Two-dimensional recursive filter design method based on improved particle swarm optimization algorithm

Abstract

The invention belongs to the field of circuits and systems, and particularly relates to a two-dimensional recursive filter design based on an improved particle swarm optimization algorithm. Comprising the following steps: 1) performing mathematical analysis on a transfer function of a two-dimensional recursive filter to establish a model; 2) finding out an optimal parameter set by using an improved particle swarm optimization algorithm, and enabling the sum of squares of errors between the amplitude response of the designed filter and the amplitude response of an ideal filter to be minimum; 3) integrating a dynamic inertia weight strategy, a dynamic learning factor and a prying strategy into a particle swarm optimization algorithm; 4) setting algorithm operation parameters; 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 design method of a two-dimensional recursive filter based on an improved particle swarm optimization algorithm. Background Art

[0002] Digital signal processing (DSP) has received extensive attention due to its high reliability, low physical size, accuracy, flexibility, and low cost. Among them, digital filters are key components. Digital filters can be divided into two types: infinite impulse response (IIR) and finite impulse response (FIR). IIR filters achieve good frequency selectivity and amplitude-frequency response with a lower order, so they are widely used in fields such as communication, control, speech, image processing, medicine, and radar, and generally provide better performance and lower computational cost than FIR filters.

[0003] There are mainly two types of methods for designing IIR filters: transformation techniques and optimization techniques. Transformation techniques design by converting digital filters into analog filters, but they require a large amount of prior knowledge and have low design efficiency; optimization techniques directly solve for the optimal coefficients under a determined optimization criterion, have high design efficiency, and can effectively solve the design problem of IIR filters. Traditional gradient-based design methods are prone to falling into local optimal solutions and cannot find the global optimal solution when dealing with the non-linear and multimodal error surfaces of IIR filters.

[0004] To solve this problem, researchers have adopted various heuristic and evolutionary optimization algorithms, such as genetic algorithms (GA), particle swarm optimization (PSO), and differential evolution algorithms (DE). However, these methods still face the problem of local optimality. Therefore, the improved particle swarm optimization algorithm proposed in this paper performs excellently in global search. Thus, 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] To solve the problems existing in the prior art, the present invention proposes an improved particle swarm optimization algorithm, which adopts a dynamic inertia weight strategy, dynamic learning factors, and a somersault strategy to improve the population diversity and search speed of the particle swarm optimization algorithm. Applying it to the design of a two-dimensional recursive filter, the optimal parameter set of the filter is obtained, 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 d (ω1, ω2) of the ideal two-dimensional filter.

[0006] The design of a two-dimensional recursive filter based on the improved particle swarm optimization algorithm described in the present invention includes the following steps:

[0007] 1) Analyze the ideal desired frequency amplitude response \(M\) of the two-dimensional recursive filter to be designed on the two-dimensional plane \((\omega_1, \omega_2)\). For the model structure of the two-dimensional recursive filter, assign random initial values to the parameters of the model. d (\omega_1, \omega_2)\), and for the model structure of the two-dimensional recursive filter, assign random initial values to the parameters of the model.

[0008] 2) Calculate the error between the amplitude response \(M(\omega_1, \omega_2)\) of the grid point frequencies on the two-dimensional plane under the model parameters and the desired response \(M\) d (\omega_1, \omega_2)\);

[0009] 3) Use the improved particle swarm optimization algorithm to search for the optimal parameters of the two-dimensional recursive filter; set the initial coefficients of the algorithm: the maximum number of iterations \(T\), the initial population \(N\), the dynamic inertia weight, and the adaptive learning factor;

[0010] 4) Combine the \(p\)-th power sum of the errors calculated in step 2) as the objective function, and run the optimization algorithm to obtain the model parameter group that can minimize the objective function;

[0011] In the design of the two-dimensional recursive filter based on the improved particle swarm optimization algorithm, the sum of the squared errors between the amplitude response \(M(\omega_1, \omega_2)\) of the grid point frequencies on the two-dimensional plane and the desired response \(M\) d (\omega_1, \omega_2)\) is used as the objective function, as follows:

[0012]

[0013] where \(J\) is the objective function, \(N_1\), \(N_2\) represent the number of grids, indicating that the actual amplitude response and the desired amplitude response of the filter are composed of \((2\times N_1 + 1)\times(2\times N_2 + 1)\) grids. \(\{a\) ij , b\) k , c\) k , d\) k , H_0\}\) represents the coefficients of the filter, that is, the parameters to be identified by the algorithm. \(\omega_1=\frac{\pi k_1}{N_1}\), \(\omega_2=\frac{\pi k_2}{N_2}\), \(p\) is a positive integer (usually \(p = 2\)).

