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

Abstract

The application discloses a kind of MIMO radar single mode waveform design based on complex circle flow network, belong to waveform design field.The application provides a kind of complex circle flow network based on model, utilize the advantage of two gradient descent models to learn in limited data.The waveform design problem is described as unconstrained quadratic fractional order problem on complex circle flow, in order to solve the problem, gradient descent algorithm is expanded as network layer on complex circle flow, and adaptive learning step length is learned;In addition, for the discrete phase used in actual system, the application designs a soft quantization LCCM network, wherein a low-resolution non-uniform quantizer is designed to quantize the phase, and the quantizer relies on a uniquely designed soft step function, which combines learnable parameters to allow adaptive fine-tuning of the decision region.The application reduces the computational cost without relaxation on the one hand, and improves the efficiency of the transmitter by using a low-resolution non-uniform quantizer for phase quantization on the other hand.

Description

Technical Field

[0001] This invention belongs to the field of radar technology, and specifically relates to a method for designing single-mode waveforms for MIMO radar based on complex circular manifold networks. Background Technology

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

[0003] For model-based methods, problem relaxation is often 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 using the Alternating Directional Multiplier Method (ADMM) to make the problem easier to handle. 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 a single-mode waveform design method based on quadratic optimization and proposes a well-known power-like method (PML) to solve this problem. As further work, the paper "G. Cui, Y. Fu, X. Yu, and J. Li, 'Local ambiguity function shaping via unimodular sequence design,' IEEE Signal Processing Letters, vol. 24, no. 7, pp. 977–981, 2017" proposes a single-mode sequence for shaping the local ambiguity function through four optimizations and an accelerated PML method to solve the problem. 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 radar communication systems,' IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 6, pp. 1378–1392, 2021" proposes a gradient projection (GP)-based method, which first relaxes the constant modulus constraint and then projects the relaxed solution onto the complex circular manifold (CCM).To reduce computational cost, the paper "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” proposes a Riemann conjugate gradient method that updates the conjugate parameters and step size via line search. However, the performance of the line search method may be limited because it is not always useful for simultaneously updating the conjugate parameters and step size.

[0004] Besides the optimization methods mentioned above, evolutionary algorithms have also been used for waveform design in various works. For example, genetic algorithms (GA) have been used to design phase-coded waveforms, orthogonal frequency division multiplexing (OFDM) radar pulses, and discrete frequency and linear frequency modulation (LFM) coded waveforms in the spatial domain. However, the selection of parameters such as crossover rate and mutation rate has a significant impact on the quality of the solution. 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 residual networks. This method achieves this by learning a large number of weight coefficients of the network, which usually requires a lot of computational resources. Generally speaking, the performance of these methods may be unsatisfactory in the common case of limited training data. In recent years, radar signal processing technology based on deep unfolded 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 the 2022 IEEE 12th Sensor Array and Multichannel Signal Processing Workshop (SAM), pp. 291–295, 2022, introduces a custom model-based deep architecture based on the Neumann series inversion lemma to find mismatch filters (MMFs), reducing computational complexity compared to matrix inversion. However, this method may not be suitable for single-mode waveform design. To address this issue, the paper "S. Khobahi, A. Bose, and M. Soltanalian, 'Deep radarwaveform design for efficient automotive radar sensing,'" in the 2020 IEEE 11th Sensor Array and Multichannel Signal Processing Workshop (SAM), pp. 1–5, 2020, proposes a model-based deep architecture inspired by power-law (PML) algorithms.It can learn network architecture parameters through a random walk-based training strategy. However, random walk-based models rely on probability transitions and may require a large number of samples to obtain better results.

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

[0006] The purpose of this invention is to provide a single-mode waveform design method for MIMO radar based on complex circular manifold networks, so as to improve the signal-to-interference-plus-noise ratio (SINR) and the efficiency of the transmitter.

