Radar waveform optimization method, system and equipment under Riemannian manifold constraint and medium

Abstract

The invention relates to the technical field of radar waveform optimization, and particularly discloses a radar waveform optimization method, system, equipment and medium under Riemannian manifold constraint, and the method comprises the steps: constructing an original expression used for representing a non-convex optimization problem related to a radar waveform; converting the original expression into a target expression in a Riemannian manifold space, and solving the target expression to obtain an optimal parameter of the target expression; and optimizing the radar waveform by using the optimal parameter. According to the method, the convergence speed of radar waveform optimization can be improved, the better interference suppression performance is achieved, and the output signal-to-interference ratio gain is improved.

Description

Background Art

[0002] Currently, the classical radar waveforms can be mainly divided into three types: Linear Frequency Modulation (LFM), Non-Linear Frequency Modulation (NLFM), and phase-coded waveforms. As a signal with a large time-bandwidth product, the LFM signal has become the most widely used signal. However, its variable parameters often only include bandwidth, chirp rate, etc., and the degree of freedom in waveform design is too low; NLFM waveforms include stepped frequency chirp signals, etc. In addition to the low degree of freedom in waveform design, their signal processing flow is also relatively complex. Phase-coded signals design the initial phase of each symbol through different coding rules to meet different application requirements, and have been widely used in the fields of radar and even communication. Due to its high design freedom, the phase-coded waveform has also become a commonly used waveform for the optimal design of the transmitted waveform of cognitive radar.

[0003] Generally speaking, the design process of radar waveforms is as Figure 2 shown. This design process is a typical closed-loop feedback system. This loop system starts from the waveform generator. By irradiating the radar working environment and targets, the echoes with a large amount of environmental information are processed in the receiver. The processing results can be used as prior information to directly guide the next adaptive design of the transmitted waveform, so as to obtain more accurate and detailed echo information related to the targets and the environment. As an adaptive closed-loop system, compared with the classical transmitted waveforms, the optimized waveforms have higher robustness. Radar waveform optimization under Riemannian manifold constraints is an important means to break through the performance bottleneck of traditional radar signal processing and improve the anti-jamming performance of radar. However, the current radar waveform optimization under Riemannian manifold constraints is difficult to solve the problem of suppressing interference in the near-range ring and design radar waveforms with better anti-jamming performance.

[0004] Therefore, there is an urgent need to provide a technical solution to solve the above problems. Summary of the Invention

[0005] To solve the above technical problems, the present invention provides a method, system, device, and medium for optimizing radar waveforms under Riemannian manifold constraints.

[0006] In the first aspect, the present invention provides a method for optimizing radar waveforms under Riemannian manifold constraints. The technical solution of this method is as follows:

[0007] Construct an original expression for characterizing the non-convex optimization problem related to radar waveforms;

[0008] Convert the original expression into an objective expression in the Riemannian manifold space, and solve the objective expression to obtain the optimal parameters of the objective expression;

[0009] Optimize the radar waveform using the optimal parameters.

[0010] The beneficial effects of a radar waveform optimization method under Riemannian manifold constraint of the present invention are as follows:

[0011] The method of the present invention can not only improve the convergence speed of radar waveform optimization, but also has better interference suppression performance and improves the output signal-to-interference ratio gain.

[0012] On the basis of the above solution, a radar waveform optimization method under Riemannian manifold constraint of the present invention can also be improved as follows.

[0013] In an alternative way, the steps of constructing the original expression for characterizing the non-convex optimization problem related to the radar waveform include:

[0014] Based on the slow-time ambiguity function, and combining the minimum interference power criterion and the constant modulus constraint condition, construct the original expression for characterizing the non-convex optimization problem related to the radar waveform.

[0015] In an alternative way, the steps of converting the original expression into an objective expression in the Riemannian manifold space include:

[0016] Utilize the geometric properties of the constant modulus constraint to convert the original expression into the objective expression in the Riemannian manifold space.

[0017] In an alternative way, the steps of solving the objective expression to obtain the optimal parameters of the objective expression include:

[0018] Utilize the Riemannian conjugate gradient algorithm to solve the objective expression to obtain the optimal parameters of the objective expression.

[0019] In a second aspect, the present invention provides a radar waveform optimization system under Riemannian manifold constraint, and the technical solution of the system is as follows:

[0020] Including: a construction module, an operation module and an optimization module;

[0021] The construction module is used to: construct the original expression for characterizing the non-convex optimization problem related to the radar waveform;

[0022] The operation module is used to: convert the original expression into an objective expression in the Riemannian manifold space and solve the objective expression to obtain the optimal parameters of the objective expression;

[0023] The optimization module is used to: optimize the radar waveform using the optimal parameters.

[0024] The beneficial effects of a radar waveform optimization system under Riemannian manifold constraint of the present invention are as follows:

[0025] The system of the present invention can not only improve the convergence speed of radar waveform optimization, but also has better interference suppression performance and enhances the output signal-to-interference ratio gain.

[0026] On the basis of the above solution, a radar waveform optimization system under Riemannian manifold constraint of the present invention can also be improved as follows.

