Single-bit radar communication integrated transmitted waveform design method based on symbol-level precoding

Abstract

According to the invention, a radar communication integrated transmitted waveform design problem under single-bit quantization is researched, and the communication performance of each user is fairly ensured by constraining the minimum instantaneous received signal-to-noise ratio of each user by using a symbol-level precoding technology; the power concentration degree of a transmitted beam is improved by minimizing the integral side lobe and main lobe ratio of a radar transmitted beam pattern so as to improve the detection performance, and the design problem is finally represented as a fractional quadratic programming problem under a large amount of discrete constraints. In order to solve the problem and obtain a single-bit integrated transmitted waveform, the invention provides a non-convex optimization algorithm based on fractional programming, an alternating direction multiplier method, a gradient projection method and other technologies for effective solution. Experimental results show that the transmitting waveform designed by the invention can be adaptive to a single-bit low-resolution digital-to-analog converter assembly, and has good radar transmitting beam pattern performance and multi-user communication performance while the structure of a transmitting system is remarkably simplified.

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and specifically discloses a method for designing a single-bit radar communication integrated transmission waveform based on symbol-level precoding. Background Art

[0002] In radar detection and multi-user communication applications, the use of large-scale transmit arrays can achieve better target detection and parameter estimation performance, greater communication capacity and reliability. However, one of the major challenges for the deployment of large-scale transmit arrays is the huge hardware cost and power consumption caused by high-resolution digital-to-analog converters (DACs). In this regard, a feasible solution is to use low-resolution (e.g., single-bit) DACs components for quantization, in exchange for a certain degree of performance loss in hardware cost and power consumption.

[0003] In order to adapt the single-bit DACs component, the corresponding transmit waveform design process needs to solve complex discrete constraints, which makes the current single-bit transmit waveform design challenging. In this regard, 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 with an approximate 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 area, and directly designs the single-bit transmit waveform with the integrated sidelobe to mainlobe ratio of the radar transmit power as the performance indicator, and uses the alternating direction multiplier method to deal with the single-bit discrete constraints in the problem. However, it should be pointed out that the current single-bit transmit waveform design methods focus on the separate design of radar or communication scenarios, and rarely consider the dual-function collaborative design of radar and communication. So far, 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 combining binary discrete particle swarm optimization-simulated annealing hybrid algorithm under the alternating minimization framework. However, these design methods adopt a block-level precoding method based on second-order statistics in communication, which is difficult to compensate for the serious signal distortion caused by single-bit quantization and cannot fully reflect the user's communication symbol detection performance.

[0004] To this end, the present invention considers combining the nonlinear characteristics of single-bit quantization itself and adopts a more sophisticated nonlinear symbol-level precoding method to compensate for the loss 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 a1. 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 beamattem 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] In order to adapt the single-bit low-resolution DACs component and realize the dual-function collaborative design of radar and communication, the present invention proposes a single-bit radar communication integrated transmission waveform design method based on symbol-level precoding. By introducing the symbol-level precoding technology into the single-bit transmission 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 by the present invention is to minimize the integrated sidelobe to mainlobe ratio of the radar transmission beam diagram under the constraints of symbol-level precoding and single-bit discrete feasible domain, improve the target detection performance by improving the power concentration of the transmission beam, and establish a single-bit radar communication integrated transmission waveform design problem model. A non-convex optimization algorithm based on fractional programming, alternating direction multiplier method, gradient projection method and other technologies is proposed to effectively solve the design problem. The solution process includes the following steps: Step 1, using Dinkelbach transformation to convert the fractional objective function into an integer objective function; Step 2, rewriting the original problem into a real-valued problem through matrix transformation; Step 3, introducing auxiliary variables to equivalently express the discrete constraints as equality constraints plus a set of continuous constraints; Step 4, using the alternating direction multiplier method framework to alternately optimize the auxiliary variables and the original transmission waveform.

