Anti-interference waveform design method based on directional diagram matching under joint constraint

Abstract

The invention discloses an anti-interference waveform design method based on directional diagram matching under joint constraint. The method comprises the following steps: establishing an MIMO cognitive radar system which adopts a uniform linear array to carry out antenna layout; constructing a transmitting directional diagram model; establishing a problem model of directional diagram matching design based on a minimum mean square error criterion under the condition that the waveform constant modulus constraint and the similarity constraint are met; for the optimization problem under the joint constraint, simplifying the similarity constraint by adopting a Lagrange multiplier method to obtain the optimization problem only having the constant modulus constraint; the geometric characteristics of the non-convex constraint are mined, and the constant modulus constraint and the Riemannian manifold are associated in nature, so that the constrained non-convex optimization problem is converted into the unconstrained convex optimization problem in the manifold; and solving the optimization problem model on the manifold based on an optimization algorithm, and obtaining an anti-interference waveform. According to the method, the optimal transmitted waveform can be generated to resist specific interference, and the transmitted waveform with better fuzzy characteristics is obtained while the matching efficiency of the transmitted beam directional diagram is remarkably improved.

Description

Technical Field

[0001] The present invention belongs to the field of cognitive radar anti-interference, and specifically relates to an anti-interference waveform design method based on pattern matching under joint constraints, especially for optimizing the anti-interference performance of cognitive radar transmission pattern matching waveform design in scenarios where target and interference directions are known. Background Art

[0002] With the widespread deployment and continuous expansion of multiple services such as satellite communications, radio broadcasting, television signal transmission, radar detection, telecommunications networks and navigation systems, the demand for limited spectrum resources has increased significantly. The contradiction between demand and supply has become increasingly prominent, resulting in modern radar systems facing increasingly complex and highly intelligent interference environments. Therefore, improving the anti-interference capability of radars and developing radar anti-interference technology has become an urgent problem. Faced with increasingly complex interference environments, traditional radars mainly process data at the receiving end, rarely use external target and environmental information, lack the ability to adapt to targets and environments, and face major bottlenecks in improving anti-interference performance. To address this problem, the concept of cognitive radar (CR) came into being. Compared with traditional radars, cognitive radars can use prior information about targets and environments to adaptively optimize radar operating modes, transmission waveforms and receiving processing methods, breaking through the framework of traditional radars working in preset operating modes and having the ability to adapt to more complex interference environments.

[0003] Existing research mainly focuses on the anti-interference of the transmitter of cognitive radar, especially in the waveform design. Among the research on the design of the transmitter waveform, the constant modulus waveform design problem based on the transmission pattern has attracted the most attention. However, the research on this type of problem is mainly limited to finding the optimal solution using the relaxed convex optimization method and the alternating iterative optimization method. This method can gradually approach the optimal solution, but the computational complexity is high, the required computing resources are large, and the degree of freedom of the problem model and the actual engineering problems are not fully considered. At the same time, in practical applications, there are multiple constraints on the transmission waveform, but most of the current research only considers the constant modulus constraint and ignores the fuzzy characteristics of the waveform. Therefore, it is necessary to further study and design the anti-interference waveform problem model and optimization algorithm with multiple constraints at the same time, so as to reduce the computational complexity and improve the anti-interference performance and fuzzy characteristics of the transmission waveform. Summary of the invention

[0004] Purpose of the invention: The purpose of the present invention is to propose an anti-interference waveform design method based on pattern matching under joint constraints. By minimizing the mean square error between the actual designed pattern and the expected pattern, an optimal transmission waveform is generated to combat specific interference while satisfying specific joint constraints. While significantly improving the transmission beam pattern matching efficiency, a transmission waveform with better fuzzy characteristics is obtained.

[0005] Technical solution: The anti-interference waveform design method based on directional pattern matching under joint constraints described in the present invention comprises the following steps:

[0006] (1) Establish a t MIMO cognitive radar system with 10 transmit antennas;

[0007] (2) Construct a transmission pattern model to achieve matching design of the transmission pattern to resist interference;

[0008] (3) Based on the transmission pattern model, a problem model of pattern matching design based on the minimum mean square error criterion is established under the condition of satisfying the waveform constant modulus constraint and similarity constraint;

[0009] (4) For the problem model of pattern matching design based on the minimum mean square error criterion, the Lagrange multiplier method is used to simplify the similarity constraints and obtain an optimization problem model with only constant modulus constraints;

[0010] (5) Aiming at the nonlinear characteristics of the optimization problem model with only constant modulus constraints, the geometric characteristics of non-convex constraints are explored, and the constant modulus constraints are associated with the Riemann manifold in terms of properties, so that the constrained non-convex optimization problem can be transformed into an unconstrained optimization problem on the manifold;

[0011] (6) An optimization algorithm RPM-CG is proposed to solve the unconstrained optimization problem on the manifold and obtain the anti-interference waveform.

