MIMO radar waveform design for high-precision doa estimation with one-bit dac

Abstract

The application discloses a MIMO radar waveform design for high-precision DOA estimation of a one-bit DAC, relates to the technical field of radars, and comprises the following steps: under far-field conditions, a waveform design problem is established for a MIMO radar equipped with two one-bit DAC transmitting antennas; variable separation and problem conversion are performed on the waveform design problem; a one-bit alternating optimization framework is proposed to solve the converted waveform design problem; in the optimization framework, a low-complexity ADMM method is proposed; and a closed-form solution of a waveform matrix is obtained in each iteration according to KKT conditions. The spatial spectrum formed by the waveform design provided by the application has a sharper spectrum peak at a target position, so that higher resolution can be obtained; the waveform design method provided by the application has higher DOA estimation precision, and the RMSE curve is closer to the theoretical lower bound.

Description

Technical Field

[0001] This application relates to the field of radar technology, and in particular to a MIMO radar waveform design for high-precision DOA estimation using a one-bit DAC. Background Technology

[0002] Beamforming technology is a key technology in wireless communication and radar systems, with wide applications. By creating wide nulls, interference can be effectively suppressed. Furthermore, power loss issues can be addressed through phase-only constraints and constant mode constraints (CMC). Therefore, beamforming designs with only phase variations under constant mode constraints have attracted particular attention.

[0003] Due to the constant modulus constraint, this problem becomes non-convex and NP-hard, making it difficult to solve directly. Solving the problem by relaxing the constant modulus constraint requires matrix inverse calculations, which incurs significant computational costs in each iteration. Summary of the Invention

[0004] This application provides a MIMO radar waveform design for high-precision DOA estimation using a one-bit DAC, in order to solve the problems in the prior art.

[0005] In a first aspect, this application provides a MIMO radar waveform design for high-precision DOA estimation using a one-bit DAC, including:

[0006] Under far-field conditions, the waveform design problem is established for the transmitting antenna of a MIMO radar equipped with two one-bit DACs.

[0007] Variable separation and problem transformation are performed on waveform design problems.

[0008] A one-bit alternating optimization framework is proposed to solve the waveform design problem after transformation. In the optimization framework, a low-complexity ADMM method is proposed, which obtains the closed-form solution of the waveform matrix in each iteration based on the KKT conditions.

[0009] Furthermore, the two one-bit DACs are used to convert the I / Q signals of the transmitting antenna.

[0010] Furthermore, the MIMO radar is a centralized MIMO radar in which both the transmitting array and the receiving array are uniform linear arrays with half-wavelength antenna spacing.

[0011] Among them, the problem of designing MIMO radar transmit waveforms based on high-precision DOA estimation using a one-bit DAC is modeled:

[0012] Consider a MIMO radar equipped with a one-bit DAC. Specifically, each transmitting antenna is equipped with two one-bit DACs to convert the I / Q signals.

[0013] The symbol set of the transmitted signal can be represented as

[0014] Where e T N represents the total energy of the transmitted signal. t Let L be the number of transmit antennas and L be the sequence length. Therefore, the transmitted waveform must satisfy... It can also be equivalently represented as:

[0015]

[0016] The design problem of MIMO radar transmit waveform based on a one-bit DAC can be expressed as:

[0017]

[0018] sttr{SS H}=e T

[0019]

[0020] in D is a diagonal matrix composed of target amplitudes, A t It is a matrix composed of the transmission guide vector, and S is the transmission waveform matrix.

[0021] The detailed modeling process is as follows:

[0022] First, we define the following:

[0023] Bold lowercase and bold uppercase letters represent vectors and matrices, respectively; the transpose, conjugate, and conjugate transpose operations are represented by (·). T ,(·) * ,(·) H express;

[0024] They represent the complex number field and the real number field, respectively;

[0025] diag([x1,x2,…,x N ]) represents a diagonal matrix, where the diagonal elements are x1, x2, ..., x3. N ;

[0026] Represents the expected value of a random variable; tr{·},vec(·),(·) -1 These represent the trace, column vectorization, and inverse of the matrix, respectively.

[0027] These represent the real and imaginary parts of a complex-valued scalar / vector / matrix, respectively.

[0028] ||·||F ,|·| represent the 2-norm of a matrix and the element-wise modulo of a vector, respectively;

[0029] ⊙ represents the Hadamard product; matrix I,1(0) represents the identity matrix and the matrix of all 1s(0), respectively.

