MIMO radar single-mode waveform design based on complex circle manifold network

Abstract

The invention discloses MIMO radar single-mode waveform design based on a complex circle manifold network, and belongs to the field of waveform design. The invention provides a model-based complex circle manifold network, which utilizes the advantages of two gradient descent models to learn limited data. According to the method, a waveform design problem is expressed as an unconstrained quadratic fractional order problem on a complex circle manifold, in order to solve the problem, a gradient descent algorithm is used as network layer expansion on the complex circle manifold, and a step length is adaptively learned; besides, a soft quantization LCCM network is designed for discrete phases adopted in an actual system, a low-resolution non-uniform quantizer is designed to quantize the phases, and the quantizer depends on a uniquely-designed soft ladder function, combines learnable parameters and is allowed to adaptively finely adjust a decision area. On one hand, relaxation is not needed, the calculation cost is reduced, on the other hand, the low-resolution non-uniform quantizer is adopted for phase quantization, and the efficiency of the transmitter is improved.

Description

Technical Field

[0001] The invention belongs to the field of radar technology, and in particular relates to a MIMO radar single-mode waveform design method based on a complex circular manifold network. Background Art

[0002] Waveform design is a key technology in MIMO radar systems to achieve interference suppression and improve signal-to-interference-noise ratio to ensure target detection performance. In order to ensure the efficiency of transmitter use, constant modulus waveforms are preferred in practical implementations. Therefore, the problem of waveform design with constant modulus constraints (CMC) has received widespread attention.

[0003] For model-based methods, problem relaxation is usually applied. For example, in long sequence processing, the paper "Q.Li, C.Li, and J.Lin, "Constant modulus secure beamforming for multicast massive MIMO wiretap channels," IEEE Transactions on Information Forensics and Security, vol. 15, pp. 264–275, 2020." introduces auxiliary variables by alternating direction multiplier method (ADMM) to make the problem tractable. The paper "M.Soltanalian and P.Stoica, "Designing unimodular codes via quadratic optimization," IEEE Transactions on Signal Processing, vol. 62, no. 5, pp. 1221–1234, 2014." focuses on the single-mode waveform design method based on quadratic optimization and proposes a well-known power-like method (PML) to solve the problem. As a further work, the paper "G.Cui, Y.Fu, X.Yu, and J.Li, "Local ambiguity function shaping via unimodular sequence design," IEEE SignalProcessing Letters, vol.24, no.7, pp.977–981, 2017." designed a unimodular sequence for local ambiguity function shaping through four optimizations, and proposed an accelerated PML method to solve the problem. In order to further improve the performance of the algorithm, the paper "CGTsinos, A.Arora, S.Chatzinotas, and B.Ottersten, "Joint transmitwaveform and receive filter design for dual-function radarcommunicationsystems," IEEE Journal of Selected Topics in Signal Processing, vol.15, no.6, pp.1378–1392, 2021." proposed a method based on gradient projection (GP), which first relaxes the constant modulus constraint and then projects the relaxed solution onto the complex circular manifold (CCM).In order to reduce the amount of calculation, the paper "J.Hu, W.Zhang, H.Zhu, K.Zhong, W.Xiong, Z.Wei, and Y.Li, "Constant modulus waveform design for MIMO radar via manifold optimization," Signal Processing, vol.190, p.108322, 2022" proposed a Riemann conjugate gradient method that updates the conjugate parameters and step size through line search. However, due to the fact that the line search method is not always useful for updating the conjugate parameters and step size at the same time, its performance may be limited.

