Low complexity target dod-doa and doppler frequency joint estimation algorithm based on space-time nested sampling

Abstract

The application provides a low-complexity target DOD-DOA and Doppler frequency joint estimation algorithm based on space-time nested sampling, and the joint estimation algorithm comprises the following steps: step 1: configuring a bistatic MIMO radar system into a space-time nested sampling model, and sampling a received signal {x q (l)} by using the space-time nested sampling model; q Step 2: performing multi-stage delay sampling on the received signal {x q (l)} to obtain {y(l)}; step 3: performing matched filtering on the received signal {y(l)} to obtain {y(t)}; step 4: solving a target echo signal covariance matrix R; step 5: vectorizing and de-redundantizing the target signal covariance matrix, and obtaining a new observation signal according to the obtained observation signal; step 6: performing three-dimensional Toeplitz matrix iterative reconstruction on the new observation signal to obtain an equivalent covariance matrix R xx in a virtual domain; and step 7: solving DOD-DOA and Doppler frequency parameters of the target by using an improved multi-dimensional ESPRIT algorithm.

Description

Technical Field

[0001] This invention belongs to the field of radar signal processing technology, specifically relating to a low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling. Background Technology

[0002] Bistatic multiple-input multiple-output (MIMO) radars possess potential advantages in counter-reconnaissance, anti-jamming, anti-stealth, and counter-anti-radiation missiles, and have received widespread attention and research in recent years. Bistatic MIMO radars can fully utilize spatiotemporal information to jointly estimate parameters such as the direction of departure (DOD), direction of arrival (DOA), and Doppler frequency of a target. After acquiring these parameters, MIMO radars can perform cross-location and tracking of the target. Most existing algorithms apply traditional super-resolution algorithms to jointly estimate the target's DOD, DOA, and Doppler frequency based on conventional array radar structures, such as the two-dimensional MUSIC algorithm, the multi-dimensional ESPRIT algorithm, and some of their improvements. However, when performing multi-dimensional parameter joint estimation for multiple targets, these algorithms either have extremely high complexity, making them difficult to implement, or their accuracy is unsatisfactory. To reduce the algorithm complexity, reference [1] proposed a dimension reduction MUSIC algorithm (RD MUSIC), which transforms the two-dimensional spectral peak search into two one-dimensional spectral peak searches, thus greatly reducing the algorithm complexity and completing the joint estimation of the target's DOD and DOA. All of the above algorithms can achieve the joint estimation of the target's DOD and DOA, but they do not consider the joint estimation of Doppler frequency. As the research progresses, scholars have successively extended the joint estimation of the transmit and receive angles to the joint estimation of the transmit and receive angles and Doppler frequency. The increase in the number of parameters to be estimated leads to a further increase in the complexity of the algorithm, and it is becoming increasingly difficult to ensure the accuracy of the estimated parameters. References [2] and [3] constructed a spatiotemporal model of the received signal and used the multidimensional ESPRIT algorithm to achieve the joint estimation of the target's transmit and receive angles and Doppler frequency. However, the complexity and accuracy of this algorithm are not high, and the parameter estimation requires an additional pairing algorithm. Reference [4] proposes a joint estimation algorithm for bistatic MIMO radar transmit / receive angle and Doppler frequency based on parallel factor analysis. This algorithm makes full use of all information at the receiver, uses the least squares iterative method to obtain the target to be estimated value, and can achieve automatic parameter pairing. Compared with the multidimensional ESPRIT algorithm, this algorithm has lower complexity and higher parameter estimation accuracy. Reference [5] extends the uniform linear array to a uniform rectangular array and uses tensor decomposition technology to propose a four-dimensional angle and Doppler frequency joint estimation algorithm. This algorithm also does not require parameter pairing and multidimensional search, but the convergence speed of this algorithm is slow and it is easy to converge to a local optimum. It needs to find the optimal solution through multiple sets of initial values, and the algorithm complexity is high.In view of the parallel factor method, which requires the emission coefficient as prior knowledge, reference [6] proposes a quadrlinear decomposition algorithm that does not require this prior knowledge. This algorithm obtains the transmit and receive angles and Doppler frequencies of the target through the quadrlinear alternating least squares (QALS) method without the need for spectral peak search and additional parameter pairing.

[0003] In order to obtain higher parameter estimation accuracy and higher degree of freedom, sparse arrays have been introduced into the joint estimation of target DOD, DOA and Doppler frequency. For example, considering the need to obtain larger array aperture, lower mutual coupling effect and higher array degree of freedom, minimum redundancy array, nested array and coprime array have been proposed one after another. Reference [7] proposes a joint estimation algorithm for target transmit and receive angle and Doppler frequency based on minimum redundancy array MIMO radar. This algorithm uses the virtual array formed by minimum redundancy array and non-uniform delay sampling to realize the second expansion of the time domain and spatial domain aperture degree of freedom. Under the same array element number and delay level conditions, the estimation performance of this algorithm is better than the quadlinear decomposition algorithm and the multidimensional ESPRIT algorithm and can estimate more targets. Reference [8] proposes a joint estimation algorithm for target multi-parameters of coprime array MIMO radar based on tensor decomposition. This algorithm uses the van der Mund structure of extended virtual manifold matrix and virtual snap effect to effectively improve the parameter estimation accuracy. However, this algorithm requires additional noise suppression algorithm and is prone to local optima and multiple iterations, resulting in high algorithm complexity. In order to solve the problem of high algorithm complexity caused by the spatiotemporal "sliding window" operation when realizing the second expansion of spatiotemporal degrees of freedom, Reference [9] proposed a method to directly construct a third-order tensor from the data after the second expansion of spatiotemporal degrees of freedom and then perform least squares iteration. This method avoids the spatiotemporal "sliding window" processing and reduces the algorithm complexity. However, the uncertainty and local optimum trap in the virtual array element redundancy removal and least squares iteration are still major drawbacks of this algorithm. Summary of the Invention

