CN103888104B - Method and system for designing FIR digital filter - Google Patents

Abstract

Provided are a method and system for designing an FIR digital filter. The method comprises the steps that modeling is carried out on the FIR digital filter according to a filter design need to obtain a mathematic model of the filter; the parameter limiting conditions of the filter are detailed according to the mathematic model to obtain a condition weighting model; the condition weighting model is solved through a genetic algorithm and a least square method to obtain an optimal filter coefficient; the FIR digital filter meeting actual filter needs is acquired according to the filter coefficient. According to the technical scheme, filter design with different properties can be obtained, the optimal filter design coefficient is obtained, and the optimal FIR digital filter is designed.

Description

FIR digital filter design method and system

Technical Field

The invention relates to the field of digital filter design, in particular to a method and a system for designing an FIR digital filter.

Background

In a wireless communication system, a received signal of a terminal is generally mixed with noise and some unwanted signal components, and needs to be filtered by a filter. Therefore, the filter has very important application value in the technical field of modern signal processing and electronic application. The traditional analog filter has the defects of complex design, huge structure, more original components and the like. With the wide application of the large-scale integrated circuit technology in the field of filter design, the development and application of digital filters become mainstream. Compared with an analog filter, the digital filter has the advantages of high precision, good flexibility, convenience for large-scale integration and the like, and the current research focus is mainly on the optimization design of the digital filter.

Based on the actual performance of the digital filter, the digital filter plays a very important role in the fields of signal processing, geological exploration, digital communication, image transmission, adaptive control and the like, and the optimized design of the digital filter has great practical significance. Since an ideal filter is non-causal, it is physically impractical for real-time signal processing applications, and in practical designs, designers typically design causal filters to approximate the ideal frequency response characteristics.

At present, a Finite Impulse Response (FIR) filter is mainly designed by a window function method, a frequency sampling method, a Chebyshev (Chebyshev) approximation method, and the like. The window function method is to use a window function to cut off unit impulse response of an ideal filter infinite in time, so that the designed filter approaches the performance requirement of the ideal filter. However, the window function method is highly required for designing the filter according to the type of the window function and is inefficient. When designing a FIR filter using a frequency sampling method, the desired frequency response is typically divided into frequency bins at equal intervals. Meanwhile, in order to weaken the side lobe, the frequency section of the transition band of the filter needs to be optimized. Frequency sampling is performed by sampling at frequency and approximating the ideal filter frequency response at the sampling points by interpolation. The chebyshev approximation is designed to uniformly distribute the weighted approximation error between the ideal frequency response and the actual frequency response to the pass band and stop band of the filter and minimize the maximum error of the filter.

Either FIR filter design approach is an approximation to an ideal filter. To more conveniently characterize the approximation of a designed filter to an ideal filter, some scholars have developed a mean square error minimization criterion, which is essential to minimize the error energy of the actual resulting filter frequency response from the ideal filter frequency response.

The prior art solutions for FIR digital filter optimization design based on the mean square error minimization criterion are as follows:

firstly, a transition zone sample value is determined through a genetic algorithm, and a traditional table look-up method is replaced, so that an optimal solution is obtained at a higher speed. And secondly, establishing an optimization model of the window function weight according to the design requirement of expected frequency characteristics, and solving an optimized value through a fast self-adaptive genetic algorithm.

With respect to the first scheme, it is desirable to obtain an optimized solution at a faster speed by replacing the original table look-up method with a genetic algorithm. However, the genetic algorithm cannot guarantee that the solution is the optimal solution, the algorithm implementation has certain complexity, the solving time is long, the engineering design work cannot be directly guided, the finally obtained solution is not necessarily the optimal solution, and the precision is low. The second scheme is mainly characterized in that a genetic algorithm is used for solving parameters of the window function, and problems in the filter design method based on the window function exist in research, including selection of the window function, long time for optimizing and setting the parameters and the like.

In summary, the existing FIR filter design technology cannot automatically adjust the performance requirements of the main lobe and the side lobe according to the design requirements of the actual filter, is difficult to obtain a filter meeting the actual requirements, and is difficult to obtain an optimal solution in the face of minimization problem, so that a corresponding optimal filter design coefficient cannot be given.

Disclosure of Invention

Therefore, it is necessary to provide a method and a system for designing an FIR digital filter, which can automatically adjust the performance requirements of the main lobe and the side lobe according to the actual design requirements of the filter to obtain a filter meeting the actual requirements, and can quickly obtain an optimal solution, thereby providing a corresponding optimal filter design coefficient.

