A method of shaping a nuclear pulse of arbitrary waveform

Abstract

The application discloses a kind of arbitrary waveform nuclear pulse conversion methods, for the trapezoidal shaping of arbitrary waveform nuclear pulse signal.First, the arbitrary waveform nuclear pulse signal is regarded as the impulse response of rational fraction transfer function system.Second, rational fraction transfer function is decomposed into multiple first-order inertia link and the form of sum, then, multiple first-order inertia link and the form of sum is carried out Laplace inverse transformation into time domain expression, and then its time domain expression is discretized and transformed, then the parameter of discretized expression is obtained using particle swarm algorithm, finally, the transfer function of trapezoidal shaping and shaping recurrence formula are obtained, the trapezoidal shaping of arbitrary waveform nuclear pulse signal is realized to a certain extent, the symmetry of trapezoidal pulse is improved, the accurate acquisition of pulse amplitude is facilitated, the energy resolution of system is improved and the counting rate is improved, which has great significance for improving the precision of radioactivity measurement.

Description

Technical Field

[0001] This invention relates to a pulse waveform transformation method. Background Technology

[0002] Digital pulse shaping technology has become an important method for shaping nuclear pulse signals, which facilitates the identification of nuclear signals using digital signal processing methods and greatly improves the performance of nuclear instruments. Commonly used digital pulse shaping algorithms mainly include trapezoidal shaping, Gaussian shaping, and... Trapezoidal forming and peak forming are two main methods for shaping nuclear pulses. Trapezoidal forming is not only easy to implement but also beneficial for identifying stacked pulses, while simultaneously considering the important parameters of energy resolution and count rate. Therefore, in practical applications, the trapezoidal forming algorithm is the most commonly used algorithm in digital pulse shaping. In recent years, domestic and international research has been conducted on trapezoidal forming methods for nuclear pulses, acquisition and identification of trapezoidal pulses, and parameter estimation. However, these studies all assume that the signal before trapezoidal forming is a negative exponential nuclear pulse signal; that is, the algorithms studied focus on how to shape a negative exponential nuclear pulse signal into a trapezoidal pulse. However, due to differences in the signal conditioning circuits of the detector's front-end analog system—such as cascaded RC networks for RC feedback preamplifiers, cascaded CR-RC networks for switch-reset preamplifiers, etc.—the different numbers of cascaded RC or CR stages, the diversity of C and R connection methods, the presence of other distributed capacitances and resistances, and the different detector response times, the transfer functions of the front-end analog system vary. This results in diversity in the nuclear pulse waveform output by the detector front-end, leading to a decrease in the symmetry of the trapezoidal pulses obtained from the actual measured nuclear pulse shaping, reducing the accuracy of the acquired pulse amplitude, and consequently affecting the qualitative and quantitative analysis of the nuclear energy spectrum. Therefore, gradient morphing of arbitrary waveform kernel pulse signals remains a challenging problem. Based on previous research, this invention investigates a gradient morphing method for arbitrary waveform kernel pulse signals to improve system energy resolution and increase the count rate. Summary of the Invention

[0003] The purpose of this invention is to disclose a pulse waveform transformation method for trapezoidal shaping of arbitrary waveform nuclear pulse signals. This method achieves trapezoidal shaping of arbitrary waveform nuclear pulse signals to a certain extent, improves the symmetry of the trapezoidal pulse, facilitates accurate acquisition of pulse amplitude, and helps improve system energy resolution and increase count rate, which is of great significance for improving the accuracy of radioactivity measurement.

[0004] The present invention performs gradient morphing on arbitrary waveform nuclear pulse signals through steps ① to ⑥.

[0005] Step ① Treat the arbitrary waveform nuclear pulse signal as the impulse response of a rational fractional transfer function system.

[0006] Step ② decomposes the transfer function in ① into a rational form consisting of multiple first-order inertial elements.

[0007] Step ③ performs an inverse Laplace transform on the form of the sum of multiple first-order inertial elements in step ② to obtain a time-domain expression.

[0008] Step ④ involves discretizing and performing a Z-transform on the time-domain expression in step ③.