[0004] In addition to the above optimization methods, evolutionary algorithms have been used in various works for waveform design. For example, genetic algorithms (GAs) have been used to design phase-coded waveforms in the spatial domain, orthogonal frequency division multiplexing (OFDM) radar pulses, and discrete frequency and linear frequency modulation coded waveforms. However, the selection of parameters such as crossover rate and mutation rate has a great impact on the quality of the solution. In order to reduce the selection of multiple parameters, the paper "J. Hu, Z. Wei, Y. Li, H. Li, and J. Wu, "Designing unimodular waveform(s) for MIMO radar by deep learning method," IEEE Transactions on Aerospace and Electronic Systems, vol. 57, no. 2, pp. 1184–1196, 2021." considers a purely data-driven deep neural network (Res-Net) method based on a residual network, which is implemented by learning a large number of weight coefficients of the network, which usually requires a lot of computing resources. In general, the performance of these methods may not be satisfactory in the common case of limited training data. In recent years, radar signal processing technology based on deep unfolding networks has just emerged. The paper "S. Khobahi, A. Mostajeran, M. Emadi, P. Wang, and M. Soltanalian, "Guaranteed deep learning for reliable radar signal processing," in 2022 IEEE 12th Sensor Array and Multichannel Signal Processing Workshop (SAM), pp. 291–295, 2022." introduces a customized model-based deep structure based on the Neumann series inversion lemma to find the mismatched filter (MMF), which reduces the computational complexity compared to matrix inversion. However, this method may not be applicable to single-mode waveform design. To address this issue, the paper "S. Khobahi, A. Bose, and M. Soltanalian, "Deep radar waveform design for efficient automotive radar sensing," in 2020 IEEE 11th Sensor Array and Multichannel Signal Processing Workshop (SAM), pp. 1–5, 2020." proposes a model-based deep architecture inspired by the power method (PML) algorithm.It can learn network architecture parameters through a random walk-based training strategy. However, random walk-based models rely on probabilistic transfer and may require a large number of samples to achieve better results.

[0005] In general, model-based relaxation methods, DNNN-based methods with poor interpretability, and randomly trained deep unfolded networks may result in limited waveform design performance, but they all have unique advantages, so it is necessary to study a waveform design method that integrates these unique advantages. Summary of the invention

[0006] The object of the present invention is to provide a MIMO radar single-mode waveform design method based on a complex circular manifold network to improve the signal-to-interference-and-noise ratio (SINR) and the efficiency of the transmitter.

[0007] The technical solution adopted by the present invention is:

[0008] The single-mode waveform design of MIMO radar based on complex circular manifold network includes the following steps:

[0009] Objective function construction steps:

[0010] Based on N R receive antennas and N T The SINR objective function is constructed based on the signal-to-interference-to-noise ratio (SINR) of a co-located MIMO radar with N transmitting antennas. The constraint of the objective function is that each vector element s(l) of the transmitted signal vector s satisfies |s(l)|=1, where the transmitted signal vector s is N T ×M transmit waveform matrix S column vectorization; transmit signal vector s vector element index l = 1, 2, ..., MN T , M is the number of snapshots; construct a complex circular manifold based on s and its constraints Transform the constructed objective function into a complex circular manifold Unconstrained quadratic fraction problem on , which can be solved by gradient descent algorithm;

[0011] Steps to construct a complex circular manifold network:

[0012] The complex circular manifold network is used to learn the step size of the gradient descent algorithm. The complex circular manifold network includes several network layers (for example, K is defined to represent the number of network layers). In each layer, the step size is converted into a learnable network parameter. The calculation processing performed in each layer includes: projecting the Euclidean gradient onto the complex circular flow tangent space to obtain the manifold gradient; descending on the tangent space based on the manifold gradient to obtain a descent result corresponding to the descent step size of the current layer; tracing back the descent result of the tangent space to the complex circular flow tangent space to obtain the solution result of the current layer with respect to the transmitted signal vector s; and the complex circular manifold network calculates the loss function based on the objective function during training;

[0013] Steps to obtain MIMO radar single-mode waveform design results:

[0014] Obtain MIMO radar parameters and signal source parameters, initialize the transmission signal vector s and the network parameters of the complex circular manifold network, and set the training end conditions;

[0015] Based on the corresponding value of the transmitted signal vector s, the input data of the complex circular manifold network is obtained for forward propagation. The initial value of the input data of the complex circular manifold network is the initialized transmitted signal vector s; the input data for the next training forward propagation is the output result of the previous forward propagation; and the corresponding loss function value is calculated based on each forward propagation result; then, based on the current loss function value, reverse calculation is performed to optimize the step size corresponding to each network layer of the complex circular manifold network; when the training end condition is met, the MIMO radar single-mode waveform design result is obtained based on the most recent output result of the complex circular manifold network.