[0027] In an optional manner, the construction module is specifically used for:

[0028] Based on the slow-time ambiguity function, combined with the minimum interference power criterion and the constant modulus constraint condition, construct the original expression representing the non-convex optimization problem related to the radar waveform.

[0029] In an optional manner, the step of converting the original expression into a target expression in the Riemannian manifold space in the operation module includes:

[0030] Utilize the geometric properties of the constant modulus constraint to convert the original expression into the target expression in the Riemannian manifold space.

[0031] In an optional manner, the step of solving the target expression in the operation module to obtain the optimal parameters of the target expression includes:

[0032] Utilize the Riemannian conjugate gradient algorithm to solve the target expression to obtain the optimal parameters of the target expression.

[0033] In the third aspect, the technical solution of an electronic device of the present invention is as follows:

[0034] It includes a memory, a processor, and a program stored on the memory and running on the processor. When the processor executes the program, it implements the steps of the radar waveform optimization method under Riemannian manifold constraint of the present invention.

[0035] In the fourth aspect, the technical solution of a computer-readable storage medium provided by the present invention is as follows:

[0036] Instructions are stored in the computer-readable storage medium. When the computer-readable storage medium reads the instructions, it causes the computer-readable storage medium to execute the steps of the radar waveform optimization method under Riemannian manifold constraint of the present invention.

[0037] The above description is only an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention, it can be implemented according to the content of the specification. And in order to make the above and other objects, features and advantages of the present invention more obvious and understandable, the specific embodiments of the present invention are hereinafter specifically exemplified. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The drawings are only used to illustrate the embodiments and are not considered to be a limitation of the present invention. Moreover, throughout the drawings, the same reference numerals are used to represent the same components. In the drawings:

[0039] Figure 1 is a schematic flow chart of an embodiment of a radar waveform optimization method under Riemannian manifold constraint of the present invention;

[0040] Figure 2 is a flow chart of radar waveform optimization design;

[0041] Figure 3 is a schematic diagram of a continuous-time radar waveform;

[0042] Figure 4 is a interference distribution diagram of the target and adjacent range rings;

[0043] Figure 5 is a echo pulse diagram of the target and adjacent range rings;

[0044] Figure 6 is a schematic diagram of a range-Doppler cell;

[0045] Figure 7 is a schematic diagram of curves, tangent spaces, tangent planes, contractions and vector transfers on a complex circular manifold;

[0046] Figure 8 is one of the schematic diagrams of the iterative process of the RSD algorithm on a complex circular manifold;

[0047] Figure 9 is the second of the schematic diagrams of the iterative process of the RCG algorithm on a complex circular manifold;

[0048] Figure 10 is a schematic diagram of the desired ambiguity function;

[0049] Figure 11 is a schematic diagram of the change of the objective functions of the RSD and RCG algorithms with the number of iterations;

[0050] Figure 12 is a schematic diagram of the change of the gradient norms of the RSD and RCG algorithms with the number of iterations;

[0051] Figure 13 is a schematic diagram of the relationship between the output SIR and the coding length;

[0052] Figure 14 STAF diagrams corresponding to different waveforms;

[0053] Figure 15 Schematic diagram of the average response on different range cells;

[0054] Figure 16 Schematic structural diagram of an embodiment of a radar waveform optimization system under Riemannian manifold constraint according to the present invention;

[0055] Figure 17 Schematic structural diagram of an embodiment of an electronic device according to the present invention. Detailed implementation manners

[0056] Hereinafter, exemplary embodiments of the present invention will be described in more detail with reference to the drawings. Although the exemplary embodiments of the present invention are shown in the drawings, it should be understood that the present invention can be implemented in various forms and should not be limited by the embodiments set forth herein.

[0057] Figure 1 The flowchart of an embodiment of a radar waveform optimization method under Riemannian manifold constraint provided by the present invention is shown. The radar waveform optimization method under Riemannian manifold constraint can be executed by an electronic device such as a terminal device or a server. Among them, the terminal device can be any fixed or mobile terminal such as a user equipment (UE), a mobile device, a user terminal, a terminal, a cellular phone, a cordless phone, a personal digital assistant (PDA), a handheld device, a computing device, a vehicle-mounted device, a wearable device, etc. The server can be a single server or a server cluster composed of multiple servers. Any electronic device can implement the radar waveform optimization method under Riemannian manifold constraint by a processor calling computer-readable instructions stored in a memory. As Figure 1 shown, it includes the following steps:

[0058] S1. Construct an original expression for characterizing a non-convex optimization problem related to the radar waveform.

[0059] S2. Convert the original expression into an objective expression in the Riemannian manifold space, and solve the objective expression to obtain the optimal parameters of the objective expression.

[0060] S3. Optimize the radar waveform by using the optimal parameters.

[0061] Optionally, S1 includes:

[0062] Based on the slow-time ambiguity function, and in combination with the minimum interference power criterion and the constant modulus constraint condition, construct the original expression for characterizing the non-convex optimization problem related to the radar waveform.