[0012] Furthermore, the MIMO cognitive radar system in step (1) adopts a uniform linear array for antenna layout, and the antenna spacing is half a wavelength.

[0013] Furthermore, the implementation process of step (2) is as follows:

[0014] Each transmitting antenna transmits a separate waveform, M represents the total number of samples of the transmitted waveform, and the transmitted waveform vector at the mth sampling moment is For a far-field target located in the direction of θ, the synthetic signal is:

[0015]

[0016] In the formula, is the launch steering vector, and its expression is:

[0017]

[0018] Where, d T is the transmitting array element spacing, λ is the carrier wavelength;

[0019] x(θ)=[x(θ,1),x(θ,2),...,x(θ,M)] is a vector composed of M synthetic signals, expressed as:

[0020] x(θ)=a t H (θ)S(3)

[0021] Where S is the transmit waveform matrix;

[0022] The average power of the transmitted signal synthesized in the θ direction, that is, the transmit beam pattern of the MIMO cognitive radar, is expressed as:

[0023]

[0024] Where R represents the covariance matrix of the transmitted waveform;

[0025] At the same time, we get the direction θ p and direction θ q The spatial cross-correlation function is expressed as:

[0026]

[0027] By establishing the MIMO radar transmission pattern model as shown in equations (1) to (5), the matching design of the transmission pattern is realized to resist interference.

[0028] Furthermore, the implementation process of step (3) is as follows:

[0029] The minimum mean square error criterion is adopted, and the spatial cross-correlation of different targets is introduced to obtain the objective function:

[0030]

[0031] Among them, K is the number of discretization points; Q is the number of spatial targets; α is the scale factor; ω k and ω c They are θ k The weights of the direction and cross-correlation terms are adjusted to achieve a balance between the two optimization objectives;

[0032] Introduce constant modulus constraint, its expression is:

[0033]

[0034] Where P t Indicates the transmit power;

[0035] Introduce similarity constraints to ensure that the designed waveform is consistent with the reference waveform in terms of fuzzy characteristics:

[0036] ||S-S0|| ∞ <ε (9)

[0037] Where S0 represents the reference waveform; ε is a parameter used to adjust the degree of similarity; when ε = 0, the design waveform is the same as the reference waveform. The larger ε is, the worse the similarity between the design waveform and the reference waveform is.

[0038] Under the condition of satisfying constant modulus constraint and similarity constraint, the radar transmit beam pattern matching design problem based on minimizing mean square error criterion is expressed as:

[0039]

[0040] Furthermore, the implementation process of step (4) is as follows:

[0041] Convert the inequality constraint into an equality constraint and the similarity constraint into:

[0042]

[0043] Among them, γ represents the parameter that makes the equality valid;

[0044] Combining the similarity equality constraint with the objective function, constructing the Lagrangian function, the simplified optimization problem with only constant modulus constraints is:

[0045]

[0046] In the formula,

[0047] Furthermore, the implementation process of step (5) is as follows:

[0048] According to the optimization problem with only constant modulus constraints, the intrinsic geometric structure of the constraints is vividly expressed, and the constraints are embedded in the search space. The smooth manifold after being mapped into the solution space is expressed as:

[0049]

[0050] Decompose the smooth manifold into a single manifold structure, M α 、M β and M γ Respectively represent the search space of variables α, β and γ, that is, M S Represents the search space of the waveform matrix S, namely:

[0051]

[0052] According to the relationship between the product manifold and the single manifold and the tangent space of the single manifold, we can get the product manifold M (α,S,β,γ) The tangent space T (α,S,β,γ) M (α,S,β,γ) for:

[0053] T (α,S,β,γ) M (α,S,β,γ) =T (α,S,β,γ,t) (M α ×M S ×M β ×M γ )

