A one-bit transmission waveform design method for MIMO radar

Abstract

The present invention discloses a one-bit transmitting waveform design method for MIMO radar, and belongs to the field of radar technology. The present invention uses angle and distance information to jointly design near-field patterns; first, a one-bit waveform design problem is constructed; then, an alternating optimization framework is proposed, which iteratively optimizes auxiliary variables and the transmitting waveform matrix, and within this framework, a low-complexity algorithm based on ADMM is proposed to update the waveform matrix. The one-bit alternating optimization framework proposed by the present invention can sequentially update the auxiliary variables and the waveform matrix S, and in particular, in each iteration, it is updated by a closed-form solution, and S is updated by a one-bit alternating direction multiplier method; compared to the far-field pattern design method that can only form a peak in the angle dimension, the near-field pattern of the method of the present invention can simultaneously form a peak in the angle dimension and the distance dimension.

Description

Technical Field

[0001] The invention belongs to the technical field of radars, and in particular relates to a one-bit transmission waveform design method for a MIMO radar. Background Art

[0002] The waveform design for shaping the transmit pattern is a key technology in MIMO radar systems and is crucial to improving target detection performance. In actual radar systems, constant modulus constraints (CMC) are often applied to improve the transmit waveform to improve transmitter efficiency. For large-scale antenna arrays, low-resolution digital-to-analog converters (DACs), especially one-bit DACs, are usually equipped to reduce power consumption and hardware costs while maintaining the constant mode characteristics of the transmit waveform. Therefore, the waveform design for shaping the transmit pattern has attracted widespread attention.

[0003] Existing research mainly focuses on the shaping of far-field transmission patterns, considering only the angular dimension information. Generally, an alternating minimization (AM) framework is proposed in the literature "Z. Cheng, B. Liao, Z. He, and J. Li, "Transmit signal design for large-scale mimosystem with 1-bit dacs," IEEE Transactions on Wireless Communications, vol. 18, no. 9, pp. 4466–4478, 2019.", which uses an approximate method to obtain a high-precision solution when solving the constant modulus waveform matrix, but this inevitably introduces approximation errors. In order to solve this problem, the Projection-Descent-Retraction (PDR) method is proposed in the paper "K. Alhujaili, V. Monga, and M. Rangaswamy, "Transmit mimoradar beampattern design via optimization on the complex circle manifold," IEEE Transactions on Signal Processing, vol. 67, no. 13, pp. 3561–3575, 2019.", which transforms the problem into an unconstrained problem on a complex circle manifold. In addition, a coordinate descent (CD) framework is proposed in the paper "S. Imani and MM Nayebi, "A coordinate descent framework for beampattern design and waveform synthesis in mimo radars," IEEE Transactions on Aerospace and Electronic Systems, vol. 57, no. 6, pp. 3552–3562, 2021.", which transforms the problem into a phase optimization problem that satisfies CMC.In the paper "Z. Huang, B. Tang, C. Huang, and L. Qin, "Direct transmit waveform design to match a desired beampattern under the constant modulus constraint," Digital Signal Processing, vol. 126, p. 103486, 2022.", a method based on Majorization-Minimization (MM) is proposed, and a closed-form solution is obtained by finding a suitable alternative function in each iteration. In the paper "R. Lin, X. Shang, and J. Li, "Effective and efficient gradient based methods for low-bit discrete-phase sequence designs," Signal Processing, vol. 207, p. 108929, 2023.", a gradient-based method is proposed, which determines the appropriate descent step size through a linear search technique to ensure the reduction of the monotonic optimization metric. However, the above method only considers the far-field pattern in the angular dimension.

[0004] To meet the growing demand for sensing performance, the future sixth-generation wireless network system will develop into large-scale antenna arrays and high-frequency bands such as millimeter waves and terahertz bands. However, this trend will significantly change the electromagnetic propagation characteristics, from far-field plane wave propagation to near-field spherical wave propagation, which will inevitably lead to near-field effects. Unlike far-field sensing systems that only contain angle information, near-field sensing systems integrate angle and distance information at the same time. Summary of the invention

[0005] In view of the above technical problems existing in the prior art, the present invention proposes a MIMO radar one-bit transmission waveform design method, which has a reasonable design, overcomes the shortcomings of the prior art, and has good effects.