[0030] Consider a given N t One transmitting antenna and N r A centralized MIMO radar with multiple receiving antennas, where both the transmitting and receiving arrays are uniform linear arrays with half-wavelength antenna spacing. Assuming far-field, narrow-band conditions, the received signal can be expressed as:

[0031]

[0032] Where P represents the number of targets, α p θ p These represent the magnitude and direction of the p-th target, respectively.

[0033] and They represent at θ p The receiving guide vector and the transmitting guide vector at the location.

[0034] It is the transmitted waveform matrix.

[0035] s n (n=1,...,N t ) represents the transmission sequence of the nth transmitting antenna, L is the sequence length, and N is Gaussian white noise.

[0036] Let A t =[a t (θ1),a t (θ2),…,a t (θ P )],

[0037] A r =[a r (θ1),a r (θ2),…,a r (θ P )],

[0038] D = diag([α1, α2, ..., α P ]),

[0039] Then the received signal Y can be rewritten as:

[0040] Y = A r Z+N (2)

[0041] in

[0042] For signal model (2), there are many array signal processing methods to estimate the target direction. In this paper, the most classic MUSIC algorithm is used to estimate the target direction. The mean square error of the DOA estimation of the MUSIC algorithm is expressed as:

[0043]

[0044] Where ω p =πsin(θ) p ) is the true frequency of the p-th target, while It is the estimated frequency of the p-th target, σ 2 This refers to noise in the receiver. Furthermore, the theoretical minimum value of the DOA estimation MSE for the MUSIC algorithm is expressed as:

[0045]

[0046] Where ν p p = 1, 2, ..., P is A α,t The singular values ​​of .

[0047] To reduce hardware costs and power consumption, a MIMO radar equipped with a one-bit DAC is considered. Specifically, each transmit antenna is equipped with two one-bit DACs to convert the I / Q signals. Furthermore, the advantage of using a one-bit DAC is that the generated signal is a constant-mode signal.

[0048] The symbol set of the transmitted signal at this time can be represented as

[0049] Where e T This represents the total energy of the transmitted signal. Therefore, the transmitted waveform must satisfy... It can also be equivalently represented as:

[0050]

[0051] Based on the above analysis, the design problem (6) of the MIMO radar transmit waveform based on a one-bit DAC can be expressed as:

[0052]

[0053] sttr{SS H}=e T (6b)

[0054]

[0055] Note that problem (6) consists of a quadratic objective function (6a) with respect to S, a quadratic equality constraint (6b), and a linear inequality constraint (6c). Due to the quadratic equality constraint, problem (6) is non-convex and difficult to solve directly.

[0056] The MIMO radar transmit waveform design method based on a one-bit DAC adopted in this application is as follows:

[0057] Then, a one-bit alternation optimization (O-AltOpt) framework is used to solve it. The specific steps are as follows:

[0058] To effectively solve the problem (6), the problem is first transformed using the variable separation technique. Then, a one-bit alternating direction multiplier (O-AltOpt) optimization framework is proposed to solve the transformed problem. In this optimization framework, the auxiliary variables, the emission waveform matrix, and the Lagrange multipliers are updated sequentially. In particular, a one-bit alternating direction multiplier (O-ADMM) method with low computational complexity is proposed in the O-AltOpt framework to design the emission waveform. According to the KKT (Karush-Kuhn-Tucher) conditions, a closed-form solution can be obtained with low computational complexity in each iteration.

[0059] First, by separating the variables, problem (6) can be transformed into problem (7):

[0060]

[0061] stZ=A α,t S (7b)

[0062] tr{SS H}=e T (7c)

[0063]

[0064] Z is the introduced auxiliary variable. The augmented Lagrangian function of problem (7) can be expressed as:

[0065]

[0066] Where ρ is the penalty parameter and γ is the Lagrange multiplier.

[0067] Secondly, an O-AltOpt optimization framework is proposed to solve the problem (7), in which Z, S, and γ are updated sequentially in the (k+1)th iteration as follows:

[0068]

[0069] γ k+1 =γ k +Zk+1 -A α,t S k+1 (9c)

[0070] The solutions to problems (9a) and (9b) are given below.

