Manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection

Abstract

The present invention discloses a manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection. First, the method establishes a waveform design problem model with the goal of minimizing the Euclidean norm of the autocorrelation function sidelobe level under the constant modulus constraint; then, the Euclidean norm objective function is converted into a series of quartic function objective functions; thereafter, by introducing variables, the Lagrange multiplier method is used to reduce the quartic function to a quadratic optimization problem; finally, the ADMM framework is used to construct subproblems, and the Riemann conjugate gradient method is used to solve the constant modulus constrained quadratic function minimization problem. The phase-coded waveform design method proposed in the present invention can significantly reduce the peak sidelobe level and effectively improve the ability to distinguish adjacent UAV targets.

Description

Technical Field

[0001] The present invention belongs to the technical field of radar signal processing, and specifically relates to a manifold-optimized low sidelobe constant modulus waveform design method for low-altitude unmanned aerial vehicle detection. Background Art

[0002] Drones often have the ability to collect signals and intercept interference, which poses a serious threat to the safety of radars. The side lobes of strong scatterers such as lighthouses and high-rise buildings in cities can easily cover up drone targets, affecting radar detection. In order to improve radar detection performance, accurately distinguish multiple nearby drone targets, and reduce the shielding effect of pulse compression side lobes of large targets on small targets, low side lobe emission waveforms are the key to solving the problem. At the same time, considering the full utilization of radar transmitter power, the constant mode characteristics of the design waveform are also very important in practical applications.

[0003] Manifold optimization algorithms are often used in the field of radar transmission signal optimization. For example, in MIMO antenna systems, a new projection, descent, and shrinkage update strategy is used, which allows monotonic cost function improvements while maintaining feasibility on complex circular manifolds. For example, for the radar transmission waveform design problem under the background of colored noise, a constant modulus waveform design method based on the stationary phase principle is proposed. The above studies involve radar waveform design, but none of these studies involve low sidelobe constant modulus waveform design. At present, low sidelobe constant modulus waveform design is still blank, and it is difficult to effectively detect drone swarms. Summary of the invention

[0004] The purpose of the present invention is to propose a manifold optimized low sidelobe constant modulus waveform design method for low altitude UAV detection.

[0005] The technical solution adopted by the present invention is a manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection, comprising:

[0006] Step 1: Taking the Euclidean norm of the autocorrelation function in the sidelobe area as the objective function and imposing constant modulus constraints on the radar transmit waveform, a waveform optimization model based on the integrated sidelobe level is constructed;

[0007] Step 2: Introduce auxiliary variables to equivalently express the minimization problem of the waveform optimization model as a constant modulus and non-convex quadratic constraint minimization problem, wherein the auxiliary variables are determined according to the emission waveform;

[0008] Step 3: According to the autocorrelation function of the sidelobe, a scaled augmented Lagrangian function is constructed, and based on the alternating direction multiplier method algorithm, the constant modulus and non-convex quadratic constraint minimization problem is decomposed into several non-convex quadratic constraint sub-problems;

[0009] Step 4: Use the Riemann conjugate gradient algorithm to solve several non-convex quadratic constraint sub-problems to obtain the optimal transmission waveform.

[0010] Preferably, the specific method of establishing the Euclidean norm minimization problem model of minimizing the autocorrelation function of the sidelobe area under the constant modulus constraint is:

[0011] Taking the Euclidean norm of the autocorrelation function in the minimization sidelobe area as the objective function and the constant modulus constraint of the emission model as 1 as the constraint condition, the waveform optimization model based on the integrated sidelobe level is constructed as follows:

[0012] ,

[0013] in, Indicates the length The phase-coded transmit waveform, The transmitted waveform Code elements, represents the autocorrelation function of the sidelobe, represents the selection matrix, of OK The column elements satisfy , represents the Dirac function, Indicates the position of the code element.

[0014] Preferably, the specific method of introducing auxiliary variables to equivalently express the minimization problem of the Euclidean norm minimization problem model as a constant modulus and non-convex quadratic constraint minimization problem is:

[0015] Introducing auxiliary variables , the optimization model is equivalently transformed into:

[0016] ,

[0017] In the formula, Indicates the length is The phase-coded transmit waveform, express The i-th code element of ; Represents the conjugate transpose operation.

