Method for generating optimal wide-spaced frequency hopping sequence based on genetic algorithm

Abstract

The present application relates to the field of communication technology, and relates to a method for generating an optimal wide-interval frequency hopping sequence based on a genetic algorithm, comprising the following steps: generating an initial population by using Bernoulli chaos mapping; calculating the maximum Hamming autocorrelation function value of each individual in the initial population or the excellent population obtained in the last iteration, and calculating the fitness of each individual; calculating the probability of each individual being selected as a parent population based on the fitness of each individual, and selecting individuals meeting the requirements as the parent population based on the probability; randomly selecting two individuals from the parent population each time, and adding the individual with the lowest Hamming autocorrelation to the excellent population obtained in the current iteration; or performing recombination operation and / or mutation operation on the selected two individuals, and then adding the individual with the lowest Hamming autocorrelation to the excellent population obtained in the current iteration. The present application uses a genetic algorithm to generate an optimal wide-interval frequency hopping sequence, and can fully utilize the characteristics of global search and adaptive optimization to quickly optimize in a larger search space.

Description

Technical Field

[0001] This invention relates to the field of communication technology, and in particular to a method for generating optimal wide-interval frequency hopping sequences based on a genetic algorithm. Background Technology

[0002] Frequency hopping communication is an important spread spectrum communication technology. Its core idea is to control the carrier frequency of the transmitted signal to rapidly hop between multiple frequencies using pseudo-random sequences. This rapid frequency change gives the communication system good anti-interference, anti-interception, and multiple access capabilities, and it is widely used in mobile communications, ultra-wideband communications, radar systems, Bluetooth, and other fields. An ideal frequency hopping sequence should have good randomness and low Hamming correlation to reduce interference and ensure the reliability of the communication link. To further improve the performance of frequency hopping sequences, the concept of wide-interval frequency hopping sequences has been proposed. This means that the interval between adjacent frequency hopping frequencies should be large enough to effectively resist narrowband interference, tracking interference, and wideband blocking interference, thereby further improving the system's anti-interference capability.

[0003] Existing wide-interval frequency hopping sequence design methods mainly fall into two categories: one is based on frequency allocation, such as intermediate frequency (IF) removal and dual-frequency methods; the other is based on frequency point correction, using adjacent or related frequency points as references to calculate and meet the wide-interval requirements. Subsequently, Ren et al. proposed constructing wide-interval frequency hopping sequences using combinatorial algebra. In 2022, Li et al. proposed two long-period wide-interval frequency hopping sequence construction schemes. Wang et al. proposed a method for constructing wide-interval frequency hopping sequences based on OC sequences. Zhou et al. and Zeng et al. constructed wide-interval frequency hopping sequences using finite fields. These construction methods have fixed parameters, limited applicability, and difficulty in simultaneously achieving multiple performance indicators. Summary of the Invention

[0004] The purpose of this invention is to provide a method for generating optimal wide-interval frequency hopping sequences based on genetic algorithms. By using genetic algorithms to generate optimal wide-interval frequency hopping sequences, the characteristics of global search and adaptive optimization can be fully utilized to quickly find the optimal sequence within a large search space.

[0005] To achieve the above-mentioned objectives, the embodiments of the present invention provide the following technical solutions:

[0006] The method for generating optimal wide-interval frequency hopping sequences based on genetic algorithms includes the following steps.

[0007] Step 1: Generate several initial frequency hopping sequences using Bernoulli chaotic mapping, and generate an initial population by performing frequency point mapping and wide interval processing on the initial frequency hopping sequences.

[0008] Step 2: Calculate the maximum Hamming autocorrelation function value of individuals in the initial population or the superior population obtained in the previous iteration, construct the fitness function, and calculate the fitness of individuals;

[0009] Step 3: Based on the individual's fitness, calculate the probability that the individual will be selected as the parent population, and select individuals that meet the requirements as the parent population based on the probability.

[0010] Step 4: Randomly select 2 individuals from the parent population each time, and directly add the individual with the lowest Hamming autocorrelation to the superior population obtained in this iteration; or perform recombination and / or mutation operations on the 2 selected individuals, and add the individual with the lowest Hamming autocorrelation to the superior population obtained in this iteration.

[0011] Step 5: Return to step 2 until the preset maximum number of iterations is reached or an individual in the superior population satisfies the maximum Hamming autocorrelation function.

[0012] Furthermore, step 1 specifically includes the following steps:

[0013] Step 1-1: Generate several initial frequency hopping sequences using Bernoulli chaotic mapping:

[0014] in, This represents the control parameters for the m-th initial frequency hopping sequence; This represents the frequency point with index k in the m-th initial frequency hopping sequence; This represents the frequency point with index k+1 in the m-th initial frequency hopping sequence; k = 0, 1, ..., K-1; K is the length of the initial frequency hopping sequence;

[0015] Generate M initial frequency hopping sequences Z using Bernoulli chaotic mapping (1) Z (2) ,...,Z (M) m = 1, 2, ..., M, where M is the total number of initial frequency hopping sequences;

[0016] Steps 1-2 employ an interval sampling strategy, sampling from each initial frequency hopping sequence Z. (m) A chaotic frequency hopping sequence of length n is generated by extracting 1 frequency point every 5 bits. , where Y (m) Let m be the chaotic frequency hopping sequence corresponding to the m-th initial frequency hopping sequence. For Y (m) The i-th frequency point in the sequence, i=0,1,…,n-1; can yield M chaotic frequency hopping sequences Y. (1) ,Y (2) ,...,Y (M) ;

[0017] Steps 1-3, convert the chaotic frequency hopping sequence Linear mapping to discrete character sets q represents the frequency slot, generating the corresponding frequency point. ; To round down, the m-th chaotic frequency hopping sequence Y is formed. (m) Corresponding frequency hopping sequence candidate individuals ;

[0018] Steps 1-4, if Then calculate ,use Replace the original Where d is the minimum frequency interval, shift is a random variable, and mod is the modulo operation; M candidate individuals X of frequency hopping sequences that meet the wide interval requirement can be obtained. (m) That is, the initial population X (1) ,X (2) ,...,X (M) .

