CN112084741A - Digital all-pass filter design method based on hybrid particle swarm algorithm - Google Patents

Digital all-pass filter design method based on hybrid particle swarm algorithm Download PDF

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CN112084741A
CN112084741A CN202010952125.XA CN202010952125A CN112084741A CN 112084741 A CN112084741 A CN 112084741A CN 202010952125 A CN202010952125 A CN 202010952125A CN 112084741 A CN112084741 A CN 112084741A
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pass filter
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CN112084741B (en
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赵禹
叶芃
孟婕
杨扩军
张凯
张涛
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a digital all-pass filter design method based on a hybrid particle swarm algorithm, which comprises the steps of firstly constructing a phase-frequency response of a digital all-pass filter based on second order cascade connection according to design requirements, and then constructing a cost function based on a target phase-frequency response and an error value formed by the phase-frequency response of the digital all-pass filter and the target phase-frequency response; and finally, searching a global optimal solution in a value range by utilizing a globally searched particle swarm algorithm to obtain a rough solution, triggering from the rough solution, and performing accurate search by utilizing a Levenbergmarquardt algorithm to optimize the positions of poles of the second-order all-pass filters, so that the mean square error minimization of the phase-frequency response of the designed all-pass filters and the target phase-frequency response is realized, and the design of the filter coefficients is realized.

Description

Digital all-pass filter design method based on hybrid particle swarm algorithm
Technical Field
The invention belongs to the technical field of all-pass filters, and particularly relates to a digital all-pass filter design method based on a hybrid particle swarm algorithm.
Background
In practice, when a signal is transmitted in a system, a certain phase distortion, especially a nonlinear distortion of the phase, may be generated, and different phase shifts may be added to signals of different frequencies, which may have a large influence on the output of the system, and in a severe case, the signal may be distorted. Compensation of the phase is an essential part for systems where strict linearity of the phase is required. The all-pass filter can change the phase characteristics of signals and effectively solve the problem of phase nonlinear distortion. The all-pass filter has the same gain for all frequency components input in the system, does not attenuate signals of any frequency, but changes the phase characteristics of the input signal. The all-pass filter plays an important role in application occasions such as phase compensation, group delay equalization and the like to meet the requirement of a system on linear phase-frequency response.
The design problem of the digital all-pass filter can be reduced to the problem of nonlinear optimization, and the traditional design algorithms include swarm intelligence optimization algorithms, such as Particle Swarm Optimization (PSO), and gradient-based nonlinear optimization algorithms, such as levenberg marquardt algorithm (LM). However, in the optimization process of designing the digital all-pass filter, the optimization speed of the PSO algorithm is obviously reduced after the global optimal solution is approached, and even the PSO algorithm is premature. The LM algorithm belongs to an unconstrained optimization problem, although the LM algorithm has a high optimization speed, the designed digital all-pass filter is easy to be unstable due to the fact that the optimization range is not limited, and besides, the LM algorithm is very sensitive to the initial value of optimization iteration.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a digital all-pass filter design method based on a hybrid particle swarm algorithm, which converts the design problem of filter coefficients into the problem of nonlinear function optimization and then searches and optimizes the filter coefficients from rough to precise by using different optimization algorithms so as to realize the design of the filter coefficients.