[0063] Optionally, the step of converting the original expression into a target expression in the Riemannian manifold space in S2 includes:

[0064] Converting the original expression into the target expression in the Riemannian manifold space by using the geometric characteristics of the constant modulus constraint.

[0065] Optionally, the step of solving the target expression to obtain the optimal parameters of the target expression in S2 includes:

[0066] Solving the target expression by using the Riemannian conjugate gradient algorithm to obtain the optimal parameters of the target expression.

[0067] The following embodiments are used to illustrate the present solution:

[0068] S101. The phase-coded waveform has low peak power and good ambiguity function characteristics, and has excellent performance in anti-reconnaissance, anti-jamming, etc., which is a waveform system that is mainly considered in radar waveform design. As Figure 3 shown, in an actual radar system, a continuous-time transmitted waveform is often considered, and its expression is:

[0069]

[0070] In formula (1): s(n) ∈ s, s represents the slow-time coding sequence used to modulate the pulses therein; represents the set of complex numbers with dimensions of m × n, represents the set of complex numbers with dimensions of N × 1, and N represents the slow-time coding length, that is, the number of pulses within the coherent accumulation interval; T r is the pulse repetition interval representing the transmitted pulse, that is, equal to the PRI. is determined by the fast-time coding sequence x = [x(0), x(1),..., x(M - 1)] T and M represents the dimension of the fast-time coding of the radar waveform. It can be further expressed as:

[0071]

[0072] In formula (2): p(t) is the sub-pulse in the fast-time dimension of the transmitted waveform, and T p represents the sub-pulse duration period, and T p < T r . For the case of M = 1, the radar transmitted waveform will be simplified to the slow-time coding waveform s = [s(0), s(1),..., s(N - 1)], and at this time, there are only changes between pulses and pulses.

[0073] S102. In an actual application scenario, the radar transmitter transmits N coherent slow-time coded signals s to the area to be detected. As Figure 4 shown, when range ambiguity exists, the radar echo pulses not only contain the target echoes within the non-ambiguous range, but also include the interference echoes of clutter blocks within adjacent range rings. The size of each range ring is Δr = cT r / 2, where c represents the electromagnetic wave propagation speed. Due to the existence of range ambiguity, for a target with a delay of T d , the time delay of the clutter block in the p-th adjacent range ring that interferes with the target is T d + p*T r . As Figure 5 shown, from the perspective of the radar, the echoes of the target and the interference scatterers will be aliased together, causing serious interference to the target detection within the non-ambiguous range ring. In the subsequent content of radar waveform design, this embodiment is dedicated to optimizing the coding sequence s to achieve interference suppression for adjacent range rings and obtain better detection performance.

[0074] S103. Assume that the radar system transmits an N-dimensional coherent pulse signal s to the observation environment, and the transmitted waveform returns to the radar receiver after being modulated by the detected target and the observation environment. The waveform at the receiving end is converted into a baseband signal through down-conversion technology, subjected to pulse-matching filtering operation, and then sampled into a digital signal. Specifically, the observation signal v of the range-azimuth unit irradiated by the radar can be modeled as:

[0075] v = v t (s) + d(s) + n (3)

[0076] In Equation (3): v t (s) represents the received signal of the target part, d(s) represents the interference signal independent of the signal, and n represents the noise component. The specific content of the following three signals will be introduced in turn:

[0077] (1) Target component: Assume that the target detected by the radar moves on the ground at a normalized Doppler velocity . According to the Doppler model of moving targets, the received signal from the target part can be expressed as:

[0078]

[0079] T where α represents the amplitude parameter of the target echo, which is related to the transmitted waveform energy, propagation loss, and scattering coefficient of the target;

[0080] (2) Signal-independent interference component: As Figure 6 shown, the radar observation area is divided into N×L range-Doppler cells. Figure 6 In it, Δr represents the distance between adjacent range rings, which is equal to the maximum unambiguous distance. r0 is less than or equal to the maximum unambiguous distance, that is, r0≤Δr. Assume that there are N t interference scatterers in different range-Doppler cells (r, h), where r∈{0, 1, …, N−1}, l∈{0, 1, …, L−1}. Then the received signal d(s) can be expressed as:

[0081]

[0082] In Equation (5): ρ i represents the amplitude of the i-th interference scatterer, represents the Doppler frequency of the i-th interference scatterer, represents the time-domain steering vector of the i-th interference scatterer, and the transfer matrix J r The calculation expression is:

[0083]

[0084] where m and n both take values in {1, 2, …, N}. The transfer matrix is determined by the relative time delay of the scatterers in adjacent range rings. It should be noted that the interference component d(s) includes not only the clutter scatterers in adjacent range rings, but also other threatening or non-threatening targets.