[0071] Problem (9a) can be represented as:

[0072]

[0073] By defining R = A α,t S k -γ k Problem (10) can be expressed as:

[0074]

[0075] make These represent the singular value decompositions of matrices Z and R, respectively.

[0076] Among them Λ Z =diag([σ z,1 ,σ z,2 ,…,σ z,P ]), σ z,1 ≥σ z,2 ≥…σ z,P These represent the singular values ​​of matrix Z;

[0077] Similarly, Λ R =diag([σ r,1 ,σ r,2 ,…,σ r,P ]), σ r,1 ≥σ r,2 ≥…σ r,P These represent the singular values ​​of matrix R.

[0078] Then problem (11) can be simplified to:

[0079]

[0080] Note that problem (12) can be decomposed into P independent univariate problems, where the p-th problem can be represented as:

[0081]

[0082] Due to f(σ) z,p Regarding the variable σ z,p The second derivative satisfies:

[0083]

[0084] Therefore f(σ) z,p ) is about the variable σz,p The convex function. Let f(σ) be... z,p Regarding the variable σ z,p The first derivative is 0:

[0085]

[0086] Problem (15) can be solved by calculating the roots or by using MATLAB's root-finding function to obtain σ. z,p Then update Λ Z =diag([σ z,1 ,σ z,2 ,…,σ z,P ]).

[0087] In summary, Z k+1 The update formula is as follows:

[0088]

[0089] Problem (9b) can be represented as:

[0090]

[0091] definition Problem (17) can be represented as problem (18):

[0092]

[0093] sttr{SS H}=e T (18b)

[0094]

[0095] Problem (18) is nonconvex due to the existence of the quadratic equality constraint (18b), making it difficult to solve directly. To effectively address this problem, the O-ADMM method is proposed.

[0096] Specifically, by introducing the auxiliary variable A = S, problem (18) can be expressed as problem (19):

[0097]

[0098] stA=S,tr{AA H}=e T (19b)

[0099]

[0100] The augmented Lagrangian function of problem (19) can be expressed as:

[0101]

[0102] in It is the dual variable of the constraint A = S, and μ > 0 is the penalty parameter.

[0103] Consider variable substitution We can obtain the augmented Lagrangian function in scaled form:

[0104]

[0105] in This is the scaled dual variable. Note that the two types of Lagrangian functions mentioned above are equivalent within the ADMM framework. However, since the scaled augmented Lagrangian function is more concise in form, the subsequent analysis will be based on the scaled Lagrangian function.

[0106] In the (i+1)th iteration of the O-ADMM framework, S, A, and U are updated sequentially:

[0107]

[0108]

[0109] U i+1 =U i +(A i+1 -S i+1 (22c)

[0110] The set Ω S ,Ω A The definition is as follows:

[0111]

[0112] Ω A ={A|tr{AA H}=e T}(23B)

[0113] The solutions to problems (22a) and (22b) are given below.

[0114] Problem (22a) can be expressed as:

[0115]

[0116]

[0117] in It can be represented as:

[0118]

[0119] Note that problem (24) consists of a quadratic objective function and linear inequality constraints, and is a convex problem with respect to the variable S. By solving the KKT conditions of problem (24), the optimal S(n,l), n=1,…,N is obtained. t The real and imaginary parts of elements l = 1, ..., L can be represented as follows:

[0120]

[0121]

[0122] Problem (22b) can be represented as problem (27):

[0123]

[0124] sttr{AA H}=e T (27b)

[0125] Problem (27) is nonconvex due to the quadratic equality constraint problem (27b).

[0126] To solve this problem, we perform eigenvalue decomposition on matrix B: B = PπP H ,

[0127] in It is a unitary matrix composed of eigenvectors.

[0128] diagonal matrix These are the corresponding eigenvalues.

[0129] By definition and F=P H (SU), Problem (27) can be rewritten as:

[0130]

[0131]

[0132] The augmented Lagrangian function of problem (28) can be expressed as:

[0133]

[0134] Where η is the equality constraint The dual variable. Observation We can know that the optimality condition for problem (28) is:

[0135]

[0136]

[0137] We can obtain the following from equation (30a) in It is a diagonal matrix, and its nth diagonal element is μ / (2λ). n +μ+2η). In order to find We need to know the value of the dual variable η. Therefore, we will... Substituting into equation (30b), we get:

[0138]

[0139] Note that problem (31) can be solved using the bisection method, since λ n ≥0, μ>0, observing the function f(η), we can see that,

[0140] when When f(η) is monotonically increasing;

[0141] When η∈(-0.5μ-λ1,∞), f(η) is monotonically decreasing.

