An anti-jamming waveform design method based on direction pattern matching under joint constraints

Abstract

The application discloses a kind of under combined constraint based on directional diagram matching anti-interference waveform design method, establishes the MIMO cognitive radar system of antenna layout using uniform linear array;Directional diagram model is constructed;Under the condition that waveform constant modulus constraint and similarity constraint are satisfied, the problem model of directional diagram matching design based on minimum mean square error criterion is established;For optimization problem under combined constraint, the similarity constraint is simplified using Lagrange multiplier method, and only the optimization problem with constant modulus constraint is obtained;The geometric characteristics of non-convex constraint are excavated, the constant modulus constraint is associated with Riemannian manifold in nature, so that the constrained non-convex optimization problem is converted into an unconstrained convex optimization problem on the manifold;Based on optimization algorithm, the optimization problem model on the manifold is solved, and the anti-interference waveform is obtained.The application can generate optimal transmit waveform to resist specific interference, significantly improve the efficiency of transmit beam directional diagram matching, and obtain transmit waveform with better blur characteristics.

Description

Technical Field

[0001] This invention belongs to the field of cognitive radar anti-jamming, specifically involving an anti-jamming waveform design method based on pattern matching under joint constraints, and particularly optimizing the anti-jamming performance of cognitive radar transmit pattern matching waveform design in scenarios where the target and interference directions are known. Background Technology

[0002] With the widespread deployment and continuous expansion of diverse services such as satellite communication, radio broadcasting, television signal transmission, radar detection, telecommunications networks, and navigation systems, the demand for limited spectrum resources has increased significantly. This contradiction between supply and demand has become increasingly prominent, leading to increasingly complex and highly intelligent interference environments for modern radar systems. Therefore, improving radar anti-jamming capabilities and developing radar anti-jamming technologies has become an urgent problem. Faced with increasingly complex interference environments, traditional radars mainly process data from the receiving end, rarely utilizing external target and environmental information, resulting in insufficient adaptability to targets and environments and a significant bottleneck in improving anti-jamming performance. To address this issue, the concept of Cognitive Radar (CR) has emerged. Compared to traditional radar, cognitive radar can utilize prior information about targets and the environment to adaptively optimize radar operating modes, transmission waveforms, and receiving processing methods, breaking through the framework of traditional radar operating according to preset modes and possessing the ability to adapt to more complex interference environments.

[0003] Current research primarily focuses on anti-jamming at the transmitter end of cognitive radar, particularly in waveform design. Among these studies, the constant-mode waveform design problem based on the transmission pattern has received the most attention. However, research on this type of problem is mainly limited to finding the optimal solution using relaxed backconvex optimization and alternating iterative optimization methods. While these methods can gradually approximate the optimal solution, they have high computational complexity, require significant computational resources, and do not fully consider the degrees of freedom of the problem model or practical engineering issues. Furthermore, in practical applications, the transmitted waveform faces multiple constraints, but current research mostly considers only constant-mode constraints, neglecting the ambiguity characteristics of the waveform. Therefore, further research is needed to design anti-jamming waveform problem models and optimization algorithms that address multiple constraints simultaneously, improving the anti-jamming performance and ambiguity characteristics of the transmitted waveform while reducing computational complexity. Summary of the Invention

[0004] Purpose of the invention: The purpose of this 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, the optimal transmission waveform is generated to counteract specific interference under the premise of satisfying specific joint constraints. This significantly improves the transmission beam pattern matching efficiency while obtaining a transmission waveform with better ambiguity characteristics.

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

[0006] (1) Establish a system equipped with N t A MIMO cognitive radar system with one transmitting antenna;

[0007] (2) Construct a transmission pattern model and implement a matching design for the transmission pattern to resist interference;

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

[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 constraint and obtain the optimization problem model with only constant modulus constraint.

[0010] (5) To address the nonlinear characteristics of optimization problem models with only constant modulus constraints, we explore the geometric properties of nonconvex constraints and associate constant modulus constraints with Riemannian manifolds in terms of properties, so that the constrained nonconvex optimization problem is 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 described in step (1) uses a uniform linear array for antenna layout, with an antenna spacing of half a wavelength.