[0009] Step ⑤ uses the particle swarm optimization algorithm to obtain the parameters of the discretized expression in ④, specifically according to the following steps (1) to (5).

[0010] (1) First, the actual measured arbitrary waveform nuclear pulse signal X(mT) s It can be expressed in the following form:

[0011]

[0012] In formula (1), u(mT) s -T0) represents the step function, m = 0, 1, 2, 3…; i = 0, 1, 2, 3…n; n is the number of first-order inertial elements; k i τ is the proportionality coefficient for each first-order inertial element; i T0 represents the time constant of each first-order inertial element; T0 represents the occurrence time of the nuclear pulse.

[0013] (2) Initialize the number of first-order inertial elements of the particle swarm algorithm n = 1.

[0014] (3) Secondly, the combination of parameters contained in formula (1) is regarded as the position information x of the particles in the particle swarm algorithm. (p，n) [·], x (p，n) [·]=x (p，n) [k (p，1) k (p，2) … k (p，n) τ (p，1) τ (p，2) …τ (p，n) T 0(p，n) ] represents the position information of the p-th particle in the population, where p = 1, 2, 3…popsize; popsize is the number of particles in the population in the algorithm, and is represented by v. (p，n) [·] represents the velocity information of the p-th particle in the population, v (p，n) [·]=v (p，n) [v (p，1) v (p，2) … v (p，2n+1) ], using X (p，n) (mT s ) represents the pulse signal corresponding to the position information of the p-th particle in the population, k (p，1) k(p，2) …k (p，n) X represents (p，n) (mT s The proportionality coefficient τ of each first-order inertial element contained in the ) (p，1) τ (p，2) …τ (p，n) X represents (p，n) (mT s The time constants T of each first-order inertial element contained in the ) 0(p，n) X represents (p，n) (mT s The time of occurrence of ) v (p，1) v (p，2) … v (p，2n+1) x represents the position information of the p-th particle in the population. (p，n) The [·] section contains the search speed corresponding to each dimension parameter.

[0015] (4) Then, initialize the position and velocity information of all particles in the population, and set the pulse signal X corresponding to the particle position information. (p，n) (mT s ) and the actual measured pulse signal X(mT) s The variance of the particle swarm optimization algorithm is used as the fitness function to judge the quality of the particles. The optimal particle position information of the population to the nth order is obtained through continuous iteration of the particles. The particle swarm optimization algorithm in this step is specifically implemented in stages A, B, C, D, and E.

[0016] Population A particle parameter initialization:

[0017] Based on the total dimension of the particle search space being 2n+1, the position information x of all particles in a population with n first-order inertial elements is obtained. (p，n) [k (p，1) k (p，2) … k (p，n) τ (p，1) τ (p，2) …τ (p，n) T 0(p，n) And the velocity information v of all particles (p，n) [v (p，1) v (p，2) … v (p，2n+1) ] Perform random parameter initialization, p = 1, 2, 3...popsize; popsize is the number of particles in the population in the algorithm, and n is X. (p，n) (mT s The number of first-order inertial elements contained in ).

[0018] B. Calculate the fitness value of the particles:

[0019] The pulse signal X corresponding to the particle position information(p，n) (mT s ) and the actual measured pulse signal X(mT) s The variance of the particle is used as the fitness function to judge the quality of the particle. The fitness value of each particle in the population is calculated. By comparing the size of the fitness values, the optimal particle position information of the first generation of the population is obtained. Each particle itself is used as the optimal particle position information of the first generation individual.

[0020] C. Particle Update:

[0021] The algorithm updates the velocity and position information of each particle in the population using an update formula.

[0022] D. Update of the swarm's optimal particle and the individual's optimal particle:

[0023] The fitness value of the next generation of particles updated through step C is calculated. The fitness values ​​are compared to obtain the current generation's best particle position information and the individual best particle position information. If the current generation's best particle position information is better than the first generation's, the current generation's best particle position information is used to replace the first generation's. Otherwise, the first generation's information is retained. The update of the individual best particle position information is done in the same way.