[0007] The technical solution adopted in this invention is as follows:

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

[0009] Steps for constructing the objective function:

[0010] Based on having N R One receiving antenna and N T The objective function is constructed based on the signal-to-interference-plus-noise ratio (SINR) of a co-located MIMO radar with multiple transmit antennas. The constraint term of this objective function is that each element s(l) of the transmitted signal vector s satisfies |s(l)|=1, where the transmitted signal vector s is N. T The column vectorization of the ×M transmitted waveform matrix S; the vector element indices l = 1, 2, ..., MN of the transmitted signal vector s. T M represents the number of snapshots; a complex circular manifold is constructed based on s and its constraints. The constructed objective function is transformed into a complex circular manifold. The unconstrained quadratic fraction problem on the x-axis can be solved using the gradient descent algorithm.

[0011] Steps for constructing 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 consists of several network layers (e.g., K represents the number of network layers). In each layer, the step size is converted into a learnable network parameter. The computational processing performed in each layer includes: projecting the Euclidean gradient onto the tangent space of the complex circular manifold to obtain the manifold gradient; descent in the tangent space based on the manifold gradient to obtain the descent result corresponding to the current layer's descent step size; backtracking the descent result in the tangent space to the complex circular manifold tangent space to obtain the solution result of the current layer with respect to the emitted signal vector s; and calculating the loss function based on the objective function during training.

[0013] Steps for obtaining MIMO radar single-mode waveform design results:

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

[0015] The input data of the complex circular manifold network is obtained based on the corresponding value of the transmitted signal vector s 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. The corresponding loss function value is calculated based on each forward propagation result. Then, backpropagation 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 termination 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 quantization complex circular manifold network). The input of this 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 region. The soft quantization layer obtains the continuous phase of the solution result of the input transmitted signal vector s through inverse tangent operation, then obtains the soft quantization phase based on the soft quantization function, and finally obtains the waveform of the quantized phase through Euler transformation. 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 for the objective function is:

[0018]

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

[0020] Among them, the first auxiliary quantity Second auxiliary quantity β0 represents the signal-to-noise ratio of the target source, β q This represents the signal-to-noise ratio of the q-th interference source, where Q represents the number of interference sources. Represents a unit vector. The variance of Gaussian white noise is represented by the steering vector of the target source. Steering vector of the interference source θ0, θ q These represent the target direction of the target source and the interference direction of the interference source, respectively. r (), a t () represent the transmit steering vector and receive steering vector at the corresponding angles, respectively; (·) * 、(·) HThese represent the conjugate and conjugate transpose operations, respectively.

[0021] Furthermore, MIMO radar parameters include: receiving antenna N R N T The parameters include: transmit antenna, number of snapshots M, array steering vector, etc.; source parameters include: target direction of the target source, signal-to-noise ratio of the target source, number of interference sources, interference direction of each interference source, interference-to-noise ratio, and Gaussian white noise variance.

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

[0023] This invention designs a model-based complex circular manifold network (LCCM-Net) for single-mode beam design in MIMO radar. The designed LCCM-Net leverages the advantages of two gradient descent models for learning on limited data. Specifically, this invention formulates the waveform design problem as an unconstrained quadratic fractional problem (UQFP) on a complex circular manifold (CCM), then expands the gradient descent algorithm as a network layer on the CCM, and adaptively learns the step size. Furthermore, for the discrete phase used in practical systems, this invention also designs a soft-quantized LCCM network (QLCCM network). This network employs a low-resolution, non-uniform quantizer to quantize the phase. This quantizer relies on a uniquely designed soft step function that incorporates learnable parameters, allowing it to adaptively fine-tune the decision region, thereby further improving transmitter performance based on the obtained waveform. Attached Figure Description

[0024] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

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

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

[0027] Figure 3 These are the SINR curves of LCCM-Net with different numbers of layers as a function of the number of iterations in the embodiments.

