A single-bit radar communication integrated transmission waveform design method based on symbol level precoding

Abstract

Adopting single-bit low-resolution digital-to-analog converter (DAC) components is an effective method to reduce the hardware cost and power consumption of a transmitting system and to realize the deployment of a large-scale transmitting array. The present invention studies the design of a radar-communication integrated transmitting waveform under single-bit quantization. While the symbol-level precoding technology is used to constrain the minimum instantaneous received signal-to-noise ratio of each user to fairly guarantee the communication performance of each user, the power concentration degree of the radar transmitting beam is improved to improve the detection performance by minimizing the integrated side-lobe-to-main-lobe ratio of the radar transmitting beam pattern. The design problem finally presents a fractional quadratic programming problem under a large number of discrete constraints. In order to solve the problem to obtain a single-bit integrated transmitting waveform, the present invention proposes a non-convex optimization algorithm based on fractional programming, alternating direction method of multipliers, gradient projection method and other technologies to effectively solve the problem. The experimental results show that the transmitting waveform designed by the present invention can adapt to the single-bit low-resolution DAC components, significantly simplify the structure of the transmitting system, and have good radar transmitting beam pattern performance and multi-user communication performance.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing technology, and specifically relates to a single-bit radar communication integrated transmission waveform design method based on symbol-level precoding. Background Technology

[0002] In radar detection and multi-user communication applications, large-scale transmitter arrays can achieve better target detection and parameter estimation performance, greater communication capacity, and higher reliability. However, a major challenge in deploying large-scale transmitter arrays is the enormous hardware cost and power consumption caused by high-resolution digital-to-analog converters (DACs). One feasible solution is to use low-resolution (e.g., single-bit) DACs for quantization, trading some performance loss for a significant reduction in hardware cost and power consumption.

[0003] To adapt to single-bit DACs components, the corresponding transmit waveform design process needs to solve complex discrete constraints, which makes the current design of single-bit transmit waveforms quite challenging. In response, many literatures have proposed corresponding solutions. Literature [1] aims to minimize the difference between the synthetic beam pattern and the desired transmit beam pattern, and designs the transmit waveform by approximating the single-bit signal through an approximation function, but the final transmit waveform is not an accurate single-bit waveform. Literature [2] considers concentrating the transmit power in the desired direction while reducing the power in the sidelobe region, and directly designs the single-bit transmit waveform with the integral sidelobe-to-main lobe ratio of the radar transmit power as the performance index, and uses the alternating direction multiplier method to handle the single-bit discrete constraints in the problem. However, it should be pointed out that the current single-bit transmit waveform design methods are concentrated on the separate design of radar or communication scenarios, and rarely consider the dual-function collaborative design of radar and communication. To date, only a few single-bit radar-communication integrated design methods have been proposed. For example, Literature [3] aims to minimize multi-user interference and the weighted sum of radar waveform similarity, and proposes a single-bit waveform design method that combines a binary discrete particle swarm optimization-simulated annealing hybrid algorithm under the alternating minimization framework. However, these design methods employ block-level precoding based on second-order statistics in communication, which is difficult to compensate for the severe signal distortion caused by single-bit quantization and cannot fully reflect the user's communication symbol detection performance.

[0004] Therefore, this invention considers combining the nonlinear characteristics of single-bit quantization itself and adopts a more refined nonlinear symbol-level precoding method to compensate for the loss of degrees of freedom caused by single-bit quantization, while improving the user's communication symbol detection performance.

[0005] [1]Cheng Zi-yang, Liao Bin, He Zi-shu, et al.Transmit signal design for large-scale MIMO system with 1-bit DACs[J]. IEEE Transactions on WirelessCommunications, 2019, 18(9): 4466-4478.

[0006] [2]Wei Tong, Cheng Zi-yang, and Liao Bin.Transmit beampattern synthesis for MIMO radar with one-bit digital-to-analog converters[J].SignalProcessing, 2021, 188.

