A DOA estimation method based on a transceiving flip coprime MIMO radar structure

Abstract

The application discloses a DOA estimation method based on a transceiving flip co-prime MIMO radar structure, and the method is characterized in that: a transceiving flip co-prime MIMO radar structure is established, and coherent target and incoherent target DOA estimations are respectively performed through a sum joint array and a sum-difference joint array. In view of the problem that spatial smoothing technology cannot utilize the received information contained in discrete virtual array elements, based on the atomic norm minimization theory, after filling the virtual array elements at discontinuous points, a convex optimization model for recovering and filling the equivalent received signals of the virtual array elements is constructed, and then non-grid sparse reconstruction DOA estimation is realized, and the accuracy of the co-prime MIMO radar DOA estimation is further improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of array signal processing, and particularly relates to a DOA estimation method based on a transceiving flip coprime MIMO radar structure. BACKGROUND

[0002] DOA (Direction-of-Arrival, DOA) estimation, as an important branch of array signal processing, has a wide range of applications in radar, sonar and communication fields. The performance of DOA estimation based on a uniform linear array (ULA) is seriously degraded when dealing with "multiple" and "dense" targets, and even fails. Therefore, for large-scale target direction finding, it is necessary to expand the array aperture by increasing the number of antennas, which undoubtedly increases the system overhead and complexity. Sparse arrays (such as minimum redundancy arrays, nested arrays and coprime arrays) have significant advantages over ULA in terms of improving degrees of freedom, suppressing element mutual coupling and reducing redundancy, and can improve the DOA estimation accuracy and the number of distinguishable sources, effectively solving the underdetermined target DOA estimation problem. In addition, the great performance advantage of sparse arrays further promotes the design innovation of MIMO (Multiple-Input Multiple-Output, MIMO) radar, and after combining the two, the array structure optimization and direction finding algorithm design can be carried out from the perspective of "joint array", further improving the target estimation performance.

[0003] While the co-prime array multiple-input multiple-output (MIMO) radar takes the co-prime array as the transmitting array and receiving array of the MIMO radar respectively, it combines the advantages of the co-prime array which greatly expands the array aperture and the MIMO radar which can virtually generate a virtual array with a number of virtual array elements far more than the real array elements, and can achieve excellent estimation or detection performance in the scene with limited hardware and software conditions. However, the current co-prime MIMO radar DOA estimation has the following problems: first, most of the existing researches are based on simple co-prime arrays, and the array arrangement mode of the improved co-prime array under the MIMO system is relatively less studied, and the virtual array element expansion advantage of the co-prime MIMO radar has not been tapped. Second, there are a large number of discrete holes in the "joint array" structure of the co-prime MIMO radar. If the spatial smoothing technique is used for DOA estimation, only the middle continuous virtual array elements can be selected and the discrete virtual array elements can be discarded to ensure model matching, but the target information contained in the discrete virtual array elements will be lost, resulting in the decline of DOA estimation performance. Although the compressed sensing (CS) algorithm is not limited by the continuity of the array elements, the calculation amount will increase sharply with the refinement degree of the spatial grid points. Third, under the condition of coherent targets, the signal covariance matrix no longer satisfies the Toeplitz structure, the vectorized "sum-difference joint array" needs to estimate more non-zero elements due to the influence of the coherence coefficient, and the steering vector is transformed from Khatri-Rao product to Kronecker product, which makes it difficult to be used for coherent target DOA estimation. SUMMARY

[0004] In view of the above problems, the present application provides a DOA estimation method based on a transceiver flip co-prime MIMO radar structure, and the technical scheme adopted by the present application is as follows:

[0005] A DOA estimation method based on a transceiver flip co-prime MIMO radar structure, characterized by comprising the following steps:

[0006] Step 1: establishing the "sum joint array" and "sum-difference joint array" signal models of the sparse array MIMO radar;

[0007] Step 2: constructing a transceiver flip co-prime MIMO radar to obtain the "sum joint array" of the transceiver flip co-prime MIMO radar;

[0008] Step 3: performing difference processing on the obtained "sum joint array" of the transceiver flip co-prime MIMO radar to obtain the "sum-difference joint array" of the transceiver flip co-prime MIMO radar;

[0009] Step 4: performing non-coherent target DOA estimation based on the "sum-difference joint array" of the transceiver flip co-prime MIMO radar;

[0010] Step 5: Perform coherent target DOA estimation based on the "sum joint array" of the transmit-receive flip coprime MIMO radar.

[0011] Furthermore, the sparse array MIMO radar signal model constructed in step 1 is as follows:

[0012] 1) The expression for the received signal y(t) is:

[0013]

[0014] Where b(t)=[b1(t),b2(t),···,b M (t)] T For transmitting signals, M is the number of transmitting antennas; w(t)=[ω1(t),ω2(t),···,ω N (t)] T θ is the Gaussian white noise vector at the receiving antenna, where N is the number of receiving antennas; k From the perspective of the target, β k α is the reflection coefficient; K is the number of targets; α t (θ k ) and α r (θ k The transmit steering vector and receive steering vector of the k-th target are respectively:

[0015]

[0016]

[0017] Where, p tm ∈P t and p rm ∈P r These represent the positions of the transmitting and receiving antennas, respectively, and p t1 =p r1 =0;

[0018] 2) The expression for the received signal y(t) after matched filtering is:

[0019]

[0020] Among them, A t =[α t (θ1),α t (θ2),…,α t (θ K )] and A r =[α r (θ1),α r (θ2),…,α r (θ K)] respectively represent the stream pattern matrix of the transmitting array and the receiving array; s(t) = [β1, β2, ···, β K ] T is a vector composed of reflection coefficients; n(t) is a Gaussian white noise vector;

[0021] 3) The covariance matrix R of x(t) is:

[0022]

[0023] wherein E[·] represents the mathematical expectation; is the source covariance matrix, and diag[·] represents a diagonal matrix; represents the power of the kth source; A = A t ⊙A r ; represents the noise power; L is the number of sampling snaps;

[0024] 4) The virtual domain observation vector r is:

[0025]

[0026] wherein

[0027]

[0028] is the array stream pattern matrix of the virtual domain observation vector r; Further, the specific operation steps of step 2 include:

[0029] Step 21: The transmitting array of the transceiver flip reciprocal MIMO radar adopts an augmented spread reciprocal array, and the unit element spacing d is normalized to the wavelength λ to obtain a transmitting element position set P t is:

[0030]

[0031] wherein P t1 and P t2 respectively represent the left half and the right half of the transmitting element position set;

[0032] The receiving element position set P r is:

[0033]

[0034] wherein P r1 and P r2 respectively represent the left half and the right half of the receiving element position set; step 22: based on formula (8) and formula (9), the “sum joint array” of the transceiver flip reciprocal MIMO radar is obtained is:

[0035]

[0036] wherein, and 0≤m≤2M-1, 0≤n≤N-1;

[0037] the forward element set of its "sum-joint array" is:

[0038]

[0039] wherein then equation (11) is equivalent to:

[0040]

[0041] Further, the "sum-joint array" of the transceiver-flip-coprime MIMO radar obtained in step 22 has a range of [(M-1)(N-1), 2MN-1] for the forward continuous elements and a range of [- (2MN-1), -(M-1)(N-1)] for the backward continuous elements; the set has a total number of elements of 4MN-1.

[0042] Further, the "sum-difference joint array" of the transceiver-flip-coprime MIMO radar obtained in step 3 is:

[0043]

[0044] and the set has a range of [- (5MN-M-N-1), 5MN-M-N-1] for the continuous elements;

[0045] has a total number of elements of 11MN-3M-3N;

[0046] has a number of discrete elements of (M-1)(N-1).

[0047] Further, the specific operation steps of step 4 are:

[0048] Step 41: filling the discrete holes in the "sum-difference joint array" of the transceiver-flip-coprime MIMO radar;

[0049] Step 42: constructing a convex optimization problem for filling the virtual array element equivalent received signal, and solving the convex optimization problem to obtain the reconstructed covariance matrix;

[0050] ​Step 43: Decompose the eigenvalues ​​of the reconstructed covariance matrix and use the MUSIC algorithm to obtain the spatial target angle.

[0051] Furthermore, the specific steps of step 41 are as follows:

[0052] Step 411: Perform redundant averaging on the virtual domain observation vector r, and fill in 0s at discrete points to convert the non-uniform virtual array into a uniform virtual array, resulting in a new virtual domain observation vector r′:

[0053]

[0054] Where A′ has a dimension of (11MN-3M-3N+2l0)×K, and 2l0 is the total number of zeros to fill; e=[0,···,0,1,0,···,0] T ;

[0055] Step 412: Perform spatial smoothing on equation (34) to obtain the forward smoothing matrix R. x .

[0056] Furthermore, the specific operational steps of step 42 include:

[0057] Step 421: Using the forward smoothing matrix R x Toeplitz matrix with the smallest difference between corresponding elements As a new covariance matrix, a convex optimization problem based on the theory of minimizing atomic norms is established:

[0058]

[0059] Among them, ||·|| r f represents the atomic norm. Z (·) represents the Hadamard product, Z = z T Let z represent a binary matrix, z represent a binary vector, and δ represent the error threshold for reconstructing the covariance matrix;

[0060] Step 422: Transform the non-convex problem shown in equation (35) into the convex problem shown in equation (36):

[0061]

[0062] Where ξ is the regularization parameter;

[0063] Step 423: Transform equation (36) into the problem of minimizing the trace of a matrix:

[0064]

[0065] where μ = ξ / [l0+1+(11MN-3M-3N-1) / 2];

[0066] Step 424: solve equation (40) by CVX toolbox to obtain the reconstructed covariance matrix Further, the specific operation steps of step 5 are:

[0067] Step 51: modify the signal model of equation (4) to a coherent signal model:

[0068]

[0069] wherein, and is a non-zero complex constant;

[0070] Step 52: select the data corresponding to the virtual array element position generated by the "sum joint array" from the signal model of equation (41), and rearrange in order to obtain a new observation vector:

[0071] x2(t) = A2s(t) + n2(t) (42)

[0072] wherein A2 is a (4MN-1) × K dimensional direction matrix;

[0073] Step 53: fill in the 2(MN-M-N+1) holes in the middle section [- (M-1)(N-1), 0) and (0, (M-1)(N-1)] of the "sum joint array" of the transceiver flip coprime MIMO radar, and supplement 0 at the hole position, so as to form the output signal of the ULA structure:

[0074] x U (t) = A U s(t) + n U (t) (43)

[0075] wherein A U represents the array flow matrix corresponding to the ULA structure formed after filling in the holes, and the dimension is U × K, wherein U = 6MN-2M-2N+1;

[0076] The covariance matrix R of x U (t) is:

[0077]

[0078] wherein R r = Eps r (t)s r (t) H ] represents the power of the reference signal; L is the number of sampling shots;

[0079] Step 54: constructing a Toeplitz matrix T based on the i-th (1≤i≤U) row of the obtained covariance matrix R U is:

[0080]

[0081] where rank(T U )=(U+1) / 2;

[0082] Step 55: selecting the Toeplitz matrix T U with the minimum difference of corresponding elements as the new covariance matrix An optimization problem based on the atomic norm minimization theory is established:

[0083]

[0084] where f G (·) represents the Hadamard product, and G is a binary matrix constructed;

[0085] Step 56: converting formula (46) into a minimum problem of the trace of a matrix

[0086]

[0087] Step 57: solving formula (48) to obtain the reconstructed covariance matrix

[0088] Step 58: performing eigenvalue decomposition on the reconstructed covariance matrix , and obtaining the spatial target angle by using the MUSIC algorithm.

