Transmit waveform pattern matching method based on minimax principle and with similarity constraint and peak-to-average value constraint

Abstract

This invention discloses a method for matching transmitted waveform patterns based on the minimax principle, while also incorporating similarity and peak-to-means constraints. First, the minimax principle is applied to the objective function of the transmitted waveform design problem to avoid the problem of the least squares index overemphasizing the edge fitting of the pattern and failing to generate a flat main lobe. Second, a direct optimization algorithm design concept is adopted to avoid the performance loss of the "two-stage" approach. Finally, by considering similarity and peak-to-means constraints to update the values ​​of the transmitted waveform and shadow variables, the optimized waveform better conforms to actual engineering conditions.

Description

Technical Field

[0001] This invention belongs to the field of radar signal processing technology, specifically relating to a method for matching transmitted waveform patterns based on the minimum-maximum principle and incorporating similarity constraints and peak-to-average constraints in radar signal processing technology. Background Technology

[0002] Transmit waveform pattern matching performance is a fundamental and crucial factor in multiple-input multiple-output (MIMO) radar systems. Transmit waveform pattern matching refers to the process of optimizing the transmitted waveform to fit a specific functional waveform pattern to the output waveform. This waveform pattern has significant application value in 5G millimeter-wave communication systems and MIMO radar systems. In 5G millimeter-wave communication systems, beam training based on analog beamcodebooks is typically used to estimate the direction of the main signal ray, where each analog beamcodebook is designed with the desired shape.

[0003] In multiple-input multiple-output (MIMO) radar systems, the transmitted waveform of the antenna can be pre-programmed through simulation modeling to maximize the power at the target location. This optimization is typically achieved by pre-programming a pattern approximating a given direction. Existing beamforming techniques generally employ two methods: one is to directly match the direction pattern vector controlled by the simulated beamforming to a predefined direction pattern shape function; the other is to maximize the main lobe gain and appropriately suppress the side lobes to obtain the maximum beam gain. Both methods require the application of constraints commonly found in communication engineering, namely similarity constraints and peak-to-mean-average (PMA) constraints.

[0004] For example, P. Stoica et al. proposed a classic "two-stage" scheme to generate the transmission waveform in 2008: first, optimize the covariance matrix of the transmission waveform, and then generate the transmission waveform that meets the requirements from the covariance matrix based on the constraints. The algorithm for the first stage is referenced in [1], and the algorithm for the second stage is referenced in [2].

[0005] However, existing algorithms for generating transmitted waveforms have the following drawbacks:

[0006] (1) The waveform pattern matching index in the prior art is least squares, which focuses too much on fitting the edge of the pattern and cannot produce a flat main lobe.

[0007] (2) The process of generating the transmitted waveform from the covariance matrix is ​​an approximate operation, which results in performance loss and the generated waveform is a suboptimal solution to the overall optimization problem.

[0008] (3) The constraints considered by the algorithms in the prior art are limited and do not take into account more practical constraints, such as similarity constraints and peak mean constraints.

[0009] References:

[0010] [1] P.Stoica, J.Li, and Y.Xie, "On probing signal design for MIMO radar," IEEE Transactions on Signal Processing, vol.55, no.8, pp.4151–4161, 2007.

[0011] [2] P.Stoica, J.Li, and X.Zhu, "Waveform synthesis for diversity-basedtransmit beampattern design," IEEE Transactions on Signal Processing, vol.56, no.6, pp.2593–2598, 2008. Summary of the Invention

[0012] To address the aforementioned problems in existing technologies, the present invention aims to provide a transmission waveform pattern matching method based on the minimum-maximum principle, while also incorporating similarity constraints and peak-to-means constraints. This method avoids the problem that the least squares index overemphasizes the edge of the fitted pattern, thus failing to produce a flat main lobe. It also avoids the performance loss of the "two-segment" scheme, and the designed waveform is more in line with actual engineering conditions.

[0013] The technical solution adopted in this invention is as follows:

[0014] A method for matching emitted waveform patterns based on the minimax principle, while also incorporating similarity and peak-to-mean constraints, includes the following steps:

[0015] Step 1: Establish an initial model of the transmitted waveform to be optimized based on the main parameters of the multiple input multiple output radar system; use the minimum-maximum principle to generate a flat main lobe;

[0016] Step 2: Introduce several intermediate variables and transform the initial model of the transmitted waveform to be optimized into a transitional model through equivalent transformation;

[0017] Step 3: Initialize the variables in the transition model to obtain the intermediate model;

[0018] Step 4: Apply peak-means constraints and similarity constraints to the original variables in the intermediate model to perform update calculations and obtain the final model;

[0019] Step 5: Based on the judgment principle, compare and judge whether the final model meets the target requirements. If it meets the target requirements, use the current value of the final model in Step 4 as the final output waveform. If it does not meet the target requirements, return to Step 4 and repeat the update calculation of each original variable in the intermediate model until the target requirements are met.