[0018] Preferably, the scaled augmented Lagrangian function constructed according to the autocorrelation function of the side lobe is specifically:

[0019] ,

[0020] in, is the dual variable, is the penalty factor and , Represents the Euclidean norm operation.

[0021] Preferably, based on the alternating direction multiplier method algorithm, the constant modulus and non-convex quadratic constraint minimization problem is decomposed into several non-convex quadratic constraint sub-problems, which are:

[0022] Sub-question 1:

[0023] ,

[0024] For the variable In the The result of the iterative operation; For the variable In the The result of the iterative operation; is the dual variable In the The result of the iterative operation;

[0025] Sub-question 2:

[0026] .

[0027] Preferably, the specific method of using the Riemann conjugate gradient algorithm to solve several non-convex quadratic constraint sub-problems to obtain the optimal transmission waveform is:

[0028] Step 4.1: Set initialization conditions , , Represents the current iteration number; For variables The initial value of in this iteration, For variables The initial value of in this iteration; is the dual variable The initial value of is the iteration tolerance;

[0029] Step 4.2: Introducing auxiliary quantities , , and ignoring the constant term, subproblem 1 can be equivalently expressed as:

[0030] ,

[0031] in, represents the real part operation;

[0032] Using the Riemann conjugate gradient method to solve the equivalent subproblem 1, we obtain The optimal phase-encoded transmit waveform of the iteration ;

[0033] Step 4.3: Introducing auxiliary quantities , , and ignoring the constant term, subproblem 2 can be equivalently expressed as:

[0034] ,

[0035] Using the Riemann conjugate gradient method to solve the equivalent subproblem 2, we obtain The optimal amount of assistance for each round of iteration ;

[0036] Step 4.4: Calculate the first Dual variable of round iteration , and determine whether the iteration termination condition is met. If so, the optimal phase-coded transmission waveform obtained in the current iteration is As the final optimal waveform design result, otherwise, return to step 4.2.

[0037] Preferably, the equivalent subproblem 1 is solved using the Riemann conjugate gradient method to obtain The optimal phase-encoded transmit waveform of the iteration The specific method is:

[0038] Step 4.2.1: Set initialization conditions :

[0039] Indicates the current iteration number, For the transmission waveform The initial value in this iteration, , For variables The initial value of in this iteration, , is the iteration tolerance;

[0040] Step 4.2.2: Calculate The Riemann-Newton gradient at is:

[0041] ,

[0042] in, For function At the point The Riemann gradient at is the conjugate coefficient, is the vector projection operation, Representation variables In the The result in the iteration is is the Riemann-Newton gradient obtained in the previous iteration;

[0043] ,

[0044] is defined as:

[0045] ,

[0046] in, represents the Hadamard product, represents the real part operation;

[0047] Step 4.2.3: Use the Riemann conjugate gradient method to calculate the The transmission waveform The update is expressed as:

[0048] ,

[0049] Representation variables In the The result in the iteration; express In the cutting space The contraction operation on ,variable Indicates any dimension column vector, is the update step size, is the conjugate gradient direction;

[0050] Step 4.2.4: Update according to the Riemann conjugate gradient method , determine whether the iteration termination condition is met ; If the termination condition is met, the obtained is the optimal solution of subproblem 1 in this iteration and Endow , participate in the solution of sub-problem 2. If the termination condition is not met, return to step 4.2.2.

[0051] Preferably, the equivalent subproblem 2 is solved using the Riemann conjugate gradient method to obtain The optimal round of iteration The specific method is:

[0052] Step 4.3.1: Set initialization conditions :

[0053] Indicates the current iteration number, For variables The initial value of in this iteration, , For variables The initial value of in this iteration, , is the iteration tolerance;

[0054] Step 4.3.2: Calculate The Riemann-Newton gradient at is:

[0055] ,

[0056] Representation variables In the The result in the iterations, For variables In the The value in the iteration, For function At the point The Riemann gradient at is the conjugate coefficient, defined as:

[0057] ,

[0058] is the vector projection operation, defined as:

[0059] in, represents the Hadamard product, represents the real part operation;

[0060] Step 4.3.3: Use the Riemann conjugate gradient method to calculate the In the iterations, the variables The updates are as follows:

[0061] ,

[0062] Representation variables In the The result in the iteration is express In the cutting space The contraction operation on is the conjugate gradient direction;

[0063] Step 4.3.4: Updated according to the Riemann conjugate gradient method , determine whether the iteration termination condition is met , if the termination condition is met, the obtained is the optimal solution of subproblem 2 in this iteration and Endow , if the termination condition is not met, return to step 4.3.2.