In order to achieve the above object, the present invention provides a method for designing a digital all-pass filter based on a hybrid particle swarm algorithm, which is characterized by comprising the following steps:
(1) constructing a phase frequency response of a digital all-pass filter based on second-order cascade connection;
(1.1) according to actual requirements, setting the order N of the digital all-pass filter, wherein N is an even number, and the number of second-order sections of the corresponding digital all-pass filter is N/2, so that the Z domain expression H (Z) of the cascaded digital all-pass filter is as follows:
Figure BDA0002677350680000021
wherein Hn(Z) Z-domain expression representing the nth order digital all-pass filter, an1、an2Is the coefficient of the nth order digital all-pass filter;
(1.2) setting the pole of the nth second order section as
Figure BDA0002677350680000022
Mn、θnThe modulus and phase angle of the nth second order pole are shown as the poles of the N-order digital all-pass filter
Figure BDA0002677350680000023
Definition variable U ═ M1,θ1,M2,θ2,…,Mn,θn,…,MN/2,θN/2]The phase-frequency response of the digital all-pass filter
Figure BDA0002677350680000024
Expressed as:
Figure BDA0002677350680000025
wherein ω is [ ω ═ ω [ [ ω ]1,ω2,…,ωl,…,ωL]，ωlThe angular frequency of the digital all-pass filter is shown, and L is the number of sampled frequency points;
(2) constructing a cost function of the digital all-pass filter;
let the target phase-frequency response of the digital all-pass filter be phigoal(ω) the phase frequency response of the digital all-pass filter
Figure BDA0002677350680000026
Has a phase frequency response phi with respect to the targetgoal(omega) error value constructed cost function phierror(U) is represented as:
Figure BDA0002677350680000027
(3) searching a cost function phi by utilizing a particle swarm algorithmerror(U) a global optimal solution;
(3.1) setting the population number S of the particle swarm algorithm and the maximum iteration number K of the particle swarm algorithm;
(3.2) initializing the particle group
Figure BDA0002677350680000028
To ensure the stability of each second order section of the digital all-pass filter, the process of initializing each particle in the population needs to be satisfied
Figure BDA0002677350680000031
Rho is a constant slightly less than 1;
(3.3) particle swarm updating and iteration;
(3.3.1) calculating the updating speed of each particle;
Figure BDA0002677350680000032
wherein, Vs kIs the velocity of the s-th particle after the k-th update, C1And C2Respectively corresponding to the acceleration factor, r1And r2Is [0,1 ]]Random number of inner, zbsAnd gb is the historical optimal solution of the S-th particle and the global optimal solution of the particle swarm composed of the S particles respectively;
(3.3.2) updating the particle position;
Figure BDA0002677350680000033
wherein χ (·) is a defining function;
(3.3.3) calculating the iterated particles
Figure BDA0002677350680000034
Cost function of
Figure BDA0002677350680000035
And updating the historical optimal solution zb of the s-th particle according to the following formulasAnd a global optimal solution gb of a particle swarm of S particles;
Figure BDA0002677350680000036
Figure BDA0002677350680000037
(3.4) repeating the step (3.3) for K times in total to obtain a final global optimal solution gb;
(4) carrying out local optimal solution by utilizing a Levenberg Marquardt algorithm;
(4.1) setting the total iteration number K of the Levenberg Marquardt algorithmmax，k＝1,2,…,KmaxInitializing k to 1;
(4.2) calculating an iteration starting point X of the Levenberg Marquardt algorithm0＝[x1,θ1,x2,θ2,…,xn,θn,…,xN/2,θN/2]Wherein
Figure BDA0002677350680000038
(4.3) calculating X after the k iterationkPhase frequency response of corresponding digital all-pass filter
Figure BDA0002677350680000039
Figure BDA0002677350680000041
(4.4) calculating X after the k iterationkCorresponding residual vector Rk；
Rk＝φdesign(ω,Xk)-φgoal(ω) (9)
(4.5) calculating X after the k iterationkA corresponding cost function value;
φerror(Xk)＝Rk·Rk T (10)
(4.6) calculating X after the k iterationkCorresponding Accord matrix JkBlackplug matrix Hk＝Jk TJkAnd diagonal matrix Dk＝diag{Hk}；
(4.7) calculating an update vector after the kth iteration;
Δk＝(Hk+λ·Dk)-1·Jk·Rk (11)
(4.8) calculating a new vector X after the k iterationnew＝Xk+ΔkAnd a corresponding cost function value phierror(Xnew) Then compare phierror(Xnew) Phi and phierror(Xk) If is phierror(Xnew)＜φerror(Xk) Then let Xk+1＝XnewAnd λ ═ λ/v; if phi iserror(Xnew)≥φerror(Xk) Then let Xk+1＝XkAnd λ ═ λ × v; wherein, lambda and v are integers, lambda > v;
(4.9) circularly executing the steps (4.3) to (4.8) to sum up to KmaxThen, the iteration end point X is finally obtainedfinal；
(5) Inverse mapping;
mixing XfinalThe pole vector is then substituted into the following equation to obtain the final pole vector Ufinal；
Figure BDA0002677350680000042
(6) According to the pole vector UfinalCalculating the coefficient of a digital all-pass filter;
Figure BDA0002677350680000043
and finishing the design of the all-pass filter.