[0014] In the design of the two-dimensional recursive filter based on the improved particle swarm optimization algorithm, the transfer function of the designed two-dimensional recursive filter is:

[0015]

[0016] where \(N\) represents the dimension of the filter; \(\omega_1, \omega_2\in(-\pi, \pi)\). The design task of the two-dimensional recursive filter is equivalent to finding a transfer function \(H(z_1, z_2)\) such that \(M(\omega_1, \omega_2)\) approximates the amplitude response \(M\) of the ideal two-dimensional recursive filter d (\omega_1, \omega_2)\). Since we are dealing with first-order factors in the denominator, the stability condition is known:

[0017] |b k +c k |-1 < d k <1 - |b k -c k |, k = 1, 2, ... N

[0018] Therefore, the design problem of the two - dimensional recursive filter can be transformed into a constrained minimization problem:

[0019]

[0020] For the design of the two - dimensional recursive filter based on the improved particle swarm optimization algorithm described above, the algorithm optimization process is as follows:

[0021] The algorithm starts the optimization process by initializing the population. The individuals within the population are randomly initialized within the search space. The initialization formula for each individual is as follows:

[0022] X i = Lb + rand(Ub - Lb) i = 1, 2... n

[0023] where X i represents the initial solution of the i - th particle within the population, Ub and Lb represent the maximum and minimum values of the individuals within the search space, rand is a random vector within the range [0, 1], and n represents the population size.

[0024] In the particle swarm optimization algorithm, the convergence and exploration ability of the algorithm can be controlled by changing the inertia weight. The value of the inertia weight determines the current velocity of the particle. The larger the inertia weight, the faster the particle's velocity, which helps to improve the exploration ability of the algorithm. The smaller the inertia weight, the slower the particle's velocity, which helps to improve the local search ability of the algorithm. By introducing a dynamic inertia weight, the algorithm has a stronger global search ability in the early stage and a stronger local search ability in the later stage. The formula for the dynamic inertia weight is as follows:

[0025]

[0026] where t is the current iteration number, and T is the maximum iteration number of the algorithm;

[0027] The learning factors in the particle swarm algorithm are important parameters that guide the update of particle velocity, and are divided into individual learning factors and population learning factors. These two parameters determine the degree to which particles rely on individual experience and population experience when updating their velocities. In order to ensure the diversity of particles in the initial stage of the search and converge to the global optimal solution as soon as possible in the later stage, by analyzing the influence of the change of learning factors, the sine and cosine functions are used to dynamically adjust the parameters c1 and c2 to better balance global search and local search. The formula for the dynamic learning factor is as follows:

[0028]

[0029] Among them, c 1-start , c 1-end , c 2-start , c 2-end are the maximum and minimum values of the learning factors c1 and c2.

[0030] By introducing the somersault strategy, each particle takes the currently discovered optimal solution as a fulcrum to somersault to a new position. By defining the range of the somersault, each individual can move to any position between the symmetric positions of the current position. As the number of iterations increases, the moving range will gradually shrink, exploring as many regions of positions as possible without affecting the convergence speed of the algorithm. The formula for the somersault strategy is as follows:

[0031] X i (t + 1) = X i (t) + S·(r1·X best - r2·X i (t))

[0032] Among them, S is the coefficient that determines the somersault, r1 and r2 are random numbers in [0, 1], X i (t) is the current particle position, and X best is the currently discovered optimal position.

[0033] Compared with previous optimization algorithms, the filter designed in this scheme has the following two advantages:

[0034] 1. The amplitude response of the two-dimensional recursive filter designed in this scheme is closer to the amplitude response of the ideal two-dimensional recursive filter;

[0035] 2. The stability of the filter can be guaranteed during the design process of this scheme. Description of the Drawings

[0036] Figure 1 is the abstract drawing;

[0037] Figure 2 is the ideal amplitude response image;

[0038] Figure 3The amplitude response image for NN;

[0039] Figure 4 The amplitude response image for PSO;

[0040] Figure 5 The amplitude response image for GA;

[0041] Figure 6 The amplitude response image for the present invention;

[0042] Figure 7 The comparison chart of the identification results of different method parameters of the present invention;

[0043] Figure 8 The specific flowchart of the parameter identification of the two-dimensional recursive filter for the improved particle swarm optimization algorithm of the present invention; Specific implementation manner

[0044] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific implementation manners. The design method of a two-dimensional recursive filter based on an improved particle swarm optimization algorithm is implemented as follows:

[0045] 1) Establish a model by performing mathematical analysis on the transfer function of the two-dimensional recursive filter;

[0046] 2) Set the initial environment and parameters of the improved particle swarm optimization algorithm: the maximum number of iterations T, the initial population N, the dynamic inertia weight, and the adaptive learning factor;

[0047] 3) Take the sum of squared errors (SSE) J as the objective function and find an optimal combination in a 15-dimensional search space to minimize J;

[0048] 4) Use the matlab software to obtain the optimal parameter set of the two-dimensional recursive filter by using the improved particle swarm optimization algorithm;

[0049] 5) Compare the results of the present invention's solution with other optimization algorithms. The results are shown in Figure 6 , Figures 1 to 5 are their amplitude response images, Figure 5 is the amplitude response image of the improved particle swarm optimization algorithm. It can be seen from the figure that the amplitude response of the two-dimensional recursive filter obtained by the present solution is closest to the ideal amplitude response of the two-dimensional filter.