[0054] =T α M α ×T S M S ×T β M β ×T γ M γ (16)

[0055] Among them, S∈M S The tangent space is:

[0056]

[0057] Where Re{·} represents the real part of a complex number, ⊙ represents the Hadamard product, and * represents the conjugate operation;

[0058] Since the manifold M α 、M β and M γ is a one-dimensional Euclidean space, then its tangent space is also a one-dimensional Euclidean space, that is:

[0059]

[0060] We obtain an unconstrained optimization problem based on the Riemann manifold:

[0061]

[0062] Furthermore, the implementation process of step (6) is as follows:

[0063] The Euclidean gradient of the objective function with respect to the variable scale factor α, the Lagrange multiplier β, the parameter γ that makes the equality valid, and the waveform matrix S are derived. The Riemann gradient is characterized according to the mapping relationship between the Euclidean gradient and the Riemann gradient. ρ is selected by different criteria. k A vector transmission operator is introduced to obtain a suitable descent direction; the Armijo line search strategy is adopted to select the step size; the contraction operator is introduced to shrink the tangent vector after descent to the manifold and update the feasible solution; it is continuously iterated until the convergence criterion is reached, and the optimal transmission waveform and transmission direction diagram that meet the joint constraints and the optimization objectives are obtained.

[0064] Furthermore, ρ is selected by different criteria. k And the vector transmission operator is introduced to implement the process as follows:

[0065] Select the descent direction, define the vector transfer operator, and select ρ by different criteria k That is, get the appropriate direction d k+1 , the descent direction of the kth iteration is given by:

[0066] d k+1 =-grad L k +ρ k Trans k→k+1 (d k ) (30)

[0067] Among them, Trans k→k+1 (·) is the vector transfer operator required to perform vector operations on different tangent spaces and is calculated by the following equation:

[0068]

[0069] in,

[0070] Furthermore, the introduction of the contraction operator to contract the descending tangent vector to the manifold and update the feasible solution is implemented as follows:

[0071] Contraction Operator The expression is:

[0072]

[0073] in:

[0074]

[0075] The formula for a more feasible solution is:

[0076]

[0077] Beneficial effects: Compared with the prior art, the present invention has the following beneficial effects:

[0078] 1. The MIMO cognitive radar pattern design model established by the present invention under the condition of known target and interference direction takes the matching performance of the transmission pattern and the fuzzy characteristics of the transmission waveform as the optimization goals, and introduces the spatial cross-correlation between different targets to minimize the mean square error between the transmission pattern and the expected pattern, thereby improving the pattern matching performance and the fuzzy characteristics of the waveform;

[0079] 2. Based on the established MIMO cognitive radar pattern design model and pattern matching objective function, the present invention establishes a new optimization problem framework based on constant modulus constraints and similarity constraints to improve the radar anti-interference performance and optimize the transmission pattern and transmission waveform;

[0080] 3. Based on the establishment of the problem model, the present invention proposes an RPM-CG optimization algorithm based on the Riemann product manifold, which can solve the problem model with lower complexity and obtain better optimization effect, effectively solve the local optimal problem in the traditional algorithm, and ensure the stability and effectiveness of the algorithm under multiple constraints; compared with the existing algorithms, the present invention exhibits better directional pattern matching performance and fuzzy characteristics. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] Figure 1 is a flow chart of the present invention;

[0082] Figure 2 This is a schematic diagram of the MIMO cognitive radar transmit array model;

[0083] Figure 3 It is a geometric diagram of a Riemannian manifold and its tangent space and tangent vector;

[0084] Figure 4 It is a schematic diagram of the conjugate gradient method iteration on the Riemann manifold;

[0085] Figure 5 Flow chart of the RPM-CG algorithm proposed by the present invention;

[0086] Figure 6 Schematic diagram of simulation results of emission pattern matching of different algorithms;

[0087] Figure 7 Simulation result diagram for emission pattern matching under different similarity parameters;

[0088] Figure 8 Comparison chart of fuzzy function slices under different similarity parameters. DETAILED DESCRIPTION

[0089] The present invention is further described in detail below with reference to the accompanying drawings.

[0090] like Figure 1 As shown, the present invention proposes an anti-interference waveform design method based on directional pattern matching under joint constraints, comprising the following steps:

[0091] Step 1: Figure 2 As shown, the establishment is equipped with N t A MIMO cognitive radar system with 4 transmit antennas.