[0007] [3] Yu Xiao-you, Yang Qi, Xiao Zhu, et al. A precoding approach for dual-functional radar-communication system with one-bit DACs[J]. IEEE Journal onSelected Areas in Communications, 2022, 40(6): 1965-1977. Summary of the Invention

[0008] This invention proposes a single-bit radar-communication integrated transmit waveform design method based on symbol-level precoding to adapt to single-bit low-resolution DAC components and achieve dual-function collaborative design for radar and communication. By introducing symbol-level precoding technology into the single-bit transmit waveform design, the minimum instantaneous received signal-to-noise ratio of each user is constrained to fairly guarantee the communication performance of each user.

[0009] The technical solution adopted in this invention is to minimize the integral sidelobe-to-mainlobe ratio of the radar transmit beam pattern under the constraints of symbol-level precoding and single-bit discrete feasible region constraints. This is achieved by improving the power concentration of the transmit beam to enhance target detection performance, thus establishing a single-bit radar-communication integrated transmit waveform design problem model. A non-convex optimization algorithm based on fractional programming, alternating direction multiplier method, and gradient projection method is proposed to effectively solve this design problem. The solution process includes the following steps: Step 1, converting the fractional objective function into an integral objective function using the Tinkelbach transform; Step 2, rewriting the original problem as a real-valued problem through matrix transformation; Step 3, introducing auxiliary variables to represent the discrete constraints as equivalent constraints plus a set of continuous constraints; Step 4, alternately optimizing the auxiliary variables and the original transmit waveform using the alternating direction multiplier method framework.

[0010] Compared with existing technologies, the significant advantages of this invention are: the designed transmission waveform has only four limited discrete phases, which is compatible with single-bit low-resolution DAC components, greatly simplifying the structure of the transmission system. Furthermore, it possesses excellent radar transmission beam pattern performance and multi-user communication performance, enabling simultaneous application in radar and communication scenarios. Attached Figure Description

[0011] Figure 1 The waveform sequence phase distribution diagrams are shown under different design methods in embodiments of the present invention.

[0012] Figure 2 This is a graph showing the relationship between the average bit error rate and communication performance requirements under different design methods in embodiments of the present invention.

[0013] Figure 3 These are single-main-lobe radar transmit beam diagrams under different design methods according to embodiments of the present invention;

[0014] Figure 4 The diagram shows the transmission beam of a dual-mainlobe radar under different design methods in embodiments of the present invention. Detailed Implementation

[0015] The present invention, namely, the integrated transmission waveform design method for single-bit radar communication based on symbol-level precoding, is further described below with reference to the accompanying drawings and examples.

[0016] This invention presents a single-bit radar-communication integrated transmission waveform design method based on symbol-level precoding. Under the constraints of symbol-level precoding and single-bit discrete feasible region constraints, this method aims to minimize the integral sidelobe-to-main-lobe ratio of the radar transmission beammap. It improves target detection performance by increasing the power concentration of the transmitted beam, thus establishing a single-bit radar-communication integrated transmission waveform design problem model. A non-convex optimization algorithm based on fractional programming, alternating direction multiplier method, and gradient projection method is proposed to effectively solve this design problem. The algorithm first uses the Tinkelbach transform to convert the fractional objective function into an integral objective function. Then, it rewrites the original problem as a real-valued problem through matrix transformation. Next, it introduces auxiliary variables to represent the discrete constraints as equivalent constraints plus a set of continuous constraints. Finally, it uses the alternating direction multiplier method framework to alternately optimize the auxiliary variables and the original transmission waveform. The specific implementation steps are as follows:

[0017] Step 1: Consider a radar-communication integrated system equipped with single-bit DACs. The radar-communication dual-function transmitting array is a uniform linear array composed of N antennas. The number of transmit waveform snapshots is L. It can transmit a detection beam in the desired direction while providing communication services to K single-antenna users. The multi-user communication adopts symbol-level precoding.

[0018] Since single-bit DACs perform single-bit quantization on both the real and imaginary channels of each transmit antenna, the transmitted waveform vector... It should be located in the single-bit discrete feasible region. In this context, α is a parameter related to the antenna's transmit power. If the maximum total transmit power of the transmitting array is P... tot Assuming each antenna transmits an integrated waveform using its maximum available power, the parameter α can be specifically expressed as:

[0019]

[0020] If the element spacing of the transmitting array is half a wavelength, then the transmitting array steering vector in the spatial direction θ can be expressed as:

[0021] a(θ) = [1, e- jπsin(θ) , ..., e -jπ(N-1)sin(θ) ] T (29)

[0022] Therefore, the far-field transmitted beam pattern P(θ) in the θ direction can be specifically represented as:

[0023]

[0024] In the above formula, I L Represents an L×L dimensional identity matrix. It represents the Kronecker product.