[0004] To address the above problems, this invention comprehensively considers the advantages and disadvantages of the multidimensional ESPRIT algorithm and the parallel factor analysis method, as well as the virtual aperture expansion characteristics of the second-order nested array. This invention aims to improve the accuracy of multi-parameter estimation while reducing the complexity of the multi-parameter joint estimation algorithm, and proposes a low-complexity joint estimation algorithm for target DOD, DOA, and Doppler frequency based on spatiotemporal nested sampling. This algorithm solves the problem of high complexity in virtual redundant array element removal by using the principle of multilinear mapping; then, it uses the method of multidimensional Toeplitz matrix iterative reconstruction to reduce the computational complexity of the spatiotemporal "sliding window" operation in the algorithm of reference [7]. Next, the improved multidimensional ESPRIT algorithm is applied to the target multi-parameter estimation. This method can avoid the problem of additional parameter pairing operation required by the original multidimensional least squares iterative method and solve the problem that the orthogonal least squares iterative algorithm is prone to getting trapped in local optima and the computational load increases due to multiple iterations. Finally, simulation shows that the algorithm proposed in this paper not only has high parameter estimation accuracy but also greatly reduces the algorithm complexity and improves the efficiency of multi-parameter joint estimation.

[0005] The technical solution adopted in this invention is as follows:

[0006] A low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling is proposed, which includes the following steps:

[0007] Step 1: Configure the bistatic MIMO radar system as a spatiotemporal nested sampling model, and use the spatiotemporal nested sampling model to sample the received signal {x}. q (l)};

[0008] Step 2: Use a nested Q-level delay receiver to process the received signal {x} q (l)} undergoes multi-level delayed sampling processing, and after Q-level delayed reception, the received data is represented as a vector form, thus obtaining {y(l)};

[0009] Step 3: Use a matched filter to perform matched filtering on the delayed received signal {y(l)} to obtain the echo signal {y(t)};

[0010] Step 4: Based on the echo signal {y(t)}, establish the covariance matrix R of the target echo signal;

[0011] Step 5: The covariance matrix R of the vectorized target echo signal is used to achieve a second expansion of the spatiotemporal degrees of freedom, which is represented as a data vector z. The data vector z is then redundantly and rearranged using a multilinear mapping method to obtain the observed signal.

[0012] Step 6: For new observation signals By performing triple Toeplitz matrix iterative reconstruction, the equivalent covariance matrix R of the virtual domain is obtained.xx ;

[0013] Step 7: Apply the improved multidimensional ESPRIT algorithm to the equivalent covariance matrix R xx Eigenvalue decomposition is performed to construct expressions for the target's emission angle, reception angle, and Doppler frequency, thereby obtaining estimates of the target's emission angle, reception angle, and Doppler frequency.

[0014] In the above technical solution, the echo signal {x} after the qth level delay q The sampling process of (l)} is as follows:

[0015] Step 101: In the spatiotemporal nested sampling model, both the transmitting array and the receiving array are second-order nested arrays. The number of array elements in the transmitting array is M = M1 + M2, and the number of array elements in the receiving array is N = N1 + N2. The delay unit is configured as Q = Q1 + Q2. Let the element spacing between unit array elements be d0 = λ / 2. Then, the set of physical array element positions of the transmitting and receiving arrays is represented as:

[0016]

[0017] Step 102: Assume there are K far-field narrowband incoherent targets. θ k and f dk (k = 1, 2, ..., K) represent the target's DOD, DOA, and Doppler frequencies, respectively. The transmitting array transmits M different orthogonal pulse signals. Under the l-th (l = 1, 2, ..., L) transmitted pulse, the echo signal model at the receiving end is expressed as:

[0018]

[0019] in, Represents the Khatri-Rao product. α k T represents the scattering coefficient of the k-th target. r Let n(l) be the repetition period of the transmitted pulse, and let n(l) be a function with a mean of 0 and a variance of 1. A Gaussian white noise vector of dimension MN×1, and:

[0020]

[0021]

[0022] A r (θ)=[a r (θ1),a r (θ2),...,a r (θ K (5)

[0023] and a r (θ k The ) represent the steering vectors of the transmitting array and the receiving array, respectively, and are expressed as:

[0024]

[0025]

[0026] Step 103: The received signal x(l) is delayed and sampled using a nested Q-level delay receiver. Assuming that the target's DOD, DOA, and Doppler frequency parameters remain unchanged during the delay time, the echo signal after the q-th level delay is represented as:

[0027]

[0028] in, n q (l)=n(l+qτ), q=1,2,...,Q.

[0029] In the above technical solution, the expression for the received signal {y(l)} is:

[0030]

[0031] in,

[0032]

[0033]

[0034] c d (f dk The Doppler steering vector is the target. and A r (θ) is called the spatial steering vector.

[0035] In the above technical solution, the expression for the echo signal {y(t)} is as follows:

[0036]

[0037] in, An information vector constructed from the scattering coefficients and Doppler parameters. 3D white noise signal vector; This represents the joint steering vector of the received signal.

[0038] In the above technical solution, the covariance matrix of the target echo signal can be expressed as:

[0039] R = E[y(t)y H [(t)]=DRs D H +σ 2 I MNQ (13)

[0040] in, Let be the target covariance matrix. This represents the power level of the signal.