[0028] Figure 4 This is the curve showing how SINR changes with the number of iterations in the embodiment;

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

[0030] Figure 6 This is the curve showing how SINR changes with computation time in the embodiment;

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

[0032] Figure 8 The curves are those of the hard quantization function proposed in the embodiments and the traditional hard quantization function. Detailed Implementation

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

[0034] This invention provides a single-mode waveform design method for MIMO radar based on a complex circular manifold network (LCCM-Net). It utilizes the advantages of two gradient descent models to learn from limited data. Specifically, the waveform design optimization problem is reformulated as an unconstrained quadratic fractional problem (UQFP) on the complex circular manifold (CCM). To solve the UQFP problem, the gradient descent algorithm is used as a network layer on the CCM, and all step sizes are adaptively learned. In implementation, this invention proposes an LCCM-Net, the structure of which is as follows: Figure 1 As shown, this invention aims to solve the optimization problem on CCM without relaxation, and adaptively trains the network with a finite step size. In addition to continuous phase design, this embodiment also considers discrete phase design, proposing a soft-quantized LCCM-Net (QLCCM-Net), which designs a non-uniformly spaced low-resolution quantizer to quantize the phase. Its network structure is shown below. Figure 2 As shown.

[0035] For ease of description, let's define it as follows:

[0036] and Let represent the real and imaginary parts of a complex vector, respectively; the transpose, conjugate, and conjugate transpose operations are respectively represented by (·). T ,(·) * ,(·) H||·|| represents the vector. Norm, ⊙, represents element-wise multiplication. Indicates the deviation of Hadamard elements; Represents the expected value of a random variable. represents the complex field, and vec(·) denotes the column vectorization of a matrix.

[0037] Constructing an optimization model for single-mode waveform design in MIMO radar:

[0038] Consider a type with N R One receiving antenna and N T In a co-located MIMO radar system with multiple transmit antennas, S represents N. T The transmitted waveform matrix is ​​×M, and the waveform symbol is s. n (m), n = 1, 2, ..., N T m = 1, 2, ..., M. N for the m-th snapshot. T The waveform vector of each 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 located at angle θ0. It is noise, and it satisfies Independent and identically distributed, This represents the noise variance of the received signal. Let E[|γ0|] represent a unit vector. 2 ]=β0, Where a t (θ) is the transmission steering vector, a r (θ) is the receiving steering vector, where θ represents the angle, and they are expressed as follows:

[0041]

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

[0043]

[0044] The model aims to maximize SINR; therefore, the power expressions for the received signal and the interfering signal should be derived first. The received power can be expressed as:

[0045]

[0046] in, Let represent the waveform covariance matrix, then This indicates the power of the received signal.

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

[0048]

[0049] in, This 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-plus-noise ratio can be expressed as:

[0054]

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

[0056] To improve transmitter efficiency, constant mode constraints can be applied to the waveform. Therefore, the optimization model for single-mode waveform design in MIMO radar can be expressed as:

[0057]

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

[0059] Generally speaking, existing processing methods mainly fall into two categories: 1) classic model-based methods, which have mostly alleviated the performance degradation problem; 2) data-driven deep neural networks (DNNs) are trained with a large amount of data, but their performance is usually unsatisfactory when the training data is limited, which reduces their practical applicability.

[0060] In this embodiment of the invention, the model-based LCCM-Net method leverages the advantages of two gradient descent models, designed with limited data, and combines the strengths of a model library and a data-driven approach 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 can be solved using the LCCM-Net method proposed in this invention, where L = MN. T .

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

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

[0066] 2) Descending in the tangent space;

[0067] 3) Unwind the tangent space back into the CCM space;

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

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

[0070] In this embodiment of the invention, the four steps of the above-mentioned LCCM network are specifically as follows:

[0071] (1) In the k-th 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 equation:

[0072]

[0073] in, It includes all tangent vectors s k The tangent space of can be represented as:

[0074]

[0075] Among them, O L This represents an L-dimensional vector consisting entirely of zeros.

[0076] in, It is a projection operator, which can be represented as 'a' represents the input to the projection operator, i.e., the gradient.

[0077] in, The actual gradient can be expressed as:

[0078]

[0079] in

[0080]

[0081] in, and They are represented as follows:

[0082]

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

[0084]

[0085] in, α k The step size.