[0010] Compared with the prior art, the present invention has the following significant advantages: the designed transmission waveform has only four limited discrete phases, which is suitable for single-bit low-resolution DACs components to greatly simplify the structure of the transmission system. It also has good radar transmission beam pattern performance and multi-user communication performance, and can be applied to radar and communication working scenarios at the same time. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 1. The waveform sequence phase distribution diagram under different design methods of the embodiment of the present invention;

[0012] Figure 2 is a graph showing the relationship between the average bit error rate and the communication performance requirement under different design methods of an embodiment of the present invention;

[0013] Figure 3 1. Single main lobe radar transmission beam diagram under different design methods of the embodiments of the present invention;

[0014] Figure 4 1 and 2 are dual main lobe radar transmission beam patterns under different design methods of the embodiments of the present invention. DETAILED DESCRIPTION

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

[0016] The present invention is a method for designing a single-bit radar communication integrated transmission waveform based on symbol-level precoding. Under the constraints of symbol-level precoding constraints and single-bit discrete feasible domain constraints, this method aims to minimize the integrated sidelobe-to-mainlobe ratio of the radar transmission beam diagram, improves the target detection performance by improving the power concentration of the transmission beam, and establishes a single-bit radar communication integrated transmission waveform design problem model. A non-convex optimization algorithm based on fractional programming, alternating direction multiplier method, gradient projection method and other techniques is proposed to effectively solve the design problem. The algorithm first uses the Dinkelbach transform to convert the fractional objective function into an integer objective function, and then rewrites the original problem into a real-valued problem through matrix transformation. Auxiliary variables are introduced to equivalently express the discrete constraints as an equality constraint plus a set of continuous constraints. Finally, the alternating direction multiplier method framework is used 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 a single-bit DACs component, in which the radar communication dual-function transmitting array is a uniform linear array composed of N antennas, with the number of transmitting waveform snapshots being L. It can transmit a detection beam in the desired direction while providing communication services for K single-antenna users, and multi-user communication adopts symbol-level precoding.

[0018] Since the single-bit DACs component performs single-bit quantization on both the real and imaginary channels of each transmitting antenna, the transmitted waveform vector Should be located in the single-bit discrete feasible region χ={±α±jα} NL Where α is a parameter related to the antenna transmission power. If the maximum total transmission power of the transmitting array is P tot , assuming that each antenna transmits an integrated waveform using its maximum available power, the parameter α can be specifically expressed as:

[0019]

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

[0021] a(θ)=[1,e -jπsin(θ) ,…,e -jπ(N-1)sin(θ) ] T (2)

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

[0023]

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

[0025] Define the expected received symbol of the kth user at time l as s k [l], the signal-to-noise ratio requirement of the kth user is Ψ k , the received noise variance is Then the symbol-level precoding parameter of the kth user at time l is It can be expressed as:

[0026]

[0027] In the above formula, Φ=π / M, where M is the number of phases used in phase shift keying modulation. j=1, 2 reflects the distance between the noise-free signal received by the user and the decision boundary of the symbol it expects to receive, which can determine the communication symbol detection performance of the user.

[0028] In order to fairly guarantee the communication performance of each user, the present invention constrains the minimum instantaneous received signal-to-noise ratio of each user based on symbol-level precoding. At the same time, in order to improve the power concentration of the transmit beam and improve the target detection performance, the present invention minimizes the integral sidelobe to mainlobe ratio of the radar transmit beam pattern. Definition is the Rayleigh fading channel between the base station and the kth user, and the spatial region is divided into the main lobe region Θ m and side lobe area Θ s The optimization problem model is:

[0029]

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

[0031] Step 2: For the fractional form of the objective function in problem (5), convert it into an integer form using the Dinkelbach transformation according to the fractional programming theory, and define A(Θ) = ∫ Θ a * (θ)a T (θ)dθ, then the objective function in the integral form obtained by Dinkelbach transformation is:

[0032]

[0033] In the above formula, Ω s and Ω m They are: ξ is an iteratively updated Dinkelbach auxiliary variable. The updated value of ξ at the tth iteration is (t) Specifically:

[0034]

[0035] Then problem (5) can be equivalently expressed as:

[0036]

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

[0038]

[0039] Therefore, problem (8) can be rewritten into real-valued form:

[0040]

[0041] Step 4: Introduce auxiliary variables The discrete constraints are equivalently expressed as equality constraints plus continuous constraints, and the augmented Lagrangian function is constructed.