A FIR digital filter design method includes the following steps:

modeling the FIR digital filter according to the design requirement of the filter to obtain a mathematical model of the filter;

refining the parameter limiting conditions of the filter according to the mathematical model to obtain a condition weighting model;

solving the conditional weighting model by using a genetic algorithm and a least square method to obtain an optimal filter coefficient;

and obtaining the FIR digital filter meeting the actual filtering requirement according to the filter coefficient.

A FIR digital filter design system comprising:

the mathematical modeling module is used for modeling the FIR digital filter according to the design requirement of the filter to obtain a mathematical model of the filter;

the condition refining module is used for refining the parameter limiting conditions of the filter according to the mathematical model to obtain a condition weighting model;

the model solving module is used for solving the conditional weighting model by utilizing a genetic algorithm and a least square method to obtain an optimal filter coefficient;

and the filter design module is used for acquiring the FIR digital filter meeting the actual filtering requirement according to the filter coefficient.

According to the FIR digital filter design method and system, the side lobes and the main lobe can be subjected to weighted summation according to different actual requirements, so that filter designs with different performances can be obtained, the genetic algorithm and the least square method are combined in the algorithm solving process, the optimal filter coefficient can be globally and quickly searched, the optimal filter meeting the actual requirements is obtained, the convergence speed of the algorithm is accelerated, the searching precision of the algorithm is improved, the optimal filter design coefficient can be obtained, and the optimal FIR digital filter is designed.

Drawings

FIG. 1 is a flow chart of a method for designing an FIR digital filter according to the present invention;

FIG. 2 is a flow chart of a hybrid genetic algorithm;

FIG. 3 is a schematic structural diagram of a FIR digital filter design system according to the present invention.

Detailed Description

The following describes in detail embodiments of the FIR digital filter design method and system of the present invention with reference to the accompanying drawings.

Referring to fig. 1, fig. 1 is a flow chart of a method for designing an FIR digital filter according to the present invention; the method mainly comprises the following steps:

and step S10, modeling the FIR digital filter according to the design requirement of the filter to obtain a mathematical model of the filter.

In the step, the side lobe and the main lobe are weighted and summed mainly according to different actual requirements, so that filter designs with different performances can be obtained.

Therefore, in the design of the invention, the filter is designed in a weighting mode, so that the performance requirements of the main lobe and the side lobe can be automatically adjusted according to the actual design requirements, and the filter which meets the actual requirements can be obtained.

In an embodiment, the modeling method of step S10 may specifically be as follows:

firstly, establishing a weighted filter design model;wherein, using H_d(e^jw) Representing the ideal filter frequency response, H (e)^jw) Representing the actually obtained filter frequency response, denoted E (E)^jw) Representing the frequency response error, the specific form of the filter design model is as follows:

E(e^jw)=H_d(e^jw)-H(e^jw). （1）

calculating the mean square error e²，e²Is expressed as

$e^2=\frac{1}{2\mathrm{\π}}{\∫}_{-\mathrm{\π}}^{\mathrm{\π}}{\left|E\left(e^{\mathrm{jw}}\right)\right|}^2\mathrm{dw}=\frac{1}{2\mathrm{\π}}{\∫}_{-\mathrm{\π}}^{\mathrm{\π}}{|H_d\left(e^{\mathrm{jw}}\right)-H\left(e^{\mathrm{jw}}\right)|}^2\mathrm{dw}.---\left(2\right)$

The two formulas are processed by Fast Fourier Transform (FFT), and the result is obtained

$\left\{\begin{array}{c}{H}_d\left(e^{\mathrm{jw}}\right)=\underset{n=\∞}{\overset{\∞}{\mathrm{\Σ}}}{h}_d\left(n\right)e^{-\mathrm{jwn}}\\ H\left(e^{\mathrm{jw}}\right)=\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}h\left(n\right)e^{-\mathrm{jwn}}\end{array}\right..---\left(3\right)$

Substituting the above equation (3) into the mean square error equation (2), the expression is as follows:

$\begin{array}{c}E\left(e^{\mathrm{jw}}\right)=H_d\left(e^{\mathrm{jw}}\right)-H\left(e^{\mathrm{jw}}\right)\\ =\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}[h_d\left(n\right)-h\left(n\right)]e^{-\mathrm{jwn}}+\underset{n=N}{\overset{+\∞}{\mathrm{\Σ}}}{h}_d\left(n\right)e^{-\mathrm{jwn}.}\end{array}---\left(4\right)$

according to the formula of Passewa, can obtain

$\begin{array}{c}{e}^2=\frac{1}{2\mathrm{\π}}{\∫}_{-\mathrm{\π}}^{\mathrm{\π}}{\left|E\left(e^{\mathrm{jw}}\right)\right|}^2\mathrm{dw}\\ =\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}[h_d\left(n\right)-h\left(n\right)]e^{-\mathrm{jwn}}+\underset{n=N}{\overset{+\∞}{\mathrm{\Σ}}}{h}_d\left(n\right)e^{-\mathrm{jwn}.}\end{array}---\left(5\right)$

In the above equation, the second left sum of the equation is constant regardless of the design value h (N), N =0,1, …, N-1; to make e²At a minimum, only the first sum is required to be minimal, where only the first part of the right end of the equation needs to be considered.

For obtaining the filter coefficients, only the values of [ - π, π]Up taking M frequency sampling points w_k(k =0,1, …, M-1), resulting in a set of h (N), N =0,1, …, N-1, such that

$\underset{k=0}{\overset{M-1}{\mathrm{\Σ}}}{|\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}h\left(n\right)e^{\mathrm{jn}{w}_k-H_d\left(w_k\right)}}^2---\left(6\right)$

And the minimum value is obtained as the minimum filter design parameter value.

Further, for simple calculation and easy operation, the solution process may be converted into operations of matrix and vector, specifically as follows:

order to

$B={\left[\begin{array}{cccc}{e}^{-j{w}_1}& e^{-2j{w}_1}& \·\·\·& e^{-\mathrm{jn}{w}_1}\\ \·\·\·& \·\·\·& \·\·\·& \\ e^{-j{w}_i}& e^{-2j{w}_i}& \·\·\·& e^{-\mathrm{jn}{w}_i}\\ \·\·\·& \·\·\·& \·\·\·& \\ e^{-j{w}_M}& e^{-2j{w}_M}& \·\·\·& e^{-\mathrm{jn}{w}_M}\end{array}\right]}_{M\×N}---\left(7\right)$

Wherein ξ = [ h (0), h (1), … h (N-1)]^T，H=[H_d(w₁),H_d(w₂),…,H_d(w_M)]Then the above weighted filter design model is simplified to

$\min {\left|\right|B\·\mathrm{\ξ}-H\left|\right|}_2^2---\left(8\right)$

In the formula,represents the two norms for which x = (x)₁,x₂,…,x_N) Then, then ${\left|\right|x\left|\right|}_2^2=x_1^2+x_2^2+\·\·\·+x_N^2.$

It should be noted that the modeling method and the mathematical model thereof are based on a preferred method for establishing a weighted filter design model and a model representation form thereof, and the actual application is not limited to the technical solution set forth in the preferred embodiment.

And step S20, refining the parameter limiting conditions of the filter according to the mathematical model to obtain a condition weighting model.

In the step, the main expression form of the actual requirement for the filter design is the attention of the main lobe and the side lobe, and the parameter limitation condition of the filter is refined according to the actual requirement, so that the weighting model under the limitation condition is obtained.

In one embodiment, to improve FIR filter performance, the main lobe width of the window function is required to be as narrow as possible to obtain a narrower transition band; the relative value of the side lobe height is as small as possible, the quantity is as small as possible, so as to obtain the characteristics of small passband ripple, large stopband attenuation and stability in both the passband and the stopband, thus the actual frequency response of the filter can better approach the ideal frequency response.

The specific processing method is that firstly, a related weight function matrix is set up, and the specific format is as follows

Wherein,at this time, the weighted model of equation (8) is rewritten into

$\min {F\left|\right|A(B\·\mathrm{\ξ}-H)\left|\right|}_2^2---\left(10\right)$

In the formula,representing a two-norm with F being the objective function.

It should also be noted that the above refinement of the parameter limitation condition of the filter is based on the preferred embodiment, and the practical application is not limited to the technical solution set forth in the above preferred embodiment.

And step S30, solving the conditional weighting model by using a genetic algorithm and a least square method to obtain an optimal filter coefficient.

In this step, the genetic algorithm in the evolutionary algorithm is combined with the conjugate gradient algorithm, so that the optimal filter coefficient can be searched globally and rapidly. The algorithm fully utilizes the global search capability of the genetic algorithm and the rapid local search capability of the conjugate gradient method, accelerates the convergence speed of the genetic algorithm and improves the search precision of the genetic algorithm. Thus, optimal filter design coefficients can be obtained.