[0041] In the above technical solution, the expression for the data vector z is:

[0042]

[0043] in, This represents the signal power of K targets;

[0044]

[0045] From equation (15), we know that It can be represented as:

[0046]

[0047] In the above technical solution, the specific process of redundancy and rearrangement of the data vector z is as follows:

[0048] According to the structure of equation (16), the vector Given a spatiotemporal structure that satisfies the "sum-difference union" principle, let its set of virtual array elements and temporal virtual sampling points be denoted as . This can be expressed as:

[0049]

[0050] In equation (17), the nested array's "difference union" array It has 2M2(M1+1)-1 virtual array elements. A "difference joint" structure with 2N2(N1+1)-1 virtual array elements and time-domain sampling. There are 2Q2(Q1+1)-1 virtual sampling points. Since the "and joint" structure in equation (17) consists of three different parameters, the set The spatiotemporal joint degrees of freedom can be expressed as:

[0051]

[0052] make The range of the virtual spacetime guiding vector is from arrive

[0053] The unique virtual matrix element can be indexed by its position from the vectorized one-dimensional matrix, and its structure is the same as that of the vectorized matrix formed by the uniform matrix.

[0054] Right now:

[0055]

[0056] In the formula, h represents 3D vector, and Represents the virtual joint guidance vector, with in,

[0057]

[0058]

[0059]

[0060] In the above technical solution, for new observation signals The specific process of iterative reconstruction of the triple Toeplitz matrix is ​​as follows:

[0061] Step 601: First, the observation vector Divided into rows There are matrix blocks, each containing Row element, if [R1] i Indicates the first There are matrix blocks, among which According to matrix block [R1] i Construct the following Toeplitz matrix for the basic elements:

[0062]

[0063] Step 602: Then add the matrix blocks Divided into rows There are n matrix blocks, assuming [R0] j Indicates the first There are matrix blocks, among which With [R0] j Using the basic elements, the matrix block [R1] is... i The elements in the matrix are constructed into the following Toeplitz matrix:

[0064]

[0065] Step 603: Finally, combine the matrix blocks. Divided into rows Row element, let's say [r0] w Represents matrix block [R0] j The Middle row elements, where With [r0] w Using the basic elements, the matrix block [R0] is... j Represented as a Toeplitz matrix as follows:

[0066]

[0067] The Toeplitz process shown in equation (24) is used to process each matrix block [R1]. i Perform this Toeplitz transformation operation on all elements, and then apply the Toeplitz transformation process of equation (25) to each element in the Toeplitz matrix after the above operation [R0]. j This will yield an equivalent multi-sample snapshot signal, i.e., the virtual domain equivalent covariance matrix R. xx From equation (25), we can see that R xx It is A 3D matrix can be represented as:

[0068]

[0069] in,

[0070] In the above technical solution, the process of obtaining the target's DOD-DOA and Doppler frequency parameters includes:

[0071] Step 701: Convert the equivalent covariance matrix R xx Eigenvalue decomposition yields a signal subspace consisting of K eigenvalues. E s =WU -1 E s The former The matrix composed of row elements is E s1 ,back The matrix composed of row elements is E s2 Based on the structure of the virtual spacetime guiding vector W, the following equation holds:

[0072]

[0073] in, Φ(f) represents the generalized inverse matrix of the solution matrix. d ) is the rotation invariant factor of the Doppler domain. Combining equation (24), we know that U and U s All are complex matrices eigenvectors, and satisfying

[0074] U s =UΔJ (28)

[0075] Where Δ represents the scaling factor, J is the column permutation matrix, and the estimated value of the spatiotemporal virtual steering vector W is obtained directly through the following formula:

[0076]

[0077] The estimated value of the extended virtual spatiotemporal steering vector W is obtained.

[0078] Step 702: Obtain the estimated value Then, based on its internal structure, the joint estimate of the target's transmit angle (DOD), receive angle (DOA), and Doppler frequency is obtained through the following three expressions;

[0079]

[0080]

[0081]

[0082] The above three formulas can be used to estimate the value of the same virtual spacetime steering vector W. The estimated values ​​of the target transmit / receive angle and Doppler frequency are obtained. Since these three parameters are all estimated from the same spatiotemporal virtual steering vector, the obtained DOD, DOA and Doppler frequency will be automatically matched.

[0083] The beneficial effects of this invention are:

[0084] This invention proposes a low-complexity joint estimation algorithm for target DOD, DOA, and Doppler frequency using spatiotemporal nested sampling. This algorithm nests the transceiver array and delay unit, achieving a secondary expansion of spatiotemporal degrees of freedom. It then proposes a multilinear mapping mechanism adaptable to various sparse array structures, effectively solving the problems of complex culling and difficult rearrangement of virtual array elements. Simultaneously, addressing the high complexity and implementation difficulties of traditional spatial smoothing algorithms, a triple Toeplitz matrix iterative reconstruction algorithm is proposed to overcome the single-shot effect of the received virtual signal. Compared to the commonly used "spatiotemporal sliding window" algorithm, this method significantly reduces algorithm complexity while maintaining the same aperture degrees of freedom. Simulation results show that the proposed algorithm maintains both high spatiotemporal degree of freedom expansion capability and excellent parameter estimation performance while significantly reducing algorithm complexity. Simulation results on the algorithm's computation speed show that the proposed algorithm is nearly 456 times faster than the "spatiotemporal sliding window" algorithm, making its engineering application possible. Attached Figure Description

[0085] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the 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.

[0086] Figure 1 This is a diagram of a bistatic nested MIMO radar structure.

[0087] Figure 2 The constellation diagrams for estimating the number of targets by two algorithms are shown: (a) the multidimensional ESPRIT algorithm can identify 46 targets; (b) the algorithm proposed in this invention can identify 60 targets.