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

[0014] Each transmitting antenna transmits a separate waveform, where M represents the total number of samples of the transmitted waveform. The transmitted waveform vector at the m-th sampling time is... For a far-field target located in the θ direction, the synthesized signal is:

[0015]

[0016] In the formula, The launch steering vector is expressed as follows:

[0017]

[0018] In the formula, d T λ is the spacing between the transmitting array elements, and λ is the carrier wavelength;

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

[0020] x(θ)=a t H (θ)S(3)

[0021] In the formula, S is the transmitted waveform matrix;

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

[0023]

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

[0025] Simultaneously obtain the direction θ p and direction θ q The spatial cross-correlation function is expressed as:

[0026]

[0027] By establishing MIMO radar transmission pattern models 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] By employing the minimum mean square error criterion and introducing spatial cross-correlation of different objectives, the objective function is obtained as follows:

[0030]

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

[0032] Introducing the constant modulus constraint, its expression is:

[0033]

[0034] In the formula, P t Indicates the transmission power;

[0035] Introduce similarity constraints to ensure that the designed waveform maintains consistency with the reference waveform in terms of ambiguity characteristics:

[0036] ||S-S0|| ∞ <ε (9)

[0037] In the formula, 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; if ε is larger, the similarity between the design waveform and the reference waveform is worse.

[0038] Under the conditions of constant modulus constraint and similarity constraint, the expression for the radar transmitted beam pattern matching design problem based on the minimum mean square error criterion is as follows:

[0039]

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

[0041] Transforming inequality constraints into equality constraints, the similarity constraint is expressed as:

[0042]

[0043] Where γ represents the parameter that makes the equality hold;

[0044] By combining the similarity equality constraint with the objective function, a Lagrangian function is constructed. The simplified optimization problem with only constant modulus constraint is:

[0045]

[0046] In the formula,

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

[0048] For optimization problems with only constant modulus constraints, the inherent geometric structure of the constraints is figuratively described. Embedding the constraints into the search space, the resulting smooth manifold representation in the solution space is:

[0049]

[0050] Decompose the smooth manifold into a single manifold structure, M α M β and M γ Let represent the search spaces for variables α, β, and γ, respectively. M S The search space for the waveform matrix S is represented as follows:

[0051]

[0052] Based on the relationship between the accumulated manifold and the simple manifold, and the tangent space of each simple manifold, we obtain the accumulated manifold M. (α,S,β,γ) 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] Where S∈M S The tangent space is:

[0056]

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

[0058] Due to manifold M α M β and M γ If a space is a one-dimensional Euclidean space, then its tangent space is also a one-dimensional Euclidean space, that is:

[0059]

[0060] We obtain the unconstrained optimization problem based on Riemannian manifolds:

[0061]

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

[0063] The Euclidean gradient of the objective function with respect to the scale factor α, Lagrange multipliers β, the parameter γ that makes equality hold, and the waveform matrix S is derived. The Riemann gradient is characterized based on the mapping relationship between the Euclidean and Riemann gradients. ρ is selected using different criteria. k The vector transfer operator is introduced to obtain a suitable descent direction; the step size is selected by adopting the Armijo line search strategy; the shrinkage operator is introduced to shrink the tangent vector after descent onto the manifold and update the feasible solution; the process is iterated until the convergence criterion is reached, and the optimal emission waveform and emission pattern that satisfy the joint constraints and the optimization objective are obtained.

[0064] Furthermore, ρ is selected through different criteria. k The implementation process, which introduces a vector transfer operator, is as follows:

[0065] Choose the descent direction, define the vector transfer operator, and select ρ based on different criteria. k That is, to obtain the appropriate direction d k+1 The descent direction of the k-th iteration is given by the following equation:

[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 for vector operations on different tangent spaces, calculated by the following equation:

[0068]

[0069] in,

[0070] Furthermore, the process of introducing the shrinkage operator to shrink the decreased tangent vector onto the manifold and update the feasible solution is 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 beneficial effects of the present invention are as follows:

[0078] 1. The present invention establishes a MIMO cognitive radar pattern design model under the condition of known target and interference direction. It takes the matching performance of the transmission pattern and the fuzzy characteristics of the transmission waveform as optimization objectives. At the same time, it introduces the spatial cross-correlation between different targets to minimize the mean square error between the transmission pattern and the desired 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, this invention establishes a new optimization problem framework based on constant modulus constraints and similarity constraints to improve radar anti-jamming performance and optimize transmission pattern and transmission waveform;

[0080] 3. Based on the establishment of a problem model, this invention proposes an RPM-CG optimization algorithm based on Riemannian product manifold, which achieves a solution to the problem model with lower complexity and obtains better optimization results. It effectively solves the local optimum problem in traditional algorithms and ensures the stability and effectiveness of the algorithm under multiple constraints. Compared with existing algorithms, this invention exhibits superior orientation pattern matching performance and fuzzy characteristics. Attached Figure Description

[0081] Figure 1 This is a flowchart of the present invention;

[0082] Figure 2 This is a schematic diagram of a MIMO cognitive radar transmitting array model.