[0085] (3) Noise component: From a statistical point of view, the noise n is generally assumed to be a circularly symmetric complex Gaussian random distribution with zero mean, and the corresponding covariance matrix is:

[0086]

[0087] where E[·] represents the statistical expectation, (·) H represents the conjugate transpose, I represents the identity matrix, is the noise power. Combining Equation (3), Equation (4) and Equation (5), performing a matched filtering process on the target component, we get:

[0088]

[0089] Assume that the interference component and the noise component are independent of each other. The interference power of the output result of the matched filter is expressed as:

[0090]

[0091] Approximating the statistical mean in Equation (9) with the sample mean and ignoring the constant term Then the interference power is approximately expressed as:

[0092]

[0093] In Equation (10): N v represents the number of units after discrete processing of the normalized Doppler frequency range [-1 / 2, 1 / 2], and p(r, h) is the power distribution diagram of the interference scattering points in the range and Doppler units (r, v h ), which can be determined by the aforementioned reconnaissance and scanning methods. In addition, g s (r, v h ) represents the STAF corresponding to the transmitted waveform s, and the calculation method is as follows:

[0094]

[0095] In Equation (11): v h = -1 / 2 + h / N v , where h = 0, 1,..., N v -1. For a given range-Doppler unit (r, v h ), g s (r, v h ) gives the corresponding response values in the range and Doppler dimensions. For the optimized waveform s, the corresponding Signal to Interference Ratio (SIR) calculation expression is:

[0096]

[0097] Designing an ideal slow-time coding waveform's ambiguity function is equivalent to minimizing the interference power. By establishing a one-to-one mapping relationship:

[0098] k ∈ {0, 1,…, NN v} → (r, h) ∈ {0, 1,…, N} × {0, 1,…, N v} (13)

[0099] Equation (10) can be rewritten as:

[0100]

[0101] where K = N × N v , represents the matrix related to the interference power distribution, and diag(x) represents the matrix formed with x as the elements on the main diagonal.

[0102] S104. For most practical applications, the waveform optimization design problem becomes more complex due to the constant modulus constraint on the transmitted waveform. The reason for this limitation is that radar amplifiers usually operate under saturated conditions to avoid amplitude modulation. To design multiple transmitted waveforms without changing any hardware, the radio frequency power amplifier needs to transmit the same power level for any waveform. Therefore, incorporating the constant modulus constraint condition into the waveform design, the optimization problem can be written as:

[0103]

[0104] Among them, Equation (15) is the original expression in this embodiment. Due to the existence of the constant modulus constraint, the above optimization problem is a typical non-convex optimization problem. In this embodiment, the basic idea of Riemannian optimization is introduced, and by using the geometric properties of the constant modulus constraint, the constant modulus constraint optimization problem in the Euclidean space is transformed into an unconstrained problem on the space of the Riemannian complex circle manifold for solution.

[0105] S105. The Riemannian manifold space is a generalized concept of the Euclidean space. Approximately speaking, the neighborhood of each point on the Riemannian manifold is homeomorphic to an open set in the Euclidean space. Therefore, the manifold can be regarded as being formed by connecting countless pieces of "Euclidean space". The manifold optimization theory is often used to solve matrix optimization problems with a smooth search space. The characteristics of these optimization problems are that the objective function and the constraint conditions have a non-linear manifold structure. In recent years, with the introduction of the concept of retraction, the application of geometric and topological ideas in the Riemannian manifold in aspects such as engineering modeling and problem description has been further developed, which also makes it possible to solve some non-linear problems.

[0106] For Equation (15), under the constant modulus constraint condition, each element of the variable s to be optimized has a fixed modulus value. Therefore, the search space of the above problem can be described as a smooth manifold, that is, the complex circle manifold. By defining the constraint condition as a manifold, Equation (15) can be converted into an unconstrained optimization problem on a constrained space, that is:

[0107]

[0108] Among them, Equation (16) is the target expression in this embodiment. represents a manifold composed of the product of N complex circular rings, and is specifically defined as:

[0109]

[0110] Among them, it can be seen from Equation (17) that is an N-dimensional manifold and is an embedded submanifold of the Euclidean space ​

[0111] Transform the constant modulus constraint optimization problem (15) defined in the Euclidean space into an unconstrained optimization problem defined on the complex circular manifold Then, a large number of efficient gradient optimization algorithms in the Euclidean space can be further extended to the manifold. Before presenting the gradient-based iterative optimization algorithm on the manifold, it is necessary to define and calculate the necessary manifold geometric structures and necessary elements one by one.

[0112] S106. Tangent Space and Tangent Vector.

[0113] For a smooth manifold It can be regarded as A smooth non-linear space in Using the property that the manifold Can be locally approximated by a linear space at any point s, the corresponding tangent space Provides a vector space that approximates the manifold The elements in this vector space are tangent vectors. Figure 7 Some related concepts on the complex circular manifold are illustrated. In Figure 7 Suppose γ(t) is a smooth manifold defined on And has a smooth mapping:

[0114]

[0115] Among them, Represents the set of real numbers, Represents the mapping relationship. As a linear subspace of the Euclidean space It is generally defined as:

[0116]

[0117] The tangent vector Of the smooth curve γ passing through the point Defines the moving direction during gradient descent. Specifically, for the complex circular manifold The tangent space at any point On it can be expressed as:

[0118]

[0119] Among them, Re{·} represents the real part operation, (·) * Represents taking the conjugate, 0 N Represents the zero vector of dimension N, which can be abbreviated as 0.