[0025] Define the expected received symbol for the k-th user at time l as s. k [l], the signal-to-noise ratio requirement for the k-th user is Ψ k The received noise variance is Then the symbol-level precoding parameters of the k-th user at time l It can be represented as:

[0026]

[0027] In the above formula, Φ = π / M, where M is the number of phases used in phase shift keying modulation. It reflects the distance between the noise-free signal received by the user and the decision boundary of the expected received symbol, and can determine the user's communication symbol detection performance.

[0028] To ensure fair communication performance for each user, this invention constrains the minimum instantaneous received signal-to-noise ratio for each user based on symbol-level precoding. Simultaneously, to improve target detection performance by increasing the power concentration of the transmitted beam, this invention minimizes the integral sidelobe-to-main-lobe ratio of the radar transmitted beam pattern. Definitions For the Rayleigh fading channel between the base station and the k-th user, the spatial region is divided into main lobe regions Θ. m and side lobe region Θ s The optimization problem model is as follows:

[0029]

[0030] In the above formula, c l For I L The lth column.

[0031] Step 2: For the fractional form of the objective function in problem (32), convert it into an integral form using the Tinkelbach transform based on fractional programming theory, and define A(Θ)=∫ Θ a * (θ)a T Then, the objective function in integral form obtained after the Tinkelbach transform is:

[0032]

[0033] In the above formula, Ω s and Ω m They are respectively: ξ is an iteratively updated Tinkelbach auxiliary variable. The updated value of ξ in the t-th iteration is ξ. (t) Specifically:

[0034]

[0035] Problem (32) can then be equivalently represented as:

[0036]

[0037] Step 3: To facilitate subsequent processing, the relevant complex vectors and complex matrices are converted to real numbers using the following transformations:

[0038]

[0039] Therefore, problem (35) can be rewritten in real-valued form:

[0040]

[0041] Step 4: Introduce auxiliary variables Discrete constraints are equivalently represented as equality constraints plus continuous constraints, and an augmented Lagrangian function is constructed.

[0042]

[0043] Problem (37) can be further transformed into the following form:

[0044]

[0045] In the above formula, w is a Lagrange multiplier, and ρ > 0 is a penalty factor.

[0046] Step 5: Using a framework based on the alternating direction multiplier method, the problem is decomposed into several easily solvable subproblems. According to the alternating direction multiplier method framework, problem (39) is decomposed into several subproblems under various constraints, which are iteratively updated, with e (k) x R (k) and w (k) Let e ​​and x represent respectively. R Given the updated values ​​of w and w after the k-th iteration, the update process for the k-th iteration is as follows:

[0047]

[0048] w (k) =w (k-1) +ρ(x R (k) -e (k) (42)

[0049] The specific optimization subproblem corresponding to the update process (40) is as follows:

[0050]

[0051] Subproblem (43) contains only one equality constraint. This invention transforms it into an unconstrained problem for direct solution using the Lagrange multiplier method. The Lagrange function of subproblem (43) is defined as:

[0052]

[0053] In the above equation, η is a Lagrange multiplier. Let... We can get e (k) Iterative update formula:

[0054]

[0055] Additionally, due to e (k) Constraints also need to be satisfied Therefore, the value of η can be directly determined. Substituting the value of η into equation (45), we can obtain:

[0056]

[0057] The specific optimization subproblem corresponding to the update process (41) is as follows:

[0058]

[0059] definition For all 2KL vectors The matrix formed here restates subproblem (47) as follows:

[0060]

[0061] In the above formula, Among them, 1 2KL It is a 2KL-dimensional vector of all 1s, where ≥ indicates that each element is greater than or equal to the previous one.

[0062] Problem (48) is a convex quadratic programming problem, which is solved quickly using the gradient projection method in this invention.