[0006] In order to achieve the above object, the present invention adopts the following technical solution:

[0007] A method for designing a one-bit transmission waveform of a MIMO radar comprises the following steps:

[0008] Step 1: Construct a one-bit waveform design problem, whose objective function is to minimize the mean square error between the desired beam pattern and the designed beam pattern;

[0009] Step 2: Iteratively optimize the auxiliary variables and the emission waveform matrix through an alternating optimization framework;

[0010] Step 3: Update the waveform matrix through the ADMM algorithm.

[0011] Preferably, in step 1, a one-bit near-field transmission waveform design problem It is expressed as:

[0012]

[0013] In the formula, represents the desired beam pattern, represents the near-field steering vector, S represents the emitted waveform, and M t represents the number of antennas, N represents the number of snapshots, tr{.} represents the trace, represents the real part, represents the imaginary part; constrain tr{SS H}=1 is to ensure the transmission power is constant. It is to limit the amplitude of the real and imaginary parts of S to a combined interval;

[0014] Introducing auxiliary variables satisfy Among them, k1=1,…,K1, k2=1,…,K2; Problem Translates to:

[0015]

[0016] Preferably, step 2 specifically includes the following steps:

[0017] Step 2.1: Optimize auxiliary variables For a fixed S, with respect to the auxiliary variable The problem is expressed as:

[0018]

[0019] The (k1, k2)th sub-problem is expressed as:

[0020]

[0021] The closed-form solution is expressed as:

[0022]

[0023] Step 2.2: Optimize the waveform matrix S;

[0024] For fixed Questions about the waveform matrix S It is expressed as:

[0025]

[0026] The definitions of matrix A and matrix B are as follows:

[0027]

[0028] Preferably, step 3 specifically includes the following steps:

[0029] Step 3.1: Introduce auxiliary variable C=S and transform formula (6) into:

[0030]

[0031] The scaled Lagrangian function of formula (6) is expressed as:

[0032]

[0033] Where U is the dual variable in scaled form, ρ is the penalty parameter, and S represents the emission waveform matrix;

[0034] In the (i+1)th iteration, the transmission waveform matrix S, auxiliary variables C, and dual variables U are optimized in the following manner:

[0035]

[0036] Among them, the set Ω S and Ω C They are defined as:

[0037]

[0038] Step 3.2: Update the waveform matrix S;

[0039] For fixed C and U, the problem about the waveform matrix S is expressed as:

[0040]

[0041] in, It is expressed as:

[0042] D = 2B + ρ (C + U) (13);

[0043] The problem about the waveform matrix S contains a quadratic objective function and linear inequality constraints, which are solved element by element; therefore, according to the KKT condition, S(m,n), m=1,…,M t ,n=1,…,the real and imaginary parts of N are updated as follows:

[0044]

[0045] Step 3.3: Update auxiliary variable C;

[0046] For fixed S and U, the problem about the auxiliary variable C is expressed as:

[0047]

[0048] Perform eigenvalue decomposition on matrix A, A = VΠV H , where V is the feature matrix, is the corresponding eigenvalue; by definition and E = V H (SU), the problem is transformed into:

[0049]

[0050] The Lagrangian function of formula (17) is expressed as:

[0051]

[0052] where μ is the constraint The dual variable of ; the optimality condition of the Lagrangian function shown in formula (18) is:

[0053]

[0054] From formula (19) we get in is a diagonal matrix with the mth diagonal element being ρ / (2λ m +ρ+2μ); then, calculate the dual variable μ; Bring in have to:

[0055]

[0056] f(μ)=0 is solved by the bisection method. Specifically, by λ m ≥0, ρ>0, f(μ) It is monotonically increasing on μ∈(-0.5ρ-λ1,∞), and monotonically decreasing on μ∈(-0.5ρ-λ1,∞). When μ→±∞, f(μ)→-1; when μ→-0.5ρ-λ1 or When , f(μ)→∞; therefore, f(μ) is and there are zero points on (-0.5ρ-λ1,∞);