Specifically, in order to rapidly solve the optimal solution of the model, the genetic algorithm in the evolutionary algorithm and the traditional classical conjugate gradient algorithm are respectively improved and combined for application, and the hybrid genetic algorithm is abbreviated as the hybrid genetic algorithm. The advantages of the algorithm are obtained through corresponding technical means, and the global search capability of the genetic algorithm and the rapid convergence characteristic of the conjugate gradient method are utilized to solve. The convergence rate of the genetic algorithm is improved through the conjugate gradient algorithm, so that the genetic algorithm can be converged to an accurate solution more quickly, the conjugate gradient method is helped to jump out a local optimal solution through the genetic algorithm, the conjugate gradient method is embedded into the genetic algorithm, all individuals in a genetic algorithm population do not need to be executed with the conjugate gradient method, and only the conjugate gradient method is required to be executed on the individuals in the center of the genetic algorithm population.

In one embodiment, the method for obtaining filter coefficients by solving the model specifically comprises the following steps:

establishing a genetic algorithm model according to the weighting model; specifically, the population number is m, and the individuals are respectively a_i∈Rⁿ,i=1,2,…,m。

Setting the population center of the model as an initial search point processed by a conjugate gradient algorithm; wherein, the center is the average value of population individuals, and the expression is as follows:m is the number of the population, and the individuals are respectively a_i∈RⁿI =1,2, …, m; due to the conjugate gradient method having twoSub-terminability, i.e., for a quadratic function, the algorithm may terminate after n iterations.

Executing n steps conjugate gradient algorithm processing to obtain a result a⁽ⁿ⁾Calculating a⁽ⁿ⁾According to a⁽ⁿ⁾Generating an individual, adding the individual into the population, and replacing the individual with the minimum fitness value; in particular, for non-quadratic functions, n steps may be performed with good results, and therefore, a₀As an initial search point of the conjugate gradient method and executing n steps of the conjugate gradient method, and the obtained result is a⁽ⁿ⁾. Calculating a⁽ⁿ⁾According to a⁽ⁿ⁾An individual is generated and then added to the population to replace the individual with the smallest fitness value, thereby keeping the population size at m.

And circularly executing the next iteration processing of the genetic algorithm, the conjugate gradient algorithm and the processing steps, namely circularly executing the hybrid genetic algorithm until the optimal solution of the weighting model is solved, and setting the optimal solution as a filter coefficient.

For better clarity of the algorithm process of step S30, an example of the algorithm is described below with reference to the drawings.

Referring to fig. 2, fig. 2 is a flowchart of a hybrid genetic algorithm, which mainly includes the following steps:

step a, initializing to generate an initial population; wherein, the population scale is set as m, and the cross probability is set as p_cThe mutation probability is p_mSetting the iteration number as n;

b, calculating an individual fitness function F, namely an objective function;

c, executing a crossover operator and a mutation operator;

step d, calculating the center a of the seed group₀；

Step e, initializing the search point of the conjugate gradient algorithm, namely a₀Setting an initial search point processed by a conjugate gradient algorithm; where k =1 is set, and the number of iterations is setn₀Setting precision requirements, and executing the following algorithm processing:

step f, ifStopping the calculation; otherwise put it intoWherein, ${\mathrm{\β}}_{k-1}=\left\{\begin{array}{c}0,k=1\\ \frac{{\left|\right|\▿F\left(a^{\left(k\right)}\right)\left|\right|}^2}{{\left|\right|\▿F\left(a^{(k-1)}\right)\left|\right|}^2},k>1;\end{array}\right.$

step g, performing one-dimensional search, and solving a one-dimensional problem: max phi (a) = F (a)^(k)+αd^(k)) To obtain α_kPut a at^(k)=a^(k)+a_kd^(k)(ii) a Let k = k +1, go to execute step f, and record the calculation result as a⁽ⁿ⁾；

Step h, calculating a⁽ⁿ⁾According to a⁽ⁿ⁾Generating an individual, adding the individual into the population, replacing the individual with the minimum fitness in the population, and executing the step c;

and i, finishing iteration, outputting an optimal solution, and setting the optimal solution as a filter coefficient.

The hybrid genetic algorithm based on the invention only needs to alternately execute the genetic algorithm and the conjugate gradient method on the central individuals of the seed groups, so that the global search characteristic of the genetic algorithm and the rapid convergence characteristic of the conjugate gradient method can be fully utilized, and the algorithm can be rapidly converged to the optimal solution.