The invention aims to realize the following steps:
the invention relates to a digital all-pass filter design method based on a hybrid particle swarm algorithm, which comprises the steps of firstly constructing a phase-frequency response of a digital all-pass filter based on second order cascade connection according to design requirements, and then constructing a cost function based on a target phase-frequency response and an error value formed by the phase-frequency response of the digital all-pass filter and the target phase-frequency response; and finally, searching a global optimal solution in a value range by utilizing a globally searched particle swarm algorithm to obtain a rough solution, triggering from the rough solution, and performing accurate search by utilizing a Levenbergmarquardt algorithm to optimize the positions of poles of the second-order all-pass filters, so that the mean square error minimization of the phase-frequency response of the designed all-pass filters and the target phase-frequency response is realized, and the design of the filter coefficients is realized.
Meanwhile, the method for designing the digital all-pass filter based on the hybrid particle swarm optimization further has the following beneficial effects:
(1) although the Levenbergmarquardt algorithm can realize the optimal solution search, the Levenberg marquardt algorithm has high dependency on initial values and weak ability to search the global optimal solution. The particle swarm algorithm has global search capability, finds a global optimal region, but has the problem of local extremum, and can only find a rough value of a global optimal solution. Therefore, the Levenberg Marquardt algorithm and the particle swarm algorithm are combined, so that the global optimal rough solution can be searched, and the global optimal precise solution can also be searched;
(2) the stability of the all-pass filter is guaranteed during design, parameters are constrained during searching of an optimal solution, the unconstrained problem of the Levenbergmarquardt algorithm needs to be converted into a constrained problem, and the result of the Levenbergmarquardt algorithm is converted into a filter stable interval through a mapping function so as to guarantee the stability of the designed filter.
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FIG. 1 is a flow chart of a method for designing a digital all-pass filter based on a hybrid particle swarm optimization according to the present invention;
FIG. 2 is a block diagram of an N-order all-pass filter system based on a two-stage cascaded architecture;
FIG. 3 is a map of when an all-pass filter is stable, where FIG. 3(a) is the stable pole region; FIG. 3(b) is a mapping function curve; FIG. 3(c) is the stable pole mode length and phase angle; FIG. 3(d) is the document interval after mapping.
Detailed Description
The following description of the embodiments of the present invention is provided in order to better understand the present invention for those skilled in the art with reference to the accompanying drawings. It is to be expressly noted that in the following description, a detailed description of known functions and designs will be omitted when it may obscure the subject matter of the present invention.
Examples
FIG. 1 is a flow chart of a method for designing a digital all-pass filter based on a hybrid particle swarm optimization.
In this embodiment, as shown in fig. 1, the method for designing a digital all-pass filter based on a hybrid particle swarm algorithm of the present invention includes the following steps:
s1, constructing a phase-frequency response of the digital all-pass filter based on second-order cascade connection;
s1.1, the expression of the N-order (N is an even number) all-pass digital filter in the time domain is expressed by a constant coefficient linear difference equation as follows:
Figure BDA0002677350680000061
the expression of the system function in the z domain, namely the system function H (z) obtained by z-changing the time domain expression in (1), is as follows:
Figure BDA0002677350680000062
factorizing the numerator denominator of the formula (2) respectively, and forming a second-order factorization form of the numerator denominator, as shown in fig. 2, so as to implement the cascade structure of the digital filter, and then the Z-domain expression h (Z) of the digital all-pass filter having N/2 second-order cascades is:
Figure BDA0002677350680000063
wherein Hn(Z) Z-domain expression representing the nth order digital all-pass filter, an1、an2Is the coefficient of the nth order digital all-pass filter;
s1.2, because the pole-zero position of the all-pass filter is in one-to-one correspondence with the frequency response of the all-pass filter, the formula (3) can be rewritten into the expression form of the pole-zero
Figure BDA0002677350680000064
Wherein ξkAnd xik *The pole of the kth second order node, is the conjugate sign,
Figure BDA0002677350680000071
is the real part operation.
The cascade structure adopts a form of second-order combination, can realize various cascade orders, and does not influence the realization of the all-pass filter; by changing the coefficient of each second order section, the zero pole is flexibly and visually adjusted, so that the frequency response is convenient to adjust; meanwhile, the expansion of the filter order can be realized only by increasing the number of the cascade of the second order sections.
