A method for designing MIMO radar transmit waveform under similarity and variable mode constraints

Abstract

The application discloses a MIMO radar transmitting waveform design method under similarity and variable mode constraints. Firstly, a spatial ISLR minimization framework under waveform similarity and variable mode constraints is constructed; secondly, a high-dimensional multi-constraint non-convex optimization problem is transformed into multiple one-dimensional single-constraint optimization problems; finally, the amplitude and phase of the transmitting waveform meeting the requirements are obtained by solving the one-dimensional single-constraint optimization problems one by one, and the optimization of the MIMO radar transmitting beam pattern is realized. The method of the application can enhance the signal energy radiated by the MIMO radar in the direction of the target of interest as much as possible while meeting the transmitting waveform characteristic control, suppress the signal energy radiated by the MIMO radar in the direction of the target of interest, improve the target detection performance of the MIMO radar in the interference environment, design the amplitude and phase of the transmitting waveform by using the prior information of the target, realize the MIMO radar antenna pattern shaping, suppress the radiation energy in the sidelobe region, improve the detection probability of the target, and has important application prospects in the interference environment.

Description

Technical Field

[0001] This invention belongs to the field of MIMO radar detection technology, specifically relating to a method for designing MIMO radar transmit waveforms under similarity and variable mode constraints. Background Technology

[0002] Compared to phased array radar, multiple-input multiple-output (MIMO) radar offers greater freedom in transmission and processing, and boasts the advantage of waveform diversity. Transmit pattern shaping is an effective method to improve the performance of MIMO radar; that is, by designing the transmit waveform, capabilities in target detection, resource utilization, and interference suppression can be enhanced. Generally, waveform design for MIMO radar can be divided into three categories: the first involves a two-step strategy of solving the optimal covariance matrix and designing the transmission waveform; the second directly optimizes the transmit beam pattern; and the third optimizes the spatial integral sidelobe level ratio (ISLR).

[0003] While the two-step strategy has relatively low computational complexity, the designed antenna pattern will deviate from the desired one. Direct optimization methods, although capable of obtaining an accurate desired antenna pattern, often involve non-convex multi-constraint waveform design problems, posing a significant challenge in solving joint optimization across spatial, temporal, and frequency domains. Therefore, a multi-domain joint optimization method based on spatial ISLR has been proposed. The paper "X. Yu, H. Qiu, J. Yang, W. Wei, G. Cui, and L. Kong, Multi-spectrally constrained MIMO radar beampattern design via sequential convex approximation. IEEE Trans. Aerosp. Electron. Syst., pp. 1–1, 2022" proposes a narrowband MIMO waveform optimization method that considers spectral, energy, and peak-to-average power ratio (PAR) constraints to maximize the ISLR of the transmit antenna pattern. To accelerate algorithm convergence, a fast iterative algorithm based on ADMM is proposed. The paper "E. Raei, M. Alaee-Kerahroodi, and MB Shankar, Spatial-and range-ISLR trade-off in MIMO radar via waveform correlation optimization. IEEE Trans. Signal Process., vol. 69, pp. 3283–3298, 2021" considers both spatial and range ISLR minimization problems and applies energy, PAR, and other constraints respectively. It implements energy allocation of the transmit antenna pattern under multiple constraints based on the coordinate descent algorithm. However, both methods only consider the design of the spatial antenna pattern and ignore the inherent characteristics of the transmit waveform, resulting in a certain performance loss in target detection performance for the designed transmit waveform. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention provides a method for designing MIMO radar transmit waveforms under similarity and variable mode constraints. By utilizing prior target information, the amplitude and phase of the transmit waveform are designed, thereby achieving MIMO radar antenna pattern shaping, suppressing radiation energy in the sidelobe region, and improving the target detection probability.

[0005] The technical solution adopted in this invention is: a method for designing MIMO radar transmit waveforms under similarity and variable mode constraints, the specific steps of which are as follows:

[0006] Step 1: Construct a MIMO radar transmission system model;

[0007] Configure a co-located MIMO radar system with There are 1 transmit antenna, and the number of samples per antenna in the fast time domain is 1. Let the matrix be... This represents the set of narrowband waveforms transmitted in the baseband, i.e. .