[0142] Furthermore, as η→±∞, f(η)→-e T And when η→-0.5μ-λ1 or When f(η)→∞.

[0143] Therefore, f(η) in There exists a unique zero on (-0.5μ-λ1,∞).

[0144] After finding η using the bisection method, we can calculate... at last

[0145] On the other hand, this application provides a computer-readable storage medium, characterized in that the computer-readable storage medium stores computer-executable instructions, which, when executed by a processor, are used to implement a MIMO radar waveform design for one-bit DAC high-precision DOA estimation as described in any one of the first aspects.

[0146] This application has the following advantages and beneficial effects:

[0147] The spatial spectrum formed by the waveform design provided in this application has sharper spectral peaks at the target location, thereby achieving higher resolution. The waveform design method provided in this application has higher DOA estimation accuracy, and the RMSE curve is closer to the theoretical lower bound. Attached Figure Description

[0148] The accompanying drawings, which are included to provide a further understanding of the embodiments of this application and form part of this application, do not constitute a limitation on the embodiments of this application. In the drawings:

[0149] Figure 1 The residual term for the MIMO radar waveform design provided in an example embodiment of this application for one-bit DAC high-precision DOA estimation. The convergence curve of the outer layer iteration number.

[0150] Figure 2 The cost function value (dB) of a MIMO radar waveform design for a one-bit DAC high-precision DOA estimation provided in another example embodiment of this application is shown as a convergence curve of the number of iterations.

[0151] Figure 3 The normalized spatial spectrum is generated by the MUSIC algorithm designed for MIMO radar waveforms with one-bit DAC high-precision DOA estimation, which is another example embodiment of this application.

[0152] Figure 4 The graph shows the RMSE (rad) of the MIMO radar waveform designed for one-bit DAC high-precision DOA estimation in the embodiments of this application as a function of the number of receiving antennas. Detailed Implementation

[0153] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numbers in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application. Rather, they are merely examples of apparatuses and methods consistent with some aspects of this application as detailed in the appended claims.

[0154] Definitions:

[0155] One-bit alternating optimization (O-AltOpt)

[0156] One-bit Alternating Directions Method of Multipliers (O-ADMM)

[0157] In mathematics, the Karush-Kuhn-Tucker conditions (also known as Kuhn-Tucker, KKT conditions, Karush-Kuhn-Tucker optimization conditions, Karush-Kuhn-Tucker conditions, Kuhn-Tucker optimization conditions, or Kuhn-Tucker conditions) are necessary and sufficient conditions for an optimal solution to a nonlinear programming problem, provided certain rules are met. This is a result of the generalized Lagrange multiplier.

[0158] Constant-mode waveform design is a key technology in MIMO radar systems. It is well known that waveform design is one of the most effective methods for achieving many performance improvements, such as better target detection performance and enhanced parameter estimation. Furthermore, constant-mode constraints on the waveform ensure that the radar power amplifier operates at maximum efficiency, thereby avoiding nonlinear distortion. Therefore, constant-mode waveform design for MIMO radar has received widespread attention.

[0159] Existing research is mainly divided into two categories: (1) using an infinite-bit DAC, which may be impractical in large-scale antenna array systems due to high hardware costs and power consumption; (2) using a one-bit DAC to obtain better target detection performance, but may not be suitable for parameter estimation scenarios.

[0160] Among the first type of methods, a typical example is the Fast randomized-MUSIC algorithm proposed in the paper "LI B, WANG S, ZHANG J, et al. Fast randomized-MUSIC for mm-wave massive MIMO radars[J]. 2021, 70(2): 1952-6.", which uses a random matrix to approximate the signal subspace, reducing the complexity of signal processing while obtaining high-resolution DOA estimation accuracy. Another example is the generalized MUSIC algorithm based on polarization MIMO radar proposed in the paper "ZHENG G, SONGY, CHEN C. Height Measurement with Meter Wave Polarimetric MIMO Radar: SignalModel and MUSIC-like Algorithm[J]. Signal Processing, 2022, 190: 108344.", which uses the estimated target direction to calculate the target height in low-altitude areas. Experimental results show that the height measurement results are close to the optimal estimation performance. The paper "WU W, TANG B. Waveform design for higher-resolution localization with MIMO radar[J]. Signal Processing, 2023, 209:108999." proposes an ADMM method to solve the waveform design problem based on minimizing the mean square error of MUSIC angle estimation. Experimental results show that the waveform designed by this method has higher resolution and lower estimation error. However, all of these methods use an infinite-bit DAC, resulting in high hardware costs and system power consumption, which poses difficulties in practical large-scale antenna array applications. Existing research shows that as the number of quantization bits in the DAC decreases, hardware costs and power consumption decrease exponentially. Therefore, an effective way to solve this problem is to use a low-precision DAC, especially a one-bit DAC.