And step S40, obtaining the FIR digital filter meeting the actual filtering requirement according to the filter coefficient.

In this step, an optimal FIR digital filter meeting the actual requirements is designed mainly according to the filter coefficients calculated in the previous step.

By integrating the FIR digital filter design method, the filter is designed in a weighting mode according to different actual requirements, the side lobes and the main lobe are subjected to weighted summation, and the performance requirements of the main lobe and the side lobes can be automatically adjusted according to the actual design requirements, so that the filter designs with different performances can be obtained; in the solution algorithm, in the face of minimization, a hybrid genetic algorithm is adopted, and the genetic algorithm in the evolutionary algorithm is combined with the traditional conjugate gradient algorithm, so that the optimal filter coefficient can be globally and quickly searched. The algorithm fully utilizes the global search capability of the genetic algorithm and the rapid local search capability of the conjugate gradient method, accelerates the convergence speed of the genetic algorithm and improves the search precision of the genetic algorithm. Therefore, the optimal filter design coefficient can be obtained, and the optimal filter meeting the actual requirement can be obtained according to the filter coefficient.

Referring to fig. 3, fig. 1 is a schematic structural diagram of a FIR digital filter design system according to the present invention; the method mainly comprises the following steps:

the mathematical modeling module 10 is used for modeling the FIR digital filter according to the design requirement of the filter to obtain a mathematical model of the filter;

a condition refining module 20, configured to refine the parameter limiting condition of the filter according to the mathematical model to obtain a condition weighted model;

a model solving module 30, configured to solve the conditional weighting model by using a genetic algorithm and a least square method to obtain an optimal filter coefficient;

and the filter design module 40 is configured to obtain, according to the filter coefficient, an FIR digital filter that meets the actual filtering requirement.

In one embodiment, the mathematical modeling module 10 is further configured to:

establishing a weighted filter design model according to the input of filter design requirements:

E(e^jw)=H_d(e^jw)-H(e^jw).

in the formula, H_d(e^jw) Representing the ideal filter frequency response, H (e)^jw) Indicates the actual obtainingFilter frequency response of, E (E)^jw) Representing a frequency response error;

calculating a mean square error equation:

$e^2=\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}[h_d\left(n\right)-h\left(n\right)]e^{-\mathrm{jwn}}+\underset{n=N}{\overset{+\∞}{\mathrm{\Σ}}}{h}_d\left(n\right)e^{-\mathrm{jwn}}.$

in the formula, e²Represents the mean square error, where:

$\left\{\begin{array}{c}{H}_d\left(e^{\mathrm{jw}}\right)=\underset{n=\∞}{\overset{\∞}{\mathrm{\Σ}}}{h}_d\left(n\right)e^{-\mathrm{jwn}}\\ H\left(e^{\mathrm{jw}}\right)=\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}h\left(n\right)e^{-\mathrm{jwn}}\end{array}\right..$

obtaining a mathematical model of the filter according to a mean square error equation:

$\min {\left|\right|B\·\mathrm{\ξ}-H\left|\right|}_2^2$

wherein:

ξ=[h(0),h(1),…h(N-1)]^T，

H=[H_d(w₁),H_d(w₂),…,H_d(w_M)]

$B={\left[\begin{array}{cccc}{e}^{-j{w}_1}& e^{-2j{w}_1}& \·\·\·& e^{-\mathrm{jn}{w}_1}\\ \·\·\·& \·\·\·& \·\·\·& \\ e^{-j{w}_i}& e^{-2j{w}_i}& \·\·\·& e^{-\mathrm{jn}{w}_i}\\ \·\·\·& \·\·\·& \·\·\·& \\ e^{-j{w}_M}& e^{-2j{w}_M}& \·\·\·& e^{-\mathrm{jn}{w}_M}\end{array}\right]}_{M\×N}$

in the formula,denotes a two-norm, w_k(k =0,1, …, M-1) is represented by [ - π, π]The upper M frequency sample points, h (N), N =0,1, …, N-1 is a set such that the formula:the obtained value is the minimum filter design parameter value.

In one embodiment, the condition refinement module 20 is further configured to:

establishing a weight function matrix A related to the actual frequency response of the filter:

wherein A is a weight function matrix,

calculating a weighting model according to the weight function matrix A:

in the formula,representing a two-norm with F being the objective function.

