Parameter estimation method and system for FDA-MIMO radar range resolution

Abstract

The application discloses a FDA-MIMO radar distance deambiguity parameter estimation method and system, comprising the following steps: dividing FDA-MIMO radar transmitting array elements into two sub-arrays, setting maximum non-ambiguous distance and frequency increment for the two sub-arrays; constructing a third-order tensor model of receiving data according to the snapshot order of matched filter output data; performing tensor parallel factor decomposition on the third-order tensor model to obtain array flow pattern estimation; normalizing each column of the estimated receiving array flow pattern matrix respectively to obtain angle estimation; dividing the estimated transmitting array flow pattern matrix into two estimated sub-arrays, and performing normalization processing on each column of each estimated sub-array; and obtaining ambiguous distance estimation by using a phase average method; and solving the distance deambiguity estimation by using the Chinese remainder theorem according to the ambiguous distance estimations of the same target obtained by the two estimated sub-arrays. The application realizes multi-target distance decoupling and has high estimation precision.

Description

Technical Field

[0001] This invention relates to the field of monostatic FDA-MIMO radar range and angle estimation technology, specifically to a parameter estimation method and system for FDA-MIMO radar range deambiguation. Background Technology

[0002] The concept of MIMO (Multiple-Input Multiple-Output) technology was first proposed in the field of communications, and its application to radar was attempted in 2003. MIMO radar possesses advantages such as high degrees of freedom (DOF) and spatial diversity gain, thus attracting widespread attention from academic and industrial communities both domestically and internationally. MIMO radar can transmit uncorrelated or mutually orthogonal signals in the time domain. At the receiver, matched filters are used to separate the transmitted signals from the corresponding transmission channels, and adaptive array processing technology is used to detect targets. The echo signal, separated into multiple signals by matched filters, effectively increases the virtual aperture of the array, thereby increasing the maximum number of detectable targets, improving resolution, and significantly enhancing the accuracy of target parameter estimation.

[0003] However, conventional MIMO radar beam pointing is only angle-dependent and not range-dependent, thus failing to distinguish targets at the same range with small angular intervals. To achieve joint range-angle estimation, Antonik and Wicks first proposed the concept of Frequency Diverse Array (FDA) at an international radar conference in 2006. This concept forms beams dependent on angle, time, and range by setting small frequency increments between different transmitting elements, meaning that frequency-controlled arrays can achieve joint range-angle estimation of targets. Leveraging its range dimension advantage, FDA radar has enormous application potential in clutter suppression and target detection. However, when the transmitting array is a uniform linear array, range and angle coupling occurs, making it impossible to accurately estimate the target's position. To achieve decoupling of range and angle, researchers have proposed many methods, such as transmitting two pulses with zero and non-zero frequency increments respectively, dividing the transmitting array elements into multiple subarrays, and transmitting different frequency increments between different subarrays. Some literature uses nonlinearly varying frequency increments to achieve angle-range decoupling. Other scholars have integrated FDA and MIMO technologies to form FDA-MIMO radar, which utilizes the high degree of freedom of MIMO to achieve decoupling of range and angle, while improving the estimation performance of target parameters.

[0004] However, in range- and angle-coupled transmit steering vectors, if the target distance is too large, exceeding the phase ambiguity range, the obtained range estimate will be ambiguous. Currently, most scholars assume that the target distance does not exceed the maximum unambiguous distance. Based on this assumption, FDA-MIMO radar and MIMO radar systems share many inherent properties, so angle estimation algorithms suitable for MIMO radar can also be adapted to FDA-MIMO radar. A series of variant algorithms, such as the Capon algorithm based on beamforming, the rotation-invariant algorithm based on subspaces, and multiple signal classification, have been used for range and angle estimation in FDA-MIMO radar. Simultaneously, to fully utilize the multidimensional structure of the matched filter output in MIMO radar, algorithms based on tensor decomposition, such as higher-order tensor singular value decomposition and higher-order tensor parallel factor decomposition, have also been used for range and angle estimation in FDA-MIMO radar. Through tensor decomposition techniques, the array manifold is estimated, and the rich information carried by the multidimensional structure within the signal is mined, further improving the accuracy of parameter estimation.

[0005] As a novel radar technology that has garnered significant attention in recent years, the FDA-MIMO parameter estimation algorithm has yet to consider practical factors, such as range ambiguity at long distances, thus leaving considerable room for improvement. Xu et al. proposed a monostatic FDA-MIMO radar angle-range deambiguity estimation algorithm based on the Capon algorithm. However, this algorithm requires prior information about the target range, setting a range compensation amount based on this prior information to achieve unambiguous range estimation. In practice, obtaining prior information about the target range is difficult. In this algorithm, both range and angle estimates are obtained through a search, and the estimation accuracy is related to the set search step size; a smaller step size results in higher accuracy but also incurs a huge computational burden and fails to obtain a closed-form solution for parameter estimation. Furthermore, the Capon-based algorithm does not consider the multidimensional structure of MIMO radar, which also contributes to some degree of loss in parameter estimation accuracy. Summary of the Invention

[0006] The technical problem this invention aims to solve is that existing FDA-MIMO radar parameter estimation methods do not consider the multi-dimensional structure of MIMO radar. When prior information about target range is required, they can only achieve range decoupling for a single target and suffer from low parameter estimation accuracy. The purpose of this invention is to provide a parameter estimation method and system for FDA-MIMO radar range deambiguation. This invention considers the multi-dimensional structure of MIMO radar, enabling multi-target range decoupling without requiring prior information about target range, and improving parameter estimation accuracy.