[0008] in, express The first transmitter Sampling at each time point, and , Indicates the first One transmitter A fast sampling time, No. The first transmitter Sampling at each time point, Indicates the first The first transmitter A fast sampling time, Represents the transpose of a matrix or vector. Represents the set of complex numbers.

[0009] Assuming the transmitting array is a uniform linear array structure, then the steering vector of the transmitting antenna... for:

[0010] (1)

[0011] in, Indicates the distance between the transmitter antennas. This represents the signal wavelength. Then in The expression for the signal transmission power in the direction is:

[0012] (2)

[0013] in, This represents the conjugate transpose of a vector. The expression is:

[0014] (3)

[0015] Step 2: Construct the optimization problem;

[0016] The transmit waveform is designed by constructing a framework that minimizes the spatial ISLR under similarity and modulus constraints.

[0017] First, the antenna pattern's main lobe, side lobes, and transition band are defined by... , and Composed of a angular grid, the main lobe, side lobes, and transition zone sets of the beam pattern are represented as follows: Spatial ISLR is defined as the ratio of the antenna pattern response in the sidelobe direction to its response in the main lobe direction, and its expression is:

[0018] (4)

[0019] in, Let represent a fractional quadratic function, and and The expression is as follows:

[0020] (5)

[0021] Introduce a variable mode constraint, which is defined as follows:

[0022] (6)

[0023] in, This represents the proportional factor that controls the amplitude fluctuation of the waveform. Indicates by the first The first antenna transmitted the first Sub-pulse.

[0024] Secondly, by imposing a similarity constraint on the transmitted waveform, the optimization problem can be expressed as:

[0025] (7)

[0026] in, Represents the infinite norm, Indicates the reference waveform. This represents the similarity parameter.

[0027] Step 3: Optimize the problem solution;

[0028] For the fractional optimization problem in step two, the coordinate descent approach is used to transform the multivariable problem into a series of univariate problems before solving them.

[0029] feasible matrix This serves as the initial waveform set. Subsequently, in each iteration, the waveform set is updated item by item according to a cyclical rule. While other variables remain constant, One element is considered a unique variable, and then optimization is performed on that unique variable.

[0030] First, set , As a unique variable, the matrix The remaining variables are fixed, as shown below:

[0031] (8)

[0032] Among them, superscript and Indicates the first The updated and unupdated entries in the next iteration. The sum of the power in the undesired azimuth of the antenna pattern is written as:

[0033] (9)

[0034] The second term can be expanded as follows:

[0035] (10)

[0036] in, Representing the conjugate of vectors; using Representation matrix of The antenna pattern response at a non-desired azimuth angle is equivalent to:

[0037] (11)

[0038] in,

[0039] (12)

[0040] Similarly, the desired antenna pattern response at the azimuth angle is as follows:

[0041] (13)

[0042] in,

[0043] (14)

[0044] in, Represented as a matrix of item.

[0045] Secondly, regarding variables The optimization problem can be written in the following form:

[0046] (15)

[0047] in, Indicates reference waveform No. Line number Column elements, The expression is:

[0048] (16)

[0049] The optimization term is represented as Where r≥0 and ∈[−π, π] respectively represent The amplitude and phase. Let and ,use replace Equation (15) can be rewritten as:

[0050] (17)

[0051] in, and These represent the amplitude and phase of the reference waveform, respectively. Objective function The expression is as follows:

[0052] (18)

[0053] Step 4: Solving for the optimal waveform;

[0054] set up This is the optimized solution of equation (17).

[0055] First, using Euler's formula... and They are represented as follows:

[0056] (19)

[0057] (20)

[0058] therefore, It can be redefined as:

[0059] (twenty one)

[0060] in, and They represent The real and imaginary parts. Similarly, and They represent The real and imaginary parts. Because and ,make Equation (21) can be rewritten as:

[0061] (twenty two)

[0062] Then, by solving the following equation, we obtain... and :

[0063] (twenty three)

[0064] Will As a constant term, equation (22) can be transformed into:

[0065] (twenty four)

[0066] in:

[0067] (25)

[0068] We can obtain:

[0069] (26)

[0070] make , , The optimized amplitude can be expressed as:

[0071] (27)

[0072] Under variable mode constraints The following conditions must be met:

[0073] (28)

[0074] Then obtain The optimal solution is:

[0075] (29)

[0076] Then, As a constant, equation (22) can be transformed into:

[0077] (30)

[0078] in:

[0079] (31)

[0080] Similarly, the optimal solution to equation (30) is obtained:

[0081] (32)

[0082] in:

[0083] (33)

[0084] (34)

[0085] (35)

[0086] and The corresponding phases are respectively and .make and Under similarity constraints, The following conditions must be met:

[0087] (36)

[0088] in, , .