[0024] E. Algorithm terminates:

[0025] There are two conditions for the algorithm to terminate: the first is reaching the set maximum number of iterations, and the second is that the fitness value corresponding to the best particle in the population meets the set requirements. If the termination conditions are not met during the iteration process, the algorithm returns to step B to continue the operation.

[0026] (5) After searching with the particle swarm optimization algorithm to obtain the optimal particle position information when n equals 1, this method is used to change the value of n in step (2) in the order of n = 2, 3, 4... and repeat steps (3) to (4) to obtain the optimal particle position information of various groups with different n values. By comparing the fitness values ​​of the optimal particles of each group with different n values, when the fitness value of the optimal particle corresponding to n is less than the fitness values ​​of the optimal particles corresponding to n-1 and n+1 at the same time, the algorithm search is completed, and the optimal particle position information of the group with the smallest fitness value is obtained. Thus, the parameter acquisition of the discretized expression is completed. The optimal particle position information of the group with the smallest fitness value obtained by the search is the parameter n and k. i τ i , i = 1, 2, 3...n.

[0027] The parameters of the discretized expression are obtained through steps (1) to (5).

[0028] Step 6: Obtain the transfer function and recursive formula for the ladder formation of arbitrary waveform kernel pulse signals.

[0029] By completing steps ① to ⑥ above, we can obtain the transfer function and recursive formula for the gradient formation of arbitrary waveform kernel pulse signals.

[0030] The beneficial effects of this invention are:

[0031] In recent years, in-depth research has been conducted both domestically and internationally on trapezoidal forming methods for nuclear pulses. However, these studies all assume that the signal before trapezoidal forming is a negative exponential nuclear pulse signal, meaning that the algorithms studied all focus on how to shape a negative exponential nuclear pulse signal into a trapezoidal pulse. In reality, due to differences in the signal conditioning circuits of the detector's front-end analog system (e.g., cascaded RC networks with RC feedback preamplifiers, cascaded CR-RC networks with switch-reset preamplifiers), the different numbers of cascaded RC or CR stages, the diversity of C and R connection methods, the presence of other distributed capacitances and resistances, and the varying detector response times, the transfer functions of the front-end analog system differ. This results in diversity in the nuclear pulse waveform output by the detector front-end, leading to decreased symmetry in the trapezoidal pulse obtained from the actual measured nuclear pulse shaping, reduced accuracy of the obtained pulse amplitude, and consequently affecting the qualitative and quantitative analysis of the nuclear energy spectrum. This invention first treats arbitrary waveform nuclear pulse signals as the impulse response of a rational fractional transfer function system. Then, it uses a particle swarm optimization algorithm to identify the parameters of the arbitrary waveform nuclear pulse signal, thereby obtaining the shaping parameters in the trapezoidal forming recursive formula and ensuring the reliability of the recursive algorithm. Finally, the transfer function and recursive formula for its direct trapezoidal formation are given. This method ensures the symmetry of the trapezoidal pulse, facilitates the accurate acquisition of the pulse amplitude, improves the energy resolution of the system, and increases the count rate, which is of great significance for improving the accuracy of radioactivity measurement. Attached Figure Description

[0032] Figure 1 This is a flowchart of the method of the present invention. Detailed Implementation

[0033] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings. These embodiments are implemented based on the technical solution of the present invention and provide detailed implementation methods and processes. However, the scope of protection of the present invention is not limited to the following embodiments.

[0034] The present invention achieves gradient morphing of arbitrary waveform kernel pulse signals through the following steps ① to ⑥:

[0035] Step ① Treat the arbitrary waveform kernel pulse signal as the impulse response of a rational fractional transfer function system, and its expression is as follows:

[0036]

[0037] In formula (2), e=0, 1, 2, 3...E; r=0, 1, 2, 3...R; (E≥R), a e b r It is a real constant.

[0038] Step ② decomposes the transfer function in ① into a rational form consisting of multiple first-order inertial elements, as shown in the following expression:

[0039]

[0040] In formula (3), i = 1, 2, 3…n; k i τ is the proportionality coefficient of each first-order inertial element. i Let n be the time constant of each first-order inertial element, and n be the number of first-order inertial elements.