[0092] The MIMO cognitive radar system uses a uniform linear array (ULA), which is divided into N T There are transmitting antennas, the spacing between transmitting array elements is half a wavelength, and the total number of samples is M. Let the number of discretization points of the spatial angle θ∈[-90°,90°] be K. Specifically in this embodiment, N T=10, M=32, K=181.

[0093] Step 2: Based on the system model established in step 1, model the transmitted signal and the directional pattern.

[0094] According to the MIMO cognitive radar system model, it is assumed that the transmitting antenna can transmit a separate waveform. The transmitting waveform vector at the mth sampling moment is For a far-field target located in the direction of θ, the composite signal can be expressed as:

[0095] x(θ,m)=a t H (θ)s m ,m=1,2,...,M (1)

[0096] In the formula, is the launch steering vector, and its expression is:

[0097]

[0098] Where, d T is the transmitting array element spacing, and λ is the carrier wavelength.

[0099] Let x(θ)=[x(θ,1),x(θ,2),...,x(θ,M)] be a vector composed of M synthetic signals, which can be expressed as:

[0100] x(θ)=a t H (θ)S(3)

[0101] Where S is the transmit waveform matrix.

[0102] According to the system model and the transmission signal model, the average power synthesized by the transmission signal in the θ direction, that is, the transmission beam pattern of the MIMO cognitive radar can be expressed as:

[0103]

[0104] Where R represents the covariance matrix of the transmitted waveform.

[0105] At the same time, the direction θ can be obtained p and direction θ q The spatial cross-correlation function is expressed as:

[0106]

[0107] By establishing the MIMO radar transmission pattern model as shown in formulas (1)-(5), the matching design of the transmission pattern is realized to resist interference. Specifically in this embodiment, λ=2,d T =λ / 2.

[0108] Step 3: Based on the transmission pattern model, a problem model of pattern matching design based on the minimum mean square error criterion is established while satisfying the waveform constant modulus constraint and similarity constraint.

[0109] The minimum mean square error criterion is adopted, and the spatial cross-correlation of different targets is introduced at the same time, thereby obtaining the objective function, which is expressed as follows:

[0110]

[0111] Among them, K is the number of discretization points; Q is the number of spatial targets; α is the scale factor; ω k and ω c They are θ k The weights of the direction and cross-correlation terms can achieve a balance between the two optimization objectives by adjusting the weights. Specifically in this embodiment, Q = 3, ω k =1,ω c =5, D(θ) is the desired directional pattern:

[0112]

[0113] Among them, θ1=-40°, θ2=0°, θ3=40°, and Δθ=20°.

[0114] Considering the actual application scenarios of radar systems, the transmitter is usually set to the maximum efficiency state (saturation or near saturation state), and the constant modulus constraint is introduced, which is expressed as:

[0115]

[0116] Where P t Indicates the transmission power. Specifically, in this embodiment, P t =N T M.

[0117] Considering the designed waveform, its fuzzy characteristics may not be ideal, so a similarity constraint is introduced to ensure that the designed waveform is consistent with the reference waveform in terms of fuzzy characteristics. This constraint can be expressed as:

[0118] ||S-S0|| ∞ <ε (9)

[0119] In the formula, S0 represents the reference waveform; ε is a parameter used to adjust the degree of similarity. Specifically in this embodiment, ε=1, the reference waveform S0 is an orthogonal linear frequency modulation (LFM) waveform, and the waveform matrix element is:

[0120]

[0121] According to the pattern matching model expression, under the condition of satisfying the constant modulus constraint and similarity constraint, the radar transmit beam pattern matching design problem expression based on the minimum mean square error criterion is:

[0122]

[0123] Step 4: For the optimization problem under joint constraints, the Lagrange multiplier method is used to simplify the similarity constraints and obtain the optimization problem with only constant modulus constraints.

[0124] Convert the inequality constraint into an equality constraint, since ||·|| ∞ represents the infinite norm, which is equivalent to the maximum absolute value of the matrix elements. Then the similarity constraint can be expressed as:

[0125]

[0126] Here, γ represents the parameter that makes the equality valid.