[0064] Preferably, the update formula of the dual variable is:

[0065] ,

[0066] In the formula, For variables In the The result of the iterative operation is is the dual variable In the The result of the iterative operation is is the coefficient.

[0067] Preferably, the iteration termination condition in step 4.4 is specifically:

[0068] ,

[0069] In the formula, is the iteration tolerance, Represents the Euclidean norm operation.

[0070] Compared with the prior art, the present invention has the following significant advantages: the present invention uses the Riemann conjugate gradient method under the ADMM architecture to effectively control the side lobes and enhance the target resolution capability of UAVs during cluster flight. In addition, simulation results show that the Riemann conjugate gradient method under the ADMM architecture is used for waveform design, and the integrated side lobe level has a significant suppression effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] In order to more clearly illustrate the present technical solution, the following drawings required for use in the prior art description are briefly introduced.

[0072] Figure 1 This is the optimal transmission waveform modulus and phase change diagram when the code element length is N=100.

[0073] Figure 2 This is the autocorrelation function diagram of the transmitted waveform with a code element length of N=100.

[0074] Figure 3 This is the autocorrelation result diagram of the received waveform with a code element length of N=100.

[0075] Figure 4 This is the optimal transmission waveform modulus and phase change diagram when the code element length is N=200.

[0076] Figure 5 This is the autocorrelation function diagram of the transmitted waveform with a code element length of N=200.

[0077] Figure 6This is the autocorrelation result diagram of the received waveform with a code element length of N=200.

[0078] Figure 7 This is the optimal transmission waveform modulus and phase change diagram when the code element length is N=300.

[0079] Figure 8 This is the autocorrelation function diagram of the transmitted waveform with a code element length of N=300.

[0080] Fig. 9 This is the autocorrelation result diagram of the received waveform with a code element length of N=300.

[0081] Fig.10 It is a flow chart of the present invention. DETAILED DESCRIPTION

[0082] The present invention is further described below in conjunction with the accompanying drawings and examples, namely a manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection. Under the condition that the transmitted phase-coded waveform satisfies the constant modulus constraint, a waveform optimization model based on the integrated sidelobe level (ISL) is established, a decomposition problem method based on the alternating direction multiplier method (ADMM) is proposed, and the non-convex optimization problem is solved by the Riemann conjugate gradient method (RCG) to achieve phase-coded waveform design based on Riemann manifold optimization. This method establishes a waveform design model with the Euclidean norm of the autocorrelation function sidelobe level minimized under the constant modulus constraint as the objective function; after converting the objective function into a series of quartic function objective functions, the Lagrange multiplier method is used to reduce the quartic function to a quadratic function optimization problem by introducing auxiliary variables; finally, the Riemann conjugate gradient method is used to solve and design the optimal waveform.

[0083] like Fig.10 As shown, a manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection has the following specific implementation steps:

[0084] Step 1: Establish a Euclidean norm minimization problem model for minimizing the autocorrelation function of the sidelobe area under the constant modulus constraint, specifically:

[0085] For phased array radar systems, in order to maximize the transmission efficiency, the radar transmission waveform must satisfy the constant modulus constraint, that is:

[0086] (1)

[0087] in, is the phase-encoded transmit waveform. Code elements, , is the number of transmitted waveform symbols. To find the modulus operation. Under the constant modulus constraint, the Euclidean norm minimization problem model of minimizing the autocorrelation function of the sidelobe area is established, specifically:

[0088] (2)

[0089] In the formula, Indicates the length is The phase-coded transmit waveform, The first Code elements, represents the autocorrelation function of the sidelobe, represents the selection matrix, of OK The column elements satisfy , represents the Dirac function.

[0090] Will Substituting into (1), the optimization model is obtained:

[0091] (3)

[0092] This problem is a quartic function optimization problem.

[0093] Step 2: Define auxiliary variables , transform the optimization model into:

[0094] (4)

[0095] Indicates the length is The phase-coded transmission waveform is used, but it only participates in the calculation as an auxiliary variable; express The i-th code element of .