[0063] The basic process of solving subproblem (48) using the gradient projection method is as follows:

[0064] 1) Gradient descent. Update variables along the negative gradient direction of the objective function, causing the objective function value to gradually decrease. For subproblem (48), the objective function f(x) R The gradient of ) is specifically:

[0065]

[0066] by Represents variable x R The update value along the gradient direction in the r-th iteration can be obtained by the following formula:

[0067]

[0068] In the above formula, τ is the descent step size, which can be obtained using an inaccurate line search (such as a reverse backtracking line search).

[0069] 2) Alternating projection. After each gradient descent, the projection is performed by... and Multiple alternating projections are performed to ensure that the updated values ​​satisfy the constraint set. Even Define the projection operator This means that the variable x R Each element x R，1 x R，2 , ..., x R，2NL Projection to set Above, specifically:

[0070]

[0071] Define the projection operator This means that the variable x R Projection to set Specifically:

[0072]

[0073] The specific process of solving subproblem (48) using the gradient projection method is as follows:

[0074] 1) Input parameters: ξ, e, Ω s，R Ω m，R w, H, α, ρ, δ

[0075] 2) Initialization settings: r=0

[0076] 3) Start the iteration:

[0077] ir = r + 1

[0078] ii. τ is obtained through reverse backtracking.

[0079] iii. Obtained through equation (50):

[0080] iv. Obtained through equations (51) and (52):

[0081] 4) If That is, select the optimal x. R Stop the loop

[0082] 5) Output the real number form of the transmitted waveform: X R

[0083] After the alternating direction multiplier framework iteration is completed, single-bit projection is used to ensure that the final output fully satisfies the single-bit discrete feasible region constraint. Define the projection operator This means that the variable x R Each element x R，1 x R，2 , ..., x R，2NL Projection to a single-bit feasible region Okay, in detail:

[0084]

[0085] Finally, the real vector x R Reconstructing it into a complex vector x, specifically:

[0086] x=xR(1:NL)+jx R (NL+1:2NL) (54)

[0087] In summary, the complete process of the symbol-level precoding-based single-bit radar communication integrated transmission waveform design method is as follows:

[0088] 1) Input parameter: Θ s Θ m h k ,α, ρ, ν, ε1, ε2

[0089] 2) Initialization settings: t=0

[0090] 3) Reconstructing from equation (54), we get: x (0)

[0091] 4) Obtain ξ using equation (34) (0)

[0092] 5) t = t + 1

[0093] 6) Setting: k = 0

[0094] 7) Start iteration:

[0095] ik = k + 1

[0096] ii. Obtain e through equation (45): (k)

[0097] iii. Obtained using the gradient projection method in step 5:

[0098] iv. Obtain w using equation (42) (k)

[0099] 8) If or Then stop the loop, and obtain x using equations (53) and (54). (t)

[0100] 9) According to equation (34), we can obtain: ξ (t)

[0101] 10) Repeat steps 5-9 until |ξ (t) -ξ (t-1) |<ν means selecting the optimal x

[0102] 11) Output single-bit radar communication integrated transmission waveform: x

[0103] Example

[0104] The specific implementation scheme of the single-bit radar communication integrated transmission waveform design method based on symbol-level precoding is further illustrated through MATLAB simulation.

[0105] 1) Simulation system parameter settings

[0106] In the simulation experiment, the number of transmitting antennas N = 20, the number of transmitted waveform snapshots L = 50, and the total transmitted power P was set to 20. tot =30dBm; the radar transmit beam pattern is sampled uniformly across the entire airspace θ∈[-90°, 90°] with a sampling interval of 1°; the number of communication users K=2, and the communication channel follows the Rayleigh fading model. All users have the same signal-to-noise ratio requirement and received noise power, which is Ψ. k =12dB, The constellation symbols are from the QPSK symbol set. The algorithm parameters and iteration termination condition are set as follows: δ = 10 -5 ε1=ε2=10 -5 ν = 10 -5 .

[0107] 2) Drawing the transmitted beam pattern

[0108] To visually demonstrate the radar performance of the single-bit radar-communication integrated transmission waveform design method based on symbol-level precoding, this embodiment uses the method proposed in this invention to draw the transmission beam diagram, and compares it with the single-bit radar transmission waveform design method in reference [2] and the constant-mode radar transmission waveform design method under infinite-bit quantization in reference [4]. The horizontal axis of the transmission beam diagram represents the spatial angular range of [-90°, 90°], and the vertical axis is the normalized power of the transmitted signal at that angle based on the maximum power of the entire spatial domain, in dB.