[0016] Furthermore, the complex circular manifold network also includes a soft quantization layer (i.e., a soft quantized complex circular manifold network), the input of the soft quantization layer is the output of the last network layer used to obtain the solution result of the transmitted signal vector s, the soft quantization layer is used to control the amplitude, smoothness and non-uniform quantization area, the soft quantization layer obtains the continuous phase of the solution result of the input transmitted signal vector s through inverse tangent operation, and then obtains the soft quantization phase based on the soft quantization function, and finally obtains the waveform of the quantized phase through Euler transformation; and during training, the quantization loss function is calculated based on the objective function and the waveform of the quantized phase output by the soft quantization layer.

[0017] Furthermore, the expression of the objective function is:

[0018]

[0019] subject to|s(l)|=1

[0020] Among them, the first auxiliary The second auxiliary amount β0 represents the signal-to-noise ratio of the target source, β q represents the signal-to-noise ratio of the qth interference source, Q represents the number of interference sources, represents a unit vector, represents the Gaussian white noise variance, the guidance vector of the target source Steering vector of interference source θ0、θ q They represent the target direction of the target source and the interference direction of the interference source, respectively. r (), a t () represent the sending steering vector and receiving steering vector of the corresponding angle respectively; (·) * 、(·) HThey represent conjugate and conjugate transpose operations respectively.

[0021] Furthermore, the MIMO radar parameters include: receiving antenna N R 、N T The signal source parameters include: the target direction of the target source, the signal-to-noise ratio of the target source, the number of interference sources, the interference direction of each interference source, the interference-to-noise ratio, and the variance of Gaussian white noise.

[0022] The technical solution provided by the present invention brings at least the following beneficial effects:

[0023] The present invention designs a model-based complex circular manifold network (LCCM-Net) for realizing MIMO radar single-mode beam design. The designed complex circular manifold network takes advantage of two gradient descent models and learns with limited data; that is, the present invention expresses the waveform design problem as an unconstrained quadratic fractional problem (UQFP) on a complex circular manifold (CCM), and then unfolds the gradient descent algorithm as a network layer on the CCM, and adaptively learns the step size. In addition, for the discrete phase used in the actual system, the present invention also designs a soft quantization LCCM network (QLCCM network), in which a low-resolution non-uniform quantizer is designed to quantize the phase. The quantizer relies on a uniquely designed soft step function, which combines learnable parameters to allow it to adaptively fine-tune the decision region, thereby further improving the performance of the transmitter based on the obtained waveform. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the drawings required for use in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0025] Figure 1 This is the network structure diagram of LCCM-Net;

[0026] Figure 2 This is the network structure diagram of QLCCM-Net;

[0027] Figure 3 is a curve of SINR of LCCM-Net with different numbers of layers as the number of iterations changes in the embodiment;

[0028] Figure 4 is a curve showing the change of SINR with the number of iterations in the embodiment;

[0029] Figure 5 are beam patterns of different methods in the embodiments;

[0030] Figure 6 is a curve showing the change of SINR with calculation time in the embodiment;

[0031] Figure 7 is a curve showing the relationship between SINR and SNR in the case of discrete phase in the embodiment;

[0032] Figure 8 It is a curve of the hard quantization function proposed in the embodiment and the traditional hard quantization function. DETAILED DESCRIPTION

[0033] To make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be described in detail and completely in conjunction with the drawings in the implementation of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all of the embodiments. Generally, the components of the embodiments of the present invention described and shown in the drawings can be arranged and designed using different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the present application for protection, but merely represents selected embodiments of the present invention.

[0034] The embodiment of the present invention provides a MIMO radar single-mode waveform design based on a complex circular manifold network. It is a waveform design method based on a model-based unrelaxed complex circular manifold network (LCCM-Net), which takes advantage of the advantages of two gradient descent models when learning limited data. That is, the waveform design optimization problem is reformulated as an unconstrained quadratic fractional problem (UQFP) on a complex circular manifold CCM. In order to solve the UQFP problem, the gradient descent algorithm is expanded as a network layer on the CCM, and all step sizes are adaptively learned. In the implementation process, the embodiment of the present invention proposes an LCCM-Net, whose structure is as follows: Figure 1 As shown, the optimization problem on CCM is solved without relaxation, and a limited step size is adaptively trained through the network; in addition to the continuous phase design, the embodiment of the present invention also considers the discrete phase design, and proposes a soft quantization LCCM-Net (QLCCM-Net), in which a non-uniformly spaced low-resolution quantizer is designed to quantize the phase, and its network structure is as follows Figure 2 shown.