[0096] Step 3: Construct the scaled augmented Lagrangian function as:

[0097] (5)

[0098] in is the dual variable, is the penalty factor.

[0099] Then, under the ADMM framework, the optimization problem (4) is decomposed into several sub-problems:

[0100] Sub-question 1:

[0101] (6)

[0102] for In the The result in the iterations, for In the The result in the iteration

[0103] Sub-question 2:

[0104] (7)

[0105] For the variable In the The result of the iterative operation; For the variable In the The result of the iterative operation; is the dual variable In the The result of the iteration operation.

[0106] Step 4: Use the Riemann conjugate gradient algorithm to solve several non-convex quadratic constraint sub-problems and give the optimal solution for the transmit waveform design. The specific method is:

[0107] Step 4.1: Set initialization conditions :

[0108] Step 4.2: Introducing auxiliary quantities , , and ignoring the constant term, subproblem 1 can be equivalently expressed as:

[0109] (8)

[0110] Then, the Riemann conjugate gradient method is used to solve problem (8). The specific method is:

[0111] Step 4.2.1: Set initialization conditions :

[0112] Indicates the current iteration number, For the transmission waveform The initial value in this iteration, , For variables The initial value of in this iteration, , is the iteration tolerance;

[0113] Step 4.2.2: Calculate The Riemann-Newton gradient at is:

[0114]

[0115] in, For function At the point The Riemann gradient at is the conjugate coefficient, is the vector projection operation, Representation variables In the The result in the iteration is For the The Riemann-Newton gradient obtained in the iteration;

[0116] (9)

[0117] is defined as:

[0118]

[0119] in, represents the Hadamard product, Represents the real part operation.

[0120] Step 4.2.3: Use the Riemann conjugate gradient method to calculate the The transmission waveform The update is expressed as:

[0121]

[0122] Representation variables In the The result in the iteration; express In the cutting space The contraction operation on variable Indicates any dimension column vector, is the update step size, is the conjugate gradient direction;

[0123] Step 4.2.4: Update according to the Riemann conjugate gradient method , determine whether the iteration termination condition is met ; If the termination condition is met, the obtained is the optimal solution of subproblem 1 in this iteration and Endow , participate in the solution of sub-problem 2. If the termination condition is not met, return to step 4.2.2.

[0124] Step 4.3: Introduce auxiliary quantity , , and ignoring the constant term, subproblem 2 can be equivalently expressed as:

[0125] (10)

[0126] Then, the Riemann conjugate gradient method is used to solve problem (10). The specific method is:

[0127] Step 4.3.1: Set initialization conditions :

[0128] Indicates the current iteration number, For variables The initial value of in this iteration, , For variables The initial value of in this iteration, , is the iteration tolerance;

[0129] Step 4.2.2: Calculate The Riemann-Newton gradient at is:

[0130] ,

[0131] in, For function At the point The Riemann gradient at is the conjugate coefficient, is the vector projection operation, Representation variables In the The result in the iteration is is the Riemann-Newton gradient obtained in the previous iteration;

[0132] , (11)

[0133] is defined as:

[0134] ,

[0135] in, represents the Hadamard product, represents the real part operation;

[0136] Step 4.2.3: Use the Riemann conjugate gradient method to calculate the The transmission waveform The update is expressed as:

[0137] ,

[0138] Representation variables In the The result in the iteration; express In the cutting space The contraction operation on ,variable Indicates any dimension column vector, is the update step size, is the conjugate gradient direction;

[0139] Step 4.2.4: Update according to the Riemann conjugate gradient method , determine whether the iteration termination condition is met ; If the termination condition is met, the obtained is the optimal solution of subproblem 1 in this iteration and Endow , participate in the solution of sub-problem 2. If the termination condition is not met, return to step 4.2.2.

[0140] Step 4.4: Update the ADMM dual variable rule renew , and determine whether the ADMM iteration termination condition is met If satisfied, the current transmission waveform As the optimal waveform design result, if the iteration termination condition is not met, return to step 4.1 and perform iterative update again.

[0141] In summary, the algorithm flow of this optimization problem is as follows:

[0142] 1) Input parameters: .

[0143] 2) Initialization settings: .

[0144] 3) Update alternately according to the ADMM rule , , .

[0145] 4) If , then stop the iteration, otherwise, , jump to 3).

[0146] 5) Output: Optimal waveform .