[0089] Further, the binary matrix G construction step in step 55 is:

[0090] Step 551: taking a binary vector g, so that the element values of g are one-to-one corresponding to each virtual array element in A U , and the filled virtual array element corresponds to 0, and the original virtual array element corresponds to 1;

[0091] Step 552: using the i-th row of gg T to construct a Toeplitz matrix:

[0092]

[0093] The present application has the following beneficial effects:

[0094] The application provides a DOA estimation method based on a transceiver flip reciprocal MIMO radar structure. First, a transceiver flip reciprocal MIMO radar is established, the transmitting array of which adopts an augmented extended reciprocal array, and the receiving array is obtained by flipping the transmitting array along the origin; second, the analytical expressions of the continuous virtual array element range, the discrete virtual array element number and the total virtual array element number are derived from the joint array perspective, compared with the traditional reciprocal MIMO radar, the virtual aperture expansion effect of the transceiver flip reciprocal MIMO radar is stronger, and the degree of freedom is higher; finally, the DOA estimation of coherent targets and incoherent targets is carried out from the perspectives of the sum joint array and the sum-difference joint array, and aiming at the problem of holes in the joint array, a convex optimization problem of filling the equivalent receiving signal of the virtual array element is constructed, and the DOA estimation is carried out combined with the MUSIC algorithm. Finally, the rationality of the array configuration and the effectiveness of the algorithm are verified through simulation experiments. BRIEF DESCRIPTION OF DRAWINGS

[0095] Figure 1 It is a transceiver flip reciprocal MIMO radar structure diagram;

[0096] Figure 2 It is a variable and distribution diagram;

[0097] Figure 3 It is a comparison of virtual array element expansion of different reciprocal MIMO radars;

[0098] Figure 4 (a)-(c) are comparisons of incoherent target spatial spectrum of different reciprocal MIMO radars; wherein Figure 4 (a) is a second-order reciprocal MIMO radar, Figure 4 (b) is a reciprocal subarray MIMO radar, Figure 4 (c) is a transceiver flip reciprocal MIMO radar;

[0099] Figure 5 It is a comparison of coherent target spatial spectrum of different reciprocal MIMO radars;

[0100] Figure 6 (a)-(b) are comparison diagrams of root mean square error (RMSE) of incoherent targets of different reciprocal MIMO radars; wherein Figure 6 (a) is an RMSE curve changing with signal to noise radio (SNR), Figure 6 (b) is an RMSE curve changing with fast shots;

[0101] Figure 7(a)-(b) show a comparison of the probability of success (PS) of coherent targets using different coprime MIMO radars; where Figure 7 (a) shows the curve of PS as a function of SNR. Figure 7 (b) is the curve of PS as a function of the number of snapshots. Detailed Implementation

[0102] To enable those skilled in the art to better understand the technical solutions of the present invention, the technical solutions of the present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0103] I. Constructing a Sparse Array MIMO Radar Signal Model

[0104] Assume a sparse array MIMO radar consists of M transmit antennas and N receive antennas, and the set of transmit antenna locations is P. t ={p tm |m=1,2,…,M}, and the set of receiving antenna locations is P. r ={p rn |n=1,2,…,N}. If there are K narrowband far-field incoherent targets in space, and the angle and reflection coefficient of the k-th target (1≤k≤K) are θ and θ, respectively. k and β k Then the received signal y(t) is:

[0105]

[0106] In the formula, b(t) = [b1(t), b2(t), ..., b M (t)] T For transmitting signals; w(t)=[ω1(t),ω2(t),···,ω N (t)] T α is the Gaussian white noise vector at the receiving antenna; t (θ k ) and α r (θ k The transmit steering vector and receive steering vector of the k-th target are respectively:

[0107]

[0108]

[0109] Where, p tm ∈P t and p rm ∈P r These represent the positions of the transmitting and receiving antennas, respectively, and p t1 =p r1 =0; λ is the signal wavelength.

[0110] The transmitted signals are orthogonal, i.e., R b = E[b(t)b(t H )] = I M The matched filtering of the received signal y(t) can be obtained as follows:

[0111]

[0112] where A t = [α t (θ1), α t (θ2), …, α t (θ K )] and A r = [α r (θ1), α r (θ2), …, α r (θ K )] represent the steering matrices of the transmitting array and the receiving array, respectively; s(t) = [β1, β2, …, β K ] T is a vector of reflection coefficients; and n(t) is a Gaussian white noise vector.

[0113] The covariance matrix R of x(t) is:

[0114]

[0115] where is the covariance matrix of the sources; represents the power of the kth source; A = A t ⊙ A r ; represents the noise power; and L is the number of sampling snapshots.

[0116] The virtual domain observation vector r can be obtained by dequantizing R as follows:

[0117]

[0118] where B is the direction matrix of the virtual domain observation vector r, and:

[0119]

[0120] As can be seen from equation (7), the virtual element positions corresponding to B are essentially composed of the "sum-difference joint array" of the physical element positions.

[0121] II. Transmit-receive flip reciprocal MIMO radar DOA estimation based on hole filling

[0122] The application provides a DOA estimation method based on a transceiving flip reciprocal MIMO radar structure, which comprises the following steps: firstly, establishing a transceiving flip reciprocal MIMO radar; secondly, deducing analytical expressions of a continuous virtual array element range, a discrete virtual array element number and a total virtual array element number from a joint array perspective; finally, carrying out DOA estimation of coherent targets and non-coherent targets from the perspectives of a sum joint array and a sum-difference joint array respectively, and constructing a convex optimization problem of filling equivalent receiving signals of virtual array elements, and combining a MUSIC algorithm to carry out DOA estimation.

