DOAEstimation of a Novel Generalized NestedMIMORadar with High Degrees of Freedom and Hole-Free Difference Coarray

A novel generalized nested multiple-input multiple-output (MIMO) radar for direction of arrival (DOA) estimation is proposed in this paper.+e proposed structure utilizes the extended two-level nested array (ENA) as transmitter and receiver and adjusts the interelement spacing of the receiver with an expanding factor. By optimizing the array element configuration, we can obtain the best number of elements of the transmitter and receiver and the attainable degrees of freedom (DOF). Compared with the existing nested MIMO radar, the proposedMIMO array configuration not only has closed-form expressions for sensors’ positions and the number of maximum DOF, but also significantly improves the array aperture. It is verified that the sum-difference coarray (SDCA) of the proposed nested MIMO radar can get higher DOF without holes. MUSIC algorithm based on Toeplitz matrix reconstruction is employed to prove the rationality and superiority of the proposed MIMO structure.


Introduction
Multiple-input multiple-output (MIMO) radar [1][2][3], with good space, frequency, and waveform diversity characteristics, is widely used in array signal processing [4][5][6] in the last few years. Compared with phased array radar, it has significant advantages in signal detection, parameter estimation [7], direction finding accuracy, spatial resolution [6], and antijamming capabilities, etc. However, the traditional MIMO radar usually adopts a uniform linear array (ULA) as transmitter and receiver, whose interelement spacing is equal to and no more than half wavelength. Hence, there are some problems in the direction of arrival (DOA) estimation for MIMO radar, such as mutual coupling of array elements [8] and limited aperture of virtual array elements [9].
In order to enhance the upper limit of degrees of freedom (DOF) and the flexibility of layout as well as reducing mutual coupling of physical sensors, sparse arrays such as the minimum redundancy array (MRA) [10,11], coprime array (CPA) [12], and nested array (NA) [13][14][15][16][17] have been explored for DOA estimation and joint multiparameter estimation. In addition, sparse arrays combined with MIMO radar can further increase DOF through the sum-difference coarray (SDCA) [18], so as to improve the accuracy of direction finding and angle resolution ability.
For the purpose of improving sensor utilization, the minimum redundancy MIMO radar [19] designs the optimal array spacing by optimizing the number of virtual array elements, whereas it requires complex computational search and lacks closed-form expressions of DOF. e coprime MIMO radar generally uses part [20] or whole CPA [21] as the transmitting array and the receiving array. Li et al. [20] combined it with the real-value ESPRIT algorithm to DOA estimation, but did not consider the virtual array expansion of the echo signal model. erefore, its DOF is limited by the number of physical sensors. Shi et al. [22] defined the generalized sum-difference coarray (GSDC) and simultaneously derived the closed-form expressions of the total number of virtual array elements of the generalized two-level coprime MIMO radar, which can obtain O(M 2 N 2 ) DOF by O(M + N) physical sensors. e obtained DOF are much higher than that in [20,21] with O(MN) DOF. Unfortunately, there are holes in the SDCA of the abovementioned coprime MIMO radars, so multiple signal classification (MUSIC) [23] and estimation of signal parameters via rotational invariance technique (ESPRIT) algorithms [24] cannot be firsthand applied to these array structures.
Nested MIMO radar [25,26] has closed-form expressions of positions and the number of virtual array elements and overcomes the defects of the minimum redundancy MIMO radar and coprime MIMO radar. Qin et al. [25] exploited nested subarrays as transmitting and receiving arrays to DOA estimation of mixed coherent and uncorrelated targets. Zheng et al. [26]  To further enhance DOF, this paper adopts the extended two-level nested array (ENA) [15] to construct a new generalized nested MIMO radar. Firstly, the whole ENA is used as the transmitting array and receiving array of MIMO radar. Next, an expanding factor is employed to increase the receiving array spacing, and closed-form expressions of DOF and the best physical array element configuration are derived. Afterwards, Toeplitz matrix reconstruction [29] based on the MUSIC algorithm is employed to exploit the superiority of the proposed array configuration.
To be more specific, the main contributions of this paper are as follows: (a) e optimal array element configuration structure of ENA is deduced and higher degrees of freedom are obtained. Besides, the difference coarray (DCA) is a ULA without holes. (b) A new generalized nested MIMO radar based on ENA is constructed, and the optimal sensors' positions and the maximum DOF are derived, which can obtain O(G 4 ) DOF with O(G) sensors. Meanwhile, the SDCA is a ULA without holes. Its DOF is much higher than the existing nested MIMO radars in [25][26][27][28].