[0161] In the second category of methods, a typical example is the paper "CHENG Z,HE Z,LIAO B. Target Detection Performance of Collocated MIMO Radar With One-Bit ADCs[J]. IEEE Signal Processing Letters, 2019, 26(12): 1832-6." which derives a one-bit likelihood ratio detector and evaluates its asymptotic performance. To suppress interference, the paper "DENG M,CHENG Z,WU L, et al. One-Bit ADCs / DACs Based MIMO Radar: Performance Analysis and Joint Design[J]. IEEE Transactions on Signal Processing, 2022, 70: 2609-24." proposes an alternating waveform and filter design to maximize the quantization signal-to-interference-plus-noise ratio (Q-SINR) and evaluates the performance loss of a one-bit MIMO radar. To reduce correlated sidelobes, the paper "LIN R, SHANG X, LI JJS P. Effective and Efficient Gradient Based Methods for Low-Bit Discrete-Phase Sequence Designs[J]. 2023:108929." proposes a gradient-based method to design low-bit sequences with low correlated sidelobes. Experimental results show that the designed sequences can achieve good correlation. However, while the above method utilizes a one-bit DAC to obtain better target detection performance, it may not be suitable for parameter estimation.

[0162] The purpose of this application is to provide a MIMO radar transmit waveform design method based on a one-bit DAC to improve DOA estimation accuracy. This application considers using a one-bit DAC for waveform design to achieve higher DOA estimation accuracy. It assumes the use of the MUSIC algorithm for DOA estimation, focusing on improving DOA estimation accuracy. First, assuming prior knowledge about the target direction and amplitude is available, a one-bit transmit waveform design problem based on minimizing the mean square error of DOA estimation is proposed. Then, a one-bit alternating optimization (O-AltOpt) framework is proposed to solve the non-convex problem. In this optimization framework, auxiliary variables, the transmit waveform matrix, and Lagrange multipliers are updated sequentially. Furthermore, a one-bit alternating direction multiplier method (O-ADMM) with low computational complexity is proposed to design the transmit waveform.

[0163] Tables 1 and 2 summarize the pseudocode for the O-ADMM algorithm and the O-AltOpt optimization method, respectively. Detailed implementation methods are described later.

[0164] Table 1 O-ADMM Method Pseudocode

[0165]

[0166] Table 2 O-AltOpt Optimization Framework Pseudocode

[0167]

[0168]

[0169] Example:

[0170] Based on the content of the instruction manual, verification was carried out, and the waveform was optimized using existing and proposed solutions.

[0171] The simulation experimental conditions are set as follows: Consider a uniform linear array with N transmitting antennas. t =6; Number of targets P=2, target positions are respectively at θ1=-θ 3dB ×180 / π, θ2=θ 3dB ×180 / π, where θ 3dB =0.891λ / (N) r λ / 2)=1.782 / N r The target amplitudes are α1 = 0.8 and α2 = 0.6; the sequence length is L = 100; and the transmitted signal energy is e. T =N t L = 600; Noise power σ in the receiver2 =1. In the O-AltOpt optimization framework, the penalty parameter is... Iteration stopping threshold ξ = 10 -4 In the O-ADMM method, the penalty parameter μ = 1, and the maximum number of iterations N = 300.

[0172] Existing solutions:

[0173] 1. The constant-mode waveform obtained by the ADMM method, i.e., the quantization bit depth is infinite: Reference: "WU W, TANG B. Waveform design for higher-resolution localization with MIMO radar[J]. Signal Processing, 2023, 209:108999."