[0035] For the convenience of description, the following definitions are given first:

[0036] and denote the real and imaginary parts of the elements of a complex vector respectively; transpose, conjugate, and conjugate transpose operations are respectively represented by (·) T ,(·) * ,(·) Hrepresents; ||·|| represents the vector norm, ⊙ represents the element-wise multiplication operation, represents the Hadamard element deviation; represents the expectation of a random variable, represents the complex field, and vec(·) represents the column vectorization of the matrix.

[0037] Construct an optimized model for single-mode waveform design of MIMO radar:

[0038] Consider a system with N R receive antennas and N T For a co-located MIMO radar system with 10 transmitting antennas, S is defined as N T ×M transmit waveform matrix, the waveform symbol is s n (m), n = 1, 2, ..., N T , m = 1, 2, ..., M. N of the mth snapshot T The waveform vector of the transmitting antenna is Where M is the number of snapshots, and Therefore, the received signal can be expressed as:

[0039] x m =γ0A(θ0)s m +d m +v m (1)

[0040] where γ0 is the reflection coefficient of the target at angle θ0, is noise, which satisfies Independent and identically distributed, represents the noise variance of the received signal, represents a unit vector. Assume that E[|γ0| 2 ]=β0, where a t (θ) is the sending steering vector, a r (θ) is the receiving steering vector, θ represents the angle, which are respectively expressed as:

[0041]

[0042] in, is the information of Q interfering scatterers, let θ q is the qth interference, γ q is the corresponding reflection coefficient, where E[|γ q | 2 ]=β q Therefore, d m It can be expressed as:

[0043]

[0044] The model goal is to maximize SINR, so the power expressions of the received signal and the interference signal should be obtained first. The received power can be expressed as:

[0045]

[0046] in, represents the waveform covariance matrix, then Indicates the power of the received signal.

[0047] In addition, formula (4) can be further transformed into:

[0048]

[0049] in, represents a unit vector.

[0050] Since the interfering scatterers are uncorrelated, the power of the received interference can be expressed as:

[0051]

[0052] In the above formula, A0 is equivalent to A(θ0), A q Equivalent to A(θ q ).

[0053] Therefore, the signal to interference and noise ratio can be expressed as:

[0054]

[0055] Among them, the auxiliary represents a unit vector, represents the noise variance of the transmitted signal.

[0056] In order to improve the efficiency of the transmitter, a constant modulus constraint can be imposed on the waveform. Therefore, the optimization model for the single-mode waveform design of MIMO radar can be expressed as:

[0057]

[0058] Where s(l) represents the lth element of the column vector s of the transmit waveform matrix S, i.e. l ​​= 1, 2, ..., MN T .

[0059] Generally speaking, existing processing methods mainly include two categories, namely: 1) classical model-based methods, most of which alleviate the problem of performance degradation; 2) data-driven deep neural networks (DNNs) trained by large amounts of data, whose performance is usually unsatisfactory when the training data is limited, reducing their practical applicability.

[0060] In the embodiment of the present invention, the model-based LCCM-Net method is designed under limited data by utilizing the advantages of two gradient descent models, and jointly utilizing the advantages of the model library and data-driven methods in this work.

[0061] Note that the constant modulus constraint satisfies the complex circular manifold, which can be expressed as:

[0062]

[0063] Therefore, the optimization problem can be reformulated as The unconstrained quadratic fraction problem on can be solved by the LCCM-Net method proposed in the embodiment of the present invention, where L = MN T .

[0064] In the embodiment of the present invention, the LCCM-Net method unfolds the gradient descent algorithm as a network layer. For each layer, the step size is converted into a learnable network variable, where all step sizes are adaptively adjusted by the network. The LCCM network mainly includes four steps, namely:

[0065] 1) Project the Euclidean gradient into the CCM tangent space;

[0066] 2) descending in the tangent space;

[0067] 3) Pull the solution of the tangent space back to the CCM space;

[0068] 4) Learn the step size through the network.

[0069] It is worth noting that since deep learning based on the TensorFlow platform cannot directly process complex matrix operations, the embodiment of the present invention converts complex matrix operations into equivalent real matrix operations.