[0083] Figure 3 A geometrical schematic diagram of a Riemannian manifold and its tangent space and tangent vectors;

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

[0085] Figure 5 Here is a flowchart of the RPM-CG algorithm proposed in this invention;

[0086] Figure 6 A schematic diagram of simulation results for matching the emission pattern of different algorithms;

[0087] Figure 7 Simulation results for matching emission patterns under different similarity parameters;

[0088] Figure 8 Comparison of fuzzy function slices under different similarity parameters. Detailed Implementation

[0089] The present invention will now be described in further detail with reference to the accompanying drawings.

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

[0091] Step 1: As Figure 2 As shown, a system equipped with N is established. t A MIMO cognitive radar system with one transmitting antenna.

[0092] The MIMO cognitive radar system employs a uniform linear array (ULA), which is divided into N... T There are 3 transmitting antennas, with an element spacing of half a wavelength, and a total sampling rate of M. Let the spatial angle θ∈[-90°, 90°] have K discretization points. 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 radiation pattern.

[0094] Based on the MIMO cognitive radar system model, it is assumed that the transmitting antenna can transmit individual waveforms. The transmitted waveform vector at the m-th sampling time is... For a far-field target located in the θ direction, the synthesized signal can be represented as:

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

[0096] In the formula, The launch steering vector is expressed as follows:

[0097]

[0098] In the formula, d T λ represents the spacing between the transmitting array elements, and λ represents the carrier wavelength.

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

[0100] x(θ)=a t H (θ)S(3)

[0101] In the formula, S is the transmitted waveform matrix.

[0102] Based on the system model and the transmitted signal model, the average power of the synthesized transmitted signal in the θ direction, i.e., the transmitted beam pattern of the MIMO cognitive radar, can be expressed as:

[0103]

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

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

[0106]

[0107] By establishing a 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, and under the conditions of constant waveform modulus constraint and similarity constraint, establish a problem model for pattern matching design based on the minimum mean square error criterion.

[0109] By employing the minimum mean square error criterion and introducing spatial cross-correlation of different objectives, the objective function is obtained, and its expression is as follows:

[0110]

[0111] Where K is the number of discretization points; Q is the number of spatial targets; α is the scale factor; ω k and ω c θ k By adjusting the weights of the direction and cross-correlation terms, a balance can be achieved between these two optimization objectives. Specifically, in this embodiment, Q = 3, ω... k =1, ω c =5, D(θ) is the desired radiation pattern:

[0112]

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

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

[0115]

[0116] In the formula, P t This indicates the transmission power, specifically P in this embodiment. t =N T M.

[0117] Considering that the designed waveform may have undesirable ambiguity characteristics, a similarity constraint is introduced to ensure that the designed waveform maintains consistency with the reference waveform in terms of ambiguity 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, and the reference waveform S0 is an orthogonal linear frequency modulation (LFM) waveform, with the waveform matrix elements as follows:

[0120]

[0121] Based on the pattern matching model expression, and under the conditions of constant modulus constraint and similarity constraint, the expression for the radar transmitted beam pattern matching design problem based on the minimum mean square error criterion is as follows:

[0122]

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

[0124] Transforming inequality constraints into equality constraints, since ||·|| ∞ Let the infinite norm be denoted as 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 hold.

[0127] By combining the similarity equality constraint with the objective function, a Lagrangian function is constructed, which is then used as the new objective function to obtain a simplified optimization problem:

[0128]

[0129] In the formula,

[0130] Step 5: Addressing the nonlinear characteristics of the problem model from Step 4, explore the geometric properties of the nonconvex constraints. Associate the constant modulus constraint with the Riemannian manifold in terms of properties, transforming the constrained nonconvex optimization problem into an unconstrained convex optimization problem on the manifold. A geometrical diagram of the Riemannian manifold and its tangent space and tangent vector is shown below. Figure 3 As shown.