[0123] 1. A transceiving flip reciprocal MIMO radar structure

[0124] Figure 1 It is a structural schematic diagram of the transceiving flip reciprocal MIMO radar. The transmitting array adopts an augmented spread reciprocal array, wherein the ULA of the left half part comprises 2M antennas with an interval of Nd, the ULA of the right half part comprises N antennas with an interval of Md, and the antennas coinciding with each other are regarded as zero points; the receiving antenna positions can be obtained by flipping the transmitting antennas along the origin. d = λ / 2 is the unit array element interval, and λ is the signal wavelength. Therefore, the total physical array element number is 4M + 2N - 2.

[0125] After normalizing the unit array element interval d to the wavelength λ, the transmitting array element position set P t is obtained.

[0126]

[0127] Wherein, P t1 and P t2 respectively represent the transmitting array element position set of the left half part and the right half part.

[0128] The receiving array element position set P r is obtained.

[0129]

[0130] Wherein, P r1 and P r2 respectively represent the receiving array element position set of the left half part and the right half part.

[0131] Definition 1: The sum joint array of the transceiving flip reciprocal MIMO radar is obtained from formula (8) and formula (9) is obtained.

[0132]

[0133] Wherein, And 0 ≤ m ≤ 2M - 1, 0 ≤ n ≤ N - 1.

[0134] Theorem 1: The sum joint array of the transceiver-flipped coprime MIMO radar has the following properties:

[0135] (a) The set has the range of [(M-1)(N-1), 2MN-1] for the positive continuous elements and the range of [- (2MN-1), -(M-1)(N-1)] for the negative continuous elements.

[0136] (b) The set has the total number of elements of 4MN-1.

[0137] Proof of Theorem 1:

[0138] (a) According to equation (10), the set of positive elements of the sum joint array of the transceiver-flipped coprime MIMO radar is:

[0139]

[0140] Since equation (11) is equivalent to:

[0141]

[0142] Next, it needs to be proved that there exist 0≤m≤2M-1 and 0≤n≤N-1 such that traverses all integers from (M-1)(N-1) to 2MN-1.

[0143] Multiplying both ends of 0≤n≤N-1 by M gives:

[0144] 0≤Mn≤M(N-1) (13)

[0145] Combining equation (13) and we have:

[0146] (M-1)(N-1)-M(N-1)≤Nm≤2MN-1 (14)

[0147] Simplifying equation (14) gives:

[0148] -1+1 / N≤m≤2M-1 / N (15)

[0149] Since N≥1, we have 0≤m≤2M-1. Therefore, the set has the range of [(M-1)(N-1), 2MN-1] for the positive continuous elements. It is easy to know from equation (10) that the set is positive-negative symmetric structure, so the set has the range of [- (2MN-1), -(M-1)(N-1)] for the negative continuous elements.​

[0150] (b) According to the above conclusion, the discrete element ranges of are [0, (M-1)(N-1)] and [2MN-1, N(2M-1)+M(N-1)], respectively. Since M and N are coprime, the total number of discrete elements in is (M-1)(N-1).

[0151] Therefore, the total number of elements in the "sum joint array" of the transceiver flip coprime MIMO radar is:

[0152]

[0153] Q.E.D.

[0154] According to Theorem 1, there are multiple discrete points in the SCA structure of the transceiver flip coprime MIMO radar, and difference processing can complete the discrete holes and further improve the number of virtual array elements.

[0155] Definition 2: Further difference processing of the "sum joint array" of the transceiver flip coprime MIMO radar can obtain its corresponding "sum-difference joint array" set as:

[0156]

[0157] Theorem 2: The "sum-difference joint array" of the transceiver flip coprime MIMO radar has the following characteristics:

[0158] (a) The continuous element range in the set is [- (5MN-M-N-1), 5MN-M-N-1];

[0159] (b) The total number of elements in the set is 11MN-3M-3N;

[0160] (c) The number of discrete elements in the set is (M-1)(N-1).

[0161] Proof of Theorem 2:

[0162] (a) Take the forward element in the set as an example. According to equation (17), we have:

[0163]

[0164] where

[0165] Combining Theorem 1(a), we have and​ Figure 1 is a distribution diagram of the variable Δ1, Figure 2

[0166] Therefore, the variable Δ1 is composed of nine parts, in which the elements contained in Δ2 and Δ4 are the same, the elements contained in Δ3 and Δ7 are the same, and the elements contained in Δ6 and Δ8 are the same. The variable of each part is analyzed as follows:

[0167]

[0168] Both Δ5 and Δ9 are in the discrete segment, and the elements contained in Δ5 and Δ9 are the same. Therefore, the sum of Δ5 and Δ9 is a discrete variable.

[0169]

[0170]

[0171]

[0172] Therefore, the sum of Δ4 and Δ2 is a continuous variable, and the value range of the elements is [(M-1)(N-1), 3MN-M-N].

[0173]

[0174] Both Δ5 and Δ9 are in the discrete segment, and the elements contained in Δ5 and Δ9 are the same. Therefore, the sum of Δ5 and Δ9 is a discrete variable.

[0175]

[0176] Therefore, the sum of Δ4 and Δ2 is a discrete variable.

[0177]

[0178] In the formula, Δ4 and Δ2 contain the same range of continuous elements, which is [(M-1)(N-1), 3MN-M-N].

[0179]

[0180] Both Δ5 and Δ9 are in the discrete segment, and the elements contained in Δ5 and Δ9 are the same. Therefore, the sum of Δ5 and Δ9 is a discrete variable.