[0070] In the embodiment of the present invention, the four steps of the above LCCM network are specifically:

[0071] (1) In the kth layer of the network (i.e., the k-th layer in the figure), the gradient is projected onto the tangent space to obtain the manifold gradient. This process can be given by the following formula:

[0072]

[0073] in, is the sum of all tangent vectors s k The tangent space of , which can be expressed as:

[0074]

[0075] Among them, O L represents an L-dimensional all-zero vector.

[0076] in, is the projection operator, which can be expressed as a represents the input of the projection operator, that is, the gradient

[0077] in, is the actual gradient, which can be expressed as:

[0078]

[0079] in

[0080]

[0081] in, and Respectively expressed as:

[0082]

[0083] (2) The descent process in the tangent space can be expressed as:

[0084]

[0085] in, α k is the step length.

[0086] (3) Backtrack to CCM to obtain the next solution, which can be expressed as:

[0087]

[0088] in,

[0089] (4) In non-convex problems, the current fixed step size α k Or the known Armijo line search method of updating the step size is sensitive to initialization, resulting in increased calculation time. To address this problem, an embodiment of the present invention proposes an LCCM-Net that adaptively adjusts the step size, in which the gradient descent algorithm is unfolded, the step size is converted into a learnable parameter that depends on each layer, and the learnable parameters of all layers are quickly adjusted through an optimization process to reduce the calculation time. At the same time, LCCM-Net provides adaptability to customize the step size, and has the potential to obtain a global optimal solution.

[0090] LCCM-Net can calculate the loss function by forward propagation when the initial trainable network variables are provided. Subsequently, the optimizer can train the network variables by minimizing the loss function by back propagation. Figure 1 As shown, LCCM-Net includes the following modules:

[0091] (1) Trainable network parameter module: This module consists of the step size of each layer α = [α1, ..., α k , ..., α k ] T Variable composition.

[0092] (2) Forward propagation module: Calculate the continuous solution through the constructed descending layer During calculation, the actual gradient is obtained according to formula (12): Based on the gradient projection function Proj(·) represented by formula (10), the manifold gradient is obtained Then, based on the tangent space descent function Des(·) represented by formula (15), we can obtain Finally, based on the backtracking function Retr(·) represented by formula (16), we get It should be noted that Figure 1 In order to simplify the expression, the relevant calculation results do not carry the superscript “r”, and K represents the number of network layers of LCCM-Net.

[0093] The loss function is calculated through the forward propagation module and can be expressed as:

[0094]

[0095] in,

[0096] (3) Back propagation module: After obtaining the loss, reverse calculation is performed to optimize the loss of all trainable variables. In this embodiment, the Adam optimizer is used. Once the training process stops, the final result of the optimized variable can be easily obtained from the network parameters and their respective transformations. Among them, the stopping condition of the training can be set to the convergence of the loss value or the maximum number of training times. For example, the square of the Frobenius norm of the difference between the loss values ​​obtained from the last two trainings is less than or equal to the preset convergence threshold.

[0097] In actual systems, due to hardware limitations, low-resolution discrete phase waveform sequences are often used. Traditional methods usually use hard quantization functions with uniformly distributed interval thresholds to quantize continuous phases, which may lead to unexpected performance losses. At the same time, the non-differentiability of the hard quantization function hinders the back propagation process of the gradient. In order to solve these problems, the embodiment of the present invention designs a soft quantization function with a non-uniformly distributed interval threshold, learns from the optimal loss function by adaptively learning the decision threshold, and then jointly trains the step size and decision threshold, that is, the embodiment of the present invention also proposes a soft quantization complex circular manifold network combining LCCM-Net and soft quantization function, referred to as QLCCM-Net.