[0109] [4]Cheng Zi-yang, Han Chun-lin, Liao Bin, et al. Communication-awarewaveform design for MIMO radar with good transmit beampattern[J]. IEEE Transactions on Signal Processing, 2018, 66(21): 5549-5562.

[0110] 3) Measurement indicators

[0111] In this invention, it is necessary to measure the radar performance of the transmitted waveform. In addition to visually displaying the differences by plotting the transmitted beam diagram, the radar transmitted beam performance of the transmitted waveform is numerically evaluated using the integral sidelobe-to-mainlobe ratio of the radar transmitted beam diagram. The specific expression is as follows:

[0112]

[0113] Where, Θ s For the side lobe region, Θ m The main lobe region. Under the same initial conditions, the smaller the integral sidelobe-to-main lobe ratio, the better the radar's transmit beam performance.

[0114] 4) Results Analysis

[0115] This invention provides four sets of example simulations. Figure 1 Phase distribution diagrams of waveform sequences for three different transmit waveform design methods. Figure 2 This is a graph showing the relationship between average bit error rate and communication performance requirements under different design methods. Figure 3 Transmit beam patterns of single-main-lobe radars under different design methods, where the main lobe region is defined as Θ. m = [-10°, 10°], with the side lobe region being Θ. s = [-90°, -10°]∪[10°, 90°]. Figure 4 Transmit beam patterns of dual-mainlobe radars under different design methods, where the main lobe region is defined as Θ. m = [-40°, -30°]∪[30°, 40°], with the sidelobe region being Θ. s = [-90°, -40°]∪[-30°, 30°]∪[40°, 90°].

[0116] Depend on Figure 1It can be seen that the phase distribution of the constant modulus waveform sequence is between [-π, π], while the phase of the single-bit waveform sequence, including that of this invention, is only located in four discrete phases {-3π / 4, -π / 4, π / 4, 3π / 4}. Compared with the constant modulus waveform, the requirement for DAC resolution is greatly reduced, which can significantly simplify the hardware structure of the transmitter and reduce the hardware cost and power consumption of the transmission array.

[0117] Depend on Figure 2 It can be seen that the radar transmission waveforms proposed in references [2, 4] have no communication function at all, while the symbol-level precoding technology used in this invention allows downlink users to decode the transmitted symbols from the received signals. Meanwhile, as the user's signal-to-noise ratio requirement Ψ increases, the user's bit error rate will further decrease. Furthermore, it can be found that increasing the number of users K only has a minor impact on communication performance, and this impact gradually decreases as Ψ increases.

[0118] Depend on Figure 3 It can be seen that the single-bit integrated waveform proposed in this invention, under the condition of having communication function, can also concentrate the power of the transmitted beam in the desired main lobe region Θ. m Meanwhile, compared to the single-bit radar waveform in reference [2], the transmit beam pattern performance of the present invention only shows a slight decrease, and the integral sidelobe-to-main-lobe ratio only increases by 1.06 dB. This indicates that the present invention utilizes symbol-level precoding technology, sacrificing only a slight loss in radar performance to achieve its application in communication scenarios.

[0119] Depend on Figure 4 It can be seen that even with increased complexity in the radar transmit beam shape, this invention can still form a good radar detection beam. Furthermore, compared to the -4.5dB integrated sidelobe-to-main-lobe ratio of a single-bit radar waveform, this invention achieves a -3.96dB integrated sidelobe-to-main-lobe ratio under these conditions, further reducing the performance gap between radar systems.

[0120] In summary, the method proposed in this invention exhibits excellent overall performance. Compared to traditional constant-mode transmission waveforms, it is compatible with single-bit low-resolution DAC components, significantly simplifying the hardware structure of the transmission system. Furthermore, compared to traditional radar transmission waveforms, this invention, based on symbol-level precoding technology, provides multi-user communication capabilities, allowing simultaneous application in radar and communication scenarios, thus demonstrating high application value and broad application prospects.