[0089] Then we get The optimal solution:

[0090] (37)

[0091] In conclusion, No. The optimal solution in the next iteration is .

[0092] Finally, optimize all entries at least once using the above method. After that, determine whether the algorithm meets the stopping condition. or If the conditions are not met, repeat the above steps to obtain the optimal transmission waveform of the MIMO radar by iteratively updating all variables in the waveform set.

[0093] in, and They represent the first Second and third The optimized waveform obtained in the next iteration To represent a sufficiently small constant, Indicates the number of iterations.

[0094] The beneficial effects of this invention are as follows: First, the method constructs a spatial ISLR minimization framework under waveform similarity and variable mode constraints. Second, it transforms the high-dimensional multi-constraint non-convex optimization problem into multiple one-dimensional single-constraint optimization problems. Finally, by solving each one-dimensional single-constraint optimization problem sequentially, the amplitude and phase of the transmitted waveform that meet the requirements are obtained, thus optimizing the transmitted beam pattern of the MIMO radar. While satisfying the control of the transmitted waveform's inherent characteristics, this method maximizes the signal energy radiated by the MIMO radar in the azimuth of the target of interest and suppresses the signal energy radiated in azimuths other than the detected target, improving the target detection performance of the MIMO radar in jammed environments. By utilizing prior target information, the amplitude and phase of the transmitted waveform are designed to shape the MIMO radar antenna pattern, suppressing the radiation energy in the sidelobe region and increasing the target detection probability. This method has significant application prospects in jammed environments. Attached Figure Description

[0095] Figure 1 This is a flowchart of a MIMO radar transmit waveform design method under similarity and variable mode constraints according to the present invention.

[0096] Figure 2 This is a flowchart of the optimization problem-solving algorithm in an embodiment of the present invention.

[0097] Figure 3 These are antenna radiation patterns with different similarity parameters in embodiments of the present invention.

[0098] Figure 4 This is a fuzzy function graph of different similarity parameters in an embodiment of the present invention.

[0099] Figure 5 These are antenna radiation patterns with different modal constraints in embodiments of the present invention.

[0100] Figure 6 The waveform amplitude diagrams are for different variable mode constraints in the embodiments of the present invention. Detailed Implementation

[0101] In this embodiment, the method of the present invention is verified through simulation experiments. All steps and results are verified on the MATLAB simulation platform. The method of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0102] To facilitate the description of this invention, some terms used in the invention are defined and explained as follows:

[0103] Airspace ISLR refers to the ratio of the sum of the sidelobe levels of all antenna patterns in a MIMO radar to the sum of the main lobe levels of all antenna patterns. The main lobe of the antenna pattern refers to the azimuth where radiated energy is concentrated, while the sidelobe refers to the azimuth where unwanted radiated energy is emitted.

[0104] like Figure 1 The flowchart of a MIMO radar transmit waveform design method under similarity and variable mode constraints of the present invention is shown below. The specific steps are as follows:

[0105] Step 1: Construct a MIMO radar transmission system model;

[0106] In this embodiment, the transmitting array is a uniform linear array (ULA) structure, employing a co-located MIMO radar system. There are 1 transmit antenna, and the number of samples per antenna in the fast time domain is 1. The system simulation parameters used in this embodiment are shown in Table 1.

[0107] Table 1

[0108]

[0109] matrix This represents the set of narrowband waveforms transmitted in the baseband, i.e. .

[0110] in, express The first transmitter Sampling at each time point, and , Indicates the first One transmitter A fast sampling time, No. The first transmitter Sampling at each time point, Indicates the first The first transmitter A fast sampling time, Represents the transpose of a matrix or vector. Represents the set of complex numbers.