[0019] Furthermore, step 2 specifically involves:

[0020] Construct an objective function f(X) that aims to maximize the individual Hamming autocorrelation function. (m) ):

[0021] Among them, X (m) The m-th individual in the initial population or the superior population obtained in the previous iteration; For X (m) Hamming autocorrelation function; For frequency point delay;

[0022] When the objective function f(X) (m) When the value is the maximum Hamming autocorrelation function, the fitness function is defined as:

[0023] Among them, Fit(f(X) (m) For individual X (m) The fitness of the parameter c is the lower bound constraint term for the Hamming autocorrelation function value.

[0024] Furthermore, step 3 specifically involves:

[0025] Calculate the probability that an individual is selected as a parent in the population:

[0026] Among them, P m This represents the probability that the m-th individual is selected;

[0027] Set the probability threshold to ,like Then select individual X (m) As the parent population;

[0028] Alternatively, you can call the np.random.choice function based on the probability P of M individuals. m Select the parent population from M individuals.

[0029] Furthermore, step 4, which involves randomly selecting two individuals from the parent population each time and performing a recombination operation on the selected two individuals, includes:

[0030] Two individuals are randomly selected from the parent population consisting of N individuals to serve as the parent sequence. and parental sequence m1∈N, m2∈N, where N is the number of individuals in the parent population. Parental sequence The i-th frequency point in For the mother generation sequence The i-th frequency point in the sequence; the frequency points in the two sequences are discretely recombined, intermediately recombined, or linearly recombined to obtain two offspring sequences, denoted as follows: and , For offspring sequence The i-th frequency point in For offspring sequence The i-th frequency point in the spectrum, i=0,1,…,n-1;

[0031] The discrete recombination process is as follows:

[0032] At the frequency point at the i-th position, if Choose ,but choose ;if Choose ,but choose ;

[0033] The intermediate recombination process is as follows:

[0034] Among them, the scaling factor and All satisfied , These are limit parameters;

[0035] The linear recombination process is as follows:

[0036] Among them, the scaling factor and All satisfied .

[0037] Furthermore, step 4, which involves randomly selecting two individuals from the parent population each time and performing a mutation operation on the selected two individuals, includes:

[0038] Two individuals are randomly selected from the parent population consisting of N individuals to serve as the parent sequence. and parental sequence m1∈N, m2∈N, where N is the number of individuals in the parent population. Parental sequence The i-th frequency point in For the mother generation sequence The i-th frequency point in the equation; set the mutation probability P. s ; Generate a random number e in the range (0,1), and combine the random number e with P s For comparison, if e≤P s Then, for any frequency point in the parent sequence, use the following formula respectively. and any frequency point in the parent generation sequence Apply Gaussian perturbation:

[0039] in, The frequency points in the parent sequence before applying the Gaussian perturbation; The frequency points in the parent sequence after applying a Gaussian perturbation; The frequency points in the parent generation sequence before applying Gaussian perturbation; The frequency points in the parent generation sequence after applying a Gaussian perturbation; is the step size factor; rand() is a random number that follows a Gaussian normal distribution [0,1).

[0040] Furthermore, in step 4, after the step of randomly selecting two individuals from the parent population each time and performing mutation operations on the selected two individuals, the following step is also included:

[0041] The frequency point after applying Gaussian perturbation is obtained through modular arithmetic. The loop is mapped to the range [0, q), where r equals 1 or 2:

[0042] in, Indicates the frequency at which the Gaussian perturbation is applied; express Frequency points after modulo operation, used Replace the original This results in the parent sequence X after applying a Gaussian perturbation. (m1) and parental sequence X (m2) ;

[0043] Then calculate the parent sequence X after applying Gaussian perturbation. (m1) and parental sequence X (m2) The Hamming autocorrelation function is used to select individuals with smaller Hamming autocorrelation functions and add them to the superior population obtained in this iteration.

[0044] Furthermore, in step 4, the step of randomly selecting two individuals from the parent population each time and performing mutation operations on the selected two individuals is as follows:

[0045] A reference sequence of length J is generated using a linear feedback shift register, where the frequency points in the reference sequence are either 0 or 1.

[0046] The parent sequence X of length n (m1) The value of each frequency point is converted into 8 bits to form a binary sequence X. (m1) Its length becomes 8n;

[0047] The binary sequence X (m1) Perform a bitwise XOR operation with the reference sequence, where 8n ≤ J. If the frequency values ​​at the same position in both sequences are equal, then the frequency value at that position is 0; if the frequency values ​​at the same position in both sequences are not equal, then the frequency value at that position is 1. This generates a binary sequence B = {b1, b2, ..., bn} of length 8n that contains only 0s and 1s. 8n}, b h Equal to 0 or 1, h = 1, 2, ..., 8n;

[0048] Reconstruct the bits of binary sequence B according to the w bits, q=2 w In a binary sequence B of length 8n, every w bits are truncated to form a new frequency point, and only the first n frequency points are retained. In this way, B is restored to decimal form to obtain the parent sequence X. (m1) The corresponding new frequency hopping sequence W1={a0,a1,...,a2} of length n n-1};

[0049] Similarly, the parent generation sequence X is obtained in the same way. (m2) The corresponding new frequency hopping sequence W2={a0,a1,...,a1} of length n n-1};

[0050] Then calculate the Hamming autocorrelation function of W1 and W2 respectively, and add the frequency hopping sequence with the smaller Hamming autocorrelation function to the superior population obtained in this iteration.

[0051] Furthermore, following step 4, the method further includes the following steps:

[0052] For each individual in the superior population, a wide-interval treatment is applied again. Let any individual in the superior population be represented as W = {a0, a1, ..., a...} n-1}, a i Let i be the i-th frequency hopping point in W, i = 0, 1, ..., n-1. If two adjacent frequency points in this individual have Then calculate d represents the minimum frequency interval, shift represents a random variable, and mod represents modulo operation. Replace the original Thus, a frequency hopping sequence W that meets the wide spacing requirement is obtained.