Claims

1. A method for designing a single-bit radar communication integrated transmission waveform based on symbol-level precoding, characterized by: Based on fractional programming, alternating direction multiplier method, and gradient projection method, a single-bit integrated radar communication transmit waveform design method is proposed. This method is applicable to the single-bit integrated transmit waveform design problem, which aims to minimize the integral sidelobe-to-main lobe ratio of the radar transmit beam pattern under constraints of single-bit discrete feasible region and symbol-level precoding. The method first uses the Tinkelbach transform to convert the fractional objective function into an integral objective function; secondly, it rewrites the original problem as an equivalent real-valued problem through matrix transformation; thirdly, it introduces auxiliary variables to represent the discrete constraints as equivalent constraints plus a set of continuous constraints; and finally, it uses the alternating direction multiplier method framework to alternately optimize the auxiliary variables and the original transmit waveform. The specific implementation steps are as follows: Step 1: To fairly guarantee the communication performance of each user, this invention constrains the minimum instantaneous received signal-to-noise ratio for each user based on symbol-level precoding. Simultaneously, to improve the power concentration of the transmitted beam and thus enhance target detection performance, this invention minimizes the integral sidelobe-to-main lobe ratio of the radar transmitted beam pattern. The dual-function radar communication transmit array is defined as a uniform linear array composed of N antenna elements, with L being the number of transmitted waveform snapshots. The transmitted waveform vector, For a single-bit discrete feasible region, α is a parameter related to the antenna transmit power. Let P(θ) be the Rayleigh fading channel between the base station and the k-th user, and let P(θ) be the far-field transmit beam pattern in the spatial direction θ, dividing the spatial region into the main lobe region Θ. m and side lobe region Θ s The optimization problem model is as follows: In the above formula, Let c be the symbol-level precoding parameters for the k-th user at time l. t Let I be an L×L dimensional identity matrix. L The lth column, Indicates the Kronecker product; Step 2: For the fractional form of the objective function in problem (1), according to the theory of fractional programming, use the Tinkelbach transform to convert it into an integral form. Then, problem (1) is equivalently expressed as: In the above formula, ξ is an auxiliary variable for iteratively updating Tinkelbach; Step 3: To facilitate subsequent processing, the relevant complex vectors and complex matrices are converted to real numbers, and problem (2) is rewritten in equivalent real-valued form: In the above formula, x R , and Corresponding to x and h respectively k and The real-valued form; Step 4: Introduce auxiliary variables Discrete constraints are equivalently represented as equality constraints plus a set of continuous constraints, constructing an augmented Lagrangian function. Problem (3) can be further transformed into the following form: In the above formula, w is a Lagrange multiplier, and ρ > 0 is a penalty factor; Step 5: Using a framework based on the alternating direction multiplier method, the problem is decomposed into several easily solvable subproblems; among them, the subproblem of optimizing the auxiliary variable e is solved using the Lagrange multiplier method: optimizing the transmission waveform x R The subproblems are solved using the gradient projection method.

2. The method for designing integrated single-bit radar communication transmission waveforms based on symbol-level precoding according to claim 1, characterized in that: In step 1, the radar communication dual-function transmitting array can transmit a detection beam in the desired direction while providing communication services to K single-antenna users, wherein the multi-user communication adopts a symbol-level precoding method. If the maximum total transmission power of the transmitting array is P tot Assuming each antenna transmits an integrated waveform using its maximum available power, the parameter α can be specifically expressed as: If the element spacing of the transmitting array is half a wavelength, then the transmitting array steering vector in the spatial direction θ can be expressed as: a(θ)=[1,e -jπsin(θ) ,…,And -jπ(N-1)sin(θ) ] T (6) Therefore, the far-field transmitted beam pattern P(θ) in the θ direction can be specifically represented as: Define the expected received symbol for the k-th user at time l as s. k [l], the signal-to-noise ratio requirement for the k-th user is Ψ k The received noise variance is Then the symbol-level precoding parameters of the k-th user at time l It can be represented as: In the above formula, Φ = π / M, where M is the number of phases used in phase shift keying modulation; It reflects the distance between the noise-free signal received by the user and the decision boundary of the expected received symbol, and can determine the user's communication symbol detection performance.