[0020] Furthermore, in the first step, a known waveform of the same length as the detection sequence of the transmitted waveform to be optimized is specified as the reference waveform. Based on the upper bound of the Euclidean distance between the transmitted waveform to be optimized and the reference waveform, the total number of transmitting antennas of the multi-input multi-output radar system, the detection sequence length of the multi-input multi-output radar system, the total energy of the transmitted waveform to be optimized, and the upper bound of the energy of each element modulus, the function formula of the transmitted waveform pattern set is obtained in combination with the total number of target azimuth angles, and the initial model of the transmitted waveform to be optimized is established.

[0021] Furthermore, in the first step, the length of the transmitted waveform to be optimized is a complex vector of the total number of transmitting antennas of the multi-input multi-output radar system and the length of the detection sequence of the multi-input multi-output radar system.

[0022] Furthermore, in the first step, the total energy of the transmitted waveform to be optimized is the sum of the squares of the moduli of each element.

[0023] Furthermore, after introducing several intermediate variables in the second step, such as the pattern scale transformation variable of the transmitted waveform to be optimized, the shadow variable of the transmitted waveform to be optimized, and intermediate auxiliary variables, the initial mathematical model is converted into a transitional model by decoupling variables and performing an equivalent transformation based on the Lagrange function.

[0024] Furthermore, in the third step, the transmit waveform to be optimized and the shadow variable in the transition model are initialized to the same value, and the intermediate model is obtained by initializing the original variables and dual variables in the transition model.

[0025] Furthermore, in the fourth step, the update values ​​of the pattern scale transformation variable, the shadow variable of the transmitted waveform to be optimized, and the intermediate auxiliary variable in the transition model of the transmitted waveform to be optimized are solved according to the corresponding function formulas, as well as the update value of the transmitted waveform; the update values ​​of each intermediate variable and the update value of the transmitted waveform are all taken as the optimal solutions of their respective function formulas.

[0026] Furthermore, in the fifth step, the Lagrange dual variable is updated based on the updated value obtained in the fourth step, and the result of the Lagrange dual variable is compared to determine whether the target requirement is met.

[0027] Furthermore, the function formula for the upper bound of the Euclidean distance between the transmitted waveform to be optimized and the reference waveform includes the number of iterations. The result obtained after updating the Lagrange dual variable is also a function formula that includes the number of iterations. By comparing the value of the number of iterations with the preset critical value, it is determined whether the current updated value of the transmitted waveform meets the target requirements.

[0028] Finally, the preset threshold value is 100. If the number of iterations is greater than or equal to 100, the current updated value of the transmitted waveform is used as the final output transmitted waveform; otherwise, the number of iterations is incremented by 1, and then the process returns to step four to repeatedly solve for the updated values ​​of each intermediate variable and the updated value of the transmitted waveform, and to update the Lagrange dual variable until the number of iterations is greater than or equal to 100.

[0029] The beneficial effects of this invention are as follows:

[0030] A transmit waveform pattern matching method based on the minimax principle, combined with similarity constraints and peak-to-means constraints, is proposed. First, the minimax principle is applied to the objective function of the transmit waveform design problem to avoid the problem that the least squares index focuses too much on fitting the edge of the pattern and cannot generate a flat main lobe. Second, a direct optimization algorithm design concept is adopted to avoid the performance loss of the "two-stage" scheme. Finally, by considering similarity constraints and peak-to-means constraints to update the values ​​of the transmit waveform and shadow variables, the optimized waveform is made to better conform to the actual engineering conditions. Attached Figure Description

[0031] Figure 1 This is a schematic diagram illustrating the process principle of the emission waveform pattern matching method based on the minimum-maximum principle and combined with similarity constraints and peak-to-average constraints according to Embodiment 1 of the present invention. Detailed Implementation

[0032] 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. All other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] like Figure 1 As shown, the overall inventive concept of this invention is dedicated to improving the traditional "two-stage" waveform generation algorithm, and proposes a transmission waveform pattern matching method that generates a flat main lobe, achieves direct optimization, and considers similarity constraints and peak-to-average constraints.