[0042]

[0043] Question (10) is further transformed into the following form:

[0044]

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

[0046] Step 5: Use a framework based on the alternating direction multiplier method to decompose the problem into multiple sub-problems that are easy to solve. According to the framework of the alternating direction multiplier method, problem (12) is decomposed into multiple sub-problems under various constraints and updated iteratively. (k) , x R (k) and w (k) Represent e, x respectively R and the updated values ​​of w after the kth iteration, then the kth iteration update process is:

[0047]

[0048]

[0049] w (k) =w (k-1) +ρ(x R (k) -e (k) ) (15)

[0050] The optimization sub-problem corresponding to the update process (13) is as follows:

[0051]

[0052] Subproblem (16) contains only one equality constraint. The present invention converts it into an unconstrained problem and solves it directly by Lagrange multiplier method. The Lagrangian function of subproblem (16) is defined as:

[0053]

[0054] In the above formula, η is the Lagrange multiplier. You can get e (k) The iterative update formula is:

[0055]

[0056] In addition, due to the (k) Constraints must also be met Therefore, the value of η can be directly determined. Substituting the value of η into equation (18), we can obtain:

[0057]

[0058] The optimization sub-problem corresponding to the update process (14) is as follows:

[0059]

[0060] definition For all 2KL vectors The matrix composed of can be reformulated as follows:

[0061]

[0062] In the above formula, Among them, 1 2KL is a 2KL-dimensional all-one vector, and ≥ means element-wise greater than or equal to.

[0063] Problem (21) is a convex quadratic programming problem, and the present invention uses the gradient projection method to quickly solve it.

[0064] The basic process of using the gradient projection method to solve subproblem (21) is as follows:

[0065] 1) Gradient descent. Update the variable along the negative gradient direction of the objective function so that the objective function value gradually decreases. For subproblem (21), the objective function f(x R ) is specifically:

[0066]

[0067] by Represents variable x R The updated value along the gradient direction at the rth iteration can be obtained by the following formula:

[0068]

[0069] In the above formula, τ is the descent step size, which can be obtained by using an inexact line search (such as a reverse backtracking line search).

[0070] 2) Alternate projection. After each gradient descent, and Multiple alternating projections are performed to ensure that the updated value satisfies the constraint set. Right now Defining the projection operator Indicates that the variable x R Each element x R，1 , x R，2 , …, x R,2NL Projection to a collection Specifically:

[0071]

[0072] Defining the projection operator Indicates that the variable x R Projection to a collection Specifically:

[0073]

[0074] The specific process of using the gradient projection method to solve subproblem (21) is as follows:

[0075] 1) Input parameters: ξ, e, Ω s，R ,Ω m，R , w, H, ɑ, ρ, δ

[0076] 2) Initialization settings: r=0

[0077] 3) Start iteration:

[0078] ir=r+1

[0079] ii. Through reverse backtracking line search, we can get: τ

[0080] iii. Through formula (23), we can obtain:

[0081] iv. Through equation (24) and equation (25), we can obtain:

[0082] 4) If That is, select the best x R Stop the loop

[0083] 5) Output the real number form of the transmitted waveform: x R

[0084] After the alternating direction multiplier framework is iterated, a single-bit projection is used to ensure that the final output result fully satisfies the single-bit discrete feasible domain constraint x R ∈χ R Defining the projection operator Indicates that the variable x R Each element x R，1 , x R,2 , …, x R，2NL Projection to the single-bit feasible domain χ R Specifically:

[0085]

[0086] Finally, the real vector x R Reconstructed into a complex vector x, specifically:

[0087] x=x R (1:NL)+jx R (NL+1:2NL) (27)

[0088] In summary, the complete process of the single-bit radar communication integrated transmission waveform design method based on symbol-level precoding is as follows:

[0089] 1) Input parameters: Θ s , Θ m ,h k ,α, ρ,v,ε1,ε2

[0090] 2) Initialization settings:

[0091] 3) Reconstructed by formula (27): x (0)

[0092] 4) Through formula (7), we can obtain: (0)

[0093] 5) t=t+1

[0094] 6) Setting: k = 0

[0095] 7) Start iteration:

[0096] ik=k+1

[0097] ii. Through formula (18), we can obtain: e (k)

[0098] iii. Obtained by the gradient projection method in step 5:

[0099] iv. Obtained by formula (15): w(k)

[0100] 8) If or Then stop the loop and use equations (26) and (27) to obtain: (t)

[0101] 9) According to formula (7), we can obtain: (t)

[0102] 10) Repeat 5-9 until |ξ (t) -ξ (t-1) |<v means to select the best x

[0103] 11) Output single-bit radar communication integrated transmission waveform: x

[0104] Example

[0105] Through MATLAB simulation, the specific implementation scheme of the single-bit radar communication integrated transmission waveform design method based on symbol-level precoding of the present invention is further explained.