[0098] In order to facilitate the back propagation process of the gradient, in the embodiment of the present invention, a quantization layer is designed using a soft quantization method, that is, a non-differentiable function is approximated as a soft differentiable function. Specifically, a soft differentiable method is designed to convert the sum of the shifted hyperbolic tangents into a soft differentiable quantity. This function can be used to control the amplitude, smoothness, and non-uniform quantization area, and can be expressed as:

[0099]

[0100] Among them, tanh() is the hyperbolic tangent function, the amplitude set {a i} is associated with the phase following the quantization. It should be noted that the phase is uniformly quantized according to the number of quantization bits B, so a i =a=π / 2 b , b represents the quantization bit identifier. The set {c i} represents the smoothing coefficient that determines the smoothness of the soft quantization function. i →∞, the function Q(φ) can well approximate the non-differentiable function. In addition, assuming that all c i The values ​​are all constants and can be represented by c, which can be considered as a manually adjustable parameter rather than a learnable parameter. i} represents the quantization region threshold of the quantization function. During the training phase, only {ρ i} as learnable parameters.

[0101] In the embodiment of the present invention, Figure 2 As shown, QLCCM-Net consists of three modules: trainable network parameters, forward propagation, and back propagation, which are set as follows:

[0102] 1) Trainable network parameter module: This module consists of the step size variables α = [α1, ..., α k , ..., α K ] T and the quantization area threshold ρ.

[0103] 2) Forward propagation module: The network first uses the LCCM-Net gradient descent layer to calculate the continuous solution s K+1 Then use the soft quantization layer to calculate the quantization solution The final calculation about In the soft quantization layer, the continuous phase φ of the waveform is first obtained by the inverse tangent operation arg(·) K+1 Then, the soft quantization function Q(·) is used to calculate the soft quantization phase Finally, the quantized phase waveform is obtained through Euler transformation The real-valued form of the quantization loss function is expressed as:

[0104]

[0105] 3) Backward propagation module: Using the quantized loss, perform backward calculations to optimize the loss of all trainable variables, where the implementation can be implemented using the Adam optimizer.

[0106] Once the training process is over, the final softmax solution can be easily obtained from the network parameters and their respective transformations However, the phase obtained by this method is not strictly discrete, so it is necessary to further quantize the output (the output of the network layer K) into a strictly discrete phase based on the actual phase resolution. To this end, the embodiment of the present invention obtains the quantization region threshold {ρ i}Construct a hard quantization function, which can be expressed as:

[0107]

[0108] Where ρ1≤…≤ρ B-1 , the decision region of the hard function of each step function is given by ρ i In fact, when c i →∞, Q(φ) can be transformed into a hard function Q H (φ).

[0109] Example

[0110] In order to verify the solution proposed in the embodiment of the present invention, the waveform is optimized by using the existing solution and the solution proposed in the embodiment of the present invention.

[0111] The simulation experiment conditions are set as follows: Transmitting antenna N T = 16, number of snapshots M = 16, target direction θ0 = 15° and interference direction θ q =[-40°:-30°]. The signal-to-noise ratio β0 of the target source is 10 dB, while the interference-to-noise ratio β of each interference source is q is 30dB. In addition, the Gaussian white noise variance is σ v 2 = 0dB. For fair comparison, the initialization of all methods is set to orthogonal linear frequency modulation (LFM) waveform.

[0112] Among them, the existing solutions used for comparison in this embodiment are:

[0113] 1)Dinkel-ADMM method: Reference "Q.Li, C.Li, and J.Lin, "Constant modulussecure beamforming for multicast massive MIMO wiretap channels," IEEETransactions on Information Forensics and Security, vol.15, pp.264–275, 2020.";

[0114] 2)Dinkel-PML method: Reference "M.Soltanalian and P.Stoica, "Designing unimodular codes via quadratic optimization," IEEE Transactions on SignalProcessing, vol.62, no.5, pp.1221–1234, 2014.";

[0115] 3)Dinkel-MM acceleration method: Reference "L.Wu, P.Babu, and DPPalomar, "Transmitwaveform / receive filter design for MIMO radar with multiple waveformconstraints," IEEE Transactions on Signal Processing, vol.66, no.6, pp.1526–1540, 2018.";

[0116] 4) PDR method (fixed step size): Reference: K.Alhujaili, V.Monga, and M.Rangaswamy, “Transmit MIMO radar beampattern design via optimization on the complexcircle manifold,” IEEE Transactions on Signal Processing, vol.67, no.13, pp.3561–3575, 2019.