[0053] Compared with existing technologies, the beneficial effects of this invention are as follows: In frequency-hopping communication systems, the design of the frequency-hopping sequence is crucial, directly affecting the anti-interference performance and spectrum utilization efficiency of the communication system. This invention employs a genetic algorithm to generate the optimal wide-interval frequency-hopping sequence, fully utilizing its global search and adaptive optimization features to quickly find the optimal sequence within a large search space. Specifically, the frequency-hopping sequence is encoded as an "individual" in the genetic algorithm, and iterative optimization is performed using steps such as population initialization, fitness evaluation, selection, recombination, and mutation to ultimately generate the optimal wide-interval frequency-hopping sequence that meets the conditions. Through its high flexibility and powerful global search capabilities, the genetic algorithm can more deeply explore the rich and diverse sequence parameter space, better meeting the communication needs in complex electromagnetic environments. With the transition from traditional methods to intelligent optimization methods, intelligent technologies such as genetic algorithms provide strong technical support for achieving high efficiency and reliability in future communication systems. These methods, through dynamic optimization and adjustment of sequence design, can better adapt to complex and ever-changing communication environments. Attached Figure Description

[0054] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0055] Figure 1 This is a flowchart of the method of the present invention;

[0056] Figure 2 A schematic diagram of a wide-interval frequency hopping sequence;

[0057] Figure 3 This is a schematic diagram of the Bernoulli chaotic mapping of the present invention. Detailed Implementation

[0058] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0059] It should be noted that similar reference numerals and letters in the following figures denote similar items; therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures. Furthermore, in the description of this invention, the terms "first," "second," etc., are used only for distinguishing descriptions and should not be construed as indicating or implying relative importance, or suggesting any such actual relationship or order between these entities or operations.

[0060] Example:

[0061] The purpose of this scheme is to obtain the optimal wide-interval frequency hopping sequence through the iteration of a genetic algorithm. Before describing the technical solution, the embodiments first introduce the definition of "optimal wide-interval frequency hopping sequence".

[0062] Suppose a discrete character set of size q q represents the frequency slot. From discrete character sets Selecting L characters from the given information to construct a frequency hopping sequence X={x(0),x(1),...,x(L-1)}, we have... Let i = 0, 1, ..., L-1; and let the length or period of the frequency hopping sequence X be L. Let the set of M frequency hopping sequences of length L be denoted as [equation missing]. For sequence sets The Hamming correlation function between any two frequency-hopping sequences X={x(0),x(1),...,x(L-1)} and Y={y(0),y(1),...,y(L-1)} is:

[0063] in, For frequency point delay, Calculate according to modulo L; Let x(k) represent the Hamming correlation function between frequency hopping sequences X and Y; x(k) is the k-th frequency point in frequency hopping sequence X. The first in the frequency hopping sequence Y One frequency point.

[0064] when hour, ;otherwise When X=Y, The Hamming autocorrelation function of the frequency-hopping sequence X can be abbreviated as: .

[0065] For any frequency hopping sequence And X≠Y, let:

[0066] Where H(X) represents the maximum Hamming autocorrelation function of the frequency hopping sequence X; H(X,Y) represents the maximum Hamming cross-correlation function between the frequency hopping sequences X and Y.

[0067] If all frequency points in the frequency hopping sequence X={x(0),x(1),...,x(L-1)} satisfy... Then X is called a wide-interval frequency hopping sequence with a minimum frequency interval of d. For example... Figure 2 As shown, the gray area represents the interference frequency band, and the blue area represents the frequency point of the current frequency hopping sequence. The wide-interval frequency hopping sequence can ensure that two adjacent frequency points in the time domain do not fall into the same interference frequency band at the same time, which is beneficial for combating narrowband interference, broadband blocking interference, tracking interference and multipath fading.

[0068] If the maximum Hamming autocorrelation function H(X) of the frequency hopping sequence X satisfies the lower bound established by the following formula (constructed by Peihua Li), then X is called the "optimal wide-interval frequency hopping sequence":

[0069] in, As intermediate parameters, we have ; This is the floor rule. This is the floor operator.

[0070] To obtain the optimal wide-interval frequency hopping sequence, this invention uses a genetic algorithm and Bernoulli chaotic mapping to generate an initial population. The generated initial population is then used to generate a wide-interval frequency hopping sequence using a random shift substitution method. The autocorrelation function of the wide-interval frequency hopping sequence is used as the objective function to generate the fitness function. After selection, recombination, and mutation operations, the optimal wide-interval frequency hopping sequence that meets the conditions is generated iteratively step by step.

[0071] This invention is achieved through the following technical solutions, such as... Figure 1 As shown, a method for generating optimal wide-interval frequency hopping sequences based on a genetic algorithm includes the following steps:

[0072] Step 1: Generate several initial frequency hopping sequences using Bernoulli chaotic mapping, and then generate the initial population by performing frequency point mapping and wide interval processing on the initial frequency hopping sequences.

[0073] Step 1-1: Generate several initial frequency hopping sequences using Bernoulli chaotic mapping.

[0074] To enhance the randomness of the generated wide-interval frequency hopping sequence and accelerate the convergence speed of the genetic algorithm, this scheme uses Bernoulli chaotic mapping to generate the initial frequency hopping sequence. The specific formula for Bernoulli chaotic mapping is as follows:

[0075] in, This represents the control parameters for the m-th initial frequency hopping sequence; This represents the frequency point with index k in the m-th initial frequency hopping sequence; This represents the frequency point with index k+1 in the m-th initial frequency hopping sequence.

[0076] This scheme utilizes Bernoulli chaotic mapping to generate M initial frequency hopping sequences (m=1,2,...,M; e.g., M=400). When generating the first initial frequency hopping sequence (m=1), control parameters are first set. and the first frequency point ; and ; Next, K frequency points are calculated according to the above formula, k=0,1,...K-1, thus forming the first initial frequency hopping sequence with the frequency point mapping values ​​in the range (0,1) and possessing ergodicity and randomness. Following the same method, M initial frequency hopping sequences Z can be obtained. (1) Z (2) ,...,Z (M) It should be noted that when generating the next initial frequency hopping sequence, the following settings are configured: and It is variable, but the length of each initial frequency hopping sequence is K.

[0077] Steps 1-2: For each initial frequency hopping sequence, select the frequency of a fixed point to generate a chaotic frequency hopping sequence.