[0106] 1) Simulation system parameter settings

[0107] In the simulation experiment, the number of transmitting antennas N is set to 20, the number of transmitting waveform snapshots L is set to 50, and the total transmitting power P is set to tot =30dbm; the radar transmission beam pattern is sampled at an interval of 1° and is uniformly sampled in the entire spatial domain θ∈[-90°, 90°]; the number of communication users K=2, and the communication channel obeys the Rayleigh fading model The signal-to-noise ratio requirements and received noise power of all users are the same, both are Ψ k =12dB, The constellation symbol is the QPSK symbol set. The algorithm-related parameters and iteration termination conditions are set as follows: δ = 10 -5 , ε1=ε2=10 -5 , v = 10 -5 .

[0108] 2) Transmit beam diagram drawing

[0109] In order to intuitively display 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 by the present invention to draw a transmission beam diagram, and compares it with the single-bit radar transmission waveform design method in document [2] and the constant modulus radar transmission waveform design method under infinite bit quantization in document [4]. The horizontal axis of the transmission beam diagram represents the spatial angle range of [-90°, 90°], and the vertical axis is the normalized power of the transmission signal at this angle based on the maximum power of the entire spatial domain, in dB.

[0110] [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.

[0111] 3) Metrics

[0112] In the present invention, it is necessary to measure the radar performance of the transmit waveform. In addition to drawing the transmit beam diagram to intuitively display the difference, the integrated sidelobe to mainlobe ratio of the radar transmit beam diagram is also used to numerically evaluate the radar transmit beam performance of the transmit waveform. The specific expression is:

[0113]

[0114] Among them, Θ s is the side lobe area, Θ m Under the same initial conditions, the smaller the integrated sidelobe main lobe ratio is, the better the radar's transmit beam performance is.

[0115] 4) Result analysis

[0116] The present invention provides four groups of example simulations. Figure 1 Phase distribution diagram of waveform sequence for three different transmission waveform design methods. Figure 2 This is a graph showing the relationship between the average bit error rate and communication performance requirements under different design methods. Figure 3 The single main lobe radar transmit beam diagram under different design methods, where the main lobe area is set to Θ m =[-10°, 10°], the side lobe area is Θ s =[-90°, -10°]∪[10°, 90°]. Figure 4 The dual main lobe radar transmit beam diagram under different design methods, where the main lobe area is set to Θ m =[-40°, -30°]∪[30°, 40°], the side lobe area is Θ s =[-90°,-40°]∪[-30°,30°]∪[40°,90°].

[0117] Depend on Figure 1It can be seen that the phase of the constant modulus waveform sequence is distributed between [-π, π], while the phase of the single-bit waveform sequence included in the present invention is only located in four groups of discrete phases {-3π / 4, -π / 4, π / 4, 3π / 4}. Compared with the constant modulus waveform, the demand for DACs resolution is greatly reduced, which can significantly simplify the hardware structure of the transmitter and reduce the hardware cost and power consumption of the transmitting array.

[0118] Depend on Figure 2 It can be seen that the radar transmission waveform proposed in the literature [2, 4] does not have any communication function, while the symbol-level precoding technology adopted by the present invention allows downlink users to decode the transmission symbols from the received signal. At the same time, as the user signal-to-noise ratio requirement Ψ increases, the user's bit error rate will be further reduced. In addition, it can be found that the increase in the number of users K will only have a slight impact on the communication performance, and this impact will gradually decrease as Ψ increases.

[0119] Depend on Figure 3 It can be seen that the single-bit integrated waveform proposed in the present invention can also concentrate the power of the transmit beam in the desired main lobe area θ under the condition of having communication function. m At the same time, compared with the single-bit radar waveform in reference [2], the transmit beam pattern performance of the present invention only slightly decreases, and the integrated sidelobe to mainlobe ratio value only increases by 1.06 dB. This shows that the present invention uses symbol-level precoding technology and only slightly loses radar performance in exchange for its application in communication scenarios.

[0120] Depend on Figure 4 It can be seen that when the complexity of the radar transmission beam shape increases, the present invention can still form a good radar detection beam. At the same time, compared with the integrated sidelobe to main lobe ratio of -4.5dB of the single-bit radar waveform, the present invention has an integrated sidelobe to main lobe ratio of -3.96dB under this condition, and the gap between radar performance is further reduced.

[0121] In summary, the method proposed in the present invention has good comprehensive performance. Compared with the traditional constant modulus transmission waveform, it can be adapted to the single-bit low-resolution DACs component, which can significantly simplify the hardware structure of the transmission system. At the same time, compared with the traditional radar transmission waveform, the present invention has a multi-user communication function based on the symbol-level precoding technology, and can be applied to radar and communication working scenarios at the same time, with high application value and broad application prospects.