[0088] Figure 3 The mean square error (RMSE) of parameter estimation varies with signal-to-noise ratio; (a) RMSE of DOD-DOA varies with signal-to-noise ratio; (b) RMSE of Doppler frequency varies with signal-to-noise ratio.

[0089] Figure 4 The resolution performance of four algorithms is compared: (a) the algorithm proposed in this invention; (b) the ULA-PARAFAC algorithm; and (c) the multidimensional ESPRIT algorithm. Detailed Implementation

[0090] 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 clearly and completely described below with reference to the accompanying drawings.

[0091] This embodiment specifically provides a low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling.

[0092] The joint estimation algorithm includes the following steps:

[0093] Step 1: Configure the bistatic MIMO radar system as a spatiotemporal nested sampling model, and use the spatiotemporal nested sampling model to sample the received signal {x}. q (l)};

[0094] Step 2: Use a nested Q-level delay receiver to process the received signal {x} q (l)} undergoes multi-level delayed sampling processing, and after Q-level delayed reception, the received data is represented as a vector form, thus obtaining {y(l)};

[0095] Step 3: Use a matched filter to perform matched filtering on the delayed received signal {y(l)} to obtain the echo signal {y(t)};

[0096] Step 4: Based on the echo signal {y(t)}, establish the covariance matrix R of the target echo signal;

[0097] Step 5: The covariance matrix R of the vectorized target echo signal is used to achieve a second expansion of the spatiotemporal degrees of freedom, which is represented as a data vector z. The data vector z is then redundantly and rearranged using a multilinear mapping method to obtain the observed signal.

[0098] Step 6: For new observation signals By performing triple Toeplitz matrix iterative reconstruction, the equivalent covariance matrix R of the virtual domain is obtained. xx ;

[0099] Step 7: Apply the improved multidimensional ESPRIT algorithm to the equivalent covariance matrix R xx Eigenvalue decomposition is performed to construct expressions for the target's emission angle, reception angle, and Doppler frequency, thereby obtaining estimates of the target's emission angle, reception angle, and Doppler frequency.

[0100] In step 1, the received signal {x} q The sampling process of (l)} is as follows:

[0101] Step 101: As Figure 1 As shown, in the spatiotemporal nested sampling model, both the transmitting array and the receiving array are second-order nested arrays. The number of elements in the transmitting array is M = M1 + M2, and the number of elements in the receiving array is N = N1 + N2. The delay unit is configured as Q = Q1 + Q2. Let the element spacing between units be d0 = λ / 2. Then, the set of physical element positions of the transmitting and receiving arrays can be represented as:

[0102]

[0103] Step 102: Assume there are K far-field narrowband incoherent targets. θ k and f dk (k = 1, 2, ..., K) represent the target's DOD, DOA, and Doppler frequencies, respectively. The transmitting array transmits M different orthogonal pulse signals. Under the l-th (l = 1, 2, ..., L) transmitted pulse, the echo signal model at the receiving end can be expressed as:

[0104]

[0105] in, Represents the Khatri-Rao product. α k T represents the scattering coefficient of the k-th target. r Let n(l) be the repetition period of the transmitted pulse, and let n(l) be a function with a mean of 0 and a variance of 1. A Gaussian white noise vector of dimension MN×1, and:

[0106]

[0107]

[0108]

[0109] and a r (θ k The ) represent the steering vectors of the transmitting array and the receiving array, respectively, which can be expressed as:

[0110]

[0111]

[0112] Step 103: The received signal x(l) is delayed and sampled using a nested Q-level delay receiver. Assuming that the target's DOD, DOA, and Doppler frequency parameters remain unchanged during the delay time, the echo signal after the q-th level delay can be expressed as:

[0113]

[0114] in, n q (l)=n(l+qτ), q=1,2,...,Q.

[0115] In step 2, {x q (l)} After Q-level delay, the received data is received and represented as a vector form, i.e.

[0116]

[0117] in,

[0118]

[0119]

[0120] c d (f dk The Doppler steering vector is the target. and A r (θ) is called the spatial steering vector.

[0121] Obviously c d (f dk ) data structure model and and A rLike (θ), it has a Vandermonde structure, which we call the target's Doppler steering vector (also known as the time-domain steering vector). and A r (θ) is called the spatial steering vector. Therefore, equation (9) can be considered as the space-time signal model of a bistatic MIMO radar. After transmitting L pulses at the transmitter and delaying the receiver by Q levels, the total echo signal after the matched filter can be expressed as:

[0122]

[0123] in, An information vector constructed from the scattering coefficients and Doppler parameters. 3D white noise signal vector; This represents the joint steering vector of the received signal.

[0124] Since the delay series Q and the array position both use nested sampling, the sampling series can also be extended by vectorizing the covariance matrix, thereby improving the parameter estimation accuracy of the Doppler frequency. From equation (12), the covariance matrix of the target echo signal can be expressed as:

[0125]

[0126] in, Let be the target covariance matrix. This represents the power level of the signal.