[0034] First, the minimum-maximum principle is used to avoid the problem that the least squares index focuses too much on the edge of the fitted pattern and cannot produce a flat main lobe. Second, the algorithm design concept of direct optimization is adopted to avoid the performance loss of the "two-stage" scheme. Finally, by considering similarity constraints and peak-to-means constraints, the designed waveform is more in line with the actual engineering conditions.

[0035] The overall planning scheme is as follows:

[0036] A method for matching emitted waveform patterns based on the minimax principle, while also incorporating similarity and peak-to-mean constraints, includes the following steps:

[0037] Step 1: Establish an initial model of the transmitted waveform to be optimized based on the main parameters of the multiple input multiple output radar system; use the minimum-maximum principle to generate a flat main lobe;

[0038] Step 2: Introduce several intermediate variables and transform the initial model of the transmitted waveform to be optimized into a transitional model through equivalent transformation;

[0039] Step 3: Initialize the variables in the transition model to obtain the intermediate model;

[0040] Step 4: Apply peak-means constraints and similarity constraints to the original variables in the intermediate model to perform update calculations and obtain the final model;

[0041] Step 5: Based on the judgment principle, compare and judge whether the final model meets the target requirements. If it meets the target requirements, use the current value of the final model in Step 4 as the final output waveform. If it does not meet the target requirements, return to Step 4 and repeat the update calculation of each original variable in the intermediate model until the target requirements are met.

[0042] Furthermore, in the first step, a known waveform of the same length as the detection sequence of the transmitted waveform to be optimized is specified as the reference waveform. Based on the upper bound of the Euclidean distance between the transmitted waveform to be optimized and the reference waveform, the total number of transmitting antennas of the multi-input multi-output radar system, the detection sequence length of the multi-input multi-output radar system, the total energy of the transmitted waveform to be optimized, and the upper bound of the energy of each element mode, combined with the total number of target azimuth angles, the function expression of the transmitted waveform pattern set is obtained, and the initial model of the transmitted waveform to be optimized is established.

[0043] Furthermore, in the first step, the length of the transmitted waveform to be optimized is a complex vector of the total number of transmitting antennas of the multi-input multi-output radar system and the length of the detection sequence of the multi-input multi-output radar system.

[0044] Furthermore, in the first step, the total energy of the transmitted waveform to be optimized is the sum of the squares of the moduli of each element.

[0045] Furthermore, after introducing several intermediate variables in the second step, namely the pattern scale transformation variable of the transmitted waveform to be optimized, the shadow variable of the transmitted waveform to be optimized, and intermediate auxiliary variables, the initial mathematical model is converted into a transitional model by decoupling variables and performing an equivalent transformation based on the Lagrange function.

[0046] Furthermore, in the third step, the transmit waveform to be optimized and the shadow variable in the transition model are initialized to the same value, and the intermediate model is obtained by initializing the original variables and dual variables in the transition model.

[0047] Furthermore, in the fourth step, the update values ​​of the pattern scale transformation variable, the shadow variable of the transmitted waveform to be optimized, and the intermediate auxiliary variable in the transition model of the transmitted waveform to be optimized are solved according to the corresponding function formulas, as well as the update value of the transmitted waveform; the update values ​​of each intermediate variable and the update value of the transmitted waveform are all taken as the optimal solutions of their respective function formulas.

[0048] Furthermore, in the fifth step, the Lagrange dual variable is updated based on the updated value obtained in the fourth step, and the result of the Lagrange dual variable is compared to determine whether the target requirement is met.

[0049] Furthermore, the function formula for the upper bound of the Euclidean distance between the transmitted waveform to be optimized and the reference waveform includes the number of iterations. The result obtained after updating the Lagrange dual variable is also a function formula that includes the number of iterations. By comparing the value of the number of iterations with the preset critical value, it is determined whether the current updated value of the transmitted waveform meets the target requirements.

[0050] Finally, the preset threshold value is 100. If the number of iterations is greater than or equal to 100, the current updated value of the transmitted waveform is used as the final output transmitted waveform; otherwise, the number of iterations is incremented by 1, and then the process returns to step four to repeatedly solve for the updated values ​​of each intermediate variable and the updated value of the transmitted waveform, and to update the Lagrange dual variable until the number of iterations is greater than or equal to 100.