3. The method for designing integrated single-bit radar communication transmission waveforms based on symbol-level precoding according to claim 1, characterized in that: In step 2, the integral objective function obtained through the Tinkelbach transform is specifically as follows: In the above formula, Ω s and Ω m They are respectively: in The updated value of the Tinkelbach auxiliary variable ξ in the t-th iteration. (t) Specifically:

4. The method for designing integrated single-bit radar communication transmission waveforms based on symbol-level precoding according to claim 1, characterized in that: In step 3, the specific transformations for converting complex vectors and complex matrices to real numbers are as follows:

5. The method for designing a single-bit radar communication integrated transmission waveform based on symbol-level precoding according to claim 1, characterized in that: In step 4, the augmented Lagrange function Specifically:

6. The method for designing integrated single-bit radar communication transmission waveforms based on symbol-level precoding according to claim 1, characterized in that: In step 5, based on the alternating direction multiplier method framework, problem (4) is decomposed into multiple sub-problems under various constraints, which are iteratively updated, with e (k) x R (k) and w (k) Let e ​​and x represent respectively. R Given the updated values ​​of w and w after the k-th iteration, the update process for the k-th iteration is as follows: w (k) =w (k-1) +ρ(x R (k) -e (k) )。 (15) 7. The method for designing integrated single-bit radar communication transmission waveforms based on symbol-level precoding according to claim 6, characterized in that: The specific optimization subproblem corresponding to the update process (13) is as follows: Subproblem (16) contains only one equality constraint. This invention transforms it into an unconstrained problem for direct solution using the Lagrange multiplier method. The Lagrange function of subproblem (16) is defined as: In the above formula, η is a Lagrange multiplier; let We can get e (k) Iterative update formula: Additionally, due to e (k) Constraints also need to be satisfied Therefore, the value of η can be directly determined; substituting the value of η into equation (18), we can obtain:

8. The method for designing integrated single-bit radar communication transmission waveforms based on symbol-level precoding according to claim 6, characterized in that: The specific optimization subproblem corresponding to the update process (14) is as follows: definition For all 2KL vectors The matrix formed here restates subproblem (20) as follows: In the above formula, Among them, 1 2KL For a 2KL-dimensional vector of all ones, ≥ indicates that each element is greater than or equal to: Problem (21) is a convex quadratic programming problem, which is solved using the gradient projection method; The basic process of solving subproblem (21) using the gradient projection method is as follows: 1) Gradient descent; update variables along the negative gradient direction of the objective function, so that the value of the objective function gradually decreases; for subproblem (21), the objective function f(x) R The gradient of ) is specifically: by Represents variable x R The update value along the gradient direction in the r-th iteration can be obtained by the following formula: In the above formula, τ is the descent step size, which can be obtained using a reverse backtracking search: 2) Alternating projection; after each gradient descent, the projection will be... exist and Multiple alternating projections are performed to ensure that the updated values ​​satisfy the constraint set. Right now Define the projection operator This means that the variable x R Each element x R，1 x R，2 , ..., x R，2NL Projection to set Above, specifically: Define the projection operator This means that the variable x R Projection to set Above, specifically: The specific process of solving subproblem (21) using the gradient projection method is as follows: 1) input parameter:ξ,e,Ω s，R ,Oh m，R ,w,H,a,p,d 2) Initialization settings: 3) Start the iteration: i.r＝r+1 ii. Obtained through reverse backtracking: τ iii. Obtained through equation (23): iv. Obtained through equations (24) and (25): 4) If That is, select the optimal x. R Stop the loop 5) Output the real number form of the transmitted waveform: x R .

9. The method for designing integrated transmit waveforms for single-bit radar communication based on symbol-level precoding according to claim 6, characterized in that: After the alternating direction multiplier method framework completes its iterations, single-bit projection is used to ensure that the final output fully satisfies the single-bit discrete feasible region constraint. Define the projection operator This means that the variable x R Each element x R，1 x R，2 , ..., x R，2NL Projection to a single-bit feasible region Above, specifically: Finally, the real vector x R Reconstructing it into a complex vector x, specifically: x=x R (1:NL)+jx R (NL+1:2NL)。 (27)。

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