[0111] Then the steering vector of the transmitting antenna for:

[0112] (38)

[0113] in, Indicates the distance between the transmitter antennas. This represents the signal wavelength. Then in The expression for the signal transmission power in the direction is:

[0114] (39)

[0115] in, This represents the conjugate transpose of a vector. The expression is:

[0116] (40)

[0117] Because LFM has good Doppler tolerance and low autocorrelation sidelobe level, this embodiment sets the similarity reference waveform as a set of orthogonal LFM waveforms, the mathematical expression of which is:

[0118] (41)

[0119] in, .

[0120] It can be obtained and orthogonal sets .

[0121] Step 2: Construct the optimization problem;

[0122] This embodiment designs waveforms by constructing an optimization framework that minimizes spatial ISLR under similarity and variable mode constraints.

[0123] First, the main lobe, side lobes, and transition band of the antenna pattern are respectively composed of... , and Composed of a angular grid, the main lobe, side lobes, and transition zone sets of the beam pattern are represented as follows: Spatial ISLR is defined as the ratio of the beam pattern response in the undesired direction (sidelobe) to its response in the desired direction (main lobe), and its expression is:

[0124] (42)

[0125] in, Let represent a fractional quadratic function, and and The expression is as follows:

[0126] (43)

[0127] To overcome the limitation on MIMO radar transmit waveform optimization caused by constant mode constraints, this embodiment introduces a waveform variable mode constraint to increase the design freedom of the waveform, appropriately allowing the waveform amplitude to fluctuate within a certain range, thereby improving the design freedom of the transmit waveform. This constraint can be expressed as:

[0128] (44)

[0129] in, This represents the proportional factor that controls the amplitude fluctuation of the waveform. Indicates by the first The first antenna transmitted the first Sub-pulse.

[0130] Secondly, to compensate for the performance loss of the transmitted waveform caused by antenna pattern shaping, a similarity constraint is imposed on the transmitted waveform. The optimization problem is then expressed as:

[0131] (45)

[0132] in, Represents the infinite norm, This represents a reference waveform and possesses excellent characteristics. This represents a similarity parameter used to limit the similarity of waveforms.

[0133] In equation (45), yes Equation (45) is a fractional-order quadratic multivariable function with a non-convex modulus constraint. Therefore, Equation (45) is a multi-constraint high-dimensional non-convex optimization problem.

[0134] Step 3: Optimize the problem solution;

[0135] For the fractional optimization problem in step two, the coordinate descent approach is used to transform the multivariable problem into a series of univariate problems before solving them. The processing flow is as follows: Figure 1 As shown.

[0136] The algorithm flow for solving the optimization problem in this embodiment is as follows: Figure 2 As shown, the matrix This serves as the initial waveform set. Subsequently, in each iteration, the waveform set is updated item by item according to a cyclical rule. While other variables remain constant, One element is considered a unique variable, and then optimization is performed on that unique variable.

[0137] First, set , As a unique variable, the matrix The remaining variables are fixed, as shown below:

[0138] (46)

[0139] Among them, superscript and Indicates the first The updated and unupdated entries in the next iteration. The sum of the power in the undesired azimuth of the antenna pattern can be written as:

[0140] (47)

[0141] The second term can be expanded as follows:

[0142] (48)

[0143] in, Representing the conjugate of vectors; using Representation matrix of The antenna pattern response at a non-desired azimuth angle is equivalent to:

[0144] (49)

[0145] in,

[0146] (50)

[0147] Similarly, the desired antenna pattern response at the azimuth angle is as follows:

[0148] (51)

[0149] in,

[0150] (52)

[0151] in, Represented as a matrix of item.

[0152] Secondly, regarding variables The optimization problem can be written in the following form:

[0153] (53)

[0154] in, Indicates reference waveform No. Line number Column elements, The expression is:

[0155] (54)

[0156] The optimization term is represented as Where r≥0 and ∈[−π, π] respectively represent The amplitude and phase. Let and ,use replace Equation (53) can be rewritten as:

[0157] (55)

[0158] in, and These represent the amplitude and phase of the reference waveform, respectively. Objective function The expression is as follows:

[0159] (56)

[0160] Step 4: Solving for the optimal waveform;

[0161] set up This is the optimized solution of equation (17).