[0007] This invention is achieved through the following technical solution:

[0008] In the first aspect, the present invention provides a parameter estimation method for range deblurring of FDA-MIMO radar, and the method includes:

[0009] Divide the transmitting array elements of the FDA-MIMO radar into two sub-arrays, and set the maximum unambiguous range and frequency increment for the two sub-arrays respectively;

[0010] According to the set maximum unambiguous range and frequency increment, construct a third-order tensor model of the received data according to the snapshot order of the output data of the monostatic FDA-MIMO radar matching filter;

[0011] Perform tensor parallel factor decomposition on the third-order tensor model to obtain the estimation of the received array manifold matrix and the transmitting array manifold matrix;

[0012] Normalize each column of the estimated received array manifold matrix and then calculate the phase; according to the phase, use the least squares method to obtain the estimation of the angle;

[0013] Divide the estimated transmitting array manifold matrix into two estimated sub-arrays, and perform normalization processing on each column of each estimated sub-array; and use the phase averaging method to process the normalized estimated sub-arrays to obtain the estimation of the ambiguous range;

[0014] For the ambiguous range estimations of the same target obtained from the two estimated sub-arrays, use the Chinese Remainder Theorem to traverse and solve to obtain the deblurred estimation of the range.

[0015] Further, dividing the transmitting array elements of the FDA-MIMO radar into two sub-arrays and setting the maximum unambiguous range and frequency increment for the two sub-arrays respectively includes:

[0016] At the transmitting end of the FDA-MIMO radar, divide the transmitting array elements into two sub-arrays: the first transmitting sub-array and the second transmitting sub-array;

[0017] Set the first maximum unambiguous range for the first transmitting sub-array and the second maximum unambiguous range for the second transmitting sub-array; where: the first maximum unambiguous range and the second maximum unambiguous range are relatively prime;

[0018] According to the first maximum unambiguous range, set the first frequency increment for the first transmitting sub-array; according to the second maximum unambiguous range, set the second frequency increment for the second transmitting sub-array.

[0019] Further, the first transmitting sub-array is equipped with M1 transmitting array elements. In the first transmitting sub-array, the transmitting frequency of the m1th (0 < m1 ≤ M1) transmitting array element is where f0 is the carrier frequency of the first transmitting array element, and Δf1 is the frequency increment between two adjacent transmitting array elements in the first transmitting sub-array;

[0020] The second transmitting subarray is equipped with M2 transmitting array elements. In the second transmitting subarray, the transmitting frequency of the m2-th (M1 < m2 ≤ M1 + M2) transmitting array element is where f0 is the carrier frequency of the first transmitting array element, Δf1 is the frequency increment between adjacent two transmitting array elements in the first transmitting subarray, M1 is the number of array elements in the first transmitting subarray, and Δf2 is the frequency increment between adjacent two transmitting array elements in the second transmitting subarray.

[0021] Furthermore, the third-order tensor model includes the following six definitions regarding tensor operations:

[0022] Definition 1, tensor n-mode unfolding: The n-mode unfolding matrix of an N-order tensor can be expressed as X N where I1, I2… I N represent the dimensional sizes of the tensor in the 1st, 2nd, …, Nth dimensions. The (i n , j)-th element X (n) (i n , j) consists of where i n ∈(1, 2, …, I n ), i1∈(1, 2, …, I1 ) , i2∈(1, 2, …, I2), i N ∈(1, 2, …, I N ) and there is

[0023] Definition 2, rank-one tensor: If a third-order tensor can be written as then this third-order tensor is a rank-one tensor. Among them, and are factor matrices; represents the vector outer product, and where represents the element at the (i1, i2, i3) position of the tensor , represents the i1-th (0 < i1 ≤ I1) element of the vector a, represents the i2-th (0 < i2 < I2) element of the vector b, represents the i3-th (0 < i3 ≤ I3) element of the vector c.

[0024] Definition 3, parallel factor decomposition (trilinear decomposition) of a tensor: The trilinear decomposition of a tensor decomposes the tensor into the sum of several rank-one tensors. For a third-order tensor where I1, I2 and I3 represent the tensor The dimensional sizes in the 1st, 2nd, and 3rd dimensions, and its trilinear decomposition can be expressed as: is equivalent to where represents the element at the (i1, i2, i3) position of the tensor , a r , b r and c r are respectively the and r-th (0 < r ≤ R) columns of the factor matrices, and are respectively the i1-th (0 < i1 ≤ I1), i2-th (0 < i2 ≤ I2), and i3-th (0 < i3 ≤ I3) elements of a r , b r and c r , and R is the number of columns of the factor matrices.

[0025] Definition 4, Uniqueness of Tensor PARAFAC Decomposition: For a third-order tensor , the necessary and sufficient condition for the uniqueness of its PARAFAC decomposition is: where is the Kruskal rank of the n-mode unfolding matrix of the tensor, and R represents the number of columns of the factor matrix.

[0026] Definition 5, n-mode Product of a Tensor and a Matrix: The n-mode product of an N-order tensor and a matrix is where, and there is . Among them, is the element of the tensor at the (i1, i2,..., i n-1 , j n , i n+1 ,..., i N ) position, is the element of the tensor at (i1, i2,..., i n-1 , i n , i n+1 ,..., i N ) position, X(j n , i n ) is the element of the matrix X at (j n , i n ) position, and there is (0 < i n ≤ I n , n = 1, 2,..., N)(0 < j n < Jn ) .

[0027] Definition 6, Properties of the n-mode product of tensors: Nth order tensors (I1, I2, ..., I N The pattern product (where each taper is a value) has the following main properties: Among them, matrix

[0028] Furthermore, based on the set maximum unambiguous range and frequency increment, and following the snapshot order of the output data from the monostatic FDA-MIMO radar matched filter, a third-order tensor model of the received data is constructed, specifically as follows:

[0029] Based on the set maximum unambiguous range and frequency increment, and following the snapshot order of the output data from the monostatic FDA-MIMO radar matched filter, the data are sequentially stacked along the third dimension of the third-order tensor to obtain the third-order tensor model of the received data. Where N is the number of receiving array elements, M = M1 + M2 is the number of transmitting true elements, and the element at position (n, m, l) is:

[0030]

[0031] Where K is the number of targets, N is the number of receiving array elements, M = M1 + M2 is the number of transmitting array elements, L is the number of snapshots, and B(n, k) is the receiving array manifold matrix. The element at position (n, k); A(m, k) is the emission array manifold. The element at the (m, k)th position; For the source signal term, This is a noise tensor. Based on the properties of tensors, It can also be written in the following form: in, It is a unit tensor of K×K×K.