Let the pole of the nth second order section be
Figure BDA0002677350680000072
Mn、θnThe modulus and phase angle of the nth second order pole are shown as the poles of the N-order digital all-pass filter
Figure BDA0002677350680000073
Definition variable U ═ M1,θ1,M2,θ2,…,Mn,θn,…,MN/2,θN/2]The phase-frequency response of the digital all-pass filter
Figure BDA0002677350680000074
Expressed as:
Figure BDA0002677350680000075
wherein ω is [ ω ═ ω [ [ ω ]1,ω2,…,ωl,…,ωL]，ωlThe angular frequency of the digital all-pass filter is shown, and L is the number of sampled frequency points;
s2, constructing a cost function of the digital all-pass filter;
let the target phase-frequency response of the digital all-pass filter be phigoal(ω) the phase frequency response of the digital all-pass filter
Figure BDA0002677350680000076
Has a phase frequency response phi with respect to the targetgoal(omega) error value constructed cost function phierror(U) is represented as:
Figure BDA0002677350680000077
to design inThe all-pass filter has the best approximation effect and its cost function phierrorThe value of (U) is small enough to make the frequency response of the designed filter as consistent as possible with the frequency response of the target filter. Since equation (5) is a non-linear expression for vector U, we can minimize φerrorThe problem of (U) translates into a mathematical problem of non-linear optimization.
In order to solve the problem of nonlinear optimization in the formula (6), the invention provides a filter design method based on a hybrid particle swarm algorithm, which fully utilizes the global search capability of the particle swarm algorithm and the local search capability of the Levenbergmarquardt algorithm to realize the optimization of the formula (6), and comprises the following specific processes:
s3, searching cost function phi by using particle swarm optimizationerror(U) a global optimal solution;
s3.1, setting the population number S of the particle swarm algorithm and the maximum iteration number K of the particle swarm algorithm;
s3.2, initializing particle swarm
Figure BDA0002677350680000081
To ensure the stability of each second order section of the digital all-pass filter, the process of initializing each particle in the population needs to be satisfied
Figure BDA0002677350680000082
Rho is a constant slightly less than 1, and the calculation of the global optimal solution of the all-pass filter is realized under the constraint condition;
s3.3, particle swarm updating and iteration are carried out;
s3.3.1, calculating the update speed of each particle;
Figure BDA0002677350680000083
wherein, Vs kIs the velocity of the s-th particle after the k-th update, C1And C2Respectively corresponding to the acceleration factor, r1And r2Is [0,1 ]]Random number of inner, zbsAnd gb is the history of the s-th particle, respectivelyThe optimal solution and a global optimal solution of a particle swarm composed of S particles;
s3.3.2, updating the particle position;
Figure BDA0002677350680000084
wherein χ (-) is a limiting function by limiting the modulus U.M of the second order section pole of the filternThe stability of the designed filter is ensured, and the expression of χ (-) is as follows:
Figure BDA0002677350680000085
s3.3.3 calculating the iterated particles
Figure BDA0002677350680000086
Cost function of
Figure BDA0002677350680000087
And updating the historical optimal solution zb of the s-th particle according to the following formulasAnd a global optimal solution gb of a particle swarm of S particles;
Figure BDA0002677350680000088
Figure BDA0002677350680000089
s3.4, repeating the step S3.3, and iterating for K times in total to obtain a final global optimal solution gb;
s4, performing local optimal solution by utilizing a Levenberg Marquardt algorithm;
and after the optimization iteration of the particle swarm optimization is completed, the global optimal solution gb obtained by the particle swarm optimization is brought into the Levenbergmarquardt optimization for further accurate search. However, since Levenberg Marquardt algorithm is an unconstrained algorithm, it is directly adoptedThe process of seeking the optimal solution by the Levenbergmarquardt algorithm is likely to result in the modular length M of the filter polesnAnd the stability interval of the all-pass filter is exceeded, which causes the instability of the filter. Therefore, in order to ensure the stability of each second order section of the all-pass filter, a constrained optimization problem needs to be converted into an unconstrained optimization problem.
Defining a mapping function
Figure BDA0002677350680000091
As shown in fig. 3(b), for an arbitrary xn(value range- ∞ < x)n+ ∞) of MnThe value is between 0 and 1, and since the function is a monotone increasing function, a one-to-one mapping relation exists between x to F (x), F (x)n) Presence of an inverse function F-1(Mn) The expression is F-1(Mn)＝ln(Mn/(1-Mn))。F-1(Mn) Mapping values between (0,1) to the entire real number space, while F (x) can be achievedn) Mapping from the entire real space to the range of (0,1) can be achieved.