[0117] 5) RCG method: Reference "J.Hu, W.Zhang, H.Zhu, K.Zhong, W.Xiong, Z.Wei, and Y.Li, "Constant modulus waveform design for MIMO radar via manifold optimization," Signal Processing, vol.190, p.108322, 2022.";

[0118] 6) Res-Net method: Reference "J.Hu, Z.Wei, Y.Li, H.Li, and J.Wu, "Designing unimodular waveform(s) for MIMO radar by deep learning method," IEEETransactions on Aerospace and Electronic Systems, vol.57, no.2, pp.1184–1196, 2021."

[0119] Figure 3 The SINR curves of LCCM-Net with different layers are shown. The LCCM-Net proposed in the embodiment of the present invention converges to specific values ​​of different layers, and the SINR gain increases with the increase of network depth. Figure 3 As shown, the SINR gains of three layers and five layers are 9.394dB and 9.395dB respectively.

[0120] Figure 4 The comparison of SINR between the scheme proposed in this embodiment and the above existing schemes is shown. Obviously, the method proposed in this embodiment is better than the Dinkel-MM acceleration method, the Dinkel-PML method, the PDR method (fixed step size), the RCG method and the Res-Net method. More precisely, the SINR gain is 0.426dB, 1.474dB, 0.208dB, 0.653dB and 0.59dB higher than the above methods respectively.

[0121] Figure 5 A comparison of nulled transmit beam patterns is shown. In general, the proposed method has lower side lobes compared to the above-mentioned prior art schemes. A closer inspection shows that the null gain is 0.72dB, 0.8dB, and 1.1dB lower than the prior art methods, respectively.

[0122] Figure 6 The relationship between SINR and CPU time of the solution proposed in this embodiment and the above-mentioned existing solution is shown. Obviously, the method proposed in this embodiment achieves the expected performance faster.

[0123] Figure 7 The relationship between SINR and SNR in the case of discrete phase is shown. The results show that in terms of 1-bit waveform and 2-bit waveform, the method proposed in the embodiment of the present invention achieves a higher signal to interference noise ratio gain than the existing method.

[0124] Figure 8 The comparison between the proposed hard quantization function and the traditional hard quantization function is shown. It can be seen that the soft decision threshold proposed in the embodiment of the present invention is very different from the traditional method in the case of 1 bit and 2 bits.

[0125] Through the above simulation, the performance advantages of the MIMO radar single-mode waveform design method based on the complex circular manifold network disclosed in the embodiment of the present invention are verified: on the one hand, the method proposed in the embodiment of the present invention does not require relaxation and the computational cost is reduced; on the other hand, a low-resolution non-uniform quantizer is used for phase quantization, thereby improving the efficiency of the transmitter.

[0126] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

[0127] The above are only some embodiments of the present invention. For those skilled in the art, several modifications and improvements can be made without departing from the creative concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. Design of single-mode waveform for MIMO radar based on complex circular manifold network, characterized by ,, including the following steps: Objective function construction steps: Based on N R receive antennas and N T The SINR objective function is constructed based on the signal-to-interference-to-noise ratio (SINR) of a co-located MIMO radar with N transmitting antennas. The constraint of the objective function is that each vector element s(l) of the transmitted signal vector s satisfies |s(l)|=1, where the transmitted signal vector s is N T ×M transmit waveform matrix S column vectorization; transmit signal vector s vector element index l = 1, 2, ..., MN T , M is the number of snapshots; construct a complex circular manifold based on s and its constraints Transform the constructed objective function into a complex circular manifold Unconstrained quadratic fraction problem on , which can be solved by gradient descent algorithm; Steps to construct a complex circular manifold network: The complex circular manifold network is used to learn the step size of the gradient descent algorithm. The complex circular manifold network includes several network layers. In each layer, the step size is converted into a learnable network parameter. The calculation processing performed by each layer includes: projecting the Euclidean gradient onto the complex circular flow tangent space to obtain the manifold gradient; descending on the tangent space based on the manifold gradient to obtain a descent result corresponding to the descent step size of the current layer; tracing the descent result of the tangent space back to the complex circular flow tangent space to obtain the solution result of the current layer about the transmitted signal vector s; and the complex circular manifold network calculates the loss function based on the objective function during training; Steps to obtain MIMO radar single-mode waveform design results: Obtain MIMO radar parameters and signal source parameters, initialize the transmission signal vector s and the network parameters of the complex circular manifold network, and set the training end conditions; Based on the corresponding value of the transmitted signal vector s, the input data of the complex circular manifold network is obtained for forward propagation. The initial value of the input data of the complex circular manifold network is the initialized transmitted signal vector s; the input data of the next training forward propagation is the output result of the previous forward propagation; The corresponding loss function value is calculated based on the forward propagation result of each time; then, reverse calculation is performed based on the current loss function value to optimize the step size corresponding to each network layer of the complex circular manifold network; when the training end conditions are met, the MIMO radar single-mode waveform design result is obtained based on the most recent output result of the complex circular manifold network.