[0032] Furthermore, tensor parallel factorization is performed on the third-order tensor model, including:

[0033] Initialize three matrices: and Where K is the number of radar targets, N is the number of receiving array elements, M = M1 + M2 is the number of transmitting array elements, M1 is the number of transmitting array elements equipped in the first transmitting subarray, and M2 is the number of transmitting array elements equipped in the second transmitting subarray.

[0034] The alternating least squares (ALS) method is used to iteratively update B, A, and S until convergence.

[0035] The update principles for B, A, and S in the (i+1)th iteration are as follows:

[0036] Among them, symbols Represents the Kronecker product, symbol Y represents the pseudo-inverse operator; (1) Y （2) Y (3) They are data tensors The expansion matrices for modes one, two, and three; I K(1) I K(2) I K(3) Tensors The expansion matrices for the first, second, and third modes;

[0037] The estimates of B, A, and S at convergence are denoted as follows: and

[0038] Furthermore, the phase is calculated after normalizing each column of the estimated receiver array manifold matrix; based on the phase, the angle is estimated using the least squares method, including:

[0039] For the estimated receiver array manifold matrix The k-th column is normalized, and the phase is calculated and denoted as . The expression for the angle estimation of the k-th target: Among them, symbols Denotes the square of the F-norm. Let h be the quantity to be solved. r，k，2 For θ k The relevant variables to be estimated; (·) T H represents the transpose operator; r The expression is as follows: Among them, h r，k The least squares solution is θ k The estimated value for

[0040] Furthermore, the estimated transmit array manifold matrix is ​​divided into two estimation subarrays, and each column of each estimation subarray is normalized. The phase averaging method is then used to process the normalized estimation subarrays to obtain an estimate of the fuzzy distance, including:

[0041] The estimated manifold matrix of the transmission array is divided into two estimation submatrices. and First estimation subarray For the estimated transmit array manifold matrix The first M1 rows, the second estimated subarray For the estimated transmit array manifold matrix The next line M2;

[0042] Normalize the k-th column of the first estimation subarray and denote it as The ambiguity distance estimation of the k-th target obtained through the first estimation subarray has the following calculation formula:

[0043]

[0044] , where () * represents the conjugate operator, angle(a) represents the phase of the complex number a, c represents the speed of light, M1 is the number of array elements of the first transmitting subarray, represents the i1-th (0 < i1 < M1) element of represents the (i1 + 1)-th element of, Δf1 represents the frequency increment of the first transmitting subarray; is the phase information related to the angle of the k-th target obtained in the angle estimation step;

[0045] Normalize the k-th column of the second estimation subarray and denote it as The ambiguity distance estimation of the k-th target obtained through the second estimation subarray has the following calculation formula:

[0046]

[0047] , where () * represents the conjugate operator, angle(a) represents the phase of the complex number a, c represents the speed of light, M2 is the number of array elements of the second transmitting subarray, represents the i2-th (0 < i2 < M2) element of represents the (i2 + 1)-th element of, Δf2 represents the frequency increment of the second transmitting subarray; is the phase information related to the angle of the k-th target obtained in the angle estimation step.

[0048] Furthermore, for the ambiguity distance estimations of the same target obtained from the two estimation subarrays, use the Chinese Remainder Theorem to traverse and solve to achieve distance de-ambiguity, including:

[0049] Set the maximum distance ambiguity numbers as J1 and J2 for the first estimation subarray and the second estimation subarray respectively;

[0050] For the first estimation subarray Second estimation subarray According to the Chinese Remainder Theorem, traversing j1∈(0, 1, ..., J1-1) and j2∈(0, 1, ..., J2-1), the distance after unfuzzification is obtained by solving the following optimization problem; the function of the optimization problem is:

[0051] in, and Let be the maximum unambiguous distances at the corresponding frequency increments of emission subarray one and emission subarray two, respectively; c represents the speed of light; Δf1 is the frequency increment of the first emission subarray, and Δf2 is the frequency increment of the second emission subarray. After obtaining j1 and j2, denoted as and The unambiguous distance estimate of the k-th target is

[0052] Secondly, the present invention provides a parameter estimation system for FDA-MIMO radar range deambiguation, the system comprising:

[0053] The actual subarray division and setting unit is used to divide the FDA-MIMO radar transmitting array into two subarrays and set the maximum unambiguous distance and frequency increment for the two subarrays respectively.

[0054] The third-order tensor model construction unit is used to construct the third-order tensor model of the received data according to the snapshot order of the output data of the monostatic FDA-MIMO radar matched filter, based on the set maximum unambiguous distance and frequency increment.

[0055] Parallel factorization unit, used to perform tensor parallel factorization on the third-order tensor model to obtain estimates of the receiver array manifold matrix and the transmitter array manifold matrix;

[0056] An angle estimation unit is used to normalize each column of the estimated receiver array manifold matrix and calculate the phase; based on the phase, the least squares method is used to obtain the angle estimate;

[0057] The fuzzy distance estimation unit is used to divide the estimated transmit array manifold matrix into two estimation subarrays, normalize each column of each estimation subarray, and process the normalized estimation subarrays using the phase averaging method to obtain the fuzzy distance estimate.

[0058] The defuzzy unit is used to estimate the fuzzy distance to the same target obtained from two estimating subarrays. It uses the Chinese remainder theorem to solve the problem and obtain the defuzzy distance estimate.

[0059] Furthermore, the real subarray division and setting unit includes a real subarray division subunit and a setting subunit;

[0060] The real subarray is divided into sub-cells, which are used at the transmitting end of the FDA-MIMO radar to divide the transmitting array elements into two subarrays: the first transmitting subarray and the second transmitting subarray.

[0061] A sub-unit is configured to set a first maximum unambiguous distance for the first transmitting subarray and a second maximum unambiguous distance for the second transmitting subarray; a first frequency increment is set for the first transmitting subarray based on the first maximum unambiguous distance; and a second frequency increment is set for the second transmitting subarray based on the second maximum unambiguous distance; wherein the first maximum unambiguous distance and the second maximum unambiguous distance are coprime.