[0147] renew The specific process is as follows:

[0148] (1) Input parameters: ;

[0149] (2) Set initialization conditions: , ;

[0150] (3) According to the Riemann conjugate gradient method renew ;

[0151] (4) If , terminate the iteration, and ,on the contrary, Jump to (3).

[0152] renew The specific process is as follows:

[0153] (1) Input parameters: :

[0154] (2) Set initialization conditions: , ;

[0155] (3) According to the Riemann conjugate gradient method renew ;

[0156] (4) If , terminate the iteration, and ,on the contrary, Jump to (3).

[0157] renew The specific process is as follows:

[0158] (1) Input parameters: ;

[0159] (2) Set initialization conditions: ;

[0160] (3) According to the ADMM dual variable update rule renew , and jump to 4).

[0161] The present invention studies and designs the problem of minimizing the sidelobe level waveform of the autocorrelation function under the constraint of the radar emission constant modulus waveform, splits the complex non-convex optimization problem under the ADMM framework, and uses the Riemann conjugate gradient method to solve it.

[0162] The present invention studies the optimization design problem of low sidelobe constant mode transmission waveform required for low-altitude UAV detection, so as to enhance the UAV target resolution capability during cluster flight.

[0163] The simulation results show that the phase coding waveform design method proposed in the present invention can significantly reduce the peak sidelobe level and effectively improve the ability to distinguish adjacent UAV targets.

[0164] Example

[0165] Through Matlab simulation, the manifold-optimized low sidelobe constant mode waveform design method for low-altitude UAV detection of the present invention is further illustrated.

[0166] 1) Simulation system parameter settings

[0167] In this simulation, phase-encoded waveforms with different time widths are used for simulation. The waveform time widths are 100, 200, and 300 respectively. The corresponding time widths are 3.33ns, 6.67ns, and 10ns respectively. The waveform bandwidths are 3GHz, 5GHz, and 7.5GHz respectively. The system sampling rate is 30GHz, and the maximum number of iterations is , Tolerance , the dual and auxiliary variables are initialized randomly.

[0168] 2) Emission waveform drawing

[0169] To show the phase encoding waveform design results, we use the optimized transmit waveform vector , combined with phase coded modulation, the transmission waveform with different numbers of code elements is finally drawn. At the same time, targets with close intervals are designed to simulate the situation where there are drones with close intervals.

[0170] 3) Metrics

[0171] In the present invention, it is necessary to measure the effect of the final transmission waveform. In addition to the transmission waveform diagram and the autocorrelation diagram of the transmission waveform, the integrated sidelobe level (ISL), peak sidelobe level (PSL), main lobe level (MLL), and peak sidelobe ratio (PSR) of the sidelobe region waveform autocorrelation are also used as measurement indicators, which are specifically defined as follows:

[0172] (12)

[0173] The smaller the PSL and the larger the PSR, the better the performance of the designed waveform. The pulse compression results of the waveform show that there are obvious peaks at the target position, and the sidelobe levels at the adjacent positions are much lower than the amplitude at the peak, so the ability to distinguish nearby drones is stronger.

[0174] 4) Result analysis

[0175] The present invention has carried out three example simulations in total:

[0176] Figure 1The optimal transmission waveform modulus and phase change diagram designed when the transmission code element length is N=100. Figure 2 is the autocorrelation function diagram of the corresponding transmission waveform, Figure 3 The autocorrelation function diagram of the received waveform when there is a target 0.1m away.

[0177] Figure 4 The optimal transmission waveform modulus and phase change diagram designed when the transmission code element length is N=200. Figure 5 is the autocorrelation function diagram of the corresponding transmission waveform, Figure 6 The autocorrelation function diagram of the received waveform when there is a target 0.1m away.

[0178] Figure 7 The optimal transmission waveform modulus and phase change diagram designed when the transmission code element length is N=300. Figure 8 is the autocorrelation function diagram of the corresponding transmission waveform, Fig. 9 The autocorrelation function diagram of the received waveform when there is a target 0.1m away. The specific parameter indicators of the phase-coded transmission waveform designed for the experimental simulation are shown in Table 1. The specific parameters of the linear frequency modulation transmission waveform with the same time width are shown in Table 2:

[0179] Table 1 Simulation results of optimal transmission waveform

[0180] Table 2 Linear frequency modulation signal simulation results

[0181]

[0182] exist Figure 1 , 4 In 7, the phase-coded waveform obtained by the algorithm proposed in the present invention has a constant modulus length of 1, which satisfies the constant modulus constraint condition, and the initial phase of each code element is The waveform generated by the algorithm proposed by the present invention conforms to the basic characteristics of the constant modulus phase coded waveform.