[0120] S107. Riemannian Metric and Riemannian Gradient.

[0121] In Euclidean space, the gradient of the objective function is an important mathematical quantity in iterative optimization algorithms. In order to obtain the objective function f(s) in The gradient on the first definition of the normed inner product g s (ξ s ,η s ) is the Riemann metric of the tangent space, g s (ξ s ,η s ) is specifically expressed as:

[0122]

[0123] in,(·) H represents the conjugate transpose of a vector, ξ s With η s Both represent the tangent space The tangent vector on . Given a smooth complex circular manifold Specify the Riemann metric g s (ξ s ,η s ), that is, to specify the inner product of the tangent space at any point on the manifold in a smooth way. Given the Riemannian metric g s (ξ s ,η s ) of smooth complex circular manifolds This is a Riemann manifold.

[0124] The Euclidean gradient of the objective function is defined on the original high-dimensional space, while the Riemann gradient is a derivative operator defined on the tangent space. Specifically, the Riemann gradient of f(s) at a point s on the point manifold is defined as:

[0125] Df(s)[ξ s ]= <gradf(s),ξ s > (22)

[0126] Where Df(s)[ξ s ] indicates that the objective function f(s) is at point s along the direction ξ s , gradf(s) represents the Riemann gradient of the objective function f(s). Df(s)[ξ s ] is calculated as follows:

[0127]

[0128] for It is a special case of Euclidean space submanifold, and the directional derivative can be expressed as:

[0129] Df(s)[ξ s ]= <Proj s(Gradf(s)), ξ s > (24)

[0130] where Gradf(s) represents the Riemannian gradient of the objective function f(s), and Proj s (Gradf(s)) represents the orthogonal projection of Gradf(s) onto . Therefore, the computational expression for the Riemannian gradient gradf(s) is:

[0131] gradf(s) = Proj s (Gradf(s)) (25)

[0132] For a complex circular manifold, its corresponding orthogonal projection operator can be expressed as:

[0133]

[0134] For the objective function (objective expression) f(s), rewrite it as:

[0135]

[0136] where the function component Then the Euclidean gradient of f(s) is expressed as:

[0137]

[0138] For the function component f k (s), its directional derivative is defined as:

[0139] Df k (s)[ξ s = <Gradf k (s), ξ s > (29)

[0140] where ξ s represents the direction vector. Further, the directional derivative of the function component is calculated as follows:

[0141]

[0142] Substitute the function component into the above formula (30), and we get:

[0143]

[0144] According to the commutativity of the inner product <u, v> = <v, u>, Df k (s)[ξ s is expressed as:

[0145] Df k(s)[ξ s = 2<(s H Φ k s)Φ k H s + Φs(s H Φ k H s), ξ s > (32)

[0146] Therefore, the Euclidean gradient of f k (s) is expressed as:

[0147] Gradf k (s) = 2(s H Φ k s)Φ k H s + Φ k s(s H Φ k H s) (33)

[0148] Using Equation (28), we can obtain:

[0149]

[0150] Using the projection formula (26), the Riemannian gradient of f(s) is:

[0151]

[0152] S108. Contraction and vector transport.

[0153] In iterative optimization algorithms, it is often necessary to calculate the search direction and move along this direction with a specific step size. In the Riemannian optimization framework, line search is usually performed on the tangent space or a local approximation model is established to find the descent direction required for the next iteration. To keep the new iteration point still on the manifold, a contraction operator needs to be defined to map the points on the tangent space back to the original manifold.

[0154] As Figure 7 shown, the contraction operator defined on is a smooth mapping from to . From a geometric point of view, the exponential mapping is the most natural contraction operator. However, its computational complexity limits its application in many aspects. Therefore, an ideal contraction operator means a sufficient approximation of the exponential map and can move along the geodesics on the manifold with low computational cost. For the Riemannian complex circle manifold a contraction operator at the point s is defined as:

[0155]

[0156] In addition, since two vectors in different tangent spaces cannot be directly added, the vector addition operation on the manifold can be achieved through vector transfer. The vector transfer operator moves the tangent vector ξ at the point s to the point , and the specific definition is as follows:

[0157]

[0158] After defining these Riemannian geometric structures and operators, use the optimization method based on the Riemannian gradient to solve the optimization problem, that is: solve the optimal parameter s of the objective expression.

[0159] S109. Iterative optimization process.