[0051] When expressing the design problem of the transmission waveform to be optimized using the objective function, the least-maximum principle is used to avoid the problem that the least squares index focuses too much on the edge of the fitted pattern and cannot produce a flat main lobe. The direct optimization algorithm design concept is adopted to avoid the performance loss of the "two-stage" scheme. Finally, by considering similarity constraints and peak-to-means constraints, the variables and shadow variables of the transmission waveform are updated to make the optimized waveform more in line with the actual engineering conditions.

[0052] Example 1: This invention aims to design a method for matching transmitted waveform patterns based on similarity constraints and peak-to-mean constraints under the minimax principle, and operates in the following steps:

[0053] Step 1: First, establish an initial model of the multi-input multi-output radar system to be optimized through scene modeling. The input format of each parameter value of the initial model is predefined as follows:

[0054] 1. Define the number of transmitting antennas of the user-specified multiple-input multiple-output radar system as M, and its detection sequence length as N;

[0055] 2. Define the total energy of the user-specified transmit waveform as... The upper bound of the energy of each element in the transmitted waveform vector—that is, the upper bound of the magnitude of each element—is c. p ;

[0056] 3. The user specifies a reference waveform. The upper bound of the Euclidean distance between the transmitted waveform to be optimized and the reference waveform is specified as follows. α is the scaling variable of the transmitted waveform pattern to be optimized.

[0057] 4. Define the total number of target azimuth angles in the multiple-input multiple-output radar system as I, and the user-specified set of transmitted waveform patterns as follows:

[0058] 5. Define the transmit waveform to be optimized as x, which is also the final output value.

[0059] Step 2: Input initial conditions, user-specified reference waveform Equal in length to the transmitted waveform x, and x and... Euclidean distance to the upper bound Total energy of the transmitted waveform The upper bound c of the modulus of each element p ;

[0060] The user-specified multi-input multi-output radar system has a total of M transmitting antennas and a detection sequence length of N; the total energy (sum of squares of the moduli of each element) of its transmitted waveform x (a complex vector of length MN) is given. The upper bound c of the modulus of each element p Known reference waveform It has the same length as x, and x is equal to x. Euclidean distance to the upper bound The user-specified set of transmit waveform patterns is i is the traversal index satisfying 1≤i≤i, p(θ) i ) is the target azimuth angle θ i The function takes the value 0 or 1, and the total number of target azimuth angles is I.

[0061] Step 3: To generate a flat main lobe, this invention adopts the minimax principle, expressing the design problem of optimizing the transmitted waveform of the multi-input multi-output radar system according to the following objective function:

[0062]

[0063]

[0064]

[0065] Where P(θ) i x) is the target azimuth angle θ i The transmitted waveform pattern at the location.

[0066] Step 4: To decouple variables and increase variable independence, several intermediate variables are introduced. Through equivalent transformation, the initial model of the transmitted waveform to be optimized is converted into a transitional model. The design problem of optimizing the transmitted waveform of the multi-input multi-output radar system is then transformed into the following functional expression:

[0067]

[0068] Based on this, the Lagrange function can be written as follows:

[0069]

[0070] α is the scaling variable of the transmitted waveform pattern to be optimized, x is the transmitted waveform to be optimized, y is the shadow variable of the transmitted waveform to be optimized, and t is the intermediate auxiliary variable introduced during the problem transformation process. i It is the i-th element of t. Denotes any variable; ρ, η, and μ are Lagrange dual variables, where ρ is updated iteratively; η i It is the i-th element of η, and the superscript H indicates the conjugate transpose.

[0071] Step 5: Variable initialization.

[0072] The variables in the transition model are initialized to obtain the intermediate model.

[0073] Initialize the transmitted waveform x to The shadow variable y of the transmitted waveform is initialized to The transmitted waveform x and its shadow variable y are initialized to the same value; ρ and the Lagrange dual variables η and μ are initialized to 1, a zero vector of length I, and a zero vector of length MN, respectively. The iteration count s is initialized to 1.

[0074] After completing the above five steps, the original variables in the intermediate model are updated and calculated to obtain the final model; the specific steps are as follows:

[0075] Step 6: Solve the following subproblems to obtain the updated value of variable α—that is, the pattern scaling transformation variable of the transmitted waveform to be optimized:

[0076]

[0077] The updated value of variable α is the optimal solution to the subproblem. Where ∑ is the summation symbol, added from subscript 1 to I, P(θ) i x) is the target azimuth angle θ i The emitted waveform pattern at point P(θ) i ,x)=x H A(θ i )x, the superscript H represents the conjugate transpose, A(θ) i ) represents the target azimuth angle θ i The guidance matrix at the location is expressed as I N It is an N×N identity matrix. For Kronecker product, is the guiding vector, j is the imaginary unit, π is the mathematical constant pi, sin is the sine function, and the superscripts * and T represent conjugate and transpose, respectively.