[0057] Step 3.4: After solving μ by bisection, the matrix It can also be calculated and finally the auxiliary variables can be updated

[0058] Beneficial technical effects brought by the present invention:

[0059] The present invention proposes a one-bit alternating optimization (O-AltOpt) framework that can update auxiliary variables in sequence. and the waveform matrix S. In particular, in each iteration, Updated by the closed-form solution, S is updated by the one-bit alternating direction multipliers method (O-ADMM). Compared with the far-field pattern design method which can only form a peak in the angle dimension, the near-field pattern of the method of the present invention can form a peak in both the angle dimension and the distance dimension. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 It is a schematic diagram of the near-field MIMO radar system model;

[0061] Figure 2 is the convergence curve of the method proposed in the embodiment as the penalty parameter ρ changes;

[0062] Figure 3 is the near-field discrete emission pattern corresponding to the method proposed in the embodiment;

[0063] Figure 4 is the discrete transmit pattern of the gradient-based method (2bG) in the embodiment;

[0064] Figure 5 is a discrete emission pattern of a quantized Riemann conjugate gradient method (Q-RCG) in an embodiment. DETAILED DESCRIPTION

[0065] The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments:

[0066] Bold lowercase letters and bold uppercase letters represent vectors and matrices, respectively; transpose, conjugate, and conjugate transpose operations are represented by (·) T ,(·) * ,(·) H express; Represent the complex number domain and the real number domain respectively; diag([x1,x2,…,x N ]) represents a diagonal matrix, where the diagonal elements are x1,x2,…,x N ; represents the expectation of a random variable; tr{·},vec(·),(·) -1 Represent the trace, column vectorization and inverse of the matrix respectively; Represent the real and imaginary parts of a complex-valued scalar / vector / matrix, respectively; ||·|| F,|·| respectively represent the matrix binorm and the vector element-wise modulus; ⊙ represents the Hadamard product; the matrices I,1(0) respectively represent the identity matrix and the matrix of all 1s(0).

[0067] Consider a system with M t The centralized MIMO radar has a number of transmitting antennas and the length of the transmission sequence of each transmitting antenna is N. It is assumed that the transmitting antenna array is a uniform linear array, where the antenna spacing d is half a wavelength, and the radar aperture D = (M t -1)d.

[0068] Under the assumption of near field and narrow band, the steering vector a not only has an angle θ with the target k and the distance to the target r k Therefore, the expression of the steering vector is:

[0069]

[0070] in, represents the distance between the kth target and the mth antenna, which can be expressed as:

[0071]

[0072] By the Fresnel approximation, It can be approximated as:

[0073]

[0074] in is a function of angle and distance. Substituting into the steering vector expression we get:

[0075]

[0076] in The mth element of is:

[0077]

[0078] We assume that the angle and distance {θ k ,r k} is divided into and in The division intervals K1 and K2 are determined by the sampling rate. Therefore, the discrete form of the steering vector α can be expressed as:

[0079]

[0080] The discrete form of the transmission pattern can be expressed as:

[0081]

[0082] Where S is the transmit waveform matrix.

[0083] In order to ensure the transmitter efficiency and avoid nonlinear distortion of the power amplifier, the transmitted waveform is usually required to have constant mode characteristics. In addition, for large-scale MIMO radar systems, it is of great significance to minimize system power consumption and hardware cost. Therefore, we consider a MIMO radar system using a one-bit digital-to-analog converter (DAC). At the same time, using a one-bit DAC can ensure the constant mode characteristics of the waveform. For the sake of generality, we assume that the total energy of the transmitted signal is 1, that is, tr{SS H Based on this, the set of all values ​​of the transmitted signal sample points can be expressed as Therefore, the transmit waveform matrix needs to satisfy It can be equivalently expressed as:

[0084] tr{SS H}=1

[0085]

[0086] in and denote the real and imaginary parts of a complex variable, respectively, and 1 denotes a column vector whose elements are all 1.