[0078] like Figure 3As shown, the frequency point mapping values ​​of the initial frequency hopping sequence obtained through Bernoulli chaotic mapping exhibit good uniformity and randomness. To reduce the criticality between frequency hopping sequences, an interval sampling strategy is adopted, sampling from each initial frequency hopping sequence Z... (m) A chaotic frequency hopping sequence of length n is generated by extracting one frequency point every 5 bits. (That is, assuming an initial frequency hopping sequence Z of length K) (m) (n frequency points were extracted from the middle), where Y (m) Let m be the chaotic frequency hopping sequence corresponding to the m-th initial frequency hopping sequence. For Y (m) The i-th frequency point in the sequence, i=0,1,…,n-1; similarly, M chaotic frequency hopping sequences Y can be obtained. (1) ,Y (2) ,...,Y (M) .

[0079] Steps 1-3 involve frequency point mapping of the chaotic frequency hopping sequence to generate candidate individuals for the frequency hopping sequence.

[0080] chaotic frequency hopping sequence Linear mapping to discrete character sets Generate the corresponding frequency points , i=0,1,...,n-1; The floor sign is used to form the m-th chaotic frequency hopping sequence Y. (m) Corresponding frequency hopping sequence candidate individuals Since m = 1, 2, ..., M, M candidate individuals X for frequency hopping sequences can be formed. (1) ,X (2) ,...,X (M) .

[0081] Steps 1-4 involve wide-interval processing of frequency points for candidate individuals in the frequency hopping sequence to generate the initial population.

[0082] The random shift substitution method is used for each frequency hopping sequence candidate individual X. (m) The core technology for wide frequency spacing processing is X (m) A nonlinear transformation is performed to eliminate frequency points that do not meet the wide spacing requirement. Specifically, if the distance between two adjacent frequency points is less than d+1, the latter frequency point is replaced with a new frequency point using the following nonlinear transformation algorithm to ensure X... (m) All frequency points meet the wide spacing requirement:

[0083] like Then calculate ,use Replace the original Where d is the minimum frequency interval, mod is the modulo operation, and shift represents a random variable, typically taken as... Thus, M candidate frequency-hopping sequences X that satisfy the wide spacing requirement can be obtained. (m) That is, the initial population X (1) ,X (2) ,...,X (M) .

[0084] Step 2: Calculate the maximum Hamming autocorrelation function value of individuals in the initial population or the superior population obtained in the previous iteration, construct the fitness function, and calculate the fitness of individuals.

[0085] In the process of population evolution, it is necessary to minimize the Hamming autocorrelation function of individuals. To avoid the direct use of the objective function leading to super-evolved individuals dominating the evolutionary direction, this scheme adopts a proportional bound construction method to design the fitness function. Specifically, the objective function is set to minimize the maximum Hamming autocorrelation function value of an individual, and its fitness function is constructed as follows.

[0086] First, construct an objective function f(X) that aims to maximize the individual Hamming autocorrelation function. (m) ):

[0087] Among them, X (m) This represents the m-th individual in the initial population or the superior population obtained in the previous iteration; if it is the initial population, then m = 1, 2, ..., M; if it is the superior population obtained in the previous iteration, then m = 1, 2, ..., U, where U is the number of individuals in the superior population obtained in the previous iteration.

[0088] Secondly, when the objective function f(X) (m) When the value is the maximum Hamming autocorrelation function, the fitness function is defined as:

[0089] Among them, Fit(f(X) (m) For individual X (m) The fitness of the parameter c is the lower bound constraint term for the Hamming autocorrelation function value.

[0090] The design of the fitness function achieves two key functions by introducing a lower bound constraint term c: (1) ensuring that the denominator is always positive and maintaining the validity of the function; (2) compressing the fitness differences of super-evolved individuals and preventing them from prematurely dominating the population. This method transforms the problem of minimizing the objective function into a search for maximizing the fitness value, while ensuring the stability of the population evolution process.

[0091] Step 3: Based on the individual's fitness, calculate the probability that the individual will be selected as the parent population, and select individuals that meet the requirements as the parent population based on the probability.

[0092] Step 2 calculates the fitness of M individuals, and step 3 calculates the probability of each individual being selected as a parent population based on the fitness. The selection operation involves choosing qualified individuals from the initial population or the superior population obtained in the previous iteration as parent populations to participate in subsequent recombination and / or mutation operations, so as to pass on the genes of the parent population to the offspring population.

[0093] This scheme uses a proportional fitness allocation method, where the current individual X... (m) The probability of being selected is the ratio of its fitness to the sum of the fitness of all individuals in the population. If the initial population size is M or the size of the superior population obtained in the previous iteration is U, where the m-th individual X... (m) The fitness is Fit(f(X) (m) Then individual X (m) The probability of being selected is:

[0094] Among them, P m This represents the probability that the m-th individual is selected.

[0095] As one possible method for determining whether the requirements are met, a probability threshold is set as follows: ,like Then select individual X (m) As the parent population.

[0096] As another possible method to determine whether the requirements are met, the np.random.choice function is called based on the probabilities P of the M individuals. m Select the parent population from M individuals.

[0097] In other words, step 3 can select individuals that meet certain conditions from the initial population or the superior population obtained from the previous iteration as the parent population in two ways. The first condition is to set a probability threshold. The second condition is to call the `np.random.choice` function to make the selection. It should be noted that the `np.random.choice` function is a current technology that calculates the probability P for each individual. m Then, the np.random.choice function is called directly to select the parent population, and the specific function of np.random.choice will not be elaborated here.

[0098] Step 4: Randomly select 2 individuals from the parent population each time, and directly add the individual with the lowest Hamming autocorrelation to the superior population obtained in this iteration; or perform recombination and / or mutation operations on the 2 selected individuals, and add the individual with the lowest Hamming autocorrelation to the superior population obtained in this iteration.

[0099] Assume that in step 3, N individuals are selected as the parent population (N<M or N<U), and in step 4, 2 individuals are randomly selected without replacement from the N parent individuals each time, which are used as the paternal sequence and the maternal sequence respectively. For any two individuals selected each time, any one of the following four methods is selected for execution until all individuals in the parent population have been selected (if N is an odd number, the last individual is directly added to the excellent population obtained in this iteration):

[0100] (A1) Directly add the paternal sequence and the maternal sequence into the excellent population obtained in this iteration;

[0101] (A2) After performing a recombination operation on the paternal sequence and the maternal sequence, two offspring sequences are obtained, then calculate the Hamming autocorrelation function of the two offspring sequences respectively, and add the sequence with the smaller Hamming autocorrelation function into the excellent population obtained in this iteration;

[0102] (A3) After performing a mutation operation on the paternal sequence and the maternal sequence, calculate the Hamming autocorrelation function of the paternal sequence and the maternal sequence respectively, and add the sequence with the smaller Hamming autocorrelation function into the excellent population obtained in this iteration;

[0103] (A4) After sequentially performing a recombination operation and a mutation operation on the paternal sequence and the maternal sequence, two offspring sequences are obtained, then calculate the Hamming autocorrelation function of the two offspring sequences respectively, and add the sequence with the smaller Hamming autocorrelation function into the excellent population obtained in this iteration.