[0183] exist Figure 2 , 5 In , 8, the transmit waveform is autocorrelated and the main lobe and side lobe levels with obvious differences are obtained. The peak side lobe levels are 14.47dB, 22.28dB, and 25.28dB, respectively. The peak side lobe ratios are 25.53dB, 23.74dB, and 24.26dB, respectively. When the symbol length gradually increases, the peak side lobe ratio is still large, indicating that the transmitted waveform performance is good.

[0184] exist Figure 3 , 6 In 9, there are two drone targets with a distance of only 0.1m between them. Figure 1 ,4 , 7, the optimal waveform of the transmission is obtained, the echo signal is processed by pulse compression, and the autocorrelation function diagram is obtained. From the results, it can be seen that the target at a distance of 0.1m forms two pulse peaks, and there is an obvious amplitude difference between the peak and the side lobe. At the same time, the distance difference corresponding to the peak is 0.1m, which meets the simulation settings. Therefore, the transmission waveform can clearly distinguish the two targets.

[0185] In Table 1 and Table 2, for transmission signals with the same time width, the transmission waveform designed by the method of the present invention has a lower integrated sidelobe level, a lower peak sidelobe level, and a higher peak sidelobe level, and its performance is significantly superior to that of conventional linear frequency modulation signals.

[0186] In summary, the phase-coded waveform design method based on Riemann manifold optimization of the present invention has good performance, the proposed method has a low sidelobe level, and similar targets can be distinguished. This shows that the phase-coded waveform design method based on Riemann manifold optimization is an effective constant modulus waveform design method for low-altitude UAV detection.

Claims

1. A manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection, characterized in that: include: Step 1: Taking the Euclidean norm of the autocorrelation function in the sidelobe area as the objective function and imposing constant modulus constraints on the radar transmit waveform, a waveform optimization model based on the integrated sidelobe level is constructed; Step 2: Introduce auxiliary variables to equivalently express the minimization problem of the waveform optimization model as a constant modulus and non-convex quadratic constraint minimization problem, wherein the auxiliary variables are determined according to the emission waveform; Step 3: According to the autocorrelation function of the sidelobe, a scaled augmented Lagrangian function is constructed, and based on the alternating direction multiplier method algorithm, the constant modulus and non-convex quadratic constraint minimization problem is decomposed into several non-convex quadratic constraint sub-problems; Step 4: Use the Riemann conjugate gradient algorithm to solve several non-convex quadratic constraint sub-problems to obtain the optimal transmission waveform.

2. The manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection according to claim 1 is characterized in that: The specific method for establishing the Euclidean norm minimization problem model of minimizing the autocorrelation function of the sidelobe area under the constant modulus constraint is: Taking the Euclidean norm of the autocorrelation function in the minimization sidelobe area as the objective function and the constant modulus of the emission model as 1, i.e., the constant modulus constraint as the constraint condition, the waveform optimization model based on the integrated sidelobe level is constructed as follows: ， in, Indicates the length is The phase-coded transmit waveform, The transmitted waveform Code elements, represents the autocorrelation function of the sidelobe, for The conjugate transpose of represents the selection matrix, of OK The column elements satisfy , represents the Dirac function, Indicates the position of the code element.

3. The manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection according to claim 2 is characterized in that: The specific method of using auxiliary variables to express the minimization problem of the Euclidean norm minimization problem model as a constant modulus and non-convex quadratic constraint minimization problem is: Introducing auxiliary variables , the optimization model is equivalently transformed into: ， In the formula, Indicates the length is The phase-coded transmit waveform, express The i-th code element of ; Represents the conjugate transpose operation.

4. The manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection according to claim 3 is characterized in that: The scaled augmented Lagrangian function constructed based on the autocorrelation function of the side lobe is: ， in, is the dual variable, is the penalty factor and , Represents the Euclidean norm operation.