[0086] (3) Backtracking to the CCM to obtain the next solution can be represented as:

[0087]

[0088] in,

[0089] (4) In non-convex problems, the step size α is currently fixed. k The known Armijo line search method for updating the step size is sensitive to initialization, leading to increased computation time. To address this issue, this invention proposes an adaptive step-size adjustment method, LCCM-Net. In this method, the gradient descent algorithm is expanded, the step size is converted into learnable parameters dependent on each layer, and the learnable parameters of all layers are quickly adjusted through an optimization process to reduce computation time. Simultaneously, LCCM-Net provides adaptability to customize the step size, potentially leading to a globally optimal solution.

[0090] Given initial trainable network variables, LCCM-Net can compute a loss function through forward propagation. Subsequently, the optimizer trains the network variables by minimizing the loss function through backpropagation. Figure 1 As shown, LCCM-Net includes the following modules:

[0091] (1) Trainable network parameter module: This module consists of the step size α = [α1, ..., α2] of each layer. k , ..., α k ] T Composition of variables.

[0092] (2) Forward propagation module: calculates continuous solutions through the constructed descent layer. In the calculation, the actual gradient is obtained according to formula (12). The manifold gradient is obtained based on the gradient projection function Proj(·) characterized by formula (10). Then, based on the tangent space descent function Des(·) characterized by formula (15), we obtain... Finally, based on the backtracking function Retr(·) represented by formula (16), we obtain It should be noted that, Figure 1 To simplify the expression, the relevant calculation results do not include 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) Backpropagation Module: After obtaining the loss, backpropagation 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 variables can be easily obtained from the network parameters and their respective transformations. The stopping condition for training can be set to the convergence of the loss value or the reaching of the maximum number of training iterations. For example, the square of the Frobenius norm of the difference between the two most recent training loss values ​​is less than or equal to a preset convergence threshold.

[0097] In practical systems, due to hardware limitations, low-resolution discrete phase waveform sequences are often used. Traditional methods typically employ hard quantization functions with uniformly distributed interval thresholds to quantize continuous phases, which may lead to unexpected performance losses. Furthermore, the non-differentiability of hard quantization functions hinders the backpropagation of gradients. To address these issues, this invention proposes a soft quantization function with non-uniformly distributed interval thresholds. By adaptively learning the decision threshold and training towards the optimal loss function, and then jointly training the step size and decision threshold, this invention also proposes a soft-quantized complex circular manifold network combining LCCM-Net and the soft quantization function, abbreviated as QLCCM-Net.

[0098] To facilitate the backpropagation of gradients, this embodiment of the invention employs a soft quantization method to design a quantization layer, approximating a non-differentiable function as a soft-differentiable function. Specifically, a soft-differentiable method is designed to convert the sum of shifted hyperbolic tangents into a soft-differentiable quantity. This function can be used to control amplitude, smoothness, and non-uniform quantization regions, and can be expressed as:

[0099]

[0100] Where tanh() is the hyperbolic tangent function, and the amplitude set {a i This is related to the phase of the quantization. It's important to note that the phase is uniformly quantized based on the number of quantization bits B; therefore, 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. When c i As we approach infinity, the function Q(φ) can approximate non-differentiable functions very well. Furthermore, it is assumed that all c... i The values ​​are all constants, denoted by c. This c can be considered a manually adjustable parameter, rather than a learnable parameter. The set {ρ} i} represents the quantization region threshold of the quantization function. During the training phase, only {ρ} can be used. i} as a learnable parameter.

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

[0102] 1) Trainable network parameter module: This module consists of the step size variable α = [α1, ..., α2] for each layer. k , ..., α K ] T It consists of the quantization region 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, the computational cost is reduced using a soft quantization layer. Finally, calculate about The loss function. In the soft quantization layer, the continuous phase φ of the waveform is first obtained through the inverse tangent operation arg(·). K+1 Then, the soft-quantized phase is calculated using the soft-quantization function Q(·). Finally, the waveform of the quantized phase is obtained through Euler transformation. The real-valued form of the quantization loss function is:

[0104]

[0105] 3) Backpropagation module: Utilizes quantized loss to perform backward computation to optimize the loss of all trainable variables, which can be implemented using the Adam optimizer.