[0127] Combining the similarity equality constraint with the objective function to construct the Lagrangian function, and using it as the new objective function, we can get the simplified optimization problem:

[0128]

[0129] In the formula,

[0130] Step 5: Based on the nonlinear characteristics of the problem model in step 4, we explore the geometric characteristics of non-convex constraints, associate the constant modulus constraints with the Riemann manifold in terms of properties, and transform the constrained non-convex optimization problem into an unconstrained convex optimization problem on the manifold. The geometric diagram of the Riemann manifold and its tangent space and tangent vector is shown in the figure below. Figure 3 shown.

[0131] According to the non-convex multivariable optimization problem, the intrinsic geometric structure of the constraints is vividly expressed, and the constraints are embedded in the search space. The smooth manifold after being mapped into the solution space is expressed as:

[0132] M (α,S,β,γ) =M α ×M S ×M β ×M γ

[0133] ={(α,S,β,γ)∈M (α,S,β,γ) :α∈M α ,S∈M S ,β∈M β ,γ∈M γ} (14)

[0134] in, M S Represents the search space of the waveform matrix S, expressed as:

[0135]

[0136] According to the established manifold structure, the corresponding tangent space structure and inner product space are defined, and the product manifold M can be obtained. (α,S,β,γ) The tangent space T (α,S,β,γ) M (α,S,β,γ) for:

[0137]

[0138] Among them, S∈M S The tangent space is:

[0139]

[0140] Manifold M α 、M β and M γ The tangent space of is a one-dimensional Euclidean space, that is:

[0141]

[0142] According to the constructed Riemann product manifold structure and non-convex optimization problem, the unconstrained convex optimization problem based on Riemann manifold is obtained:

[0143]

[0144] Step 6: Figure 4 As shown, a corresponding RPM-CG optimization algorithm is proposed to solve the optimization problem model on the established manifold, and based on the solution results, an anti-interference waveform with good fuzzy characteristics while ensuring the performance of directional pattern matching is obtained.

[0145] like Figure 5 As shown, the RPM-CG optimization algorithm is proposed to solve the problem model. In order to develop the PRM-CG optimization algorithm, it is necessary to calculate the Riemann gradient of the objective function. First, the Euclidean gradient of each component is calculated:

[0146] Grad α L=Grad α f(α,S) (20)

[0147] Grad β L=g (21)

[0148] Grad γ L=-2βγ (22)

[0149] Grad S L=GradS f(α,S)-βh (23)

[0150] Among them, Grad α f(α,S) is the Euclidean gradient of function f(α,S) with respect to α, Grad S f(α,S) is the Euclidean gradient of function f(α,S) with respect to S; h is the partial derivative of g with respect to S, as shown in the following formula:

[0151]

[0152] The Riemann gradient of each component can be obtained by orthogonally projecting the Euclidean gradient into the corresponding tangent space:

[0153] grad i L=Proj i (Grad i L)=Grad i L,i=α,β,γ (27)

[0154] grad S L=Proj S (Grad S L)=Grad S L-Re{Grad S L⊙S *}⊙S (28)

[0155] For the product manifold M (α,S,β,γ) , the Riemann gradient grad L(α,S,β,γ) can be expressed as:

[0156] grad L(α,S,β,γ)=(grad α L,grad S L,grad β L,grad γ L) (29)

[0157] In the formula, grad α L.grad β L and grad γ L are the Riemann gradients of the objective function at α, β, and γ respectively; S L represents the objective function in S∈M S The Riemann gradient at .

[0158] Then choose the appropriate descent direction, define the vector transfer operator, and select ρ by different criteria. k The appropriate direction d can be obtained k+1 , the descent direction of the kth iteration is given by:

[0159] dk+1 =-grad L k +ρ k Trans k→k+1 (d k ) (30)

[0160] Among them, Trans k→k+1 (·) is the vector transfer operator required to perform vector operations on different tangent spaces and is calculated by the following equation:

[0161]

[0162] in,

[0163] Use Armijo line search strategy to find the appropriate step size μ k , this strategy ensures that the cost of each iteration does not increase.

[0164] The contraction operator is introduced to contract the tangent vector to the manifold and update the feasible solution. The expression is:

[0165]

[0166] in,

[0167]

[0168] Therefore, the formula for a more feasible solution is:

[0169]

[0170] The performance of the RPM-CG algorithm is compared with the SQP algorithm, the Cyclic Algorithm (CA), the Maximization-Minimization (MM) based algorithm and the Nonlinear-ADMM algorithm under the CM constraint. Figure 6 Transmitting patterns designed for different algorithms. Except for the CA algorithm, the matching performance of other algorithms in the main lobe is basically the same. The sidelobe matching performance of the RPM-CG algorithm and the MM algorithm is better than that of the Nonlinear-ADMM and SQP algorithms, and is closer to the desired beam pattern.