[0041] Step ③ performs an inverse Laplace transform on the sum of multiple first-order inertial elements in step ② to obtain a time-domain expression, which is as follows:

[0042]

[0043] In formula (4), t represents time, and the other parameters are the same as in formula (2).

[0044] Step ④ Discretize and perform Z-transform as follows:

[0045] Discretization

[0046]

[0047] Z-transform

[0048]

[0049] In formula (5), T s The sampling period is represented by m = 0, 1, 2, 3..., and the other parameters are the same as those in formula (3). Formula (6) is the Z-transform formula corresponding to formula (5).

[0050] Step ⑤ uses the particle swarm optimization algorithm to obtain the parameters of the discretized expression in ④, specifically according to the following steps (1) to (5).

[0051] (1) First, the actual measured arbitrary waveform nuclear pulse signal X(mT) s It can be expressed in the following form:

[0052]

[0053] In formula (7), u(mT) s -T0) represents the step function, m = 0, 1, 2, 3…; i = 1, 2, 3…n; n is the number of first-order inertial elements; ki τ is the proportionality coefficient for each first-order inertial element; i T0 represents the time constant of each first-order inertial element; T0 represents the occurrence time of the nuclear pulse.

[0054] (2) Initialize the number of first-order inertial elements of the particle swarm algorithm n = 1.

[0055] (3) Secondly, the combination of parameters contained in formula (7) is regarded as the position information x of the particles in the particle swarm algorithm. (p，n) [·], x (p，n) [·]=x (p，n) [k (p，1) k (p，2) … k (p，n) τ (p，1) τ (p，2) …τ (p，n) T 0(p，n) ] represents the position information of the p-th particle in the population, where p = 1, 2, 3…popsize; popsize is the number of particles in the population in the algorithm, and is represented by v. (p，n) [·] represents the velocity information of the p-th particle in the population, v (p，n) [·]=v (p，n) [v (p，1) v (p，2) … v (p，2n+1) ], using X (p，n) (mT s ) represents the pulse signal corresponding to the position information of the p-th particle in the population, k (p，1) k (p，2) … k (p，n) X represents (p，n) (mT s The proportionality coefficient τ of each first-order inertial element contained in the ) (p，1) τ (p，2) …τ (p，n) X represents (p，n) (mT s The time constants T0 of each first-order inertial element contained within it. (p，n) X represents (p，n) (mT s The time of occurrence of ) v (p，1) v (p，2) … v (p，2n+1) x represents the position information of the p-th particle in the population. (p，n) The [·] section contains the search speed corresponding to each dimension parameter.

[0056] (4) Then, initialize the position and velocity information of all particles in the population, and set the pulse signal X corresponding to the particle position information. (p，n) (mTs ) and the actual measured pulse signal X(mT) s The variance of the particle swarm optimization algorithm is used as the fitness function to judge the quality of the particles. The optimal particle position information of the population is obtained through continuous iteration of the particles. The particle swarm optimization algorithm in this step is specifically implemented in stages A, B, C, D, and E.

[0057] A. Population particle parameter initialization:

[0058] Based on the total dimension of the population particle search space of 2n+1, the particle position information x (p，n) [·], x (p，n) [·]=x (p，n) [k (p，1) k (p，2) … k (p，n) τ (p，1) τ (p，2) …τ (p，n) T 0(p，n) Particle velocity information v (p，n) [·], v (p，n) [·]=v (p，n) [v (p，1) v (p，2) … v (p，2n+1) Perform random parameter initialization, where the particle position information x (p，n) [·] and speed information v (p，n) The initial range of the parameters [·] is determined by the actual measured nuclear pulse characteristics. The more initial population particles there are, i.e., the larger the popsize, the greater the probability of random initialization being close to the optimal solution. However, if the value is too large, it will affect the running speed of the algorithm. The value of popsize can be determined according to the waveform of the nuclear pulse.