[0174] 2. The waveform obtained by the ADMM method is directly quantized into a one-bit waveform: Reference: "WU W, TANG B. Waveform design for higher-resolution localization with MIMO radar[J]. Signal Processing, 2023, 209:108999."

[0175] 3. One-bit waveform obtained by the gradient-based 2bG method: Reference: "LIN R, SHANG X, LI JJ SP. Effective and Efficient Gradient Based Methods for Low-Bit Discrete-PhaseSequence Designs[J]. 2023:108929."

[0176] 4. One-bit waveform obtained by a one-dimensional search method based on coordinate descent (CD): Reference: "AN D, HU J, ZHONG K, et al. RIS Assisted Multiple User Interference Mitigation via an Accelerated Coordinate Descent Method[J]. IEEE Transactions on Cognitive Communications and Networking, 2023, 9(1):159-69."

[0177] Figure 1 The residual terms in the proposed method are shown. The convergence curve follows the number of outer iterations. It can be seen that as the number of outer iterations increases to 30, the residual term... The iteration eventually converged to 7.029e-05, satisfying the stopping condition.

[0178] Figure 2 The convergence curve of the cost function value (dB) with the number of iterations is shown. It can be seen that the cost function curve of the proposed method converges to a fixed value. As expected, using an infinite-bit DAC results in a lower convergence cost function value compared to using a one-bit DAC, due to its higher degrees of freedom. Among the methods using a one-bit DAC, the proposed method exhibits the best performance; specifically, the convergence value of the cost function of the proposed method is 1.976 dB lower than the CD method and 0.736 dB lower than the 2bG method.

[0179] Figure 3 The normalized spatial spectrum generated by the MUSIC algorithm is shown, where the number of receiving antennas N r =6, indicating that the spatial spectral performance of the ADMM method using an infinite-bit DAC is better than that of the method using a one-bit DAC. Furthermore, in the method using a one-bit DAC, the proposed method produces a spatial spectrum with sharper peaks at the target location, thus achieving higher resolution.

[0180] Figure 4 The graph shows the variation of RMSE (rad) with the number of receiving antennas, where N is the number of receiving antennas. r = [4,6,8,10,12]. It can be seen that the RMSE value gradually decreases as the number of receiving antennas increases. As expected, the RMSE curve of the ADMM method using an infinite-bit DAC is closer to the theoretical lower bound of equation (4). In the method using a one-bit DAC, the RMSE curve of the proposed method is closer to the theoretical lower bound than the RMSE curves of the 2bG method and the CD method, that is, the proposed method has higher DOA estimation accuracy.

[0181] The above simulations verify the performance advantages of the MIMO radar transmit waveform design method based on a one-bit DAC disclosed in this invention: on the one hand, the waveform designed by the proposed method can achieve higher resolution, and on the other hand, the RMSE curve of the proposed method is closer to the theoretical lower bound.

[0182] It should also be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes that element.

[0183] The above are merely embodiments of this application and are not intended to limit the scope of this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of the claims of this application.

[0184] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of this application are indicated by the following claims.

[0185] It should be understood that this application is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of this application is limited only by the appended claims.

Claims

1. A method for designing MIMO radar waveforms with high-precision DOA estimation using a one-bit DAC, characterized in that, include: Under far-field conditions, the waveform design problem is established for the transmit antennas of a MIMO radar equipped with two one-bit DACs. Each transmit antenna uses two one-bit DACs to convert the I / Q signals respectively, forming a waveform in the symbol set. In constrained MIMO radar, a waveform design problem is established with the objective of minimizing the mean square error of the DOA estimation using the MUSIC algorithm. Represents the total energy of the transmitted signal. One transmitting antenna, The sequence length is solved using a two-layer iterative architecture that nests an outer one-bit alternating optimization (O-AltOpt) framework with an inner one-bit alternating directional multiplier method (O-ADMM). The element-wise closed-form solution of the waveform matrix is ​​obtained in each iteration of O-ADMM based on the KKT conditions.

2. The MIMO radar waveform design method for high-precision DOA estimation using a one-bit DAC as described in claim 1, characterized in that, The MIMO radar is a centralized MIMO radar in which both the transmitting and receiving arrays are uniform linear arrays with half-wavelength antenna spacing.