[0127] In practical signal processing, the mean is often used to approximate the expectation to obtain the signal's covariance matrix, i.e.:

[0128]

[0129] To fully utilize the advantages of virtual array element aperture expansion and Q-order non-uniform delay sampling, it is necessary to vectorize equations (12) and (13) to achieve a second expansion of the spatiotemporal degrees of freedom, namely:

[0130]

[0131] in, This represents the signal power of K targets;

[0132]

[0133] From equation (15), we know that It can be represented as:

[0134]

[0135] From the structure of equation (16), it can be seen that the vector Given a spatiotemporal structure that satisfies the "sum-difference union" principle, let its set of virtual array elements and temporal virtual sampling points be denoted as . This can be expressed as:

[0136]

[0137] In equation (17), the nested array's "difference union" array It has 2M2(M1+1)-1 virtual array elements. A "difference joint" structure with 2N2(N1+1)-1 virtual array elements and time-domain sampling. There are 2Q2(Q1+1)-1 virtual sampling points. Since the "and joint" structure in equation (17) consists of three different parameters, the set The spatiotemporal joint degrees of freedom can be expressed as:

[0138]

[0139] make The range of the virtual spacetime guiding vector is from arrive

[0140] Redundancy removal and rearrangement of multilinear mappings

[0141] The data corresponding to the spatiotemporal virtual steering vector is selected from the data vector in equation (14) and rearranged to form a new observation vector. This process requires data redundancy removal and rearrangement of the vectorized data. Conventional redundancy removal and rearrangement methods usually employ left multiplication of multiple selection matrices. As the number of estimation parameters increases and the virtual aperture expands, the left multiplication operation will greatly increase the complexity of the algorithm. In view of this, the present invention adopts a multilinear mapping method. The core idea of ​​this method is to record the position of the virtual array element of the "and joint" structure through the principle of multilinear mapping. Then, the unique virtual array element can be indexed from the vectorized one-dimensional matrix according to its position, and the structure is the same as that of the vectorized matrix formed by the uniform matrix.

[0142] Right now:

[0143]

[0144] In the formula, h represents 3D vector, and Represents the virtual joint guidance vector, with in,

[0145]

[0146]

[0147]

[0148] Then the observed vector This can be equivalent to a single-shot received signal from a traditional uniform linear array MIMO radar system, where the number of transmit array elements is... The number of elements in the receiving array is The equivalent number of time-domain samples is

[0149] Combining equation (13), it can be seen that by using a spatiotemporal nested sampling structure combined with vectorized received signal covariance matrix operations, the spatiotemporal degrees of freedom are realized from MNQ=(M1+M2)(N1+N2)(Q1+Q2) to An extension of.

[0150] Joint estimation of transmit / receive angle and Doppler frequency

[0151] 1.1 Iterative Reconstruction of Covariance Matrix from 3D Toeplitz Matrix

[0152] Due to the new observation vector This is equivalent to a single-sample snapshot signal, whose correlation statistics suffer from rank deficiency, therefore the vector needs to be... Smoothing is performed to obtain an equivalent multi-sample snapshot signal, which is then converted into a data matrix with a covariance structure.

[0153] make The new observation vector is 3D matrix, i.e., matrix have Okay. Next, we will use the method of three-stage Toeplitz matrix iteration to reconstruct the observation vector into a... Equivalent covariance matrix of virtual domain.

[0154] Step 601: First, the observation vector Divided into rows There are matrix blocks, each containing Row element, if [R1] i Indicates the first There are matrix blocks, among which According to matrix block [R1] i Construct the following Toeplitz matrix for the basic elements:

[0155]

[0156] Step 602: Then add the matrix blocks Divided into rows There are n matrix blocks, assuming [R0] j Indicates the first There are matrix blocks, among which With [R0] j Using the basic elements, the matrix block [R1] is... i The elements in the matrix are constructed into the following Toeplitz matrix:

[0157]

[0158] Step 603: Finally, combine the matrix blocks. Divided into rows Row element, let's say [r0] w Represents matrix block [R0] j The Middle row elements, where With [r0] w Using the basic elements, the matrix block [R0] is... j Represented as a Toeplitz matrix as follows:

[0159]

[0160] The Toeplitz process shown in equation (24) is used to process each matrix block [R1]. i Perform this Toeplitz transformation operation on all elements, and then apply the Toeplitz transformation process of equation (25) to each element in the Toeplitz matrix after the above operation [R0]. j This will yield an equivalent multi-sample snapshot signal, i.e., the virtual domain equivalent covariance matrix R. xx From equation (25), we can see that R xx It is A 3D matrix can be represented as:

[0161]

[0162] in,

[0163] 1.2 Joint Estimation and Solution of Objective Parameters

[0164] The single-shot effect and rank loss problem are overcome by using the triple Toeplitz matrix iterative reconstruction method, and the equivalent covariance matrix R of the virtual domain is obtained. xx Clearly, its structure is the same as that of the traditional covariance matrix, therefore it is also suitable for solving the objective parameters using subspace algorithms. Below, we use an improved multidimensional ESPRIT algorithm to solve for the objective parameters. This method avoids the local optimum trap problem and the uncertainty of the number of iterations during convergence in the alternating least squares iterative method, and also solves the problem that parameters cannot be automatically paired when solving the problem in the traditional multidimensional ESPRIT algorithm.

[0165] Step 701: Convert the equivalent covariance matrix R xx Eigenvalue decomposition yields a signal subspace consisting of K eigenvalues. E s =WU -1 E s The former The matrix composed of row elements is E s1 ,back The matrix composed of row elements is E s2 Based on the structure of the virtual spacetime guiding vector W, the following equation holds:

[0166]

[0167] in, Φ(f) represents the generalized inverse matrix of the solution matrix. d ) is the rotation invariant factor of the Doppler domain. Combining equation (24), we know that U and U s All are complex matrices eigenvectors, and satisfying

[0168] U s =UΔJ (28)

[0169] Where Δ represents the scaling factor, J is the column permutation matrix, and the estimated value of the spatiotemporal virtual steering vector W is obtained directly through the following formula:

[0170]

[0171] The estimated value of the extended virtual spatiotemporal steering vector W is obtained.