5. The manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection according to claim 4 is characterized in that: Based on the alternating direction multiplier method algorithm, the constant modulus and non-convex quadratic constraint minimization problem is decomposed into several non-convex quadratic constraint sub-problems, namely: Sub-question 1: ， For the variable In the The result of the iterative operation; For the variable In the The result of the iterative operation; is the dual variable In the The result of the iterative operation; Sub-question 2: 。 6. The manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection according to claim 5 is characterized in that: The specific method of using the Riemann conjugate gradient algorithm to solve several non-convex quadratic constraint sub-problems and obtain the optimal transmission waveform is as follows: Step 4.1: Set initialization conditions , , Represents the current iteration number; For variables The initial value of in this iteration, For variables The initial value of in this iteration; is the dual variable The initial value of is the iteration tolerance; Step 4.2: Introducing auxiliary quantities , , and ignoring the constant term, subproblem 1 can be equivalently expressed as: ， in, represents the real part operation; Using the Riemann conjugate gradient method to solve the equivalent subproblem 1, we obtain The optimal phase-encoded transmit waveform of the iteration ; Step 4.3: Introducing auxiliary quantities , , and ignoring the constant term, subproblem 2 can be equivalently expressed as: ， Using the Riemann conjugate gradient method to solve the equivalent subproblem 2, we obtain The optimal amount of assistance for each round of iteration ; Step 4.4: Calculate the first Dual variable of round iteration , and determine whether the iteration termination condition is met. If so, the optimal phase-coded transmission waveform obtained in the current iteration is As the final optimal waveform design result, otherwise, return to step 4.

2.

7. The manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection according to claim 6 is characterized in that: Using the Riemann conjugate gradient method to solve the equivalent subproblem 1, we obtain The optimal phase-encoded transmit waveform of the iteration The specific method is: Step 4.2.1: Set initialization conditions : Indicates the current iteration number, For the transmission waveform The initial value in this iteration, , For variables The initial value of in this iteration, , is the iteration tolerance; Step 4.2.2: Calculate The Riemann-Newton gradient at is: ， in, For function At the point The Riemann gradient at is the conjugate coefficient, is the vector projection operation, Representation variables In the The result in the iteration is is the Riemann-Newton gradient obtained in the previous iteration; ， is defined as: ， in, represents the Hadamard product, represents the real part operation, express conjugation of; Step 4.2.3: Use the Riemann conjugate gradient method to calculate the The transmission waveform The update is expressed as: ， Representation variables In the The result in the iteration; express In the cutting space The contraction operation on ,variable Indicates any dimensional column vector, is the update step size; Step 4.2.4: Update according to the Riemann conjugate gradient method , determine whether the iteration termination condition is met ; If the termination condition is met, the obtained is the optimal solution of subproblem 1 in this iteration and Endow , participate in the solution of sub-problem 2. If the termination condition is not met, return to step 4.2.

2.

8. The manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection according to claim 6 is characterized in that: Using the Riemann conjugate gradient method to solve the equivalent subproblem 2, we obtain The optimal round of iteration The specific method is: Step 4.3.1: Set initialization conditions : Indicates the current iteration number, For variables The initial value of in this iteration, , For variables The initial value of in this iteration, , is the iteration tolerance; Step 4.3.2: Calculate The Riemann-Newton gradient at is: ， Representation variables In the The result in the iterations, For variables In the The value in the iteration, For function At the point The Riemann gradient at is the conjugate coefficient, defined as: ， is the vector projection operation, defined as: in, represents the Hadamard product, represents the real part operation; Step 4.3.3: Use the Riemann conjugate gradient method to calculate the In the iterations, the variables The updates are as follows: ， Representation variables In the The result in the iteration is express In the cutting space Contraction operation on ; Step 4.3.4: Updated according to the Riemann conjugate gradient method , determine whether the iteration termination condition is met , if the termination condition is met, the obtained is the optimal solution of subproblem 2 in this iteration and Endow , if the termination condition is not met, return to step 4.3.

2.

9. The manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection according to claim 6 is characterized in that: The update formula of the dual variable is: ， In the formula, For variables In the The result of the iterative operation is is the dual variable In the The result of the iteration operation.

10. The manifold-optimized low sidelobe constant modulus waveform design method for low-altitude UAV detection according to claim 6, characterized in that: The specific iteration termination condition in step 4.4 is: ， In the formula, is the iteration tolerance.

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