[0104] After all individuals in the parent population in step 4 have been selected, step 4 is ended, and the excellent population under this iteration is obtained. The excellent population contains at least one individual, then the process proceeds to step 5. The above four methods are respectively introduced below.

[0105] (A1) Two individuals are randomly selected from the parent population composed of N individuals to serve as the paternal sequence and the maternal sequence , m1∈N, m2∈N, N is the number of individuals in the parent population, is the i-th frequency point in the paternal sequence , is the i-th frequency point in the maternal sequence ; the paternal sequence X (m1) and the maternal sequence X (m2) are directly placed into the excellent population obtained in this iteration.

[0106] (A2) Two individuals are randomly selected from the parent population composed of N individuals to serve as the paternal sequence and the maternal sequence Let m1∈N, m2∈N; Discrete recombination, intermediate recombination, or linear recombination are performed on the frequency points of the two sequences to obtain two child sequences, denoted as follows: and , For offspring sequence The i-th frequency point in For offspring sequence The i-th frequency point in.

[0107] Specifically, this scheme has three recombination methods: discrete recombination, intermediate recombination, and linear recombination. When performing recombination, one of these methods can be selected with a 1 / 3 probability.

[0108] a. Discrete recombination: For each frequency point in the offspring sequence, a frequency point at the same position in the parent and mother sequences is randomly selected with a probability of 0.5. The inheritance rule for each frequency point is as follows:

[0109] In other words, at the frequency point at the i-th position, if Choose ,but choose ;if Choose ,but choose .

[0110] b. Intermediate recombination: The frequency points of the offspring sequence are calculated based on the frequency points at the same positions in the parent and mother sequences. The calculation formula is as follows:

[0111] Among them, the scaling factor and All satisfied , As a limit parameter, it is generally taken as... .

[0112] c. Linear recombination: Similar to intermediate recombination, the difference is that linear recombination uses a fixed scaling factor, i.e.:

[0113] Among them, the scaling factor and All satisfied , As a limit parameter, it is generally taken as... .

[0114] This scheme adopts a hybrid recombination strategy. During each recombination in the evolutionary process, discrete recombination, intermediate recombination, or linear recombination are dynamically called with an equal probability (1 / 3) selection mechanism to achieve multi-dimensional fusion and combination optimization of gene information (i.e. frequency points).

[0115] Finally, calculate the offspring sequence C separately. (1) and C (2) The Hamming autocorrelation function is used to select offspring sequences with lower Hamming autocorrelation values ​​to add to the superior population obtained in this iteration. For example, if the offspring sequence C... (1) The Hamming autocorrelation function is compared to the offspring sequence C. (2) If the Hamming autocorrelation function is small, then the offspring sequence C (1) Add the superior population obtained in this iteration, and remove the offspring sequence C. (2) .

[0116] (A3). Randomly select 2 individuals from the parent population consisting of N individuals as the parent sequence. and parental sequence m1∈N, m2∈N; the parent sequence X is perturbed by applying a Gaussian perturbation or an XOR perturbation. (m1) and parental sequence X (m2) The mutation operation is performed to obtain two mutated parent and mother sequences.

[0117] This scheme offers two mutation operations: Gaussian perturbation and XOR perturbation. Only one of these operations needs to be selected when performing the mutation operation. The two mutation operations will be described in detail below.

[0118] a. Gaussian perturbation.

[0119] Set the mutation probability P s (e.g., P) s =0.01), randomly generate a random number e in the range (0,1), and combine the random number e with P. s For comparison, if e≤P s Then, according to the following formula, the parent sequence X is processed respectively. (m1) any frequency point and parental sequence X (m2) any frequency point Apply Gaussian perturbation:

[0120] in, The frequency points in the parent sequence before applying the Gaussian perturbation; The frequency points in the parent sequence after applying a Gaussian perturbation; The frequency points in the parent generation sequence before applying Gaussian perturbation; The frequency points in the parent generation sequence after applying a Gaussian perturbation; `step` is the step size factor (controlling the frequency adjustment amplitude); `rand()` is a random number following a Gaussian normal distribution [0,1). This Gaussian perturbation mechanism helps to achieve fine-grained local search while overcoming the limitations of a fixed step size. It should be noted that the frequency points in the parent sequence subjected to Gaussian perturbation... and frequency points in the parent sequence subjected to Gaussian perturbation The positions i are not necessarily the same.

[0121] Gaussian perturbation can be applied to only one frequency point in each parent sequence and one frequency point in each parent sequence. Furthermore, to avoid the value of the frequency point after applying Gaussian perturbation exceeding the discrete character set... The range of frequencies after applying Gaussian perturbation is determined through modulo operation. The loop is mapped to the range [0, q), where r equals 1 or 2 to ensure the validity of the frequency points.

[0122] in, Indicates the frequency at which the Gaussian perturbation is applied; express The frequency points after modulo operation (r=1 represents the parent sequence, r=2 represents the mother sequence) are used... Replace the original This results in the parent sequence X after applying a Gaussian perturbation. (m1) and parental sequence X (m2) .

[0123] Finally, calculate the parent sequence X after applying Gaussian perturbation. (m1) and parental sequence X (m2) The Hamming autocorrelation function is used to select sequences with lower Hamming autocorrelation values ​​to add to the superior population obtained in this iteration. For example, the parent sequence X after applying Gaussian perturbation. (m1) The Hamming autocorrelation function is less than that of the parent sequence X. (m2) If the Hamming autocorrelation function is obtained, then the parent sequence X after applying Gaussian perturbation is selected. (m1) The superior population obtained in this iteration is added to the parent sequence X after applying Gaussian perturbation, while the parent sequence X is removed. (m2) .