[0172] Step 702: Obtain the estimated value Then, based on its internal structure, the joint estimate of the target's transmit angle (DOD), receive angle (DOA), and Doppler frequency is obtained through the following three expressions;

[0173]

[0174]

[0175]

[0176] The above three formulas can be used to estimate the value of the same virtual spacetime steering vector W. The estimated values ​​of the target transmit / receive angle and Doppler frequency are obtained. Since these three parameters are all estimated from the same spatiotemporal virtual steering vector, the obtained DOD, DOA and Doppler frequency will be automatically matched.

[0177] Comparison of computing speeds

[0178] To illustrate the superiority of the proposed algorithm in terms of computational complexity, its computational speed is compared with that of the "spatiotemporal sliding window" method, as both algorithms can perform secondary expansion of spatiotemporal degrees of freedom and have similar resolution and parameter estimation accuracy. Assume the configuration of the transmit and receive arrays and delay units of the bistatic nested MIMO radar is M1=2, M2=3, N1=2, N2=3, Q1=2, Q2=3, i.e., M=N=Q=5. Assume the number of spatial targets is 11 and the number of simulated snapshots is 500. The simulation platform is an Intel(R) Core(TM) i7-9750H CPU @ 2.60GHz 2.59GHz, RAM 16GB, and the software platform is MATLAB R2022a.

[0179] The simulation uses the average time taken by the algorithm in 30 simulations to demonstrate its computational speed.

[0180] Table 1 Comparison of solution speed between the "spatiotemporal sliding window" algorithm and the algorithm presented in this paper.

[0181]

[0182] Simulation experiments and results analysis

[0183] The proposed algorithm is compared with some state-of-the-art algorithms, and multiple simulation experiments are conducted to verify the advantages of the proposed algorithm in terms of the number of detectable targets, parameter estimation accuracy, and target resolution. The simulation platform is an Intel(R) Core(TM) i7-9750H CPU @ 2.60GHz 2.59GHz, 16GB RAM, and the software platform is MATLAB R2018a. The following algorithms will be used for comparison with the proposed algorithm:

[0184] 1) Multi-ESPRIT: Multidimensional ESPRIT algorithm.

[0185] 2) ULA-PARAFAC: Parallel factorization algorithm for uniform linear arrays.

[0186] 3) STSW: A multi-parameter joint estimation algorithm based on the "time sliding window" method.

[0187] 4) Proposed: Replace the joint parameter estimation algorithm of "time sliding window" with the triple Toeplitz matrix iterative reconstruction method.

[0188] Assume the transmit, receive arrays, and delay units of the bistatic spatiotemporally nested MIMO radar are configured as M1=2, M2=2, N1=2, N2=2, Q1=2, Q2=2, i.e., M=N=Q=4. We use a large number of noise-free snapshots (20,000 snapshots) to test the maximum number of detectable targets of the algorithm. Figure 2 A constellation diagram illustrating the comparison between the cumulative estimation results of 10 iterations and the preset target estimation parameters.

[0189] Figure 2 (a) indicates that the multidimensional MUSIC algorithm can estimate 46 targets. According to the literature [7], its spatiotemporal degrees of freedom are MN(Q-1) = 48, which shows that the performance of the algorithm is close to the degree of freedom.

[0190] Figure 2 (b) This indicates that the proposed algorithm can estimate 60 targets, and after the second expansion of the spatiotemporal degrees of freedom, its theoretically detectable number of targets is: It is evident that the proposed algorithm can increase the number of detectable targets, but it is still far from the theoretical level.

[0191] 1.2 Root Mean Square Error of Parameter Estimation

[0192] Assume the transmit, receive arrays, and delay units of the bistatic spatiotemporally nested MIMO radar are configured as M1=2, M2=3, N1=2, N2=3, Q1=2, Q2=3, i.e., M=N=Q=5. The mean square error (RMSE) of the joint angle estimation is calculated as follows (based on 500 Monte Carlo simulations):

[0193]

[0194]

[0195] in, and These are the results of the i-th experiment. θ k and f dk The estimated value, assuming the spatial target location is The number of snapshots was set to 200, and the SNR was evenly distributed between -10dB and 20dB.

[0196] Figure 3 The graph shows the relationship between parameter estimation error (RMSE) and signal-to-noise ratio (SNR) for the multidimensional ESPRIT algorithm, ULA-PARAFAC algorithm, STSW algorithm, and the algorithm proposed in this paper. Figure 3 (a) shows a schematic diagram illustrating the change of the joint angle estimation error of the four algorithms with the signal-to-noise ratio; Figure 3 (b) shows a schematic diagram of how the Doppler frequency estimation error of the four algorithms changes with the signal-to-noise ratio.

[0197] Depend on Figure 4 As shown in (a) and (b), the algorithm in this embodiment employs a second-order nested configuration for the transceiver array and the delayed receiver, achieving a secondary expansion of the spatiotemporal degrees of freedom. Under the same physical array element and delayer configuration, compared to the multidimensional ESPRIT and ULA-PARAFAC algorithms, the proposed algorithm has a larger aperture degree of freedom, thus its parameter estimation accuracy is significantly better than these two algorithms. For the STSW algorithm, which has the same aperture degree of freedom as the algorithm in this embodiment, the algorithm in this embodiment has similar parameter estimation accuracy. However, the proposed algorithm uses a triple Toeplitz matrix iterative reconstruction method instead of the "spatiotemporal sliding window" operation of that algorithm to overcome the single snapshot effect, greatly reducing the algorithm complexity.