[0062] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0063] This invention discloses a parameter estimation method and system for range deambiguation in FDA-MIMO radar. This invention considers the multi-dimensional structure of MIMO radar (transmit dimension, receive dimension, and time dimension), enabling multi-target range decoupling without requiring prior information about target range, and improving parameter estimation accuracy. Specifically, at the transmitter, the method first divides the transmit array elements into two subarrays, sets the maximum unambiguous range for the two subarrays (these two maximum unambiguous ranges are coprime), and then sets the frequency increment based on their respective maximum unambiguous ranges. At the receiver, the output data of the matched filter is stacked sequentially into a third-order tensor according to the number of snapshots. Then, trilinear decomposition is performed on the data tensor to obtain an estimate of the transmit and receive array manifolds. The phase of each column of the estimated receive array manifold matrix is ​​calculated, and the closed-form solution of the angle is obtained using the least squares method. The estimated transmit array manifold matrix is ​​divided into two estimation subarrays, and the phase averaging method is used for each column of each estimation subarray to obtain the ambiguity estimate of the corresponding range. Finally, for the two fuzzy distance estimates of the same target obtained from the corresponding columns of the two emission subarrays, the Chinese remainder theorem is used to solve the problem traversally to obtain the defuzzified distance estimate, thus achieving distance defuzzification. Attached Figure Description

[0064] The accompanying drawings, which are included to provide a further understanding of embodiments of the invention and form part of this application, do not constitute a limitation thereof. In the drawings:

[0065] Figure 1 This is a flowchart of the parameter estimation method for FDA-MIMO radar range deambiguation in this invention;

[0066] Figure 2 This is a structural diagram of a monostatic FDA-MIMO system based on a transmitter subarray.

[0067] Figure 3 The point map of the target is estimated in 500 simulation experiments with a signal-to-noise ratio of 0dB and a snapshot number of 50 using the method proposed in this invention.

[0068] Figure 4 This is a graph showing the result of the root mean square error of angle estimation as a function of SNR in the method proposed in this invention.

[0069] Figure 5 This is a graph showing the result of the root mean square error of distance estimation as a function of SNR in the method proposed in this invention.

[0070] Figure 6 This is a graph showing the result of the root mean square error of angle estimation in the method proposed in this invention varying with the number of snapshots.

[0071] Figure 7 This is a graph showing the result of the root mean square error of distance estimation in the method proposed in this invention varying with the number of snapshots;

[0072] Figure 8 This is a block diagram of the parameter estimation system for FDA-MIMO radar range deambiguation according to the present invention. Detailed Implementation

[0073] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the embodiments and accompanying drawings. The illustrative embodiments and descriptions of the present invention are only used to explain the present invention and are not intended to limit the present invention.

[0074] The existing FDA-MIMO radar parameter estimation method does not take into account the multi-dimensional structure of MIMO radar. When prior information about the target range is required, it can only achieve range decoupling for a single target and has problems such as low accuracy in parameter estimation.

[0075] This invention designs a parameter estimation method and system for range deambiguation of FDA-MIMO radar. Considering the multi-dimensional structure of MIMO radar (transmit dimension, receive dimension, and time dimension), this invention achieves multi-target range decoupling without requiring prior information about target range and improves parameter estimation accuracy. Specifically, at the transmitter, the method first divides the transmit array elements into two subarrays, sets the maximum unambiguous range for the two subarrays (which are coprime), and then sets the frequency increment based on their respective maximum unambiguous ranges. At the receiver, the output data of the matched filter is stacked sequentially into a third-order tensor according to the number of snapshots. Then, trilinear decomposition is performed on the data tensor to obtain an estimate of the transmit and receive array manifolds. The phase of each column of the estimated receive array manifold matrix is ​​calculated, and the closed-form solution of the angle is obtained using the least squares method. The estimated transmit array manifold matrix is ​​divided into two estimation subarrays, and the phase averaging method is used for each column of each estimation subarray to obtain the ambiguity estimate of the corresponding range. Finally, for the two fuzzy distance estimates of the same target obtained from the corresponding columns of the two emission subarrays, the Chinese remainder theorem is used to solve the problem traversally to obtain the defuzzified distance estimate, thus achieving distance defuzzification.

[0076] Example 1

[0077] like Figures 1 to 7 As shown, the present invention provides a parameter estimation method for FDA-MIMO radar range deambiguation, which includes:

[0078] The FDA-MIMO radar transmitter array is divided into two subarrays, and the maximum unambiguous range and frequency increment are set for the two subarrays respectively.

[0079] Based on the set maximum unambiguous distance and frequency increment, a third-order tensor model of the received data is constructed according to the snapshot order of the output data of the monostatic FDA-MIMO radar matched filter.

[0080] Tensor parallel factorization is performed on the third-order tensor model to obtain estimates of the receiver array manifold matrix and the transmitter array manifold matrix;

[0081] The phase is calculated by normalizing each column of the estimated receiver array manifold matrix; based on the phase, the angle is estimated using the least squares method.

[0082] The estimated transmit array manifold matrix is ​​divided into two estimation subarrays, and each column of each estimation subarray is normalized. The phase averaging method is then used to process the normalized estimation subarrays to obtain the fuzzy distance estimate.

[0083] The fuzzy distance estimates of the same target obtained from the estimation of two estimation subarrays are solved by traversing the Chinese remainder theorem to obtain the fuzzy distance estimate.

[0084] The specific implementation plan is as follows:

[0085] 1. First, we introduce six definitions of tensor operations in the third-order tensor model:

[0086] Definition 1, Tensor n-mode expansion: N-order tensor The n-mode expansion matrix can be represented as X (n) Where I1, I2…I N Tensor The dimensionality of X in the 1st, 2nd, ..., Nth dimensions. n j) elements X (n) (i n j) by Composition, where i n ∈(1, 2, ..., I) n ),i1∈(1,2,…,I1),i2∈(1,2,…,I2),i N ∈(1, 2, ..., I) N ), And there are

[0087] Definition 2, rank-one tensor: If a third-order tensor can be written as then this third-order tensor is a rank-one tensor. Herein, and are factor matrices; represents the outer product of vectors, and where denotes the element at the (i1, i2, i3) position of the tensor , denotes the i1(0 < i1 ≤ I1 ) -th element of the vector a, denotes the i2(0 < i2 < I2)-th element of the vector b, denotes the i3(0 < i3 ≤ I3)-th element of the vector c.