[0171] When the similarity parameter ε=0.5,1.0,1.5,2.0, the direction matching results are as follows Figure 7 As shown in Figure 2. When ε = 0.5, the similarity constraint is high and it is difficult for the algorithm to converge and obtain the optimal solution. When ε = 2.0, the similarity constraint does not exist and the joint constraint is equivalent to the constant modulus constraint. Figure 7In the example, when ε = 0.5, the main lobe performance is the worst, and when ε = 1.0, the side lobe performance is the worst. The influence of the similarity parameter on the ambiguity characteristics of the transmitted sequence is further observed. In order to evaluate the ambiguity characteristics of the transmitted sequence, the expression of the ambiguity function is defined, and the ambiguity function slices of the design sequence and the reference sequence are observed. The results are shown in Figure 8 As shown, it can be seen that as the similarity parameter increases, the difference between the fuzziness characteristics of the reference waveform and the designed waveform becomes greater.

[0172] The above-mentioned embodiments only express several implementation methods of the present invention, and the description is relatively specific and detailed, but it cannot be understood as limiting the scope of the invention patent. It should be pointed out that for ordinary technicians in this field, several modifications and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A method for designing an anti-interference waveform based on pattern matching under joint constraints, characterized in that: The following steps are involved: (1) Establish a t MIMO cognitive radar system with 10 transmit antennas; (2) Construct a transmission pattern model to achieve matching design of the transmission pattern to resist interference; (3) Based on the transmission pattern model, a problem model of pattern matching design based on the minimum mean square error criterion is established under the condition of satisfying the waveform constant modulus constraint and similarity constraint; (4) For the problem model of pattern matching design based on the minimum mean square error criterion, the Lagrange multiplier method is used to simplify the similarity constraints and obtain an optimization problem model with only constant modulus constraints; (5) Aiming at the nonlinear characteristics of the optimization problem model with only constant modulus constraints, the geometric characteristics of non-convex constraints are explored, and the constant modulus constraints are associated with the Riemann manifold in terms of properties, so that the constrained non-convex optimization problem can be transformed into an unconstrained optimization problem on the manifold; (6) An optimization algorithm RPM-CG is proposed to solve the unconstrained optimization problem on the manifold and obtain the anti-interference waveform.

2. The anti-interference waveform design method based on pattern matching under joint constraints according to claim 1 is characterized in that: The MIMO cognitive radar system in step (1) uses a uniform linear array for antenna layout, and the antenna spacing is half a wavelength.

3. The anti-interference waveform design method based on directional pattern matching under joint constraints according to claim 1 is characterized in that: The implementation process of step (2) is as follows: Each transmitting antenna transmits a separate waveform, M represents the total number of samples of the transmitted waveform, and the transmitted waveform vector at the mth sampling moment is For a far-field target located in the direction of θ, the synthetic signal is: x(θ,m)=a t H (i)s m ,m=1,2,...,M (1) In the formula, is the launch steering vector, and its expression is: Where, d T is the transmitting array element spacing, λ is the carrier wavelength; x(θ)=[x(θ,1),x(θ,2),...,x(θ,M)] is a vector composed of M synthetic signals, expressed as: x(θ)=a t H (i)S (3) Where S is the transmit waveform matrix; The average power of the transmitted signal synthesized in the θ direction, that is, the transmit beam pattern of the MIMO cognitive radar, is expressed as: Where R represents the covariance matrix of the transmitted waveform; At the same time, we get the direction θ p and direction θ q The spatial cross-correlation function of is expressed as: By establishing the MIMO radar transmission pattern model as shown in equations (1) to (5), the matching design of the transmission pattern is realized to resist interference.