[0181]

[0182] is in the discrete segment,​​​​​​​​​​​​ is in the continuous segment, and

[0183] (3MN-M-N)-(2MN-1) = (M-1)(N-1) (27)

[0184] According to formula (21), the sum of the two is a continuous variable, and the element value range is [3MN-M-N, 5MN-M-N-1].

[0185]

[0186] In the formula, Δ7 and Δ3 contain the same discrete elements.

[0187]

[0188] In the formula, Δ8 and Δ6 contain the same continuous element range [3MN-M-N, 5MN-M-N-1].

[0189]

[0190] In the formula, and are in the discrete segment, and both contain the same discrete elements, so the sum of the two is a discrete variable.

[0191] In summary, the continuous element range in the positive value set is [(M-1)(N-1), 5MN-M-N-1]. In addition, according to formula (21), the difference between (2MN-1) and (M-1)(N-1) can complement the continuous element (M-1)(N-1), so the continuous element range in the positive value set can be further determined as [0, 5MN-M-N-1]. Therefore, the continuous element range in the sum-difference joint array of the transceiver flip reciprocal MIMO radar is [-(5MN-M-N-1), 5MN-M-N-1].

[0192] (b) According to the analysis of Δ9 in the proof process of theorem 2(a), the maximum element in the positive value set is 2(3MN-M-N), so the number of discrete elements in does not exceed:

[0193] 2(3MN-M-N)-(5MN-M-N-1) = (M-1)(N-1) (31)

[0194] Since M and N are coprime, the total number of discrete elements in is (M-1)(N-1) / 2.

[0195] Therefore, The total number of elements in the set is:

[0196] 2(5MN-M-N-1)+2[(M-1)(N-1) / 2]+1=11MN-3M-3N (32)

[0197] (c) From the proof process of Theorem 2(b), the number of discrete elements in the set is:

[0198] (M-1)(N-1) / 2+(M-1)(N-1) / 2=(M-1)(N-1) (33)Q.E.D.

[0199] To illustrate the advantage of the virtual array expansion of the transceiver flip-coprime MIMO radar, Table 1 summarizes the total number of virtual array elements under three different coprime MIMO radar "sum-difference joint array" structures. Among them, the transceiver arrays of the second-order coprime MIMO radar adopt complete augmented coprime arrays, while the coprime subarray MIMO radar adopts two sparse uniform linear arrays as the transceiver array, so that It can be seen that the transceiver flip-coprime MIMO radar can increase the total number of virtual array elements without increasing the number of physical array elements.

[0200] Table 1 Comparison of virtual array element numbers under "sum-difference joint array" structure

[0201]

[0202] 2, Hole filling

[0203] If the radar signals of multiple targets of the MIMO sky-wave radar are coherent signals, combine Theorem 1 to estimate the DOA of the coherent targets; if the radar signals of multiple targets of the MIMO sky-wave radar are incoherent signals, combine Theorem 2 to estimate the incoherent targets.

[0204] (1) Incoherent target DOA estimation

[0205] From Theorem 2, there are still discontinuous points in the "sum-difference joint array" of the transceiver flip-coprime MIMO radar. To fully utilize all the expanded virtual array elements, the discrete holes need to be filled to estimate the DOA.

[0206] First, the virtual domain measurement vector r is redundantly averaged, and 0 is filled at the discrete points to convert the non-uniform virtual array into a uniform virtual array. At this time, the signal model formula (6) is updated as:

[0207]

[0208] In the formula, the dimension of A′ increases from (11MN-3M-3N)×K to (11MN-3M-3N+2l0)×K, where 2l0 is the total number of zeros filled; e=[0,···,0,1,0,···,0] T .

[0209] Secondly, spatial smoothing is performed on equation (34) to obtain the forward smoothing matrix R. x The positions padded with zeros do not actually receive signals. In this case, we can find a Toeplitz matrix that has the smallest difference between its corresponding elements and the matrix. As the new covariance matrix, the optimization problem based on the atomic norm minimization theory is now established as follows:

[0210]

[0211] In the formula, ||·|| r f represents the atomic norm; Z (·) represents the Hadamard product, which means that the matrix elements in the parentheses are multiplied one by one with the corresponding elements of matrix Z.

[0212] Z = zz T A is a binary matrix, z is a binary vector whose element values ​​correspond one-to-one with each virtual element in A′, that is, the filled virtual element corresponds to 0, and the original virtual element corresponds to 1; δ represents the error threshold of the reconstructed covariance matrix.

[0213] The optimization problem in equation (35) is nonconvex and can be equivalently expressed as:

[0214]

[0215] Where ξ is the regularization parameter.

[0216] when rank At that time, it can be uniquely decomposed into: using the Vandermonde decomposition method.

[0217]

[0218] Where r(θ) represents the guiding vector of the first submatrix in the spatial smoothing.

[0219] but traces satisfy:

[0220]

[0221] Therefore, the relationship between the atomic norm term and the matrix can be expressed as:

[0222]

[0223] Thus, formula (36) can be further expressed as a minimization problem of the trace of a matrix:

[0224]

[0225] In the formula, μ = ξ / [l0+1+(11MN-3M-3N-1) / 2].

[0226] The optimization problem constructed by the above formula is a typical convex optimization model, which can be solved by the CVX toolbox.

[0227] Again, the characteristic decomposition of formula (40) is carried out, and the DOA estimation is carried out in combination with the MUSIC algorithm.

[0228] The steps of the algorithm based on hole filling for the non-coherent target DOA estimation of the transceiver flip coprime MIMO radar are shown in Table 2.