[0131] Based on the nonconvex multivariable optimization problem, the inherent geometric structure of the constraints is figuratively represented. The constraints are embedded in the search space, and the resulting smooth manifold representation after mapping into the solution space is:

[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 The search space for the waveform matrix S is represented as:

[0135]

[0136] Based on the established manifold structure, by defining the corresponding tangent space structure and inner product space, the product manifold M can be obtained. (α,S,β,γ) tangent space T (α,S,β,γ) M (α,S,β,γ) for:

[0137]

[0138] Where S∈M S The tangent space is:

[0139]

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

[0141]

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

[0143]

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

[0145] like Figure 5 As shown, the RPM-CG optimization algorithm is proposed to solve the problem model. To develop the Riemann gradient of the objective function, 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 the function f(α,S) with respect to α. S f(α,S) is the Euclidean gradient of the function f(α,S) with respect to S; h is the partial derivative of g with respect to S, as shown in the following equation:

[0151]

[0152] The Riemann gradients of each component can be obtained by orthogonally projecting the Euclidean gradients onto the corresponding tangent spaces:

[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 cumulant 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 represents the Riemann gradient of the objective function at α, β, and γ, respectively; grad S L represents the objective function in S∈M. S The Riemann gradient at that location.

[0158] Next, a suitable descent direction is selected, a vector transfer operator is defined, and ρ is selected based on different criteria. k The appropriate direction d can then be obtained. k+1 The descent direction of the k-th iteration is given by the following equation:

[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 for vector operations on different tangent spaces, calculated by the following equation:

[0161]

[0162] in,

[0163] The Armijo line search strategy is used to find the appropriate step size μ. k This strategy ensures that the cost does not increase with each iteration.

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

[0165]

[0166] in,

[0167]

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

[0169]

[0170] Under CM constraints, the performance of the RPM-CG algorithm was compared with that of the SQP algorithm, the cyclic algorithm (CA), the maximization-minimization (MM) algorithm, and the Nonlinear-ADMM algorithm. Figure 6 Transmit patterns designed for different algorithms. Except for the CA algorithm, the matching performance of other algorithms within the main lobe is basically the same. The RPM-CG and MM algorithms have better sidelobe matching performance than the Nonlinear-ADMM and SQP algorithms, and are 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. When ε = 0.5, the similarity constraint is high, making it 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 results, the main lobe performance is worst when ε = 0.5, and the sidelobe performance is worst when ε = 1.0. Further observation is made of the influence of the similarity parameter on the ambiguity characteristics of the transmitted sequence. To evaluate the ambiguity characteristics of the transmitted sequence, an expression for the ambiguity function is defined, and the ambiguity function slices of the design sequence and the reference sequence are observed. The results are as follows: Figure 8 As shown, it can be seen that as the similarity parameter increases, the difference between the ambiguity characteristics of the reference waveform and the design waveform becomes greater.