[0087] The method of the present invention is as follows:

[0088] A method for designing a one-bit transmission waveform of a MIMO radar comprises the following steps:

[0089] Step 1: Construct a one-bit waveform design problem, whose objective function is to minimize the mean square error between the desired beam pattern and the designed beam pattern;

[0090] Step 3: Update the waveform matrix through the ADMM algorithm.

[0091] One-bit near-field transmission waveform design problem It is expressed as:

[0092]

[0093] question Contains a quartic objective function, a quadratic equality constraint, and a linear inequality constraint. Due to the quadratic equality constraint, the problem It is non-convex and difficult to solve directly.

[0094] To solve the problem We use the “almost equivalent” criterion to transform the quartic problem into a quadratic problem. Specifically, we introduce the auxiliary variable satisfy Then the question can be converted to:

[0095]

[0096] Step 2: Iteratively optimize the auxiliary variables and the emission waveform matrix through the alternating optimization framework; the details are as follows:

[0097] Due to the existence of coupling variable S, and quadratic equality constraints, the problem is non-convex and difficult to solve directly. Therefore, we propose a one-bit alternating optimization (O-AltOpt) framework that can update the auxiliary variables sequentially. and the waveform matrix S. In particular, in each iteration, is updated by the closed-form solution, and S is updated by the O-ADMM method.

[0098] 1. Optimize auxiliary variables

[0099] For a fixed S, with respect to the auxiliary variable The problem is expressed as:

[0100]

[0101] Note that the above problem can be decomposed into K1K2 independent sub-problems, where the (k1, k2)th sub-problem can be expressed as:

[0102]

[0103] The closed-form solution to the above problem can be expressed as:

[0104]

[0105] 2. Optimize the waveform matrix S

[0106] For fixed Questions about the waveform matrix S It can be expressed as:

[0107]

[0108] The definitions of matrix A and matrix B are as follows:

[0109]

[0110] Since the above problem contains quadratic equality constraints, the problem is non-convex and difficult to solve directly. Therefore, the O-ADMM method is proposed to solve it. Specifically, we introduce an auxiliary variable C = S and then transform the above problem into:

[0111]

[0112] The scaled Lagrangian function of formula (6) is expressed as:

[0113]

[0114] Where U is the dual variable in scaled form and ρ is the penalty parameter;

[0115] In the (i+1)th iteration, the transmission waveform matrix S, auxiliary variables C, and dual variables U are optimized in the following manner:

[0116]

[0117] Among them, the set Ω S and Ω C They are defined as:

[0118]

[0119] (1) Update the waveform matrix S;

[0120] For fixed C and U, the problem about the waveform matrix S is expressed as:

[0121]

[0122] in, It is expressed as:

[0123] D = 2B + ρ (C + U) (13);

[0124] Note that the problem about the waveform matrix S contains a quadratic objective function and linear inequality constraints, which is a convex problem and can be solved element by element. Therefore, according to the KKT condition, S(m,n), m=1,…,M t ,n=1,…,the real and imaginary parts of N can be updated as:

[0125]

[0126] (2) Update auxiliary variable C

[0127] For fixed S and U, the problem about the auxiliary variable C is expressed as:

[0128]

[0129] Due to the existence of quadratic equality constraints, the problem is non-convex and difficult to solve directly. In order to solve the above problem, the matrix A is decomposed into eigenvalues, A = VΠV H , where V is the feature matrix, is the corresponding eigenvalue. By definition and E = V H (SU), the problem can be transformed into:

[0130]

[0131] The Lagrangian function of formula (17) is expressed as:

[0132]

[0133] where μ is the constraint The dual variable of ; the optimality condition of the Lagrangian function shown in formula (18) is:

[0134]

[0135] From formula (19) we get in is a diagonal matrix with the mth diagonal element being ρ / (2λ m +ρ+2μ); then, we need to calculate the dual variable μ. Bring in have to:

[0136]