For the cascade-type all-pass filter, in order to ensure the stability of each second order section, the pole modulus should be smaller than ρ, as shown in fig. 3 (a). Converting the second order stable region into the pole modulus MnAnd pole phase θnIs a coordinate axis, as shown in FIG. 3(c), when M is presentnHas a value range of (0, rho)]To pole modulus MnBy an inverse function F-1(Mn) Is mapped to variable xnAfter, xnThe value range of (a) is expanded to the whole real number space, as shown in fig. 3(d), the problem of converting the unconstrained problem into the constrained problem is realized, meanwhile, the value of the whole real number space can be mapped between 0 and 1 through a mapping function, and a one-to-one correspondence relationship exists, so that the stability of the second order section of each all-pass filter is ensured.
The Levenbergmarquardt algorithm is a nonlinear least square algorithm, and the algorithm can modify parameters during execution to combine the advantages of the Gauss-Newton algorithm and the gradient descent method and improve the defects of the Gauss-Newton algorithm and the gradient descent method. According to the size of lambda, the step length is switched between the Newton method step length and the gradient descent method step length, and the specific optimization process is as follows:
s4.1, setting the total iteration number K of the Levenberg Marquardt algorithmmax，k＝1,2,…,KmaxInitializing k to 1;
s4.2, calculating iteration starting point X of Levenberg Marquardt algorithm0＝[x1,θ1,x2,θ2,…,xn,θn,…,xN/2,θN/2]Wherein
Figure BDA0002677350680000101
at this time xnThe value range of (1) can ensure the stability of the designed all-pass filter within the range of (-infinity, + ∞); thus, the pole value space mapping U → X of the all-pass filter is also completed;
s4.3, calculating X after k iterationkPhase frequency response of corresponding digital all-pass filter
Figure BDA0002677350680000102
Figure BDA0002677350680000103
S4.4, calculating X after k iterationkCorresponding residual vector Rk；
Rk＝φdesign(ω,Xk)-φgoal(ω) (13)
S4.5, calculating X after k iterationkA corresponding cost function value;
φerror(Xk)＝Rk·Rk T (14)
s4.6, calculating X after k iterationkCorresponding Accord matrix JkBlackplug matrix Hk＝Jk TJkAnd diagonal matrix Dk＝diag{Hk}；
S4.7, calculating an update vector after the kth iteration;
Δk＝(Hk+λ·Dk)-1·Jk·Rk (15)
s4.8, calculating a new vector X after the kth iterationnew＝Xk+ΔkAnd a corresponding cost function value phierror(Xnew) Then compare phierror(Xnew) Phi and phierror(Xk) If is phierror(Xnew)＜φerror(Xk) Then let Xk+1＝XnewAnd λ ═ λ/v; if phi iserror(Xnew)≥φerror(Xk) Then let Xk+1＝XkAnd λ ═ λ × v; wherein λ is 10000, v is 10;
s4.9, circularly executing the steps S4.3-S4.8 to sum up to KmaxThen, the iteration end point X is finally obtainedfinal；
S5, space inverse mapping;
mixing XfinalThe pole vector is then substituted into the following equation to obtain the final pole vector Ufinal；
Figure BDA0002677350680000104
S6, according to the pole vector UfinalCalculating the coefficient of a digital all-pass filter;
Figure BDA0002677350680000111
and finishing the design of the all-pass filter.
Although illustrative embodiments of the present invention have been described above to facilitate the understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the embodiments, and various changes may be made apparent to those skilled in the art as long as they are within the spirit and scope of the present invention as defined and defined by the appended claims, and all matters of the invention which utilize the inventive concepts are protected.