[0078] Step 7: Solve the following subproblems to obtain the updated value of variable t—that is, the intermediate auxiliary variable:

[0079]

[0080] The updated value of variable t is the optimal solution to the subproblem: Define an intermediate auxiliary variable h, whose i-th element h i =αp(θ) i )-P(θ i ,x)-ρη i If ||h||1≤ρ, update t to h; otherwise, define a symbolic vector a=sign(h), where Let b be a symbolic function, and define an absolute value vector b = abs(h), where Let b be an absolute value function; arrange the elements of the absolute value vector b in descending order, so that the arrangement of the elements in the vector is b. (1) ≥v (2) ≥...≥b (N) The parenthesized index indicates the largest element in the vector; calculate the maximum index value. Where k is the traversal index satisfying 1≤k≤i, and arg max represents the index of the maximum value in the set; calculate the threshold value. Update any k-th element of variable t to a k max(b k `-γ, 0}, max` means taking the larger of the two.

[0081] Step 8: Solve the following subproblems to obtain the variable x—that is, the updated value of the transmitted waveform to be optimized. In this process, the peak-to-means constraint is used for optimization and updating:

[0082]

[0083] Where l is the traversal index satisfying 1≤l≤MN, and intermediate auxiliary variables. Auxiliary matrix Auxiliary scalar λ max The matrix represents the largest eigenvalue, vec represents the matrix vectorization operation, and the auxiliary scalar ψ2 = λ max (M); This optimization problem is solved using Algorithm 2 in reference [3].

[0084] The specific details of Algorithm 2 in this reference are as follows:

[0085] Any l-th element x of variable x l Updated to Where l is the traversal index satisfying 1≤l≤MN, v l It is the l-th element of v, the arg function represents taking the argument of the complex number, min represents taking the smaller of the two, and the auxiliary variable β satisfies the equation

[0086] Step 9: Solve the following subproblems to obtain the updated value of the shadow variable y. Similarity constraints are used for optimization during this process:

[0087]

[0088] The updated value of variable y is the optimal solution to the subproblem: if it satisfies Update y to x + ρμ; otherwise, introduce an intermediate value. Update y to

[0089] Step 10:

[0090] Update ρ and the Lagrange dual variables η and μ: if the following conditions are met Update η to μ is ρ remains unchanged; otherwise, η and μ remain unchanged, and ρ is updated to 0.9ρ.

[0091] Step 11: Obtain the final transmitted waveform x value;

[0092] Determine whether the current value of x is the final transmitted waveform value based on the comparison between the iteration number s and the critical value;

[0093] If the number of iterations s is greater than or equal to the critical value of 100, then the current value of x is used as the output waveform of this method.

[0094] Otherwise, update the iteration count s to s+1 and return to step 5;

[0095] Repeat steps 6 through 11 until the number of iterations s is greater than or equal to the critical value of 100, and obtain the final transmitted waveform x value.

[0096] That is: if the number of iterations s is greater than or equal to 100, the current value of x is used as the transmitted waveform output by the method; otherwise, the number of iterations s is updated to s+1, and the operation steps of the fifth step are repeated until the condition that the number of iterations s is greater than or equal to 100 is met, and the current value of x is used as the final transmitted waveform output by the method.

[0097] This invention is not limited to the above-described optional embodiments. Anyone can derive other various forms of products under the guidance of this invention. However, regardless of any changes made in their shape or structure, any technical solution that falls within the scope of the claims of this invention shall be protected by this invention.

[0098] References:

[0099] [3] JATropp, ISDhillon, RWHeath, and T.Strohmer, “Designingstructured tight frames via an alternating projection method,” IEEE Transactions on Information Theory, vol.51, no.1, pp.188–209, 2005.