[0088] Definition 3, parallel factor decomposition (trilinear decomposition) of a tensor: The trilinear decomposition of a tensor decomposes the tensor into the sum of several rank-one tensors. For a third-order tensor where I1, I2, and I3 represent the dimensional sizes of the tensor in the 1st, 2nd, and 3rd dimensions, its trilinear decomposition can be expressed as: which is equivalent to where denotes the element at the (i1, i2, i3) position of the tensor , a r , b r , and c r are respectively the and -th columns of the factor matrices, and are respectively the i1(0 < i1 ≤ I1)-th, i2(0 < i2 ≤ I2)-th, and i3(0 < i3 ≤ I3)-th elements of a r , b r , and c r , and R is the number of columns of the factor matrix.

[0089] Definition 4, uniqueness of parallel factor decomposition of a tensor: For a third-order tensor , the necessary and sufficient condition for the uniqueness of its parallel factor decomposition is: where is the Kruskal rank of the n-mode unfolding matrix of the tensor, and R represents the number of columns of the factor matrix.

[0090] Definition 5, product of an n-mode tensor and a matrix: The n-mode product of an N-order tensor and a matrix is Among them, and there is . Among them, is a tensor at the (i1, i2, …, i n-1 , j n , i n+1 , …, i N ) element, is a tensor at (i1, i2, ···, i n-1 , i n , i n+1 , ···, i N ) element, X(jn, i n ) is the element of matrix X at (j n , i n ), and there is (0 < i n ≤I n , n = 1, 2, …, N)(0 < j n <J n ).

[0091] Definition 6, properties of the n-mode product of a tensor: The mode product of an Nth-order tensor (I1, I2, …, I N is the size of each dimension) mainly has the following properties: Among them, the matrix

[0092] 2. FDA-MIMO radar signal model based on transmitting subarrays

[0093] This invention considers a monostatic FDA-MIMO radar scenario, and its array arrangement model is as Figure 2 shown. Assume that the FDA-MIMO radar consists of M transmitting array elements and N receiving array elements. The transmitting array and the receiving array are both uniform linear arrays (ULA). The spacing between the array elements of the transmitting and receiving arrays is half wavelength. At the transmitting end of the FDA-MIMO radar, the transmitting array elements are divided into two subarrays: the first transmitting subarray and the second transmitting subarray; the first transmitting subarray is equipped with M1 transmitting array elements. In the first transmitting subarray, the transmitting frequency of the m1(0 < m1 ≤ M1)th transmitting array element is Among them, f0 is the carrier frequency of the first transmitting array element, and Δf1 is the frequency increment between two adjacent transmitting array elements in the first transmitting subarray.

[0094] The second transmitting subarray is equipped with M2 transmitting array elements. In the second transmitting subarray, the transmitting frequency of the m2(M1 < m2 ≤ M1 + M2)th transmitting array element is Where Δf2 is the frequency increment between two adjacent transmitting elements in the second transmitting subarray.

[0095] Suppose there are K far-field point targets, and the angle and distance of the k-th target are (θ). k r k If the launch steering vectors of k targets are given, then the launch steering vectors of k targets can be expressed as: In the formula, the symbol ⊙ represents the Hadamard product.

[0096] The transmit array manifold can then be represented as:

[0097] The receiving angle steering vector associated with the k-th target can be expressed as: The receiver array manifold can be represented as:

[0098] The received data of the l-th snapshot after passing through the matched filter is represented as follows:

[0099] Y l =B∑ l A T +N l l = 1, 2, ..., L, where ∑ l =diag(s l ),

[0100]

[0101] ,β k Let f be the scattering coefficient of the k-th target. d，k Let be the Doppler frequency of the k-th target, T be the pulse emission period, and L be the number of snapshots. According to the definition of a tensor, the data from multiple snapshots can be stacked into a third-order tensor. The received data of the l-th snapshot in three-dimensional space can be represented as follows: in, Tensor The l-th slice in the third dimension. Based on tensor fundamentals and properties, the received data after matched filtering can be reformulated in the following tensor form: Where B is the receiver array manifold matrix; A is the transmitter array manifold matrix; It is a K×K×K unit tensor. It is a noise tensor and satisfies S = [s1, s2, ..., s L ] TLet S be the source signal term. It can be seen that the signal model satisfies the trilinear model of tensor decomposition. Therefore, parallel factorization can be used to obtain the estimate of the corresponding array manifold, and then the estimated array manifold can be used to obtain the estimates of the angle and distance. In the above tensor formula, B represents the received dimension data, A represents the transmitted dimension data, and S represents the time dimension data.

[0102] 3. Estimation of array manifold factor

[0103] In this invention, alternating least squares (ALS) is used to estimate the factor matrix, and the objective function can be written as:

[0104]

[0105] The above optimization problem can be transformed into the following sub-optimization problems:

[0106]

[0107]

[0108]

[0109] The objective function for B can then be written as:

[0110] B i+1 The solution is:

[0111] The objective function for A can be written as:

[0112] A i+1 The solution is:

[0113] The objective function for S can then be written as:

[0114] S i+1 The solution is:

[0115] After obtaining the estimates of factor matrices B, A, and S, they are respectively denoted as... and

[0116] 4. Angle estimation

[0117] Factor matrix estimation obtained by alternating least squares (ALS) In reality, it is an estimate of the receiver array manifold matrix, which can be used as an angle estimate. Let b k If B is the k-th column, then b k phase This can be represented as [0, πsin(θ)]k ), …, (N-1)sin(θ) k )] T After obtaining an estimate of B using ALS, the estimated... The k-th column is normalized to eliminate scale ambiguity, and denoted as . but By means of The phase is obtained and denoted as . Finally, the angle can be obtained by solving the following minimum optimization problem: in, h is the variable to be estimated. r，k，2 For θ k The relevant variables to be estimated. H r The expression is as follows: Among them, h r，k The least squares solution is θ k The estimated value for When the solution of the trilinear decomposition is unique, and The columns in the text are automatically paired. This means... and The columns correspond to the receive steering vector and transmit steering vector of the same target, respectively.