[0160] When solving the unconstrained optimization problem in Euclidean space, the gradient descent algorithm is undoubtedly the simplest and most direct choice algorithm. In Euclidean space, for a smooth function f(x), the standard gradient descent algorithm starts from an initial random point and performs the following iterative process:

[0161] x k+1 = x k + μ k d k , k = 0, 1, … (38)

[0162] where d k represents the descent direction, and μ k > 0 represents the search step size. Different selection strategies for the descent direction d k will produce different gradient-based optimization algorithms. When d k is selected as the gradient Gradf(x) of f(x), the iteration becomes:

[0163] x k+1 = x k - μ k Gradf(x k ), k = 0, 1, … (39)

[0164] The above is the iterative process of the steepest descent method in Euclidean space. Similarly, for equation (16), the corresponding Riemannian Gradient Descent (RGD) algorithm performs the following iterative process:

[0165]

[0166] where d k is taken as the opposite direction of the Riemannian gradient, that is When this is the case, the Riemannian Steepest Descent (RSD) method can be obtained, and its iterative process is as follows:

[0167]

[0168] Figure 8 The iterative process of the RSD method is shown. When calculating the next iterative point of the calculation point s k , first calculate the Riemannian gradient gradf(s k ) at sk, and then take its opposite direction as the descent direction; after calculating the step size, perform gradient descent in the tangent space, and then pull the point obtained by descending on the tangent space back to the manifold surface to obtain the new iterative point s k+1 .

[0169] To ensure that f(s k ) is non-increasing during the iterative process, the RSD algorithm uses the Armijo line search method to solve for the step size required in the iterative operation. The implementation process of the Armijo line search method is shown in Table 1.

[0170] Table 1:

[0171]

[0172]

[0173] To obtain better convergence performance, the conjugate gradient method in the Euclidean space uses a linear combination of the steepest descent direction at the currently considered iterative point and the previous direction of the algorithm as the search direction at the current step. After further generalization by Hesteness, Stiefelyu, and others, this method has been widely applied in unconstrained optimization problems and has become an important class of optimization algorithms.

[0174] For the smooth function f(x) in the Euclidean space, when using the conjugate gradient method to iteratively solve its minimum value, the descent direction at each iteration is expressed as:

[0175] d k = -Gradf(x) + β k-1 d k-1 (42)

[0176] where β k has multiple calculation methods, and its iterative process is transformed into:

[0177] x k+1 = x k - μ k (Gradf(x) - β k-1 dk-1 ), k = 0, 1, … (43)

[0178] Correspondingly, for the Riemannian Conjugate Gradient (RCG), the search method in the tangent space can also be composed of the weighted combination of the negative gradient direction at the current iteration point and the direction used in the previous iteration. Since the search directions of the two consecutive iteration points are not in the same tangent space, a vector transfer operator is needed when adding them. Thus, the search direction of the RCG method is as follows:

[0179]

[0180] Parameter β k-1 is selected according to the Polark-Ribière parameter rule. It is worth noting that when β k = 0, the RCG algorithm degenerates into the RSD algorithm. The iterative process of the RCG algorithm is given in Figure 9 . Meanwhile, the iterative process of the RCG algorithm is shown in Table 2.

[0181] Table 2:

[0182]

[0183]

[0184] S109. Verify the effectiveness of the proposed method through simulation experiments. In the experiment, first set the number of normalized Doppler frequency units N v = 64, and set the number of range rings (i.e., code length) to N = 64. It is assumed in the experiment that the interference is uniformly distributed in the range-Doppler units, and the distribution scenario of the interference power is modeled as the following:

[0185]

[0186] According to the interference distribution scenario, the STAF of the corresponding optimal waveform (optimal parameters) is as Figure 10 shown.

[0187] First, analyze the convergence performance of the RSD and RCG algorithms. In the experiment, the number of iterations of both algorithms is set to 100. Figure 11 shows the variation of the loss function values of the RSD and RCG methods with the number of iterations; Figure 12The variation of the gradient norms of the RSD and RCG algorithms with the number of iterations is given. From the experimental results, it can be observed that as the iteration process progresses, the loss value of the objective function in all algorithms steadily decreases, and the RSD algorithm with a linear convergence rate converges significantly slower than the RCG algorithm. In addition, the RCG algorithm that uses the search direction information of the previous iteration can achieve better performance of the objective function.

[0188] To further verify the interference suppression performance of the optimized waveform, the output SIR values of different algorithms are compared in this embodiment. At the same time, to highlight the superiority of the Riemannian optimization algorithm, the RSD and RCG algorithms are compared with the CIAFIS algorithm. In addition, the output SIR performance of some classical phase-coded waveforms, such as random phase codes and Zad-off codes, is also considered. Figure 13 The relationship between the SIR and the coding sequence length corresponding to different design rule waveforms is given. For the waveform optimization algorithm under the Riemannian framework, the same random phase coding is used as the initial input. The iteration stops when the number of iterations reaches 100 or the gradient norm is less than 10 -6 . From the experimental results, it can be seen that as the coding length increases, the RCG algorithm can achieve the highest output SIR value.

[0189] Figure 14 (a)-(e) in it give the STAFs corresponding to the radar waveforms optimized by the CIAFIS, RSD, and RCG algorithms, as well as the random phase coding waveform and the Zad-off coding waveform. In the experiment, the coding length and the number of Doppler cells are both set to 64. From Figure 14 it can be seen that strong peaks are formed in the STAFs corresponding to all waveforms at the target position (i.e., at the range-Doppler cell (0,0)), which exactly shows that the waveforms have a strong response to the target. In addition, in the area where the interference is distributed, all the optimized waveforms form depressions, indicating the low response performance of the waveforms to interference and also showing the interference suppression performance of the waveforms. Comparing different STAFs, it can be seen that the waveform optimized by the RCG has the closest characteristics to the ideal STAF ( Figure 10 shown), and the optimization effect is the best.