In one embodiment, model solving module 30 is further configured to:

establishing a genetic algorithm model according to the weighting model;

setting the center of the seed group of the model as an initial search point processed by a conjugate gradient algorithm; wherein, the center is the average value of population individuals, and the expression is as follows:m is the number of the population, and the individuals are respectively a_i∈Rⁿ,i=1,2,…,m；

Executing n steps conjugate gradient algorithm processing to obtain a result a⁽ⁿ⁾Calculating a⁽ⁿ⁾According to a⁽ⁿ⁾Generating an individual, adding the individual into the population, and replacing the individual with the minimum fitness value;

and circularly executing the next iteration processing step of the genetic algorithm and the conjugate gradient algorithm processing step until the optimal solution of the weighting model is solved, and setting the optimal solution as a filter coefficient.

Further, the algorithm used by the model solving module 30 for solving the conditional weighting model specifically includes:

a. initializing to generate an initial population; wherein, the population scale is set as m, and the cross probability is set as p_cThe mutation probability is p_mSetting the iteration number as n;

b. calculating an individual fitness function F;

c. performing a crossover operator and a mutation operator;

d. center a of computing species group₀；

e. A is to₀Setting an initial search point processed by a conjugate gradient algorithm; where k =1 is set, and the number of iterations n is set₀Setting precision requirements, and executing the following algorithm processing:

f. if it is notStopping the calculation; otherwise put it intoWherein, ${\mathrm{\β}}_{k-1}=\left\{\begin{array}{c}0,k=1\\ \frac{{\left|\right|\▿F\left(a^{\left(k\right)}\right)\left|\right|}^2}{{\left|\right|\▿F\left(a^{(k-1)}\right)\left|\right|}^2},k>1;\end{array}\right.$

g. one-dimensional search is carried out to solve the one-dimensional optimization problem of max phi (α) = F (a)^(k)+αd^(k)) To obtain α_kPut a at^(k)=a^(k)+α_kd^(k)(ii) a Let k = k +1, go to execute step f, and record the calculation result as a⁽ⁿ⁾；

h. Calculating a⁽ⁿ⁾To generate a new population, i.e. according to a⁽ⁿ⁾Generating an individual, adding the individual into the population, replacing the individual with the minimum fitness in the population, and executing the step c;

i. and (5) finishing iteration, outputting an optimal solution, and setting the optimal solution as a filter coefficient.

The FIR digital filter design system of the present invention corresponds to the FIR digital filter design method of the present invention one to one, and the technical features and the advantageous effects described in the embodiments of the FIR digital filter design method are applicable to the embodiments of the FIR digital filter design system, which is hereby stated.

The above-mentioned embodiments only express several embodiments of the present invention, and the description thereof is more specific and detailed, but not construed as limiting the scope of the present invention. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the inventive concept, which falls within the scope of the present invention. Therefore, the protection scope of the present patent shall be subject to the appended claims.

Claims (10)

1. A method for designing an FIR digital filter is characterized by comprising the following steps:

modeling the FIR digital filter according to the design requirement of the filter to obtain a mathematical model of the filter; the mathematical model is as follows:

$\begin{array}{cc}\min & \left|\right|B\·\mathrm{\ξ}-H|{|}_2^2\end{array}$

wherein:

ξ＝[h(0),h(1),K h(N-1)]^T，

H＝[H_d(w₁),H_d(w₂),K,H_d(w_M)]

$B={\left[\begin{array}{cccc}{e}^{-{\mathrm{jw}}_1}& e^{-2{\mathrm{jw}}_1}& L& e^{-{\mathrm{jnw}}_1}\\ L& L& L& \\ e^{-{\mathrm{jw}}_i}& e^{-2{\mathrm{jw}}_i}& L& e^{-{\mathrm{jnw}}_i}\\ L& L& L& \\ e^{-{\mathrm{jw}}_M}& e^{-2{\mathrm{jw}}_M}& L& e^{-{\mathrm{jnw}}_M}\end{array}\right]}_{M\×N}$

in the formula,denotes a two-norm, w_k(k-0, 1, Λ, M-1) is represented by [ -pi, pi]The M frequency samples above, h (N), N ═ 0,1, Λ, N-1, is a set such that the formula:obtaining a filter design parameter value with the minimum value;

refining the parameter limiting conditions of the filter according to the mathematical model to obtain a condition weighting model; the weighting model is as follows:in the formula,representing a two-norm, F is an objective function, wherein,

solving the conditional weighting model by using a genetic algorithm and a least square method to obtain an optimal filter coefficient;

and obtaining the FIR digital filter meeting the actual filtering requirement according to the filter coefficient.