[0118] 5. Distance Estimation

[0119] After obtaining the transmitter array manifold matrix using the alternating least squares (ALS) method, the estimated transmitter array manifold matrix is ​​divided into two estimated submatrices. and First estimation subarray For the estimated transmit array manifold matrix The first M1 rows, the second estimated subarray For the estimated transmit array manifold matrix The next line M2;

[0120] For the first estimated subarray The k-th column is normalized and denoted as . For the first estimated subarray For the k-th emission steering vector, the two adjacent elements differ by a fixed phase, i.e. Then the fuzzy distance estimation of the k-th target It can be calculated using the following formula:

[0121]

[0122] ,in,() *represents the conjugate operator, angle(a) represents the phase of the complex number a, c represents the speed of light, M1 is the number of elements of the first transmitting subarray, represents the i1-th (0 < i1 < M1) element of represents the (i1 + 1)-th element of is the phase information related to the angle of the k-th target obtained in the angle estimation step;

[0123] Similarly, for the second estimation subarray the k-th column of is normalized and denoted as For the k-th transmitting steering vector of the second estimation subarray the difference between two adjacent elements is a fixed phase, that is Then the ambiguous distance estimation of the k-th target can be calculated by the following formula:

[0124]

[0125] , where, () * represents the conjugate operator, angle(a) represents the phase of the complex number a, c represents the speed of light, M2 is the number of elements of the second transmitting subarray, represents the i2-th (0 < i2 < M2) element of represents the (i2 + 1)-th element of is the phase information related to the angle of the k-th target obtained in the angle estimation step.

[0126] After obtaining the ambiguous distance estimations of the same target under the two estimation subarrays, the Chinese Remainder Theorem is used to traverse and solve to obtain the unambiguous estimation of the distance. Specifically, for the first estimation subarray and the second estimation subarray the maximum distance ambiguity numbers are set as J1 and J2 respectively; for the first estimation subarray and the second estimation subarray According to the Chinese Remainder Theorem, traverse j1 ∈ (0, 1,..., J1 - 1) and j2 ∈ (0, 1,..., J2 - 1), then the unambiguous distance after de-ambiguity is obtained by solving the following optimization problem; the function of the optimization problem is: ?

[0127] where, and Let be the maximum unambiguous distances at the corresponding frequency increments of emission subarray one and emission subarray two, respectively; c represents the speed of light; Δf1 is the frequency increment of the first emission subarray, and Δf2 is the frequency increment of the second emission subarray. After obtaining j1 and j2, denoted as and The unambiguous distance estimate of the k-th target is

[0128] To verify the effectiveness of the proposed method, extensive computational simulation experiments were conducted. In the simulation experiments, the maximum unambiguous range of the first transmitting subarray was set to 16981m, and the frequency increment Δf1 of the first transmitting subarray was set to 4416.71Hz; the maximum unambiguous range of the second transmitting subarray was set to 13987m, and the frequency increment of the second transmitting subarray was set to Δf2 = 5362.1219Hz. The transmission frequency of the first transmitting element was set to f0 = 3GHz. The two transmitting subarrays had 8 transmitting elements, and the number of receiving elements was set to 16. Assuming there were three far-field point targets located at (10°, 6.123km), (15°, 25.4km), and (30°, 60.3km) respectively. It can be seen that the distance of the first target is the unambiguous range, while the distances of the second and third targets are both in the ambiguous range in both transmitting subarrays. In the following simulations, the root mean square error (RMSE) is used to evaluate the estimation accuracy of the distance and angle by the algorithm. The root mean square error is defined as: Among them, T t Represents the number of Monte Carlo simulation experiments. and Representing θ k and r k The estimated value in the t-th experiment.

[0129] Figure 4 and Figure 5 The root mean square error (RMSE) plots are obtained from 1000 Monte Carlo simulations performed at different signal-to-noise ratios (SNR) with a snapshot count L=50 for estimating the target distance and angle. Figure 4 and Figure 5 In this invention, the method proposed is marked as 'the proposed method', and the Cramerro is marked as 'CRB'.

[0130] Figure 3 This is a scatter plot of 1000 simulation results with a signal-to-noise ratio of 0dB and a snapshot count of 50. Figure 3 As can be seen, the estimated values ​​of the three targets are relatively concentrated and closely approximate the true values, which fully demonstrates the effectiveness of the method proposed in this invention.

[0131] Figure 4 and Figure 5The root mean square error (RMSE) plots are generated from 1000 Monte Carlo simulations performed at different signal-to-noise ratios with a snapshot count of 50. Figure 4 and Figure 5 As can be seen, the angle and distance estimated by the method of the present invention approach the CRB, and both decrease with the increase of signal-to-noise ratio, which fully demonstrates that the proposed method has good distance and angle estimation performance.

[0132] Figure 6 and Figure 7 This is a root mean square error (RMSE) plot of target angle and distance estimation obtained from 1000 Monte Carlo simulations with different shot counts at a signal-to-noise ratio of 0dB. From... Figure 5 and Figure 6 As can be seen, the root mean square error of angle and distance estimation by the proposed method decreases with the increase of the number of snapshots.

[0133] This invention provides an unambiguous solution for angle and range in multi-target scenarios. Specifically, at the transmitting end, the transmitting array elements are divided into two subarrays. The maximum unambiguous ranges for the two subarrays are first set, and these maximum unambiguous ranges are coprime. Then, frequency increments are set based on their respective maximum unambiguous ranges. At the receiving end, the data output from the monostatic FDA-MIMO radar matched filter is stacked sequentially into a third-order tensor according to the number of snapshots. The data tensor is then subjected to trilinear decomposition to obtain an estimate of the array manifold. The phase of each column of the estimated receiving array manifold matrix is ​​normalized, and the angle estimate (i.e., the closed-form solution of the angle) is obtained using the least squares method. Based on the number of transmitting elements in each transmitting subarray, the estimated transmitting array manifold is divided into two estimation subarrays. For each column of each estimation subarray, the average phase method is used to obtain the corresponding ambiguous range estimate. Finally, the Chinese remainder theorem is used to iteratively solve for the unambiguous range estimate, thus achieving unambiguous range estimation.