[0106] Once the training process is complete, the final soft 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 the output (output of network layer K) needs to be further quantized into a strictly discrete phase based on the actual phase resolution. To this end, embodiments of the present invention use the obtained 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 for each step function is determined by ρ. i Confirmed. In fact, when c i As we approach infinity, Q(φ) can be transformed into a hard function Q. H (φ).

[0109] Example

[0110] To verify the solution proposed in the embodiments of the present invention, the waveform was optimized using existing solutions and the solution proposed in the embodiments of the present invention.

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

[0112] Among 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 beam pattern design via optimization on the complex circle 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-Nets with different numbers of layers are shown. The LCCM-Net proposed in this embodiment converges to specific values ​​for different layers, and the SINR gain increases with increasing network depth. Figure 3 As shown, the SINR gains for the three-layer and five-layer systems are 9.394dB and 9.395dB, respectively.

[0120] Figure 4 The SINR comparison between the proposed solution in this embodiment and the aforementioned existing solutions is presented. Clearly, the method proposed in this embodiment outperforms 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 0.426 dB, 1.474 dB, 0.208 dB, 0.653 dB, and 0.59 dB higher than the aforementioned methods, respectively.

[0121] Figure 5 A comparison of zero-point transmit beam patterns is presented. Generally, the method proposed in this embodiment has lower sidelobes compared to the existing methods described above. More detailed examination shows that the zero-point gain is 0.72 dB, 0.8 dB, and 1.1 dB lower than that of existing methods, respectively.

[0122] Figure 6 The relationship between SINR and CPU time is shown between the proposed solution in this embodiment and the existing solutions described above. Clearly, the method proposed in this embodiment achieves the expected performance more quickly.

[0123] Figure 7 The relationship between SINR and SNR is shown in the case of discrete phase. The results demonstrate that, for both 1-bit and 2-bit waveforms, the method proposed in this embodiment achieves a higher signal-to-interference-plus-noise ratio (SINR) gain than existing methods.

[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 this embodiment of the invention differs significantly from the traditional method in both 1-bit and 2-bit cases.

[0125] The above simulations verify the performance advantages of the MIMO radar single-mode waveform design method based on complex circular manifold networks disclosed in the embodiments of the present invention: on the one hand, the method proposed in the embodiments of the present invention does not require relaxation and reduces computational costs; on the other hand, the use of a low-resolution non-uniform quantizer for phase quantization improves 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, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

[0127] The above descriptions are merely some embodiments of the present invention. Those skilled in the art can make various modifications and improvements without departing from the inventive concept of the present invention, and these all fall within the scope of protection of the present invention.

Claims

1. A MIMO radar single-mode waveform design based on complex-torus flow network, characterized in that, The method comprises the following steps: The objective function construction step: Based on having N R One receiving antenna and N T The objective function is constructed based on the signal-to-interference-plus-noise ratio (SINR) of a co-located MIMO radar with multiple transmit antennas. The constraint term of this objective function is that each element s(l) of the transmitted signal vector s satisfies |s(l)|=1, where the transmitted signal vector s is N. T The column vectorization of the ×M transmitted waveform matrix S; the vector element indices l = 1, 2, ..., MN of the transmitted signal vector s. T M represents the number of snapshots; a complex circular manifold is constructed based on s and its constraints. The constructed objective function is transformed into a complex circular manifold. The unconstrained quadratic fraction problem on the x-axis can be solved using the gradient descent algorithm. The complex circle manifold network construction step: The complex circle manifold network is used to learn the step size of the gradient descent algorithm, and the complex circle manifold network comprises a plurality of network layers, in each layer of which, the step size is converted into a learnable network parameter, and the calculation and processing performed by each layer comprises: projecting the Euclidean gradient onto the tangent space of the complex circle flow to obtain a manifold gradient; descending in the tangent space based on the manifold gradient to obtain a descending result of the descending step size corresponding to the current layer; backtracking the descending result of the tangent space to the tangent space of the complex circle flow to obtain a solution result of the current layer with respect to the transmit signal vector s; and the complex circle manifold network performs loss function calculation based on the objective function during training; The MIMO radar single-mode waveform design result acquisition step: MIMO radar parameters and signal source parameters are acquired, the transmit signal vector s and the network parameters of the complex circle manifold network are initialized, and a training end condition is set; Input data of the complex circle manifold network is obtained based on the corresponding value of the transmit signal vector s for forward propagation, and the initial value of the input data of the complex circle manifold network is the initialized transmit signal vector s; the input data for the next forward propagation during training is the output result of the last forward propagation; And the loss function value corresponding to each forward propagation result is calculated; the step size corresponding to each network layer of the complex circle manifold network is optimized based on the current loss function value through back calculation; when the training end condition is met, the MIMO radar single-mode waveform design result is obtained based on the output result of the complex circle manifold network in the last time. The expression of the objective function is: subject to |s(l)|=1 where the first auxiliary quantity the second auxiliary quantity β0represents the signal-to-noise ratio of the target source, β q represents the signal-to-noise ratio of the qthinterfering source, Q represents the number of interfering sources, represents a unit vector, represents the Gaussian white noise variance, the steering vector of the target source the steering vector of the interfering source θ0, θ q respectively represent the target direction of the target source and the interference direction of the interfering source, a r (), a t () respectively represent the transmitting steering vector and the receiving steering vector corresponding to the angle; (·) * , (·) H respectively represent the conjugate and the conjugate transpose operation.