[0162] First, using Euler's formula... and They are represented as follows:

[0163] (57)

[0164] (58)

[0165] therefore, It can be redefined as:

[0166] (59)

[0167] in, and They represent The real and imaginary parts. Similarly, and They represent The real and imaginary parts. Because and ,make Equation (59) can be rewritten as:

[0168] (60)

[0169] Then, by solving the following equation, we obtain... and :

[0170] (61)

[0171] Will As a constant term, equation (60) can be transformed into:

[0172] (62)

[0173] in:

[0174] (63)

[0175] We can obtain:

[0176] (64)

[0177] make , , The optimized amplitude can be expressed as:

[0178] (65)

[0179] Under variable mode constraints The following conditions must be met:

[0180] (66)

[0181] Furthermore, it is possible to obtain The optimal solution is:

[0182] (67)

[0183] Then, As a constant, equation (60) can be transformed into:

[0184] (68)

[0185] in:

[0186] (69)

[0187] Similarly, the optimal solution to equation (68) can be obtained:

[0188] (70)

[0189] in:

[0190] (71)

[0191] (72)

[0192] (73)

[0193] and The corresponding phases are respectively and .make and Under similarity constraints, The following conditions must be met:

[0194] (74)

[0195] in, , .

[0196] Therefore, we can obtain The optimal solution:

[0197] (75)

[0198] In conclusion, No. The optimal solution in the next iteration is .

[0199] Finally, optimize all entries at least once using the above method. After that, determine whether the algorithm meets the stopping condition. or If the conditions are not met, repeat step four to obtain the optimal transmission waveform of the MIMO radar by iteratively updating all variables in the waveform set.

[0200] in, and They represent the first Second and third The optimized waveform obtained in the next iteration To represent a sufficiently small constant, Indicates the number of iterations.

[0201] Figure 3 These are antenna patterns with different similarity parameters, where the amplitude parameter γ is always 0.3, and the similarity parameter... The values ​​were 0, 0.5, 1, 1.5, and 2, respectively. As the similarity parameter increased, the sidelobe level of the antenna pattern decreased significantly.

[0202] Figure 4 It is a fuzzy function graph with different similarity parameters, where the amplitude parameter γ is always 0.3. Figure 4 (a) is the fuzzy function graph of LFM. Figure 4 Similarity parameters of (b)-(f) The similarity parameters are 0, 0.5, 1, 1.5, and 2, respectively. The smaller the value, the closer the optimized fuzzy function is to the reference waveform.

[0203] Figure 5 These are antenna patterns with different modal constraints, where the similarity parameter... With a constant value of 2, and amplitude parameters γ of 0, 0.1, 0.2, and 0.3, the sidelobe level of the designed antenna pattern gradually decreases as the amplitude fluctuation range increases. When γ = 0, the variable mode constraint degenerates into a constant mode constraint, at which point the sidelobe level of the beam pattern is highest.

[0204] Figure 6 These are the waveform amplitudes under different modulus constraints, where the similarity parameter... The constant value is 2. Figure 6 The amplitude parameters γ in (a)-(d) are 0, 0.1, 0.2 and 0.3 respectively. When implementing the beam pattern design, the amplitude of the designed waveform always meets the given amplitude constraint.

[0205] Therefore, it is necessary to select appropriate similarity and amplitude parameters by balancing the importance between spatial ISLR and waveform ambiguity function.

[0206] In summary, the method of this invention establishes an optimization framework with the spatial integral sidelobe level ratio (ISLR) as the objective function and waveform similarity and signal amplitude as constraints. This framework optimizes the transmit beam pattern of MIMO radar and can be used to solve the problem of signal correlation interference where the target and the interference are not in the same azimuth, thereby improving the target detection probability in interference scenarios. Its key feature is that it utilizes prior target azimuth information to design the amplitude and phase of the transmitted waveform. While satisfying waveform similarity and signal amplitude requirements, it maximizes the signal energy radiated by the MIMO radar in the azimuth of the target of interest and reduces the signal energy radiated in azimuths other than the target, thus improving the target detection performance of the MIMO radar in interference environments.