Claims

1. A method for matching transmitted waveform patterns based on the minimax principle and incorporating both similarity and peak-to-mean constraints, characterized in that: It includes the following steps: S1: Establish an initial model for the transmitted waveform to be optimized based on the main parameters of the multi-input multi-output radar system: Specify a known waveform of equal length to the detection sequence length of the transmitted waveform to be optimized as a reference waveform. Based on the upper bound of the Euclidean distance between the transmitted waveform to be optimized and the reference waveform, the total number of transmitting antennas in the multi-input multi-output radar system, the detection sequence length of the multi-input multi-output radar system, the total energy of the transmitted waveform to be optimized, and the upper bound of the energy of each element's modulus, combine this with the total number of target azimuth angles to obtain a functional expression for the transmitted waveform pattern set, thus establishing the initial model for the transmitted waveform to be optimized. The length of the transmitted waveform to be optimized is a complex vector of the total number of transmitting antennas in the multi-input multi-output radar system and the detection sequence length of the multi-input multi-output radar system. Using the minimum-maximum principle, a flat main lobe is generated, specifically expressed as the target functional expression below: in, c is the total energy of the transmitted waveform. p P(θ) is the upper bound of the energy of each element in the emitted waveform vector. i x) is the target azimuth angle θ i The transmitted waveform pattern at the location is given, where α is the scaling variable of the transmitted waveform pattern to be optimized, and x is the transmitted waveform. As the reference waveform, p(θ) i ) is the target azimuth angle θ i The function takes the value 0 or 1; For x and The upper bound of the Euclidean distance; S2: Introduce several intermediate variables and transform the initial model of the transmitted waveform to be optimized into a transitional model through equivalent transformation. The intermediate variables are the pattern scale transformation variable of the transmitted waveform to be optimized, the shadow variable of the transmitted waveform to be optimized, and the intermediate auxiliary variable. Then, through decoupling variables, the initial mathematical model is transformed into a transitional model according to the Lagrange function through equivalent transformation. S3: Initialize the variables in the transition model to obtain the intermediate model; S4: Apply peak mean constraint and similarity constraint to the original variables in the intermediate model to perform update calculations and obtain the final model; S5: Based on the judgment principle, compare and judge whether the final model meets the target requirements. If it meets the target requirements, use the current value of the final model in S4 as the final output waveform. If it does not meet the target requirements, return to S4 and repeat the update calculation of each original variable in the intermediate model until the target requirements are met.

2. The emission waveform pattern matching method based on the minimax principle and incorporating similarity and peak-to-mean constraints as described in claim 1, characterized in that: The total energy of the transmitted waveform to be optimized in S1 is the sum of the squares of the moduli of each element.

3. The emission waveform pattern matching method based on the minimum-maximum principle and incorporating similarity constraints and peak-to-means constraints as described in claim 2, characterized in that: In step S3, the transmit waveform to be optimized and the shadow variable in the transition model are initialized to the same value, and the intermediate model is obtained by initializing the original variables and dual variables in the transition model.

4. The emission waveform pattern matching method based on the minimum-maximum principle and incorporating similarity constraints and peak-to-means constraints as described in claim 3, characterized in that: In step S4, the update values ​​of the pattern scale transformation variable, the shadow variable of the transmitted waveform to be optimized, and the intermediate auxiliary variable in the transition model of the transmitted waveform to be optimized are solved according to the corresponding function formulas, as well as the update value of the transmitted waveform; the update values ​​of each intermediate variable and the update value of the transmitted waveform are all taken as the optimal solutions of their respective function formulas.

5. The emission waveform pattern matching method based on the minimum-maximum principle and incorporating similarity constraints and peak-to-means constraints as described in claim 4, characterized in that: In step S5, the Lagrange dual variable is updated based on the updated value obtained in step S4, and the result of the Lagrange dual variable is compared to determine whether the target requirement is met.

6. The emission waveform pattern matching method based on the minimum-maximum principle and incorporating similarity constraints and peak-to-means constraints as described in claim 5, characterized in that: The function formula for the upper bound of the Euclidean distance between the transmitted waveform to be optimized and the reference waveform includes the number of iterations. The result obtained after updating the Lagrange dual variable is also a function formula that includes the number of iterations. By comparing the value of the number of iterations with the preset critical value, it is determined whether the current updated value of the transmitted waveform meets the target requirements.

7. The emission waveform pattern matching method based on the minimum-maximum principle and incorporating similarity constraints and peak-to-means constraints as described in claim 6, characterized in that: The preset threshold value is 100. If the number of iterations is greater than or equal to 100, the current updated value of the transmitted waveform will be used as the final output transmitted waveform. Otherwise, increment the iteration count by 1, return to S4, and repeat the process of solving for the updated values ​​of each intermediate variable and the updated value of the transmitted waveform, updating the Lagrange dual variable until the iteration count is greater than or equal to 100.

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