[0059] B. Calculate the fitness value of the particles:

[0060] The pulse signal X corresponding to the particle position information (p，n) (mT s ) and the actual measured pulse signal X(mT) s The variance of the particle is used as the fitness function to judge the quality of the particle. The fitness value of each particle in the population is calculated. By comparing the fitness values, the optimal particle position information of the first generation of the nth-order population is obtained. Each particle itself is used as the optimal particle position information of the first generation individual. The specific implementation of this step is as follows (a)~(b):

[0061] (a) Establishing the fitness function

[0062]

[0063] Formula (8) represents the fitness value of the p-th particle in the population, i = 1, 2, 3…n; p = 1, 2, 3…popsize; popsize is the number of particles in the population in the algorithm, m = 0, 1, 2, 3…M; M is the number of discrete samples, T s The sampling period is denoted as u(mT). s -T 0(p，n) X(mT) represents the step pulse corresponding to the p-th particle in the population. s ) represents the actual measured pulse signal, X (p，n) (mT s x represents the position information of the p-th particle in the population. (p，n) The pulse signal corresponding to [·], T 0(p，n) X represents (p，n) (mT s The occurrence time of ) k (p，i) X represents (p，n) (mT s The proportionality coefficients τ of each first-order inertial element corresponding to ) (p，i) X represents (p，n) (mT s The corresponding time constants of each order;

[0064] (b) Calculation of particle fitness value

[0065] fitness (p，n) =f(x) (p，n) [·]) (9)

[0066] Formula (9) represents the fitness value of the p-th particle in the population when the number of first-order inertial elements is n, and the fitness value of each particle in the population is popsize according to the index p = 1, 2, 3... (p，n) The calculation is performed, where popsize is the number of particles in the population. By comparing the fitness values, the smaller the value, the better, thus obtaining the optimal particle position information Q of the initial generation of the population when the number of first-order inertial links is n. (p，n) The particle itself is used as the initial individual's optimal particle position information G (p，n) .

[0067] C. Particle Update:

[0068] The velocity and position information of each particle in the population are updated using the algorithm's update formula, specifically according to steps (a) to (b):

[0069] (a) Speed ​​Update

[0070] v (p，n) (θ) l+1 =wv (p，n) (θ)l +c1 rand1[G (p，n) (θ) l -x (p，n) (θ) l ]+c2rand2[Q (p，n) (θ) l -x (p，n) (θ) l (10)

[0071] (b) Location update

[0072] x (p，n) (θ) l+1 =x (p，n) (θ) l +βv (p，n) (θ) l+1 (11)

[0073] Wherein, the search space dimension θ = 1, 2, 3…2n+1; v (p，n) (θ) l+1 and x (p，n) (θ) l+1 G represents the velocity and position information of the p-th particle in the θ-th dimension during the (l+1)-th iteration when the number of first-order inertial elements is n. (p，n) (θ) l and G (p，n) (θ) l , respectively represent the optimal individual particle position and the optimal position of the population particle in dimension θ of the p-th particle after the l-th iteration update when the number of first-order inertial links is n. c1 and c2 are acceleration constants, usually taken as [0, 4]. rand1 and rand2 are random numbers between 0 and 1. β is the constraint factor for velocity update. w is the inertial factor, which represents the weight coefficient for velocity update and affects whether the particle finds the population optimal solution or gets trapped in a local optimal solution. The larger w is, the stronger the global search ability of the particle. Conversely, the smaller w is, the stronger the local search ability of the particle. In order to avoid the particle search getting trapped in a local optimal solution and causing the algorithm to fail, the linear descent weight method is adopted here. The formula (12) is used to optimize w. When the number of iterations is small, the global search ability of the particle is guaranteed and the particle is accelerated to locate the position of the global optimal solution. When the number of iterations is large in the later stage, w is reduced to ensure the particle's search ability for local solutions, thereby improving the accuracy of the algorithm.

[0074]

[0075] In formula (12), w max and w min These are the maximum and minimum values ​​of the w inertia factor, and l. count and l maxThese are the current number of iterations and the maximum number of iterations for the algorithm, respectively.

[0076] D. Update of swarm optimal particle position information and individual optimal particle position information:

[0077] The fitness value of the next generation of particles updated through step C is calculated. The fitness values ​​are compared to obtain the current generation's best particle position information and the individual best particle position information. If the current generation's best particle position information is better than the first generation, that is, the smaller the corresponding fitness value of the particle, then the current generation's best particle position information replaces the first generation's. Otherwise, the first generation is retained. The update of the individual best particle position information is the same.