[0198] 1.3 Resolution performance of target estimation

[0199] The bistatic nested MIMO radar has the following configurations for its transmit and receive arrays and delay units: M1=2, M2=3, N1=2, N2=3, Q1=2, Q2=3, i.e., M=N=Q=5, SNR=5dB, and number of snapshots L=200. There are two targets in space with similar azimuths. and

[0200] Figure 4 A constellation diagram of the resolution performance of four algorithms is presented, which is a schematic diagram comparing the cumulative estimation results of 30 iterations with the preset estimated target parameters. Figure 4 As shown in (a), (b), (c), and (d), the proposed algorithm and the STSW algorithm can clearly distinguish two similar targets because they have the same aperture degree of freedom. However, the ULA-PARAFAC algorithm and the multidimensional ESPRIT algorithm cannot effectively distinguish two targets that are spatially close.

[0201] This embodiment nests the transceiver array and delay unit, achieving a secondary expansion of the spatiotemporal degrees of freedom. It then proposes a multilinear mapping mechanism adaptable to various sparse array structures, effectively solving the problems of complex culling and difficult rearrangement of virtual array elements. Simultaneously, addressing the high complexity and implementation difficulties of traditional spatial smoothing algorithms, a triple Toeplitz matrix iterative reconstruction algorithm is proposed to overcome the single-shot effect of the received virtual signal. Compared to the commonly used "spatiotemporal sliding window" algorithm, this method significantly reduces algorithm complexity while maintaining the same aperture degrees of freedom. Simulation experiments show that the proposed algorithm maintains both high spatiotemporal degree of freedom expansion capability and excellent parameter estimation performance while significantly reducing algorithm complexity. Simulation results on the algorithm's computation speed show that the proposed algorithm is nearly 456 times faster than the "spatiotemporal sliding window" algorithm, making its engineering application possible.

[0202] The above description is only used to illustrate the technical solution of the present invention and is not intended to limit it. Any other modifications or equivalent substitutions made by those skilled in the art to the technical solution of the present invention, as long as they do not depart from the spirit and scope of the technical solution of the present invention, should be covered within the scope of the claims of the present invention.

[0203] [1]Zhang XF,Xu LY,Xu L,et al.Direction of Departure(DOD)andDirection of Arrival(DOA)Estimation in MIMO Radar with Reduced-DimensionMUSIC[J].IEEE COMMUNICATIONS LETTERS,2010,14(12):1161-1163.

[0204] [2] Liu Shuai, Zhang Gong, Liu Wenbo. Joint estimation of multidimensional parameters of bistatic MIMO radar based on spatiotemporal structure [J]. Acta Aeronautica Sinica, 2010, 31(06):1196-1203.

[0205] [3]Gong J, Lv H, Guo Y.Multidimensional Parameters Estimation for Bistatic MIMO Radar[C]. 2011 7th International Conference on WirelessCommunications, Networking and Mobile Computing, 2011:1-4.

[0206] [4] Zhang Jianyun, Zheng Zhidong, Li Xiaobo. Joint estimation algorithm for transmit / receive angle and Doppler frequency of bistatic MIMO radar [J]. Journal of Electronics and Information Technology, 2010, 32(08):1843-1848.

[0207] [5]Cheng Y, Gu H, Su W.Joint 4-D Angle and Doppler Shift Estimation viaTensor Decomposition for MIMO Array[J]. IEEE Communications Letters, 2012, 16(6):917-920.

[0208] [6] Li Jianfeng, Zhang Xiaofei. Joint estimation of angle and Doppler frequency of bistatic MIMO radar based on tetralinear decomposition [J]. Acta Aeronautica Sinica, 2012, 33(08): 1474-1482.

[0209] [7] Zheng Zhidong, Fang Fei, Yuan Honggang, Yu Yanming, Tao Huan. Joint estimation method of transmit / receive angle and Doppler frequency of bistatic MIMO radar under spatiotemporal nonuniform sampling[J]. Journal of Electronics and Information Technology, 2015, 37(09):2164-2170.

[0210] [8] Fan Jinyu, Gu Hong, Su Weimin, Chen Jinli. A method for multi-parameter estimation of coprime array MIMO radar targets based on tensor decomposition [J]. Journal of Electronics and Information Technology, 2015, 37(04):933-938.

[0211] [9] Li Yuncheng, Cui Chen, Gong Yang. Joint estimation algorithm for bistatic MIMO radar target parameters based on parallel factor analysis under spatiotemporal nonuniform sampling [J]. Journal of Air Force Engineering University (Natural Science Edition), 2018, 19(06):66-72.