[0229] Table 2 Algorithm steps

[0230]

[0231] The traditional subspace algorithm only selects continuous virtual array elements for DOA estimation, and does not use the spatial information contained in the discrete virtual array elements, and the estimation performance is poor; the CS algorithm has high accuracy, but with the dense division of the pre-defined grid points N θ , the operation amount will increase exponentially, specifically order of magnitude, which is not conducive to the application of practical engineering; the all observation values in the reconstructed covariance matrix of the DOA estimation algorithm based on hole filling are obtained by solving the global optimization problem, which ensures the accuracy of the covariance matrix reconstruction, and the calculation complexity mainly comes from the Vandermonde decomposition operation, specifically O(P 3 ) order of magnitude, wherein P is the model order and is much smaller than N θ . The algorithm has lower operation amount than the CS algorithm and higher estimation accuracy than the traditional subspace algorithm.

[0232] (2) Coherent target DOA estimation

[0233] According to theorem 1, the middle section (i.e. [- (M-1) (N-1), 0) and (0, (M-1) (N-1)]) of the “sum joint array” of the transceiver flip coprime MIMO radar has a hole, so the subspace DOA estimation algorithm based on spatial smoothing technology and Toeplitz matrix reconstruction cannot be directly used for coherent target DOA estimation, and the hole needs to be filled first to form a ULA structure.

[0234] When the signal source is coherent, the signal model corresponding to formula (4) needs to be modified as:​

[0235]

[0236] wherein, and is a non-zero complex constant;

[0237] The data corresponding to the virtual array position produced by the sum-union array is selected from the formula (41) and rearranged in order to obtain a new observation vector:

[0238] x2(t) = A2s(t) + n2(t) (42)

[0239] wherein A2 is a direction matrix of (4MN-1) x K dimension;

[0240] According to the formula (10), the virtual array range corresponding to the sum-union array of the transceiver flip coprime MIMO radar is [- (3MN-M-N), 3MN-M-N], and A2 only has 4MN-1 rows (corresponding to 4MN-1 virtual array elements), so 2(MN-M-N+1) holes need to be filled.

[0241] The output signal of the ULA structure formed after filling 0 in the hole position is:

[0242] x U (t) = A U s(t) + n U (t) (43)

[0243] wherein A U represents the direction matrix corresponding to the ULA structure formed after filling the holes, and the dimension is U x K, wherein U = 6MN-2M-2N+1.

[0244] The covariance matrix of x U (t) is:

[0245]

[0246] wherein R r = E[s r (t)s r (t) H ] represents the power of the reference signal; and L is the number of sampling snaps.

[0247] Using the i-th (1≤i≤U) row of R, a Toeplitz matrix can be constructed as follows:

[0248]

[0249] wherein rank(T U ) = (U+1) / 2.

[0250] However, the position of the 0 is actually not received signal, then can be based on the atomic norm minimization theory to find a corresponding element difference with the smallest Toeplitz matrix As a new covariance matrix:

[0251]

[0252] In the formula, f G (·) represents the Hadamard product, that is, the elements in the brackets are multiplied by the corresponding elements of the matrix G. The construction method of the matrix G is as follows:

[0253] First, take a binary vector g, whose element value is corresponding to each virtual array element of A U , that is, the filled virtual array element corresponds to 0, and the original virtual array element corresponds to 1.

[0254] Secondly, the i-th (1≤i≤U) row of gg T is used to construct the following Toeplitz matrix:

[0255]

[0256] In the formula, G is a binary matrix.

[0257] Finally, formula (46) can be expressed as a minimization problem of the trace of the matrix:

[0258]

[0259]

[0260] The matrix obtained by formula (48) is subjected to eigenvalue decomposition, and the DOA estimation is carried out in combination with the MUSIC algorithm.

[0261] The steps of the transceiver flip coprime MIMO radar coherent target DOA estimation algorithm based on hole filling are shown in Table 3:

[0262] Table 3 Algorithm steps

[0263]

[0264] Embodiment

[0265] In order to further illustrate the reliability and superiority of the method provided by the present application, the algorithm is verified by simulation experiment.

[0266] 1. Simulation experiment scene setting

[0267] Assume that the total number of array elements used for simulation is 12, i.e. M=2, N=3, and the specific configurations of the transceiving arrays of each co-prime MIMO radar are shown in Table 4.

[0268] Table 4 Configuration of transceiving arrays of each co-prime MIMO radar when the number of array elements is 12

[0269]

[0270] 2. Simulation experiment and result analysis

[0271] (1) Experiment 1: Comparison of virtual array element distribution

[0272] Figure 3 The virtual array element expansion of different co-prime MIMO radars is shown in the figure, in which the hollow symbols represent the virtual array element distribution under the "sum joint array" structure, and the solid symbols represent the virtual array element distribution under the "sum-difference joint array" structure. It can be seen from the figure that the transceiving flip co-prime MIMO radar has a larger virtual array aperture than the traditional co-prime MIMO radar, and the transceiving flip co-prime MIMO radar has more continuous virtual array elements under the "sum-difference joint array" structure.

[0273] (2) Experiment 2: Comparison of spatial spectrum

[0274] Firstly, the non-coherent target DOA estimation performance of different co-prime MIMO radars is compared. Assume that 15 non-coherent targets in space are located at [-70°:10°:70°], the search region is [-90°, 90°] with an interval of 0.01°, SNR=0dB, and L=200. From Figure 4 It can be seen that the DOA estimation algorithm based on hole filling has better angle measurement performance than the DOA estimation algorithm that discards discrete virtual array elements, because hole filling is equivalent to increasing the virtual array aperture, so that the receiving information in the discrete virtual array elements can be utilized, and all the expanded virtual array elements play a role in DOA estimation. In addition, the second-order co-prime MIMO radar has poor estimation ability for the above-mentioned 15 non-coherent targets due to its small virtual array aperture, while the number of virtual array elements generated by the co-prime subarray MIMO radar and the transceiving flip co-prime MIMO radar enables them to effectively identify the 15 non-coherent targets.