Claims (2)

1. A digital all-pass filter design method based on a hybrid particle swarm algorithm is characterized by comprising the following steps:
(1) constructing a phase frequency response of a digital all-pass filter based on second-order cascade connection;
(1.1) according to actual requirements, setting the order N of the digital all-pass filter, wherein N is an even number, and the number of second-order sections of the corresponding digital all-pass filter is N/2, so that the Z domain expression H (Z) of the cascaded digital all-pass filter is as follows:
Figure FDA0002677350670000011
wherein Hn(Z) Z-domain expression representing the nth order digital all-pass filter, an1、an2Is the coefficient of the nth order digital all-pass filter;
(1.2) setting the pole of the nth second order section as
Figure FDA0002677350670000012
Mn、θnThe modulus and phase angle of the nth second order pole are shown as the poles of the N-order digital all-pass filter
Figure FDA0002677350670000013
Definition variable U ═ M1,θ1,M2,θ2,…,Mn,θn,…,MN/2,θN/2]The phase-frequency response of the digital all-pass filter
Figure FDA0002677350670000014
Expressed as:
Figure FDA0002677350670000015
wherein ω is [ ω ═ ω [ [ ω ]1,ω2,…,ωl,…,ωL]L is the number of the sampled frequency points;
(2) constructing a cost function of the digital all-pass filter;
let the target phase-frequency response of the digital all-pass filter be phigoal(ω) the phase frequency response of the digital all-pass filter
Figure FDA0002677350670000016
Has a phase frequency response phi with respect to the targetgoal(omega) error value constructed cost function phierror(U) is represented as:
Figure FDA0002677350670000017
(3) searching a cost function phi by utilizing a particle swarm algorithmerror(U) a global optimal solution;
(3.1) setting the population number S of the particle swarm algorithm and the maximum iteration number K of the particle swarm algorithm;
(3.2) initializing the particle group
Figure FDA0002677350670000018
To ensure the stability of each second order section of the digital all-pass filter, the process of initializing each particle in the population needs to be satisfied
Figure FDA0002677350670000021
Rho is a constant slightly less than 1;
(3.3) particle swarm updating and iteration;
(3.3.1) calculating the updating speed of each particle;
Figure FDA0002677350670000022
wherein,
Figure FDA0002677350670000023
is the velocity of the s-th particle after the k-th update, C1And C2Respectively corresponding to the acceleration factor, r1And r2Is [0,1 ]]Random number of inner, zbsAnd gb is the historical optimal solution of the S-th particle and the global optimal solution of the particle swarm composed of the S particles respectively;
(3.3.2) updating the particle position;
Figure FDA0002677350670000024
wherein χ (·) is a defining function;
(3.3.3) calculating the iterated particles
Figure FDA0002677350670000025
Cost function of
Figure FDA00026773506700000210
And updating the historical optimal solution zb of the s-th particle according to the following formulasAnd a global optimal solution gb of a particle swarm of S particles;
Figure FDA0002677350670000026
Figure FDA0002677350670000027
(3.4) repeating the step (3.3) for K times in total to obtain a final global optimal solution gb;
(4) carrying out local optimal solution by utilizing a Levenberg Marquardt algorithm;
(4.1) setting the total iteration number K of the Levenberg Marquardt algorithmmax，k＝1,2,…,KmaxInitializing k to 1;
(4.2) calculate Levenberg MarquarIterative starting point X of special algorithm0＝[x1,θ1,x2,θ2,…,xn,θn,…,xN/2,θN/2]Wherein
Figure FDA0002677350670000028
(4.3) calculating X after the k iterationkPhase frequency response of corresponding digital all-pass filter
Figure FDA0002677350670000029
Figure FDA0002677350670000031
(4.4) calculating X after the k iterationkCorresponding residual vector Rk；
Rk＝φdesign(ω,Xk)-φgoal(ω) (9)
(4.5) calculating X after the k iterationkA corresponding cost function value;
φerror(Xk)＝Rk·Rk T (10)
(4.6) calculating X after the k iterationkCorresponding Accord matrix JkBlackplug matrix Hk＝Jk TJkAnd diagonal matrix Dk＝diag{Hk}；
(4.7) calculating an update vector after the kth iteration;
Δk＝(Hk+λ·Dk)-1·Jk·Rk (11)
(4.8) calculating a new vector X after the k iterationnew＝Xk+ΔkAnd a corresponding cost function value phierror(Xnew) Then compare phierror(Xnew) Phi and phierror(Xk) If is phierror(Xnew)＜φerror(Xk) Then let Xk+1＝XnewAnd λ ═ λ/v; if phi iserror(Xnew)≥φerror(Xk) Then let Xk+1＝XkAnd λ ═ λ × v; wherein, lambda and v are integers, lambda > v;
(4.9) circularly executing the steps (4.3) to (4.8) to sum up to KmaxThen, the iteration end point X is finally obtainedfinal；
(5) Inverse mapping;
mixing XfinalThe pole vector is then substituted into the following equation to obtain the final pole vector Ufinal；
Figure FDA0002677350670000032
(6) According to the pole vector UfinalCalculating the coefficient of a digital all-pass filter;
Figure FDA0002677350670000033
and finishing the design of the all-pass filter.
2. The method according to claim 1, wherein the expression of the limiting function χ (-) is:
Figure FDA0002677350670000041
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