[0078] E. Algorithm terminates:

[0079] There are two conditions for the algorithm to terminate: the first is reaching the set maximum number of iterations, and the second is that the fitness value corresponding to the best particle in the population meets the set requirements. If the termination conditions are not met during the iteration process, the algorithm returns to step B to continue the operation.

[0080] (5) After the above particle swarm optimization algorithm search, the optimal particle position information Q of the nth-order population is obtained after iteration. (p，n) Using this method, by changing the value of n in step (2) in the order of n = 2, 3, 4... and repeating steps (3) to (4), the position information of the optimal particle of each group for different n values ​​can be obtained; by comparing the fitness values ​​of the optimal particles of each group for these different n values, when the fitness value of the optimal particle corresponding to n is less than the fitness values ​​of the optimal particles corresponding to n-1 and n+1, that is, f(Q (p，n) )<f(Q (p，n-1) ), f(Q (p，n) )<f(Q (p，n+1) When the algorithm completes its search, it obtains the optimal particle position information Q of the swarm with the minimum fitness value. (p，n) The optimal particle position information Q of the nth-order population with the smallest fitness value. (p，n) That is, the parameters n and k are what we are looking for. i τ i , i = 1, 2, 3...n.

[0081] The parameters of the discretized expression are obtained through steps (1) to (5).

[0082] Step 6, obtaining the transfer function and recursive formula for the ladder formation of arbitrary waveform kernel pulse signals, is achieved through the following steps (a) to (c):

[0083] (a) The Z-transform expression for the trapezoidal pulse is as follows:

[0084]

[0085] Y in formula (13) M The amplitude of the trapezoidal pulse is flat-topped. t a Let t be the rise time of the trapezoidal pulse. b t is the sum of the rise time and the flat-top time of the trapezoidal pulse. c Let t be the total time for the trapezoidal formation. c =t a +t b T s The sampling period;

[0086] (b) Transfer function for ladder formation of arbitrary waveform kernel pulse signal:

[0087]

[0088] Forming parameters in formula (14) i=1, 2, 3…n; j=1, 2, 3…n; n c =n a +n b The remaining parameters are the same as in formula (12), where parameter k i , τ i i = 1, 2, 3…n and n are obtained in step ⑤;

[0089] (c) Obtain the recursive formula for the gradient shape of arbitrary waveform kernel pulse signals:

[0090] From formula (14), we can obtain the trapezoidal pulse signal Y(z) = X(z)H(z), that is...

[0091]

[0092] Further, the recurrence formula is obtained:

[0093]

[0094]

[0095] In formula (16) i = 0, 1, 2, ..., n; j = 0, 1, 2, ..., n-1; the remaining parameters are the same as in formula (14), where, It has n parameters The set of (d[·]) n ) is n forming parameters d i The set, V i (d[·] n ) indicates that each time from (d[·] n Take a product term consisting of i parameters with distinct indices from the n items of the set. This represents the total number of product terms obtained using the method described above. Indicates each time from Take the n items from the set that are related to V i For a product term where all parameter indices are distinct, the numbers retrieved using the two methods described above will be different each time, regardless of the order of the indices. The parameter k... i τ i The number of first-order inertial elements, n, is obtained in step ⑤.

[0096] Steps ① to ⑥ above are the methods for obtaining the transfer function and recursive formula for the ladder formation of arbitrary waveform kernel pulse signals.

[0097] As described above, the method for obtaining the transfer function and recursive formula for trapezoidal shaping of arbitrary waveform nuclear pulse signals utilizes the particle swarm optimization algorithm to identify the proportional coefficients and time constants of each stage of the detector front-end analog circuit. This ensures the consistency between the pulse trapezoidal shaping parameters and the actual nuclear pulse signal parameters, guaranteeing the reliability of the algorithm. Furthermore, by treating the pulse signal as the impulse response of a rational fractional transfer function system, the distortion problem of the pulse signal after digital shaping caused by the diversity of pulse types and waveform anomalies is overcome. This method ensures the symmetry of the trapezoidal pulse, which is beneficial for the accurate extraction of pulse amplitude and the improvement of the count rate in the energy spectrum measurement system, and is of great significance for improving energy resolution.