2. The method of claim 1, wherein the step of modeling the FIR digital filter according to the filter design requirements to obtain a mathematical model of the filter comprises:

establishing a weighted filter design model according to the input of filter design requirements:

E(e^jw)＝H_d(e^jw)-H(e^jw).

in the formula, H_d(e^jw) Representing the ideal filter frequency response, H (e)^jw) Representing the actually obtained filter frequency response, E (E)^jw) Representing a frequency response error;

calculating a mean square error equation:

$e^2=\underset{n=0}{\overset{N-1}{\Σ}}\[h_d\left(n\right)-h\left(n\right)\]e^{-jwn}+\underset{n=N}{\overset{+\mathrm{\∞}}{\Σ}}{h}_d\left(n\right)e^{-jwn}.$

in the formula, e²Represents the mean square error, where:

$\left\{\begin{array}{c}{H}_d\left(e^{jw}\right)=\underset{n=-\mathrm{\∞}}{\overset{\mathrm{\∞}}{\mathrm{\Σ}}}{h}_d\left(n\right)e^{-jwn}\\ H\left(e^{jw}\right)=\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}h\left(n\right)e^{-jwn}\end{array}\right..$

and acquiring a mathematical model of the filter according to a mean square error equation.

3. The method of claim 1, wherein the step of refining the parameter constraints of the filter according to the mathematical model to obtain a conditional weighting model comprises:

and establishing a weight function matrix A related to the actual frequency response of the filter, and calculating a weighting model according to the weight function matrix A.

4. The method of claim 1, wherein the step of solving the conditional weighting model using genetic algorithm and least squares to obtain optimal filter coefficients comprises:

establishing a genetic algorithm model according to the weighting model;

setting the population center of the model as an initial search point processed by a conjugate gradient algorithm; wherein, the center is the average value of population individuals, and the expression is as follows:m is the number of the population, and the individuals are respectively a_i∈Rⁿ,i＝1,2,K,m；

Executing n steps conjugate gradient algorithm processing to obtain a result a⁽ⁿ⁾Calculating a⁽ⁿ⁾According to a⁽ⁿ⁾Generating an individualAdding the individual into the population to replace the individual with the minimum fitness value;

and circularly executing the next iteration processing step of the genetic algorithm and the conjugate gradient algorithm processing step until the optimal solution of the weighting model is solved, and setting the optimal solution as a filter coefficient.

5. The FIR digital filter design method according to claim 4, wherein the step of solving the conditional weighting model using genetic algorithm and least squares to obtain the optimal filter coefficients specifically comprises:

a. initializing to generate an initial population; wherein, the population scale is set as m, and the cross probability is set as p_cThe mutation probability is p_mSetting the iteration number as n;

b. calculating an individual fitness function F;

c. performing a crossover operator and a mutation operator;

d. center a of computing species group₀；

e. A is to₀Setting an initial search point processed by a conjugate gradient algorithm; where k is set to 1, and the number of iterations n is set₀Setting precision requirements, and executing the following algorithm processing:

f. if it is notStopping the calculation; otherwise put it intoWherein,

g. one-dimensional search is carried out to solve the one-dimensional problem that max phi (α) is equal to F (a)^(k)+αd^(k)) To obtain α_kPut a at^(k)＝a^(k)+α_kd^(k)(ii) a Let k equal to k +1, go to execute step f, and record the calculation result as a⁽ⁿ⁾；

h. Calculating a⁽ⁿ⁾The fitness function of (a) is determined,generating a new population, i.e. according to a⁽ⁿ⁾Generating an individual, adding the individual into the population, replacing the individual with the minimum fitness in the population, and executing the step c;

i. and (5) finishing iteration, outputting an optimal solution, and setting the optimal solution as a filter coefficient.

6. A FIR digital filter design system, comprising:

the mathematical modeling module is used for modeling the FIR digital filter according to the design requirement of the filter to obtain a mathematical model of the filter; the mathematical model is as follows:

$\begin{array}{cc}\min & \left|\right|B\·\mathrm{\ξ}-H|{|}_2^2\end{array}$

wherein:

ξ＝[h(0),h(1),K h(N-1)]^T，

H＝[H_d(w₁),H_d(w₂),K,H_d(w_M)]