[0124] If the generated random number e > P s Then do not apply to the parent sequence X (m1) and parental sequence X (m2) Apply a Gaussian perturbation to the parent sequence X (m1) and parental sequence X (m2) They are directly added to the superior population obtained in this iteration.

[0125] b. XOR perturbation.

[0126] First, a reference sequence of length J (with frequency points in the reference sequence being 0 or 1) is generated using a linear feedback shift register (LFSR) as a disturbance source.

[0127] Next, the parent sequence X of length n... (m1) The value of each frequency point is converted into 8 bits to form a binary sequence X. (m1) Its length becomes 8n;

[0128] Then, the binary sequence X (m1) Perform a bitwise XOR operation with the reference sequence (8n≤J). If the frequency values ​​at the same position in both sequences are equal, the frequency value at that position is 0; if the frequency values ​​at the same position are not equal, the frequency value at that position is 1. This generates a binary sequence B={b1,b2,...,b...} of length 8n containing only 0s and 1s. 8n}, b h The values ​​are either 0 or 1, and h = 1, 2, ..., 8n, to achieve pseudo-random perturbation;

[0129] Next, according to the w-bit (q=2) w To restore the bits of the binary sequence B, that is, in the binary sequence B of length 8n, every w bits are truncated to form a new frequency point, and only the first n frequency points are retained. In this way, B is restored to decimal form to obtain the parent sequence X. (m1) The corresponding new frequency hopping sequence W1={a0,a1,...,a2} of length n n-1};

[0130] Similarly, the parent generation sequence X can also be obtained in the same way. (m2) The corresponding new frequency hopping sequence W2={a0,a1,...,a1} of length n n-1};

[0131] Finally, the Hamming autocorrelation functions of W1 and W2 are calculated separately, and the sequence with the smaller Hamming autocorrelation function is added to the superior population obtained in this iteration. For example, if the Hamming autocorrelation function of W1 is smaller than that of W2, then W1 is added to the superior population obtained in this iteration, while W2 is removed.

[0132] (A4). Randomly select 2 individuals from the parent population consisting of N individuals as the parent sequence. and parental sequence Given two sequences, m1∈N and m2∈N, perform discrete recombination, intermediate recombination, or linear recombination on the frequency points of the two sequences to obtain two offspring sequences, denoted as follows: and .

[0133] Specifically, this scheme has three recombination methods: discrete recombination, intermediate recombination, and linear recombination. When performing recombination, one of these methods can be selected with a 1 / 3 probability.

[0134] a. Discrete recombination: For each frequency point in the offspring sequence, a frequency point at the same position in the parent and mother sequences is randomly selected with a probability of 0.5. The inheritance rule for each frequency point is as follows:

[0135] In other words, at the frequency point at the i-th position, if Choose ,but choose ;if Choose ,but choose .

[0136] b. Intermediate recombination: The frequency points of the offspring sequence are calculated based on the frequency points at the same positions in the parent and mother sequences. The calculation formula is as follows:

[0137] Among them, the scaling factor and All satisfied , As a limit parameter, it is generally taken as... .

[0138] c. Linear recombination: Similar to intermediate recombination, the difference is that linear recombination uses a fixed scaling factor, i.e.:

[0139] Among them, the scaling factor and All satisfied , As a limit parameter, it is generally taken as... .

[0140] The offspring sequence C is subjected to either a Gaussian perturbation or an XOR perturbation. (1) and offspring sequence C (2) A mutation operation is performed to obtain two mutated offspring sequences.

[0141] a. Gaussian perturbation.

[0142] Set the mutation probability P s (e.g., P) s =0.01), randomly generate a random number e in the range (0,1), and combine the random number e with P. sFor comparison, if e≤P s Then, the offspring sequence C is processed according to the following formula. (1) any frequency point and offspring sequence C (2) any frequency point Apply Gaussian perturbation:

[0143] in, For the offspring sequence C (1) The frequency point before applying the Gaussian perturbation; For the offspring sequence C (1) The frequency point after applying a Gaussian perturbation; For the offspring sequence C (2) The frequency point before applying the Gaussian perturbation; For the offspring sequence C (2) The frequency point after applying a Gaussian perturbation; `step` is the step size factor; `rand()` is a random number following a Gaussian normal distribution [0,1). It should be noted that the offspring sequence C subjected to Gaussian perturbation... (1) frequency points and the offspring sequence C subjected to Gaussian perturbation (2) frequency points The positions i are not necessarily the same.

[0144] Only for the offspring sequence C (1) A frequency point and offspring sequence C (2) A Gaussian perturbation can be applied to one frequency point. Furthermore, to avoid the value of the frequency point after applying the Gaussian perturbation exceeding the discrete character set... The range of frequencies after applying Gaussian perturbation is determined through modulo operation. The loop is mapped to the range [0, q), where r equals 1 or 2 to ensure the validity of the frequency points.

[0145] in, Indicates the frequency at which the Gaussian perturbation is applied; express The frequency point after modulo operation (r=1 represents the offspring sequence C) (1) When r=2, it represents the offspring sequence C. (2) ),use Replace the original This results in the offspring sequence C after applying a Gaussian perturbation. (1) and offspring sequence C (2) .

[0146] Finally, the offspring sequence C after applying Gaussian perturbation is calculated respectively. (1)and offspring sequence C (2) The Hamming autocorrelation function is used to select sequences with lower Hamming autocorrelation values ​​to add to the superior population obtained in this iteration. For example, the offspring sequence C after applying Gaussian perturbation. (1) The Hamming autocorrelation function is less than that of the offspring sequence C. (2) If the Hamming autocorrelation function is obtained, then the offspring sequence C after applying Gaussian perturbation is selected. (1) The superior population obtained in this iteration is added to the population, while the offspring sequence C after applying Gaussian perturbation is removed. (2) .

[0147] If the generated random number e > P s Then do not apply to the offspring sequence C (1) and offspring sequence C (2) Apply a Gaussian perturbation to the offspring sequence C (1) and offspring sequence C (2) They are directly added to the superior population obtained in this iteration.

[0148] b. XOR perturbation.

[0149] First, a reference sequence of length J (with frequency points in the reference sequence being 0 or 1) is generated using a linear feedback shift register (LFSR) as a disturbance source.