[0137] f(μ)=0 can be solved by bisection method. Specifically, by λ m ≥0, ρ>0, f(μ) It is monotonically increasing on μ∈(-0.5ρ-λ1,∞), and monotonically decreasing on μ∈(-0.5ρ-λ1,∞). When μ→±∞, f(μ)→-1; when μ→-0.5ρ-λ1 or When , f(μ)→∞; therefore, f(μ) is and there are zero points on (-0.5ρ-λ1,∞);

[0138] After solving μ by bisection, the matrix It can also be calculated and finally the auxiliary variables can be updated

[0139] In summary, Tables 1 and 2 summarize the problems solved by the O-ADMM method. And solve the problem through the O-AltOpt optimization framework

[0140] Table 1O-ADMM method pseudo code

[0141]

[0142] Table 2 O-AltOpt optimization framework pseudo code

[0143]

[0144]

[0145] Example

[0146] In order to verify the proposed scheme, the waveform is optimized using the existing scheme and the proposed scheme.

[0147] The simulation experiment conditions are set as follows: We consider a uniform linear array with M transmitting antennas. t =100, transmission sequence length N = 64, transmission signal frequency f = 30GHz; angle range is -90° to 90°, evenly divided into K1 = 20 intervals, distance range is 0 to 0.2*D 2 / λ, evenly divided into K2 = 20 intervals. In the O-ADMM method, the penalty parameter ρ = 16 and the maximum number of iterations K = 300. The target location is set to That is, the ideal directional diagram is expressed as:

[0148]

[0149] In order to evaluate the convergence and transmission pattern performance of the proposed method, the following methods are compared:

[0150] (1) A gradient-based method using a one-bit DAC under the far-field assumption, referred to as the 2bG method: LIN R, SHANG X, LI JJS P. Effective and Efficient Gradient Based Methods for Low-Bit Discrete-Phase Sequence Designs[J]. 2023: 108929.

[0151] (2) The Riemann conjugate gradient method using a one-bit DAC under the far-field assumption, referred to as the Q-RCG method (the continuous waveform is quantized into an orthogonal phase-shift keying waveform in each iteration): reference "J.Hu, W.Zhang, H.Zhu, K.Zhong, W.Xiong, Z.Wei, and Y.Li, "Constant modulus waveform design for mimo radar viamanifold optimization," Signal Processing, vol. 190, p. 108322, 2022.".

[0152] Figure 1 It is a schematic diagram of the near-field MIMO radar system model.

[0153] Figure 2 The convergence curve of the objective function value as the penalty parameter ρ changes is shown. It can be seen that when the penalty parameter ρ changes, the objective function value curve generally shows a downward trend, but it is not strictly monotonic. This is because when solving non-convex problems, the ADMM algorithm can only converge to the KKT point, and it cannot be guaranteed that this point is the global optimal point. In addition, when ρ = 0.001, the objective function curve can converge to a lower point, which means that the transmission pattern matching error is minimized.

[0154] Figure 3 The normalized transmission direction formed by the transmission waveform designed by the proposed method under the near-field assumption is shown. The horizontal axis represents the number of discretized grids k1(k2), the left vertical axis represents the normalized directivity pattern in the angle dimension, and the right vertical axis represents the normalized directivity pattern in the distance dimension. It can be seen that a peak is formed at k1=12 in the angle dimension and a peak is formed at k2=8 in the distance dimension, which is consistent with the positions of the peaks of the expected discrete directivity pattern in the angle dimension and the distance dimension.

[0155] Figure 4 The normalized emission direction formed by the emission waveform designed by the 2bG method under the far-field assumption is shown. It can be seen that this method can only form a peak at the position of angle dimension k1=12, but cannot form a peak in the distance dimension.

[0156] Figure 5 The normalized emission direction formed by the emission waveform designed by the Q-RCG method under the far-field assumption is shown. It can be seen that this method can only form a peak at the angle dimension k1=12, but cannot form a peak in the distance dimension.