[0172] The embodiments described above are merely examples of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. A method for designing anti-jamming waveform based on direction pattern matching under joint constraints, characterized in that, The method comprises the following steps: (1) establishing a MIMO cognitive radar system equipped with N t transmitting antennas; (2) Constructing a transmitting direction pattern model to realize the matching design of the transmitting direction pattern to resist interference; (3) Based on the transmitting direction pattern model, a problem model of the direction pattern matching design based on the minimum mean square error criterion is established under the condition of satisfying the waveform constant modulus constraint and the similarity constraint; (4) For the problem model of the direction pattern matching design based on the minimum mean square error criterion, the similarity constraint is simplified by using the Lagrange multiplier method, and an optimization problem model with only the constant modulus constraint is obtained; (5) According to the non-linear characteristics of the optimization problem model with only the constant modulus constraint, the geometric characteristics of the non-convex constraint are excavated, the constant modulus constraint is associated with the Riemannian manifold in nature, and the constrained non-convex optimization problem is converted 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 the anti-interference waveform is obtained; The step (3) is implemented in the following manner: The minimum mean square error criterion is adopted, and the spatial cross-correlation of different targets is introduced to obtain a target function: where K is the number of discretization points; Q is the number of spatial targets; a is a scale factor; ω k and ω c are the weight values of the θ k direction and the cross-correlation term, respectively. By adjusting the weight values, a balance between the two optimization objectives is achieved. The constant modulus constraint is introduced, and its expression is: where P t denotes the transmit power; M denotes the total number of samples of the transmit waveform; The similarity constraint is introduced to ensure that the designed waveform is consistent with the reference waveform in the fuzzy characteristics: ||S-S0|| ∞ <ε (9) In the formula, S0 represents the reference waveform; ε is a parameter for adjusting the similarity degree; when ε = 0, the designed waveform is the same as the reference waveform, and if ε is larger, the similarity between the designed waveform and the reference waveform is poorer; Under the condition of satisfying the constant modulus constraint and the similarity constraint, the expression of the radar transmitting beam direction pattern matching design problem based on the minimum mean square error criterion is: The step (4) is implemented in the following manner: The inequality constraint is converted into an equality constraint, and the similarity constraint is represented as: Wherein, γ represents a parameter for making the equality hold; The similarity equality constraint is combined with the target function to construct a Lagrange function, and the simplified optimization problem with only the constant modulus constraint is: In the formulae, The step (5) is implemented in the following manner: According to the optimization problem with only the constant modulus constraint, the internal geometric structure of the constraint condition is visually represented, the constraint condition is embedded into the search space, and the smooth manifold after the mapping of the solution space is represented as: M (α,S,β,γ) = M α x M S x M β x M γ = {(a, S, β, γ) e M (α,S,β,γ) : a e M α , S e M S , β e M β , γ e M γ} (14) Decompose the smooth manifold into single manifold structure, M α , M β and M γ represent the search space of variables α, β and γ, respectively, i.e. M S represents the search space of waveform matrix S, i.e.: According to the relationship between the product manifold and the single manifold and the tangent space of each single manifold, the tangent space T (α,S,β,γ) of the product manifold M (α,S,β,γ) is obtained (α,S,β,γ) : T (a,S,β,γ) M (α,S,β,γ) = T (α,S,β,γ,t) (M α x M S x M β x M γ ) = T α M α x T S M S x T β M β x T γ M γ (16) where S ∈ M S The tangent space of S is: Wherein, 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 are one-dimensional Euclidean space, their tangent spaces are also one-dimensional Euclidean space, that is: An unconstrained optimization problem based on the Riemannian manifold is obtained: The step (6) is implemented in the following manner: The Euclidean gradient of the objective function with respect to the variable scale factor α, the Lagrange multiplier β, the parameter γ making the equality true, and the waveform matrix S is derived, and the Riemannian gradient is represented according to the mapping relationship between the Euclidean gradient and the Riemannian gradient; and the different criteria are selected for ρ k A vector transport operator is introduced to obtain a suitable descending direction; an Armijo line search strategy is adopted to select a step length; a contraction operator is introduced to contract the tangent vector after descending to the manifold and update the feasible solution; and iteration is continuously performed until a convergence criterion is reached, so that an optimal transmitting waveform and a transmitting direction pattern satisfying the joint constraint and the optimization target are obtained.

2. The method according to claim 1, wherein, The MIMO cognitive radar system in the step (1) adopts a uniform linear array for antenna layout, and the antenna interval is half a wavelength.

3. The method of claim 1, wherein, The step (2) is implemented in the following manner: Each transmit antenna transmits a separate waveform, M represents the total number of samples of the transmitted waveform, and the transmitted waveform vector at the mth sample instant is For a far-field target located in the direction of θ, the composite signal is: x(θ, m) = a t H (θ)s m m = 1, 2,..., M (1) wherein is the emission steering vector, which is expressed as In the formula, d T λ is the spacing between the transmitting array elements, and λ is the carrier wavelength; x (θ) = [x (θ, 1), x (θ, 2),..., x (θ, M)] is a vector composed of M synthesized signals, and is represented as: x(θ) = a t H (θ) S (3) In the formula, S is a transmitting waveform matrix; The average power of the synthesized signal in the θ direction, that is, the transmitting beam direction pattern of the MIMO cognitive radar is represented as: Wherein, R represents the covariance matrix of the transmitting waveform; The spatial cross-correlation function of the direction θ p and the direction θ q is represented as: By establishing the MIMO radar transmitting direction pattern model as shown in the formula (1) to (5), the matching design of the transmitting direction pattern to resist interference is realized.

4. The method of claim 1, wherein, Selecting p by different criteria k And introduce the vector transport operator to implement the process as follows: Selecting the descent direction, defining the vector transport operator, selecting p by different criteria k i.e. getting the right direction d k+1 The descent direction of the kth iteration is given by d k+1 = -grad L k + p k Trans k→k+1 (d k ) (30) where Trans k→k+1 (·) is the vector transport operator needed to perform vector operations on different tangent spaces, computed by the following equation: wherein 5. The method of claim 4, wherein, The implementation process of introducing the shrinkage operator to shrink the tangent vector after the descent to the manifold and update the feasible solution is as follows: Shrink operator The expression for the shrink operator is: Wherein: The formula for updating the feasible solution is: where μ k is the step size found using the Armijo line search strategy.

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