[0098] In the above embodiments of the present invention, the transfer function and the method of forming recursive formula for ladder formation of arbitrary waveform kernel pulse signals have been described in detail. However, it should be noted that the above description is only one embodiment of the present invention. When other types of pulse waveforms involve the use of ladder formation and parameter identification proposed herein, the present invention is still effective. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for gradient forming of arbitrary waveform kernel pulses, characterized in that, The gradient morphing of arbitrary waveform kernel pulse signals is achieved through the following steps ① to ⑥: Step ① Treat the arbitrary waveform nuclear pulse signal as the impulse response of a rational fractional transfer function system; Step ② decomposes the transfer function in ① into a rational form consisting of multiple first-order inertial elements; Step ③ Perform an inverse Laplace transform on the form of the sum of multiple first-order inertial elements in step ② to obtain a time-domain expression; Step ④ involves discretizing and performing a Z-transform on the time-domain expression in step ③; Step ⑤ uses the particle swarm optimization algorithm to obtain the parameters of the discretized expression in step ④, and is implemented as follows: (1) First, the arbitrary waveform kernel pulse measured in actual operation is discretized into a signal X(mT). s It can be expressed in the following form: In formula (1), u(mT) s -T0) represents the step function, m = 0, 1, 2, 3…; i = 1, 2, 3…n, where n is the number of first-order inertial elements; k i τ is the proportionality coefficient for each first-order inertial element; i T represents the time constant of each first-order inertial element; T0 represents the occurrence time of the nuclear pulse. (2) Initialize the number of first-order inertial elements in the particle swarm algorithm to n = 1; (3) Secondly, the combination of parameters contained in formula (1) is regarded as the position information x of the particles in the particle swarm algorithm. (p，n) [·],x (p，n) [·]=x (p，n) [k (p，1) k (p，2) … k (p，n) τ (p，1) τ (p，2) …τ (p，n) T 0(p，n) ] represents the position information of the p-th particle in the population, where p = 1, 2, 3…popsize; popsize is the number of particles in the population in the algorithm, and is represented by v. (p，n) [·] represents the velocity information of the p-th particle in the population, v (p，n) [·]=v (p，n) [v (p，1) v (p，2) …v (p，2n+1) ], using X (p，n) (mT s ) represents the pulse signal corresponding to the position information of the p-th particle in the population, k (p，1) k (p，2) … k (p，n) X represents (p，n) (mT s The proportionality coefficient τ of each first-order inertial element contained in the ) (p，1) τ (p，2) …τ (p，n) X represents (p，n) (mT s The time constants T of each first-order inertial element contained in the ) 0(p，n) X represents (p，n) (mT s The time of occurrence of ) v (p，1) v (p，2) …v (p，2n+1) x represents the position information of the p-th particle in the population. (p,n) The search speed corresponding to each dimension parameter included in [·]; (4) Then, initialize the position and velocity information of all particles in the population, and set the position information x of the p-th particle. (p，n) The pulse signal X corresponding to [·] (p，n) (mT s ) and the actual measured pulse signal X(mT) s The variance of the particle is used as the fitness function to judge the quality of the particle, and the optimal particle position information of the population is obtained through continuous algorithm iteration. The fitness function is as follows: Formula (2) represents the fitness value of the p-th particle in the population, i = 1, 2, 3…n; p = 1, 2, 3…popsize; popsize is the number of particles in the population in the algorithm, m = 0, 1, 2, 3…M; M is the number of discrete samples, T s The sampling period is denoted as u(mT). s -T 0(p，n) X(mT) represents the step pulse corresponding to the p-th particle in the population. s ) represents the actual measured pulse signal, X (p，n) (mT s x represents the position information of the p-th particle in the population. (p,n) The pulse signal corresponding to [·], T 0(p，n) X represents (p，n) (mT s The occurrence time of ) k (p,i) X represents (p,n) (mT s The proportionality coefficients τ of each first-order inertial element corresponding to ) (p,i) X represents (p,n) (mT s The time constants of each first-order inertial element corresponding to ). (5) After searching with the particle swarm optimization algorithm to obtain the optimal particle position information when n equals 1, this method is used to change the value of n in step (2) in the order of n = 2, 3, 4... and repeat steps (3) to (4) to obtain the optimal particle position information of each swarm when n has different values. By comparing the fitness values ​​of the optimal particles of each swarm when n has different values, when the fitness value of the optimal particle corresponding to n is less than the fitness values ​​of the optimal particles corresponding to n-1 and n+1, the algorithm search is completed, and the optimal particle position information of the swarm with the smallest fitness value is obtained. Thus, the parameter acquisition of the discretized expression is completed. The optimal particle position information of the swarm with the smallest fitness value obtained by the search is the parameter n and k. i τ i , i = 1, 2, 3…n; Step 6: Obtain the transfer function and recursive formula for the ladder formation of arbitrary waveform kernel pulse signals.