[0157] Through the above simulation, the performance advantage of the MIMO radar one-bit transmission waveform design method based on minimizing the near-field transmission pattern matching error disclosed in the present invention is verified: compared with the far-field pattern design method that can only form a peak in the angle dimension, the near-field pattern designed by our proposed method can form a peak in both the angle dimension and the distance dimension.

[0158] Of course, the above description is not a limitation of the present invention, and the present invention is not limited to the above examples. Changes, modifications, additions or substitutions made by technicians in this technical field within the essential scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A method for designing a one-bit transmission waveform for a MIMO radar, characterized in that: The steps include: Step 1: Construct a one-bit waveform design problem, whose objective function is to minimize the mean square error between the desired beam pattern and the designed beam pattern; Step 2: Iteratively optimize the auxiliary variables and the emission waveform matrix through an alternating optimization framework; Step 3: Update the waveform matrix through the ADMM algorithm.

2. The MIMO radar one-bit transmission waveform design method according to claim 1, characterized in that: In step 1, the one-bit near-field transmission waveform design problem It is expressed as: In the formula represents the desired beam pattern, represents the near-field steering vector, S represents the emitted waveform, and M t represents the number of antennas, N represents the number of snapshots, tr{.} represents the trace, represents the real part, represents the imaginary part; constrain tr{SS H }=1 is used to ensure constant transmission power, Used to limit the amplitude of the real and imaginary parts of S to a combined interval; Introducing auxiliary variables satisfy Among them, k1=1,…,K1, k2=1,…,K2; Problem Translates to: In the formula represents the two-norm.

3. The MIMO radar one-bit transmission waveform design method according to claim 1, characterized in that: Step 2 specifically includes the following steps: Step 2.1: Optimize auxiliary variables For a fixed S, with respect to the auxiliary variable The problem is expressed as: The (k1, k2)th sub-problem is expressed as: The closed-form solution is expressed as: In the formula express conjugation of; Step 2.2: Optimize the waveform matrix S; For fixed Questions about the waveform matrix S It is expressed as: The definitions of matrix A and matrix B are as follows:

4. The MIMO radar one-bit transmission waveform design method according to claim 2, characterized in that: Step 3 specifically includes the following steps: Step 3.1: Introduce auxiliary variable C=S and transform formula (6) into: The scaled Lagrangian function of formula (6) is expressed as: Where U is the dual variable in scaled form, ρ is the penalty parameter, and S represents the emission waveform matrix; In the (i+1)th iteration, the transmission waveform matrix S, auxiliary variables C, and dual variables U are optimized in the following manner: Among them, the set Ω S and Ω C They are defined as: Step 3.2: Update the waveform matrix S; For fixed C and U, the problem about the waveform matrix S is expressed as: in, It is expressed as: D = 2B + ρ (C + U) (13); The problem about the waveform matrix S contains a quadratic objective function and linear inequality constraints, which are solved element by element; therefore, according to the KKT condition, S(m,n), m=1,…,M t ,n=1,…,the real and imaginary parts of N are updated as follows: Step 3.3: Update auxiliary variable C; For fixed S and U, the problem about the auxiliary variable C is expressed as: Perform eigenvalue decomposition on matrix A, A = VΠV H , where V is the feature matrix, is the corresponding eigenvalue; by definition and E = V H (SU), the problem is transformed into: The Lagrangian function of formula (17) is expressed as: where μ is the constraint The dual variable of ; the optimality condition of the Lagrangian function shown in formula (18) is: From formula (19) we get in is a diagonal matrix with the mth diagonal element being ρ / (2λ m +ρ+2μ); then, calculate the dual variable μ; Bring in have to: f(μ)=0 is solved by the bisection method. Specifically, by λ m ≥0, ρ>0, f(μ) It is monotonically increasing on μ∈(-0.5ρ-λ1,∞), and monotonically decreasing on μ∈(-0.5ρ-λ1,∞). When μ→±∞, f(μ)→-1; when μ→-0.5ρ-λ1 or When , f(μ)→∞; therefore, f(μ) is and there are zero points on (-0.5ρ-λ1,∞); Step 3.4: After solving μ by bisection, the matrix It can also be calculated and eventually update the auxiliary variables

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