[0207] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A method for designing MIMO radar transmit waveforms under similarity and variable mode constraints, the specific steps of which are as follows: Step 1: Construct a MIMO radar transmission system model; Configure a co-located MIMO radar system with There are 1 transmit antenna, and the number of samples per antenna in the fast time domain is 1. Let the matrix be... This represents the set of narrowband waveforms transmitted in baseband, i.e. ; in, express The first transmitter Sampling at each time point, and , Indicates the first One transmitter A fast sampling time, No. The first transmitter Sampling at each time point, Indicates the first The first transmitter A fast sampling time, Represents the transpose of a matrix or vector. Represents the set of complex numbers; Assuming the transmitting array is a uniform linear array structure, then the steering vector of the transmitting antenna... for: （1）； in, Indicates the distance between the transmitter antennas. Indicates the signal wavelength; then in The expression for the signal transmission power in the direction is: （2）； in, This represents the conjugate transpose of a vector. The expression is: （3）； Step 2: Construct the optimization problem; The transmit waveform is designed by constructing a framework that minimizes the spatial ISLR under similarity and variable mode constraints; First, the antenna pattern's main lobe, side lobes, and transition band are defined by... , and Composed of a angular grid, the main lobe, side lobes, and transition zone sets of the beam pattern are represented as follows: Spatial ISLR is defined as the ratio of the antenna pattern response in the sidelobe direction to its response in the main lobe direction, and its expression is: （4）； in, Let represent a fractional quadratic function, and and The expression is as follows: （5）； Introduce a modulus constraint, defined by the following expression: （6）； in, This represents the proportional factor that controls the amplitude fluctuation of the waveform. Indicates by the first The first antenna transmitted the first Sub-pulse, ; Secondly, by imposing similarity constraints on the transmitted waveform, the optimization problem can be expressed as: （7）； in, Represents the infinite norm, Indicates the reference waveform. Represents similarity parameters; Step 3: Optimize the problem solution; For the fractional optimization problem in step two, the coordinate descent approach is used to transform the multivariate problem into a series of univariate problems before solving them; feasible matrix This serves as the initial waveform set; subsequently, in each iteration, the waveform set is updated item by item according to a cyclical rule; while other variables remain constant, One element is considered a unique variable, and then optimization is performed on that unique variable; First, set As a unique variable, the matrix The remaining variables are fixed, as shown below: （8）； in, , superscript and Indicates the first The updated and unupdated entries in the next iteration; the sum of the power in the undesired azimuth of the antenna pattern is written as: （9）； The second item is expanded as follows: （10）； in, Representing the conjugate of vectors; using Representation matrix of The antenna pattern response at the undesired azimuth angle is equivalent to: （11）； in, （12）； Similarly, the desired antenna pattern response at the azimuth angle is as follows: （13）； in, （14）； in, Represented as a matrix of item; Secondly, regarding variables The optimization problem can be written in the following form: （15）； in, Indicates reference waveform No. Line 1 Column elements, The expression is: （16）； The optimization term is represented as Where r≥0 and ∈[−π, π] respectively represent The amplitude and phase; let and ,use replace Equation (15) can be rewritten as: （17）； in, and Represent the amplitude and phase of the reference waveform, respectively; objective function The expression is as follows: （18）； Step 4: Solving for the optimal waveform; set up This is the optimal solution for equation (17); First, using Euler's formula... and They are represented as follows: （19）； （20）； therefore, Redefined as: （21）； in, and They represent The real and imaginary parts; similarly, and They represent The real and imaginary parts; due to and ,make Equation (21) is rewritten as: （22）； Then, by solving the following equation, we obtain... and : （23）； Will As a constant term, equation (22) is transformed into: （24）； in: （25）； We can obtain: （26）； make , , The optimized amplitude is expressed as: （27）； Under variable mode constraints The following conditions must be met: （28）； Then obtain The optimal solution is: （29）； Then, As a constant, equation (22) is transformed into: （30）； in: （31）； Similarly, the optimal solution to equation (30) is obtained: （32）； in: （33）； （34）； （35）； and The corresponding phases are respectively and ;make and Under similarity constraints, The following conditions must be met: （36）； in, , ; Then we get The optimal solution: （37）； In conclusion, No. The optimal solution in the next iteration is ; Finally, optimize all entries at least once using the above method. After that, determine whether the algorithm meets the stopping condition. or If the conditions are not met, repeat the above steps to obtain the optimal transmission waveform of the MIMO radar by iteratively updating all variables in the waveform set. in, and They represent the first Second and third The optimized waveform obtained in the next iteration To represent a sufficiently small constant, Indicates the number of iterations.

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