2. The ladder forming method for arbitrary waveform kernel pulses according to claim 1, characterized in that, The arbitrary waveform nuclear pulse signal in step ① takes the following form: In formula (3), e=0, 1, 2, 3...E; r=0, 1, 2, 3...R; E≥R, a e b r It is a real constant.

3. The ladder forming method for arbitrary waveform kernel pulses according to claim 1, characterized in that, In step ②, the transfer function is rationally decomposed into multiple first-order inertial elements in the following form: In formula (4), i = 1, 2, 3…n; k i τ is the proportionality coefficient of each first-order inertial element. i Let n be the time constant of each first-order inertial element, and n be the number of first-order inertial elements.

4. The ladder forming method for arbitrary waveform kernel pulses according to claim 1, characterized in that, In step ③, the sum of multiple first-order inertial elements is transformed by an inverse Laplace transform into a time-domain expression as follows: In formula (5), t represents time, and the other parameters are the same as in formula (4).

5. The ladder forming method for arbitrary waveform kernel pulses according to claim 1, characterized in that, In step ④, the discretization and Z-transform are performed as follows: Discretization Z-transform In formula (6), T s The sampling period is represented by m = 0, 1, 2, 3...; the other parameters are the same as those in formula (4), and formula (7) is the Z-transform formula corresponding to formula (6).

6. The ladder forming method for arbitrary waveform kernel pulses according to claim 1, characterized in that, The transfer function and recursive formula for obtaining the gradient shape of the arbitrary waveform kernel pulse signal in step ⑥ are implemented according to the following steps (a) to (c): (a) The Z-transform expression for the trapezoidal pulse is as follows: Y in formula (8) max The amplitude of the trapezoidal pulse is flat-topped. t a Let t be the rise time of the trapezoidal pulse. b t is the sum of the rise time and the flat-top time of the trapezoidal pulse. c Let t be the total time for the trapezoidal formation. c =t a +t b T s The sampling period; (b) Transfer function for ladder formation of arbitrary waveform kernel pulse signal: Forming parameters in formula (9) i=1, 2, 3…n; j=1, 2, 3…n; n c =n a +n b The remaining parameters are the same as in formula (8), where parameter k i τ i i = 1, 2, 3…n and n are obtained in step ⑤; (c) Obtain the recursive formula for the gradient shape of arbitrary waveform kernel pulse signals: From formula (9), we can derive the trapezoidal pulse signal Y(z) = X(z)H(z), that is... Further, the recurrence formula is obtained: In formula (11) j = 0, 1, 2, ..., n-1; the remaining parameters are the same as in formula (9), where, It is n parameters The set of (d[·]) n ) is n forming parameters d i The set, V i (d[·] n ) indicates that each time from (d[·] n Take a product term consisting of i parameters with distinct indices from the n items of the set. This represents the total number of product terms obtained using the method described above. Indicates each time from Take the n items from the set that are related to V i For a product term where all parameter indices are distinct, the numbers retrieved using the two methods described above will be different each time, regardless of the order of the indices. The parameter k... i τ i Both and n are obtained in step ⑤.

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