2. The MIMO radar single-mode waveform design based on complex circular manifold network as claimed in claim 1, characterized in that: The expression of the objective function is: subject to|s(l)|=1 Among them, the first auxiliary The second auxiliary amount β0 represents the signal-to-noise ratio of the target source, β q represents the signal-to-noise ratio of the qth interference source, Q represents the number of interference sources, represents a unit vector, represents the Gaussian white noise variance, the guidance vector of the target source Steering vector of interference source θ0、θ q They represent the target direction of the target source and the interference direction of the interference source, respectively. r (), a t () represent the sending steering vector and receiving steering vector of the corresponding angle respectively; (·) * 、(·) H They represent conjugate and conjugate transpose operations respectively.

3. The MIMO radar single-mode waveform design based on a complex circular manifold network as claimed in claim 1 or 2, characterized in that: MIMO radar parameters include: receiving antenna N R 、N T transmitting antennas, number of snapshots M, and array steering vector; the signal source parameters include: the target direction of the target source, the signal-to-noise ratio of the target source, the number of interference sources, the interference direction of each interference source, the interference-to-noise ratio, and the variance of Gaussian white noise.

4. The MIMO radar single-mode waveform design based on complex circular manifold network as claimed in claim 2, characterized in that: The calculations performed at each layer are as follows: (1) The specific calculation of manifold gradient is: The subscript k is used to identify the network layer of the complex circular manifold network, and s k Represents the input data of network layer k, that is, the tangent vector s k , Indicates that all tangent vectors s are included k The cutting space, Represents the projection operator, whose input is the tangent vector s k Gradient denote the real and imaginary parts of the elements of the complex vector, respectively, and ⊙ denotes the Hadamard product; Indicates the actual gradient, the intermediate amount and Respectively expressed as: (2) Based on the descent of the manifold gradient in the tangent space, the descent result corresponding to the descent step length of the current layer is obtained as follows: in, Indicates the actual decline result, the decline amount α k is the step size of network layer k; (3) Trace the descending result of the tangent space back to the complex circular flow tangent space: in, Represents the solution result of the transmitted signal vector s output by the network layer k.

5. The MIMO radar single-mode waveform design based on complex circular manifold network as claimed in claim 3, characterized in that: The loss function of the complex circular manifold network during training is: in, 6. The MIMO radar single-mode waveform design based on a complex circular manifold network as claimed in claim 1 or 2, characterized in that: The complex circular manifold network also includes a soft quantization layer, the input of which is the output of the last network layer for obtaining the solution result of the transmitted signal vector s, the continuous phase of the input of the soft quantization layer is obtained by inverse tangent operation, the soft quantization phase is obtained based on the soft quantization function, and finally the waveform of the quantized phase is obtained by Euler transformation; During training, the quantization loss function is calculated based on the objective function and the waveform of the quantization phase output by the soft quantization layer.

7. The MIMO radar single-mode waveform design based on complex circular manifold network as claimed in claim 6, characterized in that: The specific soft quantization function is: Among them, φ represents the input of the soft quantization function, tanh() represents the hyperbolic tangent function, B represents the number of quantization bits, and a i Amplitude means, c i represents the smoothing coefficient that determines the smoothness of the soft quantization function, ρ i Represents the quantization region threshold of the quantization function, where i is the bit number index.

8. The MIMO radar single-mode waveform design based on complex circular manifold network as claimed in claim 6, characterized in that: The waveform of the quantized phase output by the soft quantization layer is quantized into a discrete phase by a hard quantization function, wherein the expression of the hard quantization function is: Where ρ1≤…≤ρ B-1 , the decision region of the hard function of each step function is given by ρ i In fact, when c i →∞, Q(φ) can be transformed into a hard function Q H (φ).

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