Claims

1. A single-bit radar communication integrated transmission waveform design method based on symbol-level precoding, which is characterized by: Based on fractional programming, alternating direction multiplier method, gradient projection method and other technologies, a single-bit radar communication integrated transmission waveform design method is proposed. This method is applicable to the single-bit integrated transmission waveform design problem with the goal of minimizing the integral sidelobe to mainlobe ratio of the radar transmission beam pattern under the constraints of single-bit discrete feasible domain and symbol-level precoding constraints. This method first uses the Dinkelbach transform to convert the fractional objective function into an integer objective function, and then rewrites the original problem into an equivalent real-valued problem through matrix transformation. Auxiliary variables are introduced to represent the discrete constraints equivalently as equality constraints plus a set of continuous constraints. Finally, the alternating direction multiplier method framework is used to alternately optimize the auxiliary variables and the original transmission waveform. The specific implementation steps are as follows: Step 1: In order to fairly guarantee the communication performance of each user, the present invention constrains the minimum instantaneous received signal-to-noise ratio of each user based on the symbol-level precoding method. At the same time, in order to improve the power concentration of the transmit beam to improve the target detection performance, the present invention minimizes the integrated sidelobe to mainlobe ratio of the radar transmit beam pattern. Definition is the transmitting waveform vector, χ={±ɑ±jα} NL is a single-bit discrete feasible domain, ɑ is a parameter related to the antenna transmission power, is the Rayleigh fading channel between the base station and the kth user, P(θ) is the far-field transmission beam pattern in the spatial domain θ, and the spatial domain is divided into the main lobe area Θ m and side lobe area Θ s The optimization problem model is: In the above formula, is the symbol-level precoding parameter of the kth user at time l, c l is the L×L dimensional identity matrix I L The lth column of represents the Kronecker product. Step 2: For the fractional form of the objective function in problem (1), convert it into an integer form using the Dinkelbach transformation according to the fractional programming theory. Then problem (1) is equivalently expressed as: In the above formula, ξ is an iteratively updated Dinkelbach auxiliary variable. Step 3: To facilitate subsequent processing, convert the relevant complex vectors and complex matrices into real numbers and rewrite problem (2) into an equivalent real-valued form: In the above formula, x R , and χ R Corresponding to x, h respectively k and the real-valued form of χ. Step 4: Introduce auxiliary variables The discrete constraints are equivalently expressed as equality constraints plus a set of continuous constraints, and the augmented Lagrangian function is constructed. Question (3) is further transformed into the following form: In the above formula, w is the Lagrange multiplier, and ρ>0 is the penalty factor. Step 5: Use a framework based on the alternating direction multiplier method to decompose the problem into multiple easily solvable sub-problems. The sub-problem of optimizing the auxiliary variable e is solved by the Lagrange multiplier method; the sub-problem of optimizing the transmission waveform x is solved by the Lagrange multiplier method. R The sub-problem of is solved by the gradient projection method.

2. The single-bit radar communication integrated transmission waveform design method based on symbol-level precoding according to claim 1 is characterized in that: In step 1, the radar communication dual-function transmitting array is a uniform linear array composed of N antenna units, the number of transmitting waveform snapshots is L, and it can transmit a detection beam in a desired direction while providing communication services for K single-antenna users, wherein multi-user communication adopts symbol-level precoding. If the maximum total transmit power of the transmitting array is P lot , assuming that each antenna transmits an integrated waveform using its maximum available power, the parameter ɑ can be specifically expressed as: If the array element spacing of the transmitting array is half a wavelength, the transmitting array steering vector in the spatial domain θ direction can be expressed as: a(θ)=[1,e -jπsin(θ) ,…,e -jπ(N-1)sin(θ) ] T (6) Therefore, the far-field transmission beam pattern P(θ) in the θ direction can be specifically expressed as: Define the expected received symbol of the kth user at time l as s k [l], the signal-to-noise ratio requirement of the kth user is Ψ k , the received noise variance is Then the symbol-level precoding parameter of the kth user at time l is It can be expressed 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 symbol it expects to receive, and can determine the user's communication symbol detection performance.