[0275] Secondly, the coherent target DOA estimation performance of different co-prime MIMO radars is compared, and the spatial spectrum is shown in Figure 5 . Assume that 7 coherent targets in space are located at [-30°:10°:30°], the coherence coefficient between the targets is all set to 1, the search region is [-90°, 90°] with an interval of 0.01°, SNR=0dB, and L=200. Combined with Figure 3 and Figure 5It can be seen that the transceiver-flip-coprime MIMO radar with the sum-difference array structure can estimate all the targets due to its larger virtual aperture, the coprime subarray MIMO radar can only estimate five targets in [-30°:10°:30°], and the second-order coprime MIMO radar cannot estimate the targets effectively due to its smaller virtual aperture.

[0276] (3) Experiment three: RMSE comparison

[0277] Figure 6 The mean square error of the non-coherent target DOA estimation of different coprime MIMO radars is depicted. It is assumed that three non-coherent targets are located at {-10°, 0°, 10°} and the number of Monte Carlo experiments is 200. Figure 6 (a) is the curve of RMSE changing with SNR, where the number of snapshots L = 200; Figure 6 (b) is the curve of RMSE changing with the number of snapshots, where SNR = 0 dB. In general, the DOA estimation algorithm of the coprime MIMO radar based on hole filling under the sum-difference array structure can make full use of all the extended virtual elements, so it has better estimation performance than the DOA estimation algorithm of the coprime MIMO radar that discards discrete virtual elements. Under the same algorithm, the transceiver-flip-coprime MIMO radar has the smallest DOA estimation error due to its higher degree of freedom, and its performance is better than that of the coprime subarray MIMO radar, while the second-order coprime MIMO radar has the weakest estimation performance due to its lowest number of virtual elements. In addition, the estimation accuracy of the transceiver-flip-coprime MIMO radar that discards discrete virtual elements is higher than that of the second-order coprime MIMO radar and the coprime subarray MIMO radar that fill in virtual elements, because under the sum-difference array structure, the range of continuous virtual elements of the transceiver-flip-coprime MIMO radar is [-24, 24], while after filling in virtual elements, the range of continuous virtual elements of the second-order coprime MIMO radar and the coprime subarray MIMO radar is only [-18, 18] and [-23, 23].

[0278] (4) Experiment four: success probability comparison

[0279] This experiment compares the DOA estimation success probability curves of different coprime MIMO radars under the sum-difference array structure with coherent targets as an example. It is assumed that three coherent targets are located at {-10°, 0°, 10°} and the coherence coefficient between the targets is set to 1, and the number of Monte Carlo experiments is set to 200. Here, the success probability is defined as the ratio of the number of experiments with DOA estimation error within ±0.01° to the total number of experiments. Figure 7 (a) is the curve of PS changing with SNR, where L = 200; Figure 7(b) is the curve of the probability of successful DOA estimation of PS versus the number of snapshots, where SNR=0dB. It can be seen that after SNR>0dB and L>200, the probability of successful DOA estimation of the transceiver-flipping-coprime MIMO radar is close to 100%, which is much higher than that of the coprime subarray MIMO radar and the second-order-coprime MIMO radar. Therefore, by filling the holes in the "and joint array", the transceiver-flipping-coprime MIMO radar can obtain a larger virtual aperture, and thus has a higher probability of successful estimation.

[0280] The foregoing is considered as illustrative only of the principles of the application. Further, since numerous modifications and changes will readily occur to those skilled in the art, it is not desired to limit the application to the exact construction and practice described. Accordingly, all such variations and modifications are intended to be included within the scope of the application as defined in the following claims and the equivalents thereof.

Claims

1. A DOA estimation method based on a transceiving flip orthogonal MIMO radar structure, characterized in that, Comprise the following steps: Step 1: establish sparse array MIMO radar "and joint array" and "sum-difference joint array" signal model; Step 2: build transceiver flip coprime MIMO radar, get the "and joint array" of transceiver flip coprime MIMO radar; Step 3: the "and joint array" of transceiver flip coprime MIMO radar obtained is processed by difference, and the "sum-difference joint array" of transceiver flip coprime MIMO radar is obtained; Step 4: non-coherent target DOA estimation based on the "sum-difference joint array" of transceiver flip coprime MIMO radar; Step 5: coherent target DOA estimation based on the "and joint array" of transceiver flip coprime MIMO radar; wherein the "sum-difference joint array" of the transceiving flip-orthogonal MIMO radar obtained in step 3 is: and the set where the range of consecutive elements is [- (5MN-M-N-1), 5MN-M-N-1]; The total number of elements in the medium is 11MN-3M-3N; The number of middle discrete elements is (M-1)(N-1); The specific operation steps of step 4 are: Step 41: fill the discrete holes in the "sum-difference joint array" of transceiver flip coprime MIMO radar; Step 42: construct a convex optimization problem for filling the equivalent received signal of virtual array elements, and solve the convex optimization problem to obtain the reconstructed covariance matrix; Step 43: the eigenvalues of the reconstructed covariance matrix are decomposed, and the spatial target angle is solved by using the MUSIC algorithm.