3. The MIMO radar waveform design method for high-precision DOA estimation with a one-bit DAC according to claim 2, characterized in that, The waveform design problem for a MIMO radar equipped with two one-bit DAC transmitting antennas under far-field conditions includes: Constructing the signal model: For a having One transmitting antenna and A centralized MIMO radar with multiple receiving antennas receives signals as follows: （1） in Represents the number of targets. , Representing the first The magnitude and direction of each goal, and Represents respectively in The receive steering vector and the transmit steering vector at the location, It is the transmitted waveform matrix. Representing the The transmission sequence of each transmitting antenna. It is the sequence length. It is Gaussian white noise; For the received signal model: （2） in , , ， ， ， Based on the signal model, array signal processing is used to estimate the target direction, including the use of the MUSIC algorithm to estimate the target direction. The mean square error of the DOA estimation of the MUSIC algorithm is expressed as: (3) in It is the first The true frequency of each target, and It is the first The estimated frequency of each target The noise in the receiver, the theoretical minimum value of the DOA estimate MSE of the MUSIC algorithm is expressed as: (4) in for The singular values; At this point, the symbol set of the transmitted signal of a MIMO radar equipped with a one-bit DAC is represented as: ， in Represents the total energy of the transmitted signal, and the transmitted waveform satisfies This is also equivalent to: (5) Among them, matrix These represent the identity matrix and a matrix consisting entirely of 1s (0s), respectively. The design formula (6) for the MIMO radar transmit waveform based on a one-bit DAC is expressed as follows: (6a) (6b) (6c) The obtained waveform design formula (6) is derived from the information about It consists of a quadratic objective function (6a), a quadratic equality constraint (6b), and a linear inequality constraint (6c).

4. The MIMO radar waveform design method for high-precision DOA estimation with a one-bit DAC according to claim 3, characterized in that, The waveform design problem is subjected to variable separation and problem transformation, including: First, by separating the variables, formula (6) is transformed into: (7a) （7b） （7c） （7d） in It is an auxiliary variable introduced, and the augmented Lagrangian function of formula (7) is expressed as: (8) in It is a penalty parameter. It is a Lagrange multiplier.

5. The MIMO radar waveform design method for high-precision DOA estimation with a one-bit DAC according to claim 4, characterized in that, in, In solving the transformed waveform design problem using a one-bit alternating optimization framework, the auxiliary variables, the emitted waveform matrix, and the Lagrange multipliers are updated sequentially within the optimization framework, specifically including: In the In the next iteration The following will be updated in order: (9a) （9b） （9c） Formula (9a) is expressed as: (10) By definition Formula (10) is expressed as: (11) make , Representing matrices respectively and Singular value decomposition, in , Representative matrix The singular values; same, , Representative matrix singular values, Then formula (11) can be simplified to: (12) The decomposition formula (12) is There are 1 independent univariate problems, where the 1st... Each is represented as: (13) because Regarding variables The second derivative satisfies: (14) It's about variables. A convex function, let Regarding variables The first derivative is 0: (15) Solve formula (15) to obtain Then update , The update formula is as follows: (16) Formula (9b) is expressed as: (17) definition , Formula (17) is expressed as formula (18): (18a) （18b） （18c） Introducing auxiliary variables Formula (18) is expressed as formula (19): (19a) （19b） （19c） The augmented Lagrangian function of formula (19) is expressed as: (20) in It's about constraints. dual variables, It is a penalty parameter; Consider variable substitution This yields the augmented Lagrangian function in scaled form: (twenty one) in These are the scaled dual variables, in the first... iteration Updated sequentially: (22a) (22b) (22c) The set The definition is as follows: (23a) (23b) Formula (22a) is expressed as: (24a) (24b) in Represented as: (25) Solving the KKT conditions of formula (24) yields the optimal solution. The real and imaginary parts of an element are represented as follows: (26a) (26b) Formula (22b) is expressed as formula (27): (27a) (27b) For matrix Perform eigenvalue decomposition , in It is a unitary matrix composed of eigenvectors. diagonal matrix , These are the corresponding eigenvalues. definition and Formula (27) can be rewritten as formula (28): (28a) (28b) The augmented Lagrangian function of formula (28) is expressed as: (29) in It is an equality constraint The dual variable; The optimality condition of formula (28) is: (30a) (30b) We obtain the following from equation (30a) ,in It is a diagonal matrix, and its nth diagonal element is... ;Will Substituting into equation (30b), we get: (31) By solving formula (31) using the bisection method, the following can be calculated: ,at last .

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