[0150] Next, the child sequence C of length n... (1) The value of each frequency point is converted into 8 bits to form a binary sequence C. (1) Its length becomes 8n;

[0151] Then, the binary sequence C (1) Perform a bitwise XOR operation with the reference sequence (8n≤J). If the frequency values ​​at the same position in both sequences are equal, the frequency value at that position is 0; if the frequency values ​​at the same position are not equal, the frequency value at that position is 1. This generates a binary sequence B={b1,b2,...,b...} of length 8n containing only 0s and 1s. 8n}, b h The values ​​are either 0 or 1, and h = 1, 2, ..., 8n, to achieve pseudo-random perturbation;

[0152] Next, according to the w-bit (q=2) w To restore the bits of the binary sequence B, that is, in the binary sequence B of length 8n, every w bits are truncated to form a new frequency point, and only the first n frequency points are retained. In this way, B is restored to decimal form, and the offspring sequence C is obtained. (1) The corresponding new frequency hopping sequence W1={a0,a1,...,a2} of length n n-1};

[0153] Similarly, the offspring sequence C can also be obtained in the same way. (2) The corresponding new frequency hopping sequence W2={a0,a1,...,a1} of length n n-1};

[0154] Finally, the Hamming autocorrelation functions of W1 and W2 are calculated separately, and the frequency hopping sequence with the smaller Hamming autocorrelation function is selected to be added to the superior population obtained in this iteration. For example, if the Hamming autocorrelation function of W1 is smaller than that of W2, then W1 is added to the superior population obtained in this iteration, while W2 is removed.

[0155] In summary, each time two individuals are selected from the parent population of N individuals to perform any one of the methods A1, A2, A3, and A4, the resulting superior population after this iteration will have U individuals, where U ≤ N. If the two individuals selected from the parent population each time only perform the case of method A1 or A3 without applying Gaussian perturbation, then the number of individuals U in the final superior population will be equal to the number of individuals N in the parent population. However, in order to narrow down the range of superior populations after multiple iterations and select the optimal wide-interval frequency hopping sequence, in the preferred case, methods A1, A2, A3, and A4 should all be selected. Therefore, U is usually... <N。

[0156] Furthermore, a wide-interval treatment is applied to each individual in the superior population. Let any individual in the superior population be represented as W = {a0, a1, ..., a...}. n-1 If two adjacent frequency points in this individual have Then calculate ,use Replace the original Thus, a frequency-hopping sequence W that satisfies the wide-interval requirement is obtained. Furthermore, all individuals in the superior population are frequency-hopping sequences that satisfy the wide-interval requirement.

[0157] Step 5: Return to step 2 until the preset maximum number of iterations is reached or an individual in the superior population satisfies the maximum Hamming autocorrelation function.

[0158] As one possible implementation method, the maximum number of iterations is preset to be... (like The superior population obtained in step 4 is returned to step 2. When the iteration number t is greater than 1, the process continues. When the time comes, terminate the algorithm and set the first... All frequency hopping sequences obtained from the superior population are used as the optimal wide-interval frequency hopping sequences for output.

[0159] As another possible implementation, a maximum Hamming autocorrelation function bound is preset. In the t-th iteration, if the Hamming autocorrelation function of any frequency hopping sequence in the superior population is less than or equal to bound, the algorithm is terminated, and the frequency hopping sequence is used as the optimal wide-interval frequency hopping sequence for output.

[0160] Example 2:

[0161] This embodiment experimentally verifies the scheme of Embodiment 1. The optimal wide-interval frequency hopping sequence is generated according to the following parameter definitions.

[0162] Frequency slot q: q∈{7,8,...,17};

[0163] Sequence length n: n∈{q,3q+1};

[0164] As shown in Tables 1-1, 1-2, 1-3, and 1-4, when the frequency gap q=11, this scheme can find the optimal wide-interval frequency hopping sequence with sequence length n∈{q,3q+1} and wide interval d=[2,3,4] that satisfies the bound.

[0165] Table 1-1 Optimal Wide-Interval Frequency Hopping Sequences with Sequence Length n∈{11,19}

[0166] Table 1-2 Optimal Wide-Interval Frequency Hopping Sequences with Sequence Length n∈{20,26}

[0167] Table 1-3 Optimal Wide-Interval Frequency Hopping Sequences with Sequence Length n∈{27,30}

[0168] Table 1-4 Optimal Wide-Interval Frequency Hopping Sequences with Sequence Length n∈{31,34}