2. The DOA estimation method based on the transceiving flipping coprime MIMO radar structure of claim 1, wherein, The signal model constructed in step 1 is: 1) the expression of the received signal y(t) is: Where b(t)=[b1(t),b2(t),···,b M (t)] T For transmitting signals, M is the number of transmitting antennas; w(t)=[ω1(t),ω2(t),···,ω N (t)] T θ is the Gaussian white noise vector at the receiving antenna, where N is the number of receiving antennas; k From the perspective of the target, β k α is the reflection coefficient; K is the number of targets; α t (θ k ) and α r (θ k The transmit steering vector and receive steering vector of the k-th target are respectively: where p tm ∈ P t and p rm ∈ P r represent the transmitting and receiving antenna positions, respectively, and p t1 = p r1 = 0. 2) the expression of the received signal y(t) after matched filtering is: where A t = [a t (θ1), a t (θ2),..., a t (θ K )] and A r = [a r (θ1), a r (θ2),..., a r (θ K )] represent the streamer matrices of the transmitting and receiving arrays, respectively; s(t) = [β1, β2, ···, β K ] T is a vector of reflection coefficients; n(t) is a Gaussian white noise vector; 3) the covariance matrix R of x(t) is: where E[•] denotes the mathematical expectation; is the source covariance matrix, and diag[•] denotes a diagonal matrix; denotes the power of the kth source; A = A t ⊙A r ; denotes the noise power; and L is the number of sample snapshots. 4) the virtual domain observation vector r is: Wherein, an array flow matrix for a virtual domain observation vector r; 3. The DOA estimation method based on the transceiving flipping coprime MIMO radar structure of claim 2, wherein, The specific operation steps of step 2 include: Step 21: The transmitting array of the transceiving flip-orthogonal MIMO radar adopts an augmented spread orthogonal array, and the unit element spacing d is normalized to the wavelength λ to obtain a set of transmitting element positions P t is: where P t1 and P t2 denote the left and right half part of the set of transmit array element positions, respectively. Set of receive array element positions P r Is: where P r1 and P r2 denote the left and right half part of the receive array element position set, respectively. Step 22: Based on the equations (8) and (9) the "sum- joint array" of the transceiving flip- reciprocal MIMO radar is obtained is: wherein and 0≤m≤2M-1, 0≤n≤N-1; The "and joint array" of forward elements is: wherein then equation (11) is equivalent to:

4. The DOA estimation method based on the transceiving flipping coprime MIMO radar structure of claim 3, wherein, The "sum-joint array" of the transceiving flip-orthogonal MIMO radar obtained in step 22 The positive direction continuous elements range from [(M-1)(N-1), 2MN-1], and the negative direction continuous elements range from [- (2MN-1), -(M-1)(N-1)]; the set The total number of elements is 4MN-1.

5. The DOA estimation method based on transceiving flipping co-prime MIMO radar structure of claim 1, wherein, The specific operation steps of step 41 are: Step 411: do redundant average processing on the virtual domain observation vector r, and fill 0 at the discrete points to convert the non-uniform virtual array into a uniform virtual array, and obtain a new virtual domain observation vector r' as: Wherein, the dimension of A' is (11MN-3M-3N+2l0)×K, and 2l0 is the total number of filled 0; e＝[0,···,0,1,0,···,0] T ； Step 412: Spatially smoothing the equation (34) to obtain a forward smoothing matrix R x .

6. The DOA estimation method based on the transceiving flipping coprime MIMO radar structure of claim 5, wherein, The specific operation steps of step 42 include: Step 421 : The forward smoothing matrix R x Toeplitz matrix with minimal difference of corresponding elements A convex optimization problem based on the atomic norm minimization theory is established as a new covariance matrix: where || · || r denotes the atomic norm, f Z (·) denotes the Hadamard product, Z = zz T denotes a binary matrix, z denotes a binary vector, and δ denotes an error threshold for the reconstruction of the covariance matrix; Step 422: the non-convex problem shown in formula (35) is equivalent to the convex problem shown in formula (36): Wherein, ξ is the regularization parameter; Step 423: formula (36) is transformed into a minimum problem of the trace of a matrix as: Wherein, μ=ξ / [l0+1+(11MN-3M-3N-1) / 2]; Step 424: Solve equation (40) by CVX toolbox to get the reconstructed covariance matrix 7. The DOA estimation method based on the transceiving flipping co-prime MIMO radar structure of claim 3, wherein, The specific operation steps of step 5 are: Step 51: modify the signal model of formula (4) to a coherent signal model: wherein and is a non-zero complex constant; Step 52: select the data corresponding to the virtual array element position generated by the "and joint array" from the signal model of formula (41), and rearrange in order to obtain a new observation vector: x2(t)=A2s(t)+n2(t) (42) Wherein, A2 is a (4MN-1)×K direction matrix; Step 53: fill the 2(MN-M-N+1) holes in the middle section [- (M-1)(N-1), 0) and (0, (M-1)(N-1)] of the "and joint array" of transceiver flip coprime MIMO radar with 0, so that it constitutes the output signal of ULA structure: x U (t) = A U s(t) + n U (t) (43) wherein A U represents the array flow pattern matrix corresponding to the ULA structure formed after the holes are filled, and has a dimension of UxK, where U=6MN-2M-2N+1; Then we get x U The covariance matrix R of (t) is where R r = E[s r (t)s r (t) H ] represents the power of the reference signal; L is the number of sample taps. Step 54: Construct a Toeplitz matrix T based on the i-th (1≤i≤U) row of the obtained covariance matrix R U is: where rank(T U ) = (U + 1) / 2; Step 55: T U Toeplitz matrix with minimal difference of corresponding elements As the new covariance matrix, an optimization problem based on the atomic norm minimization theory is established: where f G (·) denotes the Hadamard product, G is a constructed binary matrix; Step 56: convert formula (46) into a minimum problem of the trace of a matrix Step 57: Solve equation (48) to obtain the reconstructed covariance matrix Step 58: performing eigenvalue decomposition on the reconstructed covariance matrix The spatial target angle is solved by using the MUSIC algorithm.

8. The DOA estimation method based on the transceiving flipping co-prime MIMO radar structure of claim 7, wherein, The construction steps of the binary matrix G in step 55 are: Step 551: Take a binary vector g, whose element value corresponds to each virtual array element in A U The filled virtual array element corresponds to 0, and the original virtual array element corresponds to 1. Step 552: Construct Toeplitz matrix using gg T of the ith row

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