipped with a basic membrane (tertiary membrane); at the initial stage of the algorithm, the initial solution set is allocated to the primary membrane and the secondary membranes according to a certain ratio (n1, n2) for data mining. There is a certain probability P s that a part of the solution set is sent to the tertiary membrane for local mining and returned; the algorithm designs a communication rule: after a certain number of iterations (I m ), each secondary membrane sends the better candidate solutions in the solution set to the primary membrane to adjust its search direction, enhancing the anti-deception and convergence of the overall algorithm.

4. A two-dimensional recursive filter design method based on the P system and collaborative optimization strategy according to claim 1, characterized in that For the crossover and mutation operator in step 2), select two solutions x1 and x2 (a total of floor(n1 / 2) pairs) from the first-level membrane. The crossover operator it performs is as shown in Equation (3), and the mutation operator for the candidate solution set in the first-level membrane is as shown in Equation (4): where x1 and x2 are parent individuals, c is the crossover point, x child1 and x child2 are the generated offspring; where r is the weight value, with a value range before (0, 1), and x i,j is the j-th dimensional variable of the current individual i, and x g,j is the value of the j-th dimensional variable of the global optimal solution, and x w,j is the value of the j-th dimensional variable of the local worst evaluation value. P i is the mutation probability. If the mutation probability is satisfied, a mutation operation based on the global optimal variable is performed on the variable.

5. A two-dimensional recursive filter design method based on the P system and cooperative optimization strategy according to claim 1, characterized in that For the cosine enhancement operator in step 2), this operator is mainly divided into two parts: the helix position update in the second-level membrane (the cosine update is designed for the individual movement control parameters α1 and ɑ2) is as shown in Equation (5), and the local exploitation in the third-level membrane (optimization near the individual solution) is as shown in Equation (6): Where ɑ1 = a + (1 - a)·cos(ε·π·t / T max ), α2 = (1 - a)·(1 - cos(ε·π·t / T max ), both a and ε are coefficient constants (the best effects are achieved when a = 0.7 and ε = 1), t is the current iteration number, and T max is the maximum number of iterations, β = exp{bl}·cos(2·π·b), b is a random number within [0, 1], and l = exp{3·cos(π·(T max - t + 1) / T max )}; where TF = ±1, P = (1 - t / T max ) t / Tmax .

6. A method for designing a two-dimensional recursive filter based on a P system and a collaborative optimization strategy according to claim 1, characterized in that Use the optimal parameter set obtained in step 5) as the parameters of the two-dimensional recursive filter, and implement the system of the two-dimensional filter according to the parameters, thereby completing the design of the two-dimensional recursive digital filter.

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