[0190] To further compare the clutter interference suppression effects of different waveforms, the mean value of the waveform responses on different range cells in the interference distribution area is calculated and compared in Figure 15 . From Figure 15 it can be seen that compared with the initial random phase code and the Zad-off Chu code, the optimized waveforms (optimal parameters) have lower response values in the interference presence interval. In addition, for the three optimized design radar waveforms, the mean response under the Riemannian manifold optimization framework is lower, and the waveform optimized by the RCG algorithm has better performance than the RSD algorithm.

[0191] In order to solve the problem of interference suppression in the near-range ring of pulse Doppler radar, the technical solution of this embodiment uses slow-time ambiguity function as an analysis tool, jointly considers the criterion of minimizing interference power and constant modulus constraint conditions, and establishes a non-convex optimization model related to the transmission waveform. By utilizing the geometric characteristics of the constant modulus constraint, the original constrained optimization problem is transformed into an unconstrained optimization problem on the Riemann manifold space, which can then be implemented using the conjugate gradient algorithm under the Riemann optimization framework. The implementation results show that compared with the classic fixed-system radar waveform and the radar waveform designed by the existing optimization algorithm, the waveform designed by the algorithm proposed in this article not only has a better convergence speed, but also has better interference suppression performance, and the output signal-to-interference ratio is improved by more than 5dB.

[0192] The technical solution of this embodiment can not only improve the convergence speed of radar waveform optimization, but also has better interference suppression performance and improves the output signal-to-interference ratio gain.

[0193] Figure 16 FIG. 2 is a schematic diagram showing a structure of an embodiment of a radar waveform optimization system 200 under Riemannian manifold constraints provided by the present invention. Figure 16 As shown, the system 200 includes: a construction module 210, a calculation module 220 and an optimization module 230;

[0194] The construction module 210 is used to: construct an original expression for characterizing a non-convex optimization problem related to radar waveforms;

[0195] The operation module 220 is used to: convert the original expression into a target expression on the Riemann manifold space, and solve the target expression to obtain the optimal parameters of the target expression;

[0196] The optimization module 230 is used to optimize the radar waveform using the optimal parameters.

[0197] In an optional manner, the construction module 210 is specifically used to:

[0198] Based on the slow-time ambiguity function and in combination with the interference power minimization criterion and the constant modulus constraint, the original expression for the non-convex optimization problem characterizing radar waveforms is constructed.

[0199] In an optional manner, the step of converting the original expression into a target expression on the Riemann manifold space in the operation module 220 includes:

[0200] The original expression is converted into the target expression on the Riemannian manifold space by utilizing the geometric properties of the constant modulus constraint.

[0201] In an alternative approach, the step of solving the target expression in the operation module 220 to obtain the optimal parameter of the target expression includes:

[0202] Using the Riemann conjugate gradient algorithm, solve the target expression to obtain the optimal parameter of the target expression.

[0203] The technical solution of this embodiment can not only improve the convergence speed of radar waveform optimization, but also has better interference suppression performance, improving the output signal-to-interference ratio gain.

[0204] For the above parameters and the steps for each module in the radar waveform optimization system 200 under the Riemann manifold constraint in this embodiment to achieve corresponding functions, reference can be made to the parameters and steps in the embodiment of the radar waveform optimization method under the Riemann manifold constraint in the above text, which will not be elaborated here.

[0205] As Figure 17 shown, an electronic device 300 according to an embodiment of the present invention, the electronic device 300 includes a processor 320, the processor 320 is coupled to a memory 310, and at least one computer program 330 is stored in the memory 310. The at least one computer program 330 is loaded and executed by the processor 320 so that the electronic device 300 implements any one of the above-mentioned radar waveform optimization methods under the Riemann manifold constraint. Specifically:

[0206] The electronic device 300 may vary greatly due to configuration or performance differences, and may include one or more processors 320 (Central Processing Units, CPUs) and one or more memories 310. Among them, at least one computer program 330 is stored in the one or more memories 310, and the at least one computer program 330 is loaded and executed by the one or more processors 320 so that the electronic device 300 implements any one of the radar waveform optimization methods provided in the above embodiments. Of course, the electronic device 300 may also have components such as a wired or wireless network interface, a keyboard, and an input / output interface for input / output. The electronic device 300 may further include other components for implementing device functions, which will not be elaborated here.

[0207] A computer-readable storage medium according to an embodiment of the present invention stores at least one computer program, and the at least one computer program is loaded and executed by a processor so that a computer implements any one of the above-mentioned radar waveform optimization methods under the Riemann manifold constraint.