4. The anti-interference waveform design method based on pattern matching under joint constraints according to claim 1 is characterized in that: The implementation process of step (3) is as follows: The minimum mean square error criterion is adopted, and the spatial cross-correlation of different targets is introduced to obtain the objective function: Among them, K is the number of discretization points; Q is the number of spatial targets; α is the scale factor; ω k and ω c They are θ k The weights of the direction and cross-correlation terms are adjusted to achieve a balance between the two optimization objectives; Introduce constant modulus constraint, its expression is: Where P t Indicates the transmit power; Introduce similarity constraints to ensure that the designed waveform is consistent with the reference waveform in terms of fuzzy characteristics: ||S-S0|| ∞ <ε (9) Where S0 represents the reference waveform; ε is a parameter used to adjust the degree of similarity; when ε = 0, the design waveform is the same as the reference waveform. The larger ε is, the worse the similarity between the design waveform and the reference waveform is. Under the condition of satisfying constant modulus constraint and similarity constraint, the radar transmit beam pattern matching design problem based on minimizing mean square error criterion is expressed as:

5. The anti-interference waveform design method based on pattern matching under joint constraints according to claim 1 is characterized in that: The implementation process of step (4) is as follows: Convert the inequality constraint into an equality constraint and the similarity constraint into: Among them, γ represents the parameter that makes the equality valid; Combining the similarity equality constraint with the objective function, constructing the Lagrangian function, the simplified optimization problem with only constant modulus constraints is: In the formula, 6. The anti-interference waveform design method based on pattern matching under joint constraints according to claim 1 is characterized in that: The implementation process of step (5) is as follows: According to the optimization problem with only constant modulus constraints, the intrinsic geometric structure of the constraints is vividly expressed, and the constraints are embedded in the search space. The smooth manifold after being mapped into the solution space is expressed as: M (α,S,β,γ) =M α ×M S ×M β ×M γ ={(α,S,β,γ)∈M (α,S,β,γ) :α∈M α ,S∈M S ,β∈M β ,r∈M γ } (14) Decompose the smooth manifold into a single manifold structure, M α 、M β and M γ Respectively represent the search space of variables α, β and γ, that is, M S Represents the search space of the waveform matrix S, namely: According to the relationship between the product manifold and the single manifold and the tangent space of the single manifold, we can get the product manifold M (α,S,β,γ) The tangent space T (α,S,β,γ) M (α,S,β,γ) for: T (α,S,β,γ) M (α,S,β,γ) =T (α,S,β,γ,t) (M α ×M S ×M β ×M γ ) =T α M α ×T S M S ×T β M β ×T γ M γ (16) Among them, S∈M S The tangent space is: Where Re{·} represents the real part of a complex number, ⊙ represents the Hadamard product, and * represents the conjugate operation; Since the manifold M α 、M β and M γ is a one-dimensional Euclidean space, then its tangent space is also a one-dimensional Euclidean space, that is: We obtain an unconstrained optimization problem based on the Riemann manifold:

7. The anti-interference waveform design method based on pattern matching under joint constraints according to claim 1 is characterized in that: The implementation process of step (6) is as follows: The Euclidean gradient of the objective function with respect to the variable scale factor α, the Lagrange multiplier β, the parameter γ that makes the equality valid, and the waveform matrix S are derived. The Riemann gradient is characterized according to the mapping relationship between the Euclidean gradient and the Riemann gradient. ρ is selected by different criteria. k A vector transmission operator is introduced to obtain a suitable descent direction; the Armijo line search strategy is adopted to select the step size; the contraction operator is introduced to shrink the tangent vector after descent to the manifold and update the feasible solution; it is continuously iterated until the convergence criterion is reached, and the optimal transmission waveform and transmission direction diagram that meet the joint constraints and the optimization objectives are obtained.

8. The anti-interference waveform design method based on pattern matching under joint constraints according to claim 1 is characterized in that: Select ρ by different criteria k And the vector transmission operator is introduced to implement the process as follows: Select the descent direction, define the vector transfer operator, and select ρ by different criteria k That is, get the appropriate direction d k+1 , the descent direction of the kth iteration is given by: d k+1 =-degree L k +ρ k Trans k→k+1 (d k ) (30) Among them, Trans k→k+1 (·) is the vector transfer operator required to perform vector operations on different tangent spaces and is calculated by the following equation: Among them, i=a,b,c; 9. The anti-interference waveform design method based on pattern matching under joint constraints according to claim 1 is characterized in that: The introduction of the contraction operator to contract the descending tangent vector to the manifold and update the feasible solution is implemented as follows: Contraction Operator The expression is: in: The formula for a more feasible solution is:

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