3. The single-bit radar communication integrated transmission waveform design method based on symbol-level precoding according to claim 1 is characterized in that: In step 2, the integral form objective function obtained by the Dinkelbach transformation is specifically: In the above formula, Ω s and Ω m They are: in The updated value ξ of the Dinkelbach auxiliary variable ξ at the tth iteration (t) Specifically:

4. The single-bit radar communication integrated transmission waveform design method based on symbol-level precoding according to claim 1 is characterized in that: In step 3, the specific transformation of the complex vector and the complex matrix into real numbers is:

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 Lagrangian function Specifically:

6. The single-bit radar communication integrated transmission waveform design method based on symbol-level precoding according to claim 1 is characterized in that: In step 5, according to the framework of the alternating direction multiplier method, problem (4) is decomposed into multiple sub-problems under various constraints and updated iteratively, with e (k) , x R (k) and w (k) Represent e, x respectively R and the updated values ​​of w after the kth iteration, then the kth iteration update process is: w (k) =w (k-1) +ρ(x R (k) -e (k) ) (15) 7. The single-bit radar communication integrated transmission waveform design method based on the alternating direction multiplier method framework according to claim 6 is characterized in that: The optimization sub-problem corresponding to the update process (13) is as follows: Subproblem (16) contains only one equality constraint. The present invention converts it into an unconstrained problem and solves it directly by Lagrange multiplier method. The Lagrangian function of subproblem (16) is defined as: In the above formula, η is the Lagrange multiplier. You can get e (k) The iterative update formula is: In addition, due to the (k) Constraints must also be met Therefore, the value of η can be directly determined. Substituting the value of η into equation (18), we can obtain:

8. The single-bit radar communication integrated transmission waveform design method based on the alternating direction multiplier method framework according to claim 6 is characterized in that: The optimization sub-problem corresponding to the update process (14) is as follows: definition For all 2KL vectors The matrix composed of can be reformulated as follows: In the above formula, Among them, 1 2KL is a 2KL-dimensional all-one vector, and ≥ means element-wise greater than or equal to. Problem (21) is a convex quadratic programming problem, and the present invention uses the gradient projection method to solve it. The basic process of using the gradient projection method to solve subproblem (21) is as follows: 1) Gradient descent. Update the variable along the negative gradient direction of the objective function so that the objective function value gradually decreases. For subproblem (21), the objective function f(x R ) is specifically: by Represents variable x R The updated value along the gradient direction at the rth iteration can be obtained by the following formula: In the above formula, τ is the descent step size, which can be obtained by using an inexact line search (such as a reverse backtracking line search). 2) Alternate projection. After each gradient descent, exist and Multiple alternating projections are performed to ensure that the updated value satisfies the constraint set. Right now Defining the projection operator Indicates that the variable x R Each element x R，1 , x R,2 , …, x R,2NL Projection to a collection Specifically: Defining the projection operator Indicates that the variable x R Projection to a collection Specifically: The specific process of using the gradient projection method to solve subproblem (21) is as follows: 1) input parameter:ξ,e,Ω s,R ,Oh m，R ,w,H,ɑ,ρ,δ 2) Initialization settings: 3) Start iteration: i.r＝r+1 ii. Through reverse backtracking line search, we can get: τ iii. Through formula (23), we can obtain: iv. Through equation (24) and equation (25), we can obtain: 4) If That is, select the best x R Stop the loop 5) Output the real number form of the transmitted waveform: x R。 9. The single-bit radar communication integrated transmission waveform design method based on the alternating direction multiplier method framework according to claim 6 is characterized in that: After the alternating direction multiplier method framework is iterated, single-bit projection is used to ensure that the final output result fully satisfies the single-bit discrete feasible domain constraint x R ∈χ R . Define the projection operator Indicates that the variable x R Each element x R，1 , x R,2 , …, x R，2NL Projection to the single-bit feasible domain χ R Specifically: Finally, the real vector x R Reconstructed into a complex vector x, specifically: x=x R (1:NL)+jx R (NL+1:2NL)(27) In summary, the complete process of the single-bit radar communication integrated transmission waveform design method based on symbol-level precoding is as follows: 1) input parameter:Th s ,I m ,h k ,a, p, v, ε1, ε2 2) Initialization settings: 3) Reconstructed by formula (27): x (0) 4) Through formula (10), we can obtain: (0) 5) t=t+1 6) Setting: k = 0 7) Start iteration: ik=k+1 ii. Through formula (19), we can obtain: e (k) iii. Obtained by the gradient projection method in claim 8: iv. Obtained by formula (15): w (k) 8) If or Then stop the loop and use equations (26) and (27) to obtain: (t) 9) According to formula (10), we can obtain: (t) 10) Repeat 5-9 until |ξ (t) -ξ (t-1) |<v means to select the best x 11) Output single-bit radar communication integrated transmission waveform: x.

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