[0208] Optionally, the computer-readable storage medium may be a read-only memory (ROM), a random access memory (RAM), a compact disc read-only memory (CD-ROM), magnetic tape, floppy disk, and optical data storage device, etc.

[0209] In an exemplary embodiment, there is also provided a computer program product or a computer program. The computer program product or the computer program includes computer instructions, and the computer instructions are stored in a computer-readable storage medium. The processor of the electronic device reads the computer instructions from the computer-readable storage medium, and the processor executes the computer instructions, so that the electronic device executes any one of the above-mentioned radar waveform optimization methods under the Riemannian manifold constraint.

[0210] It should be noted that the terms "first", "second", etc. in the description and claims of the present application are used to distinguish similar objects, rather than to limit a specific order or sequence. The order of use of similar objects may be interchanged appropriately, so that the embodiments of the present application described herein can be implemented in an order other than the illustrated or described order.

[0211] Those skilled in the art know that the present invention can be implemented as a system, a method, or a computer program product. Therefore, the present disclosure can be specifically implemented in the following forms: it can be completely hardware, can be completely software (including firmware, resident software, microcode, etc.), or can be a combination of hardware and software, which is generally referred to as "circuit", "module", or "system" herein. In addition, in some embodiments, the present invention can also be implemented in the form of a computer program product in one or more computer-readable media, and the computer-readable media contain computer-readable program code.

[0212] Any combination of one or more computer-readable media may be employed. The computer-readable media may be a computer-readable signal medium or a computer-readable storage medium. A computer-readable storage medium may be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination of the foregoing. More specific examples (a non-exhaustive list) of the computer-readable storage medium include: an electrical connection having one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the present application, the computer-readable storage medium may be any tangible medium that contains or stores a program which can be used by or in connection with an instruction execution system, apparatus, or device.

[0213] Although the embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art may make variations, modifications, substitutions, and alterations within the scope of the present invention to the above embodiments.

Claims

1. A radar waveform optimization method under the constraint of Riemannian manifold, characterized in that, Including: Constructing an original expression for characterizing a non-convex optimization problem related to radar waveforms; Converting the original expression into an objective expression in a Riemannian manifold space and solving the objective expression to obtain the optimal parameters of the objective expression; Optimizing the radar waveform using the optimal parameters.

2. The radar waveform optimization method under the constraint of Riemannian manifold according to claim 1, characterized in that The steps of constructing an original expression for characterizing a non-convex optimization problem related to radar waveforms include: Based on the slow-time ambiguity function, and in combination with the minimum interference power criterion and the constant modulus constraint condition, constructing the original expression for characterizing the non-convex optimization problem related to radar waveforms.

3. The radar waveform optimization method under the constraint of Riemannian manifold according to claim 1, characterized in that, The steps of converting the original expression into an objective expression in a Riemannian manifold space include: Using the geometric properties of the constant modulus constraint to convert the original expression into the objective expression in the Riemannian manifold space.

4. The radar waveform optimization method under the constraint of Riemannian manifold according to claim 1, characterized in that The steps of solving the objective expression to obtain the optimal parameters of the objective expression include: Using the Riemannian conjugate gradient algorithm to solve the objective expression to obtain the optimal parameters of the objective expression.

5. A radar waveform optimization system under the constraint of Riemannian manifold, characterized in that, Including: A construction module, an operation module, and an optimization module; The construction module is configured to: construct an original expression for characterizing a non-convex optimization problem related to radar waveforms; The operation module is configured to: convert the original expression into an objective expression in a Riemannian manifold space and solve the objective expression to obtain the optimal parameters of the objective expression; The optimization module is configured to: optimize the radar waveform using the optimal parameters.

6. The radar waveform optimization system under the constraint of Riemannian manifold according to claim 5, characterized in that, The construction module is specifically configured to: Based on the slow-time ambiguity function, and in combination with the minimum interference power criterion and the constant modulus constraint condition, construct the original expression for characterizing the non-convex optimization problem related to radar waveforms.

7. The radar waveform optimization system under the constraint of Riemannian manifold according to claim 5, characterized in that The steps in the operation module of converting the original expression into an objective expression in a Riemannian manifold space include: Using the geometric properties of the constant modulus constraint to convert the original expression into the objective expression in the Riemannian manifold space.

8. The radar waveform optimization system under the constraint of Riemannian manifold according to claim 5, characterized in that The steps in the operation module of solving the objective expression to obtain the optimal parameters of the objective expression include: Using the Riemannian conjugate gradient algorithm to solve the objective expression to obtain the optimal parameters of the objective expression.

9. An electronic device, characterized in that, The electronic device includes a processor, the processor is coupled to a memory, and at least one computer program is stored in the memory. The at least one computer program is loaded and executed by the processor so that the electronic device implements the radar waveform optimization method under Riemannian manifold constraints as described in any one of claims 1 to 4.

10. A computer-readable storage medium, characterized in that, At least one computer program is stored in the computer-readable storage medium. The at least one computer program is loaded and executed by a processor so that the computer-readable storage medium implements the radar waveform optimization method under Riemannian manifold constraints as described in any one of claims 1 to 4.

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