$B={\left[\begin{array}{cccc}{e}^{-{\mathrm{jw}}_1}& e^{-2{\mathrm{jw}}_1}& L& e^{-{\mathrm{jnw}}_1}\\ L& L& L& \\ e^{-{\mathrm{jw}}_i}& e^{-2{\mathrm{jw}}_i}& L& e^{-{\mathrm{jnw}}_i}\\ L& L& L& \\ e^{-{\mathrm{jw}}_M}& e^{-2{\mathrm{jw}}_M}& L& e^{-{\mathrm{jnw}}_M}\end{array}\right]}_{M\×N}$

in the formula,denotes a two-norm, w_k(k-0, 1, Λ, M-1) is represented by [ -pi, pi]The M frequency samples above, h (N), N ═ 0,1, Λ, N-1, is a set such that the formula:obtaining a filter design parameter value with the minimum value;

the condition refining module is used for refining the parameter limiting conditions of the filter according to the mathematical model to obtain a condition weighting model; the weighting model is as follows:in the formula,representing a two-norm, F is an objective function, wherein,

the model solving module is used for solving the conditional weighting model by utilizing a genetic algorithm and a least square method to obtain an optimal filter coefficient;

and the filter design module is used for acquiring the FIR digital filter meeting the actual filtering requirement according to the filter coefficient.

7. The FIR digital filter design system according to claim 6, wherein the mathematical modeling module is further configured to:

establishing a weighted filter design model according to the input of filter design requirements:

E(e^jw)＝H_d(e^jw)-H(e^jw).

in the formula, H_d(e^jw) Representing the ideal filter frequency response, H (e)^jw) Representing the actually obtained filter frequency response, E (E)^jw) Representing a frequency response error;

calculating a mean square error equation:

$e^2=\underset{n=0}{\overset{N-1}{\Σ}}\[h_d\left(n\right)-h\left(n\right)\]e^{-jwn}+\underset{n=N}{\overset{+\mathrm{\∞}}{\Σ}}{h}_d\left(n\right)e^{-jwn}.$

in the formula, e²Represents the mean square error, where:

$\left\{\begin{array}{c}{H}_d\left(e^{jw}\right)=\underset{n=-\mathrm{\∞}}{\overset{\mathrm{\∞}}{\mathrm{\Σ}}}{h}_d\left(n\right)e^{-jwn}\\ H\left(e^{jw}\right)=\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}h\left(n\right)e^{-jwn}\end{array}\right..$

and acquiring a mathematical model of the filter according to a mean square error equation.

8. The FIR digital filter design system according to claim 6, wherein the condition refinement module is further to:

and establishing a weight function matrix A related to the actual frequency response of the filter, and calculating a weighting model according to the weight function matrix A.

9. The FIR digital filter design system according to claim 6, wherein the model solution module is further to:

establishing a genetic algorithm model according to the weighting model;

setting the population center of the model as an initial search point processed by a conjugate gradient algorithm; wherein, the center is the average value of population individuals, and the expression is as follows:m is the number of the population, and the individuals are respectively a_i∈Rⁿ,i＝1,2,K,m；

Executing n steps conjugate gradient algorithm processing to obtain a result a⁽ⁿ⁾Calculating a⁽ⁿ⁾According to a⁽ⁿ⁾Generating an individual, adding the individual into the population, and replacing the individual with the minimum fitness value;

and circularly executing the next iteration processing step of the genetic algorithm and the conjugate gradient algorithm processing step until the optimal solution of the weighting model is solved, and setting the optimal solution as a filter coefficient.

10. The FIR digital filter design system according to claim 9, wherein the algorithm used by the model solution module to solve the conditional weighting model specifically comprises:

a. initializing to generate an initial population; wherein, the population scale is set as m, and the cross probability is set as p_cThe mutation probability is p_mSetting the iteration number as n;

b. calculating an individual fitness function F;

c. performing a crossover operator and a mutation operator;

d. center a of computing species group₀；

e. A is to₀Setting an initial search point processed by a conjugate gradient algorithm; where k is set to 1, and the number of iterations n is set₀Setting precision requirements, and executing the following algorithm processing:

f. if it is notThen stop the meterCalculating; otherwise put it intoWherein,

g. one-dimensional search is carried out to solve the one-dimensional optimization problem that max phi (α) is equal to F (a)^(k)+αd^(k)) To obtain α_kPut a at^(k)＝a^(k)+α_kd^(k)(ii) a Let k equal to k +1, go to execute step f, and record the calculation result as a⁽ⁿ⁾；

h. Calculating a⁽ⁿ⁾To generate a new population, i.e. according to a⁽ⁿ⁾Generating an individual, adding the individual into the population, replacing the individual with the minimum fitness in the population, and executing the step c;

i. and (5) finishing iteration, outputting an optimal solution, and setting the optimal solution as a filter coefficient.