[0134] This invention addresses the ambiguity in long-range FDA range estimation and the multidimensional structure (transmit dimension, receive dimension, and time dimension) of MIMO radar output data. It employs tensor-based parallel factor decomposition to estimate array flow, phase calculation combined with least squares method to estimate angle, phase averaging method to estimate ambiguous range, and Chinese remainder theorem to achieve range deambiguity, thereby obtaining high-precision angle and unambiguous range estimates.

[0135] Example 2

[0136] like Figure 8 As shown, the difference between this embodiment and Embodiment 1 is that this embodiment provides a parameter estimation system for FDA-MIMO radar range deambiguation, which uses the parameter estimation method for FDA-MIMO radar range deambiguation in Embodiment 1; the system includes:

[0137] The actual subarray division and setting unit is used to divide the FDA-MIMO radar transmitting array into two subarrays and set the maximum unambiguous distance and frequency increment for the two subarrays respectively.

[0138] The third-order tensor model construction unit is used to construct the third-order tensor model of the received data according to the snapshot order of the output data of the monostatic FDA-MIMO radar matched filter, based on the set maximum unambiguous distance and frequency increment.

[0139] The factorization unit is used to perform tensor parallel factorization on the third-order tensor model to obtain estimates of the receiver array manifold matrix and the transmitter array manifold matrix.

[0140] An angle estimation unit is used to normalize each column of the estimated receiver array manifold matrix and calculate the phase; based on the phase, the least squares method is used to obtain the angle estimate;

[0141] The fuzzy distance estimation unit is used to divide the estimated transmit array manifold matrix into two estimation subarrays, normalize each column of each estimation subarray, and process the normalized estimation subarrays using the phase averaging method to obtain the fuzzy distance estimate.

[0142] The defuzzy unit is used to estimate the fuzzy distance to the same target obtained from two estimating subarrays. It uses the Chinese remainder theorem to solve the problem and obtain the defuzzy distance estimate.

[0143] Specifically, the real subarray division and setting unit includes a real subarray division subunit and a setting subunit;

[0144] The real subarray is divided into sub-cells, which are used at the transmitting end of the FDA-MIMO radar to divide the transmitting array elements into two subarrays: the first transmitting subarray and the second transmitting subarray.

[0145] A sub-unit is configured to set a first maximum unambiguous distance for the first transmitting subarray and a second maximum unambiguous distance for the second transmitting subarray; a first frequency increment is set for the first transmitting subarray based on the first maximum unambiguous distance; and a second frequency increment is set for the second transmitting subarray based on the second maximum unambiguous distance; wherein the first maximum unambiguous distance and the second maximum unambiguous distance are coprime.

[0146] The execution process of each unit can be carried out according to the parameter estimation method flow of FDA-MIMO radar range deambiguation in Example 1, and will not be described in detail in this example.

[0147] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0148] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0149] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0150] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0151] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A parameter estimation method for FDA-MIMO radar range deambiguation, characterized in that, include: The FDA-MIMO radar transmitter array is divided into two subarrays, and the maximum unambiguous range and frequency increment are set for the two subarrays respectively. Based on the set maximum unambiguous distance and frequency increment, a third-order tensor model of the received data is constructed according to the snapshot order of the output data of the monostatic FDA-MIMO radar matched filter. Tensor parallel factorization is performed on the third-order tensor model to obtain estimates of the receiver array manifold matrix and the transmitter array manifold matrix; The phase is calculated after normalizing each column of the estimated receiver array manifold matrix. Based on the phase, the angle is estimated using the least squares method; The estimated transmit array manifold matrix is ​​divided into two estimation submatrices, and each column of each estimation submatric is normalized separately. The phase averaging method is then used to process the normalized estimation subarray to obtain an estimate of the fuzzy distance. The fuzzy distance estimates of the same target obtained from the estimation of two estimation subarrays are solved by traversing the Chinese remainder theorem to obtain the fuzzy distance estimate.

2. The parameter estimation method for FDA-MIMO radar range deambiguation according to claim 1, characterized in that, The FDA-MIMO radar transmitter array is divided into two subarrays, and maximum unambiguous range and frequency increment are set for each subarray, including: At the transmitting end of the FDA-MIMO radar, the transmitting array elements are divided into two subarrays: the first transmitting subarray and the second transmitting subarray. A first maximum unambiguous distance is set for the first transmitting subarray, and a second maximum unambiguous distance is set for the second transmitting subarray; wherein the first maximum unambiguous distance and the second maximum unambiguous distance are coprime. A first frequency increment is set for the first transmitting subarray based on the first maximum unambiguous distance; a second frequency increment is set for the second transmitting subarray based on the second maximum unambiguous distance.

3. The parameter estimation method for FDA-MIMO radar range deambiguation according to claim 2, characterized in that, The first transmitting subarray is equipped with M1 transmitting elements. In the first transmitting subarray, the transmission frequency of the m1th transmitting element is... Where f0 is the carrier frequency of the first transmitting element, and Δf1 is the frequency increment between two adjacent transmitting elements in the first transmitting subarray; <m1≤M1; The second transmitting subarray is equipped with M2 transmitting elements. In the second transmitting subarray, the transmission frequency of the m2th transmitting element is... Where Δf2 is the frequency increment between two adjacent transmitting elements in the second transmitting subarray; M1 <m2≤M1+M2。 4. The parameter estimation method for FDA-MIMO radar range deambiguation according to claim 1, characterized in that, Based on the set maximum unambiguous range and frequency increment, and following the snapshot order of the matched filter output data from the monostatic FDA-MIMO radar, a third-order tensor model of the received data is constructed, specifically as follows: Based on the set maximum unambiguous range and frequency increment, and following the snapshot order of the output data from the monostatic FDA-MIMO radar matched filter, the data are sequentially stacked along the third dimension of the third-order tensor to obtain the third-order tensor model of the received data; the third-order tensor model... The expression is: Where B is the receiver array manifold matrix; A is the transmitter array manifold matrix; and S is the source signal term. Represents the outer product of vectors; It is a K×K×K unit tensor. It is a noise tensor.