[0169] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for generating optimal wide-interval frequency hopping sequences based on a genetic algorithm, characterized in that, If the maximum Hamming autocorrelation function H(X) of the frequency hopping sequence X satisfies the lower bound established by the following formula, then X is called the optimal wide-interval frequency hopping sequence: ； in, As intermediate parameters, we have ; This is the floor rule. This is the floor operator; The method includes the following steps: Step 1: Generate several initial frequency hopping sequences using Bernoulli chaotic mapping, and generate an initial population by performing frequency point mapping and wide interval processing on the initial frequency hopping sequences. Step 1 specifically includes the following steps. Step 1-1: Generate several initial frequency hopping sequences using Bernoulli chaotic mapping: ； in, This represents the control parameters for the m-th initial frequency hopping sequence; This represents the frequency point with index k in the m-th initial frequency hopping sequence; This represents the frequency point with index k+1 in the m-th initial frequency hopping sequence; k = 0, 1, ..., K-1; K is the length of the initial frequency hopping sequence; Generate M initial frequency hopping sequences Z using Bernoulli chaotic mapping (1) Z (2) ,...,Z (M) m = 1, 2, ..., M, where M is the total number of initial frequency hopping sequences; Steps 1-2 employ an interval sampling strategy, sampling from each initial frequency hopping sequence Z. (m) A chaotic frequency hopping sequence of length n is generated by extracting 1 frequency point every 5 bits. , where Y (m) Let m be the chaotic frequency hopping sequence corresponding to the m-th initial frequency hopping sequence. For Y (m) The i-th frequency point in the sequence, i=0,1,…,n-1; can yield M chaotic frequency hopping sequences Y. (1) ,Y (2) ,...,Y (M) ; Steps 1-3, convert the chaotic frequency hopping sequence Linear mapping to discrete character sets q represents the frequency slot, generating the corresponding frequency point. ; To round down, the m-th chaotic frequency hopping sequence Y is formed. (m) Corresponding frequency hopping sequence candidate individuals ; Steps 1-4, if Then calculate ,use Replace the original Where d is the minimum frequency interval, shift is a random variable, and mod is the modulo operation; M candidate individuals X of frequency hopping sequences that meet the wide interval requirement can be obtained. (m) That is, the initial population X (1) ,X (2) ,...,X (M) ; Step 2: Calculate the maximum Hamming autocorrelation function value of individuals in the initial population or the superior population obtained in the previous iteration, construct the fitness function, and calculate the fitness of individuals; Step 2 specifically involves, Construct an objective function f(X) that aims to maximize the individual Hamming autocorrelation function. (m) ): ； Among them, X (m) The m-th individual in the initial population or the superior population obtained in the previous iteration; For X (m) Hamming autocorrelation function; For frequency point delay; When the objective function f(X) (m) When the value is the maximum Hamming autocorrelation function, the fitness function is defined as: ； Among them, Fit(f(X) (m) For individual X (m) The fitness; parameter c is the lower bound constraint term for the Hamming autocorrelation function value; Step 3: Based on the individual's fitness, calculate the probability that the individual will be selected as the parent population, and select individuals that meet the requirements as the parent population based on the probability. Step 4: Randomly select 2 individuals from the N parent individuals without replacement each time, to form the parent sequence and the mother sequence respectively; for each selected 2 individuals, any one of the following four methods can be executed, and methods A1, A2, A3, and A4 must be selected in each iteration, until all individuals in the parent population have been selected: (A1). Directly add the parent and mother sequences to the superior population obtained in this iteration; (A2). After recombination of the parent and mother sequences, two offspring sequences are obtained. The Hamming autocorrelation function of the two offspring sequences is then calculated. The sequence with the smaller Hamming autocorrelation function is added to the superior population obtained in this iteration. (A3). After performing mutation operations on the parent and mother sequences, calculate the Hamming autocorrelation function of the parent and mother sequences respectively, and add the sequences with smaller Hamming autocorrelation functions to the superior population obtained in this iteration; (A4). After performing recombination and mutation operations on the parent and mother sequences in sequence, two offspring sequences are obtained. The Hamming autocorrelation function of the two offspring sequences is then calculated. The sequence with the smaller Hamming autocorrelation function is added to the superior population obtained in this iteration. In step 4, the step of randomly selecting two individuals from the parent population each time and performing mutation operations on the selected two individuals is as follows: A reference sequence of length J is generated using a linear feedback shift register, where the frequency points in the reference sequence are either 0 or 1. The parent sequence X of length n (m1) The value of each frequency point is converted into 8 bits to form a binary sequence X. (m1) Its length becomes 8n; The binary sequence X (m1) Perform a bitwise XOR operation with the reference sequence, where 8n ≤ J. If the frequency values ​​at the same position in both sequences are equal, then the frequency value at that position is 0; if the frequency values ​​at the same position in both sequences are not equal, then the frequency value at that position is 1. This generates a binary sequence B = {b1, b2, ..., bn} of length 8n that contains only 0s and 1s. 8n }, b h Equal to 0 or 1, h = 1, 2, ..., 8n; Reconstruct the bits of binary sequence B according to the w bits, q=2 w In a binary sequence B of length 8n, every w bits are truncated to form a new frequency point, and only the first n frequency points are retained. In this way, B is restored to decimal form to obtain the parent sequence X. (m1) The corresponding new frequency hopping sequence W1={a0,a1,...,a2} of length n n-1 }; Similarly, the parent generation sequence X is obtained in the same way. (m2) The corresponding new frequency hopping sequence W2={a0,a1,...,a1} of length n n-1 }; Next, calculate the Hamming autocorrelation function of W1 and W2 respectively, and add the frequency hopping sequence with the smaller Hamming autocorrelation function to the superior population obtained in this iteration; Step 5: Return to step 2 until the preset maximum number of iterations is reached or an individual in the superior population satisfies the maximum Hamming autocorrelation function.

2. The method for generating optimal wide-interval frequency hopping sequences based on a genetic algorithm according to claim 1, characterized in that, Step 3 specifically involves, Calculate the probability that an individual is selected as a parent in the population: ； Among them, P m This represents the probability that the m-th individual is selected; Set the probability threshold to ,like Then select individual X (m) As the parent population; Alternatively, you can call the np.random.choice function based on the probability P of M individuals. m Select the parent population from M individuals.

3. The method for generating optimal wide-interval frequency hopping sequences based on a genetic algorithm according to claim 1, characterized in that, Step 4, which involves randomly selecting two individuals from the parent population each time and performing a recombination operation on the selected two individuals, includes: Two individuals are randomly selected from the parent population consisting of N individuals to serve as the parent sequence. and parental sequence m1∈N, m2∈N, where N is the number of individuals in the parent population. Parental sequence The i-th frequency point in For the mother generation sequence The i-th frequency point in the sequence; the frequency points in the two sequences are discretely recombined, intermediately recombined, or linearly recombined to obtain two offspring sequences, denoted as follows: and , For offspring sequence The i-th frequency point in For offspring sequence The i-th frequency point in the spectrum, i=0,1,…,n-1; The discrete recombination process is as follows: ； At the frequency point at the i-th position, if Choose ,but choose ;if Choose ,but choose ; The intermediate recombination process is as follows: ； Among them, the scaling factor and All satisfied , These are limit parameters; The linear recombination process is as follows: ； Among them, the scaling factor and All satisfied .

4. The method for generating optimal wide-interval frequency hopping sequences based on a genetic algorithm according to claim 1, characterized in that, Following step 4, the following steps are also included: For each individual in the superior population, a wide-interval treatment is applied again. Let any individual in the superior population be represented as W = {a0, a1, ..., a...} n-1 }, a i Let i be the i-th frequency hopping point in W, i = 0, 1, ..., n-1. If two adjacent frequency points in this individual have Then calculate d represents the minimum frequency interval, shift represents a random variable, and mod represents modulo operation. Replace the original Thus, a frequency hopping sequence W that meets the wide spacing requirement is obtained.

Images