Claims

1. A design method of a two-dimensional recursive filter based on an improved particle swarm optimization algorithm, characterized by including The following steps: 1) Analyze the ideal desired frequency amplitude response M of the two-dimensional recursive filter to be designed on the two-dimensional plane (ω1, ω2). d For the model structure of the two-dimensional recursive filter, assign random initial values to the parameters of the model. 2) Calculate the error between the amplitude response M(ω1, ω2) of the grid point frequency on the two-dimensional plane under the calculation model parameters and the expected response M d (ω1, ω2). 3) Use the improved particle swarm optimization algorithm to search for the optimal parameters of the two-dimensional recursive filter; set the initial coefficients of the algorithm: the maximum number of iterations T, the initial population N, the dynamic inertia weight, and the adaptive learning factor; 4) Combine the p-th power sum of the errors calculated in step 2) as the objective function, and run the optimization algorithm to obtain the model parameter set that can minimize the objective function; 5) Use the optimal model parameter values obtained by the optimization algorithm as the basis for selecting the hardware parameters of the resistance, capacitance, and inductance of the two-dimensional recursive filter, and complete the construction of the circuit of the two-dimensional filter.

2. A two-dimensional recursive filter design method based on an improved particle swarm optimization algorithm according to claim 1, wherein the method for dividing the two-dimensional grid is as follows: divide the frequency range [-ω 1,L , ω 1,L of ω1 into 2N1 parts to form (ω 1，L / N1)*{-N1, -(N1 - 1),..., -2, -1, 0, 1, 2, (N1 - 1), N1}, a total of (2×N1 + 1) sampling frequency points of ω1; divide the frequency range [-ω 2,L , ω 2,L of ω1 into 2N2 parts to form (ω 2，L / N2)*{-N2, -(N 2-1 ),..., -2, -1, 0, 1, 2, (N2 - 1), N2}, a total of (2×N2 + 1) sampling frequency points of ω2; when testing the filter on the two-dimensional frequency plane, the frequency sampling points on the two dimensions are orthogonal.

3. According to the method for designing a two-dimensional recursive filter based on the improved particle swarm optimization algorithm described in claim 1, during the algorithm iteration process, a dynamic inertia weight strategy, a dynamic learning factor, and a somersault strategy are introduced, which improve the population diversity and search speed of the particle swarm optimization algorithm.

4. According to the method for designing a two-dimensional recursive filter based on the improved particle swarm optimization algorithm described in claim 1, the transfer function of the designed two-dimensional recursive filter is: where N represents the dimension of the filter; {a ij , b k , c k , d k , H0} represents the coefficients of the filter, z1 = e -jω1 , z2 = e -jω2 , ω1, ω2 ∈ (-π, π); the design task of the 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 the ideal two-dimensional recursive filter; that is, minimizing the error between the ideal amplitude response M d (ω1, ω2) and the amplitude response M(ω1, ω2) to be designed; ω1 = (πk1 / N1), ω2 = (πk2 / N2) (3) where J is the objective function, which is the sum of errors between the actual amplitude response and the desired amplitude response of the designed filter at (2×N1 + 1)×(2×N2 + 1) grid frequency points, {a ij , b k , c k , d k , H0} represents the coefficients of the filter, that is, the parameters to be identified by the algorithm.

5. According to the method for designing a two-dimensional recursive filter based on the improved particle swarm optimization algorithm described in claim 1, during the algorithm iteration process, the formula for introducing the dynamic inertia weight is as follows: where t is the current iteration number, and T is the maximum iteration number of the algorithm.

6. According to the method for designing a two-dimensional recursive filter based on the improved particle swarm optimization algorithm described in claim 1, during the search process of the particle swarm algorithm, the learning factor is an important parameter guiding the update of the particle velocity, which is divided into an individual learning factor and a swarm learning factor. They determine the degree to which the particle depends on individual experience and swarm experience when updating the velocity. By introducing a dynamic learning factor, the diversity of particles is guaranteed in the initial stage of the search, and the convergence to the global optimal solution is accelerated in the later stage. The formula for the dynamic learning factor is as follows: where c 1-start and c 1-end and c 2-start and c 2-end are the maximum and minimum values of the learning factors c1 and c2.

7. According to the method for designing a two-dimensional recursive filter based on the improved particle swarm optimization algorithm described in claim 1, by introducing the somersault strategy, each particle somersaults to a new position with the currently found optimal solution as the fulcrum. By defining the range of the somersault, each individual can move to any position between the symmetric positions of the current position. As the number of iterations increases, the moving range will gradually shrink, exploring as many regions of positions as possible without affecting the algorithm convergence speed. The formula for the somersault strategy is as follows: X i (t + 1)= X i (t)+ S·(r1·X best - r2·X i (t)) (7) where S is the coefficient determining the somersault, r1 and r2 are random numbers in [0, 1], and X i (t) is the current particle position, and X best is the currently discovered optimal position.

8. According to the method for designing a two-dimensional recursive filter based on the improved particle swarm optimization algorithm described in claim 1, the optimal parameter set obtained by this algorithm is used as the parameters of the two-dimensional recursive filter, and the system implementation of the two-dimensional filter is carried out based on the parameters, thereby completing the design of the two-dimensional recursive digital filter.

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