5. The parameter estimation method for FDA-MIMO radar range deambiguation according to claim 4, characterized in that, Performing tensor parallel factorization on the third-order tensor model includes: Initialize three matrices: and Where K is the number of radar targets, N is the number of receiving array elements, M = M1 + M2 is the number of transmitting array elements, M1 is the number of transmitting array elements equipped in the first transmitting subarray, and M2 is the number of transmitting array elements equipped in the second transmitting subarray. The alternating least squares method is used to iteratively update B, A, and S until convergence; let the value of B, A, and S in the i-th iteration be B. i A i and S i Then the update principle for B, A, and S in the (i+1)th iteration is: Among them, symbols Represents the Kronecker product, symbol Y represents the pseudo-inverse operator; (1) ,Y (2) ,Y (3) They are data tensors The expansion matrices for modes one, two, and three; I K(1) ,I K(2) ,I K(3) Tensors The expansion matrices for the first, second, and third modes; The estimates of B, A, and S at convergence are denoted as follows: and 6. The parameter estimation method for FDA-MIMO radar range deambiguation according to claim 1, characterized in that, The phase is calculated after normalizing each column of the estimated receiver array manifold matrix. Based on the phase, the angle is estimated using the least squares method, including: For the estimated receiver array manifold matrix The k-th column is normalized, and the phase is calculated and denoted as . The expression for the angle estimation of the k-th target: Among them, symbols Denotes the square of the F-norm. Let h be the quantity to be solved. r，k，2 For θ k The relevant variables to be estimated; H r The expression is as follows: Among them, h r,k The least squares solution is θ k The estimated value for 7. The parameter estimation method for FDA-MIMO radar range deambiguation according to claim 1, characterized in that, The estimated transmit array manifold matrix is ​​divided into two estimation submatrices, and each column of each estimation submatric is normalized separately. The normalized estimation subarray is processed using the phase averaging method to obtain an estimate of the fuzzy distance, including: The estimated manifold matrix of the transmission array is divided into two estimation submatrices. and First estimation subarray For the estimated transmit array manifold matrix The first M1 rows, the second estimated subarray For the estimated transmit array manifold matrix The next line M2; For the first estimated subarray The k-th column is normalized and denoted as . Through the first estimation subarray The fuzzy distance estimate of the k-th target. The calculation formula is: in,() * Let represent the conjugate operator, angle(a) denotes the phase calculation over the complex number a, c represents the speed of light, and M1 is the number of elements in the first emission subarray. express The i1th element, 0 < i1 < M1; express The (i1+1)th element, Δf1 represents the frequency increment of the first transmitting subarray; The phase information related to the k-th target angle is obtained in the angle estimation step; For the second estimated subarray The k-th column is normalized and denoted as . Through the second estimation subarray The fuzzy distance estimate of the k-th target. The calculation formula is: ,in,() * Let represent the conjugate operator, angle(a) denotes the phase calculation over the complex number a, c represents the speed of light, and M2 is the number of elements in the second emission subarray. express The i2th element, 0 < i2 < M2; express The (i2+1)th element, Δf2, represents the frequency increment of the second transmitting subarray; The phase information related to the k-th target angle is obtained from the angle estimation step.

8. The parameter estimation method for FDA-MIMO radar range deambiguation according to claim 1, characterized in that, For the fuzzy range estimates of the same target obtained from two estimating subarrays, the Chinese Remainder Theorem is used to solve the fuzzy range, achieving fuzzy range resolution, including: For the first estimated subarray Second estimation subarray Let J1 and J2 be the maximum ambiguity numbers for distance, respectively; For the first estimated subarray Second estimation subarray According to the Chinese Remainder Theorem, traversing j1∈(0,1,…,J1-1) and j2∈(0,1,…,J2-1), the distance after unfuzzification is obtained by solving the following optimization problem; the function of the optimization problem is: in, These are the maximum unambiguous distances for the corresponding frequency increments of transmitting subarray one and transmitting subarray two, respectively. c represents the speed of light; Δf1 is the frequency increment of the first emission subarray, and Δf2 is the frequency increment of the second emission subarray. After obtaining... and Afterwards, recorded as and The unambiguous distance estimate of the k-th target is 9. A parameter estimation system for FDA-MIMO radar range deambiguation, characterized in that, The system includes: The actual subarray division and setting unit is used to divide the FDA-MIMO radar transmitting array into two subarrays and set the maximum unambiguous distance and frequency increment for the two subarrays respectively. The third-order tensor model construction unit is used to construct the third-order tensor model of the received data according to the snapshot order of the output data of the monostatic FDA-MIMO radar matched filter, based on the set maximum unambiguous distance and frequency increment. The factorization unit is used to perform tensor parallel factorization on the third-order tensor model to obtain estimates of the receiver array manifold matrix and the transmitter array manifold matrix. An angle estimation unit is used to normalize each column of the estimated receiver array manifold matrix and calculate the phase; based on the phase, the least squares method is used to obtain an estimate of the angle; The fuzzy distance estimation unit is used to divide the estimated transmit array manifold matrix into two estimation subarrays, normalize each column of each estimation subarray, and process the normalized estimation subarrays using the phase averaging method to obtain the fuzzy distance estimate. The defuzzy unit is used to estimate the fuzzy distance to the same target obtained from two estimating subarrays. It uses the Chinese remainder theorem to solve the problem and obtain the defuzzy distance estimate.

10. The parameter estimation system for FDA-MIMO radar range deambiguation according to claim 9, characterized in that, The real subarray division and setting unit includes a real subarray division subunit and a setting subunit; The real subarray division sub-unit is used to divide the transmitting array element into two subarrays at the transmitting end of the FDA-MIMO radar: the first transmitting subarray and the second transmitting subarray. The setting subunit is used to set a first maximum unambiguous distance for the first transmitting subarray and a second maximum unambiguous distance for the second transmitting subarray; set a first frequency increment for the first transmitting subarray based on the first maximum unambiguous distance; and set a second frequency increment for the second transmitting subarray based on the second maximum unambiguous distance; wherein the first maximum unambiguous distance and the second maximum unambiguous distance are coprime.

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