Claims

1. A low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling, characterized in that... The joint estimation algorithm includes the following steps: Step 1: Configure the bistatic MIMO radar system as a spatiotemporal nested sampling model, and use the spatiotemporal nested sampling model to sample the received signal. ; Step 2: Use a nested Q-level delay receiver to process the received signal. After performing multi-stage delayed sampling processing, the received data is received with a Q-stage delay and represented as a vector form, thus obtaining... ; Step 3: Use a matched filter to process the delayed received signal. Perform matched filtering to obtain the echo signal. ; Step 4: Based on the echo signal Establish the covariance matrix of the target echo signal. ; Step 5: Vectorize the covariance matrix of the target echo signal To achieve a second extension of the spatiotemporal degrees of freedom, that is, to represent it as a data vector. The method of multilinear mapping is used to process data vectors. Redundancy and rearrangement are performed to obtain the observed signal. ; Step 6: For new observation signals By performing triple Toeplitz matrix iterative reconstruction, the equivalent covariance matrix of the virtual domain is obtained. ; Step 7: Apply the improved multidimensional ESPRIT algorithm to the equivalent covariance matrix Eigenvalue decomposition is performed to construct expressions for the target's emission angle, reception angle, and Doppler frequency, thereby obtaining estimates of the target's emission angle, reception angle, and Doppler frequency. Received signal The sampling process is as follows: Step 101: In the spatiotemporal nested sampling model, both the transmitting array and the receiving array are second-order nested arrays, and the number of elements in the transmitting array is... The number of elements in the receiving array is The delay is configured as follows Let the element spacing of the unit array element be... The set of physical array element locations for the transmitting and receiving arrays is represented as: （1） Step 102: Assume it exists A far-field narrowband incoherent target , and , These represent the target's DOD, DOA, and Doppler frequencies, respectively, and the transmitting array transmits... The nth different orthogonal pulse signal, in the nth The echo signal model at the receiving end under the next transmitted pulse is represented as follows: （2） in, , Represents the Khatri-Rao product. , Indicates the first The scattering coefficient of a target The repetition period of the transmitted pulse. With a mean of 0 and a variance of of A Gaussian white noise vector, and: （3） （4） （5） and Let represent the steering vectors of the transmitting array and the receiving array, respectively, and express them as follows: （6） （7） Step 103: Use a nested Q-level delay receiver to process the received signal. Delayed sampling is performed, assuming that the target's DOD, DOA, and Doppler frequency parameters remain unchanged during the delay time. Then, the... The echo signal after the delay is represented as: （8） in, , ; For new observation signals The specific process of iterative reconstruction of the triple Toeplitz matrix is ​​as follows: Step 601: First, the observation vector Divided into rows There are matrix blocks, each containing row element, if Indicates the first There are matrix blocks, among which According to matrix blocks Construct the following Toeplitz matrix for the basic elements: (twenty three) Step 602: Then add the matrix blocks ( (Divided into rows) There are matrix blocks, assuming Indicates the first There are matrix blocks, among which ,by Using basic elements, divide the matrix blocks The elements in the matrix are constructed into the following Toeplitz matrix: （24） Step 603: Finally, combine the matrix blocks. , Divided into rows row element, assuming Represents a matrix block The Middle row elements, where ,by Using basic elements, divide the matrix blocks Represented as a Toeplitz matrix as follows: （25） Each matrix block is processed using the Toeplitz process shown in equation (24). Perform this Toeplitz transformation operation on all elements, and then apply the Toeplitz transformation process of equation (25) to each element of the Toeplitz matrix after the above operation. This will yield an equivalent multi-sample snapshot signal, i.e., the virtual domain equivalent covariance matrix. From equation (25), we can see that It is A 3D matrix can be represented as: （26） in, , , .

2. The low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling as described in claim 1, is characterized in that, Received signal The expression is: （9） in, （10） （11） The Doppler steering vector of the target. and It is called the spatial steering vector.

3. The low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling as described in claim 2, is characterized in that... In step 3 above, the echo signal The expression is as follows: （12） in, , An information vector constructed from the scattering coefficients and Doppler parameters. 3D white noise signal vector; This represents the joint steering vector of the received signal.

4. The low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling as described in claim 3, is characterized in that, In step 4 above, the covariance matrix of the target echo signal can be expressed as: （13） in, Let be the target covariance matrix. This represents the power level of the signal.

5. The low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling as described in claim 4, is characterized in that... In step 5 above, the data vector The expression is （14） in, , representing the signal power of K targets; (15) From equation (15), we know that It can be represented as: （16）。 6. The low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling as described in claim 5, is characterized in that... For data vectors The specific process of redundancy and rearrangement is as follows: According to the structure of equation (16), the vector Given a spatiotemporal structure that satisfies a sum-difference joint spatiotemporal structure, let the set of its virtual array elements and temporal virtual sampling points be denoted as . Then it can be expressed as: （17） In equation (17), the difference joint array of nested arrays have A virtual array element, have A virtual array element, a difference joint structure for time-domain sampling. have The set of virtual sampling points, since the joint structure in equation (17) consists of three different parameters, is therefore... The spatiotemporal joint degrees of freedom can be expressed as: （18） make , , The range of the virtual spacetime guiding vector is from arrive ; The unique virtual matrix element can be indexed by its position from the vectorized one-dimensional matrix, and its structure is the same as that of the vectorized matrix formed by the uniform matrix. Right now: （19） In the formula, express 3D vector, and Represents the virtual joint guidance vector, with ,in, (20) （21） （22）。 7. The low-complexity joint estimation algorithm for target DOD-DOA and Doppler frequency based on spatiotemporal nested sampling as described in claim 6, is characterized in that... In step 7 above, the process of obtaining the target's DOD-DOA and Doppler frequency parameters includes: Step 701: Convert the equivalent covariance matrix By performing eigenvalue decomposition, we can obtain a set of features. The signal subspace composed of eigenvalues , ,make The former The matrix composed of row elements is ,back The matrix composed of row elements is Then, according to the virtual spatiotemporal guiding vector For the structure , the following equation holds: （27） in, Denotes the generalized inverse matrix of the solution matrix. Let be the rotation invariant factor of the Doppler domain. Combining equation (27), we can see that... and All are complex matrices eigenvectors, and satisfying （28） in, Indicates the scaling factor. As a column permutation matrix, the spatiotemporal virtual steering vector can be obtained directly using the following formula. The estimated value: （29） Obtain the extended virtual spacetime steering vector The estimated value ; Step 702: Obtain the estimated value Then, based on its internal structure, the joint estimate of the target's transmit angle (DOD), receive angle (DOA), and Doppler frequency is obtained through the following three expressions; （30） （31） （32） The above three formulas can be used to obtain the same virtual spacetime guiding vector. The estimated value The estimated values ​​of the target transmit / receive angle and Doppler frequency are obtained. Since these three parameters are all estimated from the same spatiotemporal virtual steering vector, the obtained DOD, DOA and Doppler frequency will be automatically matched.

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