2. The MIMO radar single-mode waveform design based on complex-circular manifold network of claim 1, wherein, MIMO radar parameters include: receiving antenna N R N T One transmitting antenna, M snapshots, array steering vector; source parameters include: target direction of the target source, signal-to-noise ratio of the target source, number of interfering sources, interference direction of each interfering source, interference-to-noise ratio, and Gaussian white noise variance.

3. The MIMO radar single-mode waveform design based on complex-circular manifold network of claim 1, wherein, The calculation performed by each layer is specifically: (1) The specific calculation of the manifold gradient is: where the subscript k is used to identify the network layer of the complex toric manifold network, s k represents the input data of the network layer k, i.e., the tangent vector s k , represents the tangent space including all tangent vectors s k , represents the projection operator, whose input is the gradient of the tangent vector s k , represents the element real part and imaginary part of the complex vector, respectively, and represents the Hadamard product; represents the actual gradient, the intermediate quantities and are represented as follows: (2) The descending result of the descending step size corresponding to the current layer is obtained by descending in the tangent space based on the manifold gradient: wherein, denotes the real descent result, the descent amount α k is the step size of the network layer k; (3) The descending result of the tangent space is backtracked to the tangent space of the complex circle flow: wherein represents the solution of the transmitted signal vector s with respect to the network layer k output.

4. The MIMO radar single-mode waveform design based on complex-circular manifold network of claim 3, wherein, The loss function of the complex circle manifold network during training is: wherein 5. The MIMO radar single-mode waveform design based on complex-codimension network of claim 1, wherein, The complex circle manifold network further comprises a soft quantization layer, the input of the soft quantization layer is the output of the network layer used to obtain the solution result with respect to the transmit signal vector s in the last layer, the continuous phase of the input of the soft quantization layer is obtained through 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 through Euler transformation; And during training, the quantization loss function calculation is performed based on the objective function and the waveform of the quantized phase output by the soft quantization layer.

6. The MIMO radar single-mode waveform design based on complex-circular manifold network of claim 5, wherein, The soft quantization function is specifically: wherein φ represents an input of a soft quantization function, tanh() represents a hyperbolic tangent function, B represents a number of quantization bits, a i amplitude representation, c i represents a smoothing coefficient that determines a smoothness of the soft quantization function, p i represents a quantization region threshold of the quantization function, i is a bit number index.

7. The MIMO radar single-mode waveform design based on complex-circular manifold network of claim 5, wherein, The waveform of the quantized phase output by the soft quantization layer is quantized into a discrete phase through a hard quantization function, wherein the expression of the hard quantization function is: wherein p1<..., p i ..., p B-1 Each decision region of the hard function of the step function is determined by p i In fact, when c i → ∞, Q(φ) can be transformed into a hard function Q H (φ).

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