Echo Signal Model
A monostatic sparse array MIMO radar consists of a transmitter with M � M 1 + M 2 arrays and a receiver with N � N 1 + N 2 arrays. e positions of the transmitting array are located at P t � p tm |m � 1, 2, . . . , M and the positions of the receiving array are located at P r � p rn |n � 1, 2, . . . , N , respectively. e unit array element spacing d of the sensor is equal to λ/2, where λ stands for the signal wavelength. Suppose that there are K far-field uncorrelated narrowband sources from angles θ � θ k |k � 1, 2, . . . , K , and the reflection coefficient of the k-th source is β k . en, the echo signal model can be expressed as follows: T denotes the transmit signal; w(t) is an additive white Gaussian noise; and α t (θ k ) and α r (θ k ) are the transmit steering vectors and receive steering vectors of the k-th source, respectively, which can be expressed as where p tm ∈ P t and p rn ∈ P r represent the sensor positions in the transmitter and receiver, respectively, and p t1 � p r1 � 0.
Since the transmitting waveforms of MIMO radar are orthogonal to each other, i.e., the output of the generalized matched filters for the echo signal can be expressed as follows: A r � [a r (θ 1 ), a r (θ 2 ), . . . , a r (θ K )]; n(t) is an additive white Gaussian noise vector; s(t) � [β 1 , β 2 , . . . , β K ] T ; ⊗ and ⊙ denote Kronecker product and Khatri-Rao product, respectively. e covariance matrix of the echo signal can be obtained by where is the target covariance matrix, σ 2 k denotes the signal energy of the k-th target, A� A t ⊙ A r , and σ 2 n is the noise variance. e observing vector can be obtained by vectorizing R: where vec(·) represents vectorized operation; p � [σ 2 1 , σ 2 2 , . . . , σ 2 k ] T ; (·) implies the complex conjugation of the matrix.

Extended Two-level Nested Array
In this section, the configuration of the extended two-level nested array (ENA) is formulated first. en the optimal configuration structure of the physical array elements and the closed-form expressions of its DOF are derived. Figure 1, the ENA configuration maintains the basic structure of the nested array, except for increasing the first two sensors of the sparse ULA by d intervals. And the total number of sensors is equal

ENA Configuration. As shown in
erefore, the extended two-level nested array sensor location can be expressed as en, the difference coarray of ENA can be defined as When the total number of sensors is fixed to M � M 1 + M 2 , the array element optimal configuration structure can be optimized to have attainable DOF, as shown in Table 1.

Proposed Generalized Nested MIMO Radar
In this section, the entire ENA is adopted as the transmitter and receiver to construct an extended nested MIMO radar (ENA-TR). Furthermore, a generalized extended two-level nested MIMO radar (GENA-TR) based on ENA-TR is proposed and the closed-form expressions for attainable DOF are deduced. (7), the position of the transmitter with M sensors and receiver with N sensors is given by

ENA-TR. From equation
According to equation (8), the difference coarray of the transmitter and receiver are both consecutive ULAs with 2M 2 (M 1 + 1) + 1 and 2N 2 (N 1 + 1) + 1 virtual array elements, which are located at It can be known from equation (6) that the virtual element positions of B� A * ⊙ A are composed of sum-difference coarray of physical sensor positions.
erefore, the sum-difference coarray of ENA-TR is essentially the sum coarray of two difference coarrays.

GENA-TR.
e ENA-TR configuration fails to make full use of the virtual aperture expansion effect of the sumdifference coarray. By introducing the interelement spacing expansion factor, a generalized ENA-TR (GENA-TR) is established to increase DOF, as shown in Figure 2.
e interelement spacing of the receiver is enlarged with an expansion factor α, so the receiver sensor positions are located at It can be seen from set (15) that the difference coarray of the receiver is a filled ULA with interelement spacing enlarged by factor α. And the difference coarray set is given by According to equations (10) and (15), we can get the sum-difference coarray set of GENA-TR: (a) It can be known from equations (12) and (16) that S T and S α R are symmetrical sets, so l t + l α r � l t − l α r . erefore, the sum-difference coarray of GENA-TR is essentially a difference coarray of two subarrays. When 1 ≤ α ≤ 2M 2 (M 1 + 1) + 1, S GENA−TR SDCA can get consecutive virtual array elements with certain positions [13]. (b) According to equations (12) and (16), the margins of S GENA−TR SDCA can be obtained. For the left margin, For the right margin, In addition, S T and S α R are both consecutive integers, so S GENA−TR SDCA is a hole-free ULA. (c) According to proposition (b), the maximum number of e sum-difference coarray of GENA-TR can attain the maximum number of DOF, where α � 2M 2 (M 1 + 1) + 1. When the total number of physical array elements is determined to be G, the array construction problem turns to an optimization problem about the maximum number of DOF e Lagrange function of equation (20) can be expressed as where η represents the Lagrange multiplier.
Taking the partial of f with respect to M 1 , M 2 , N 1 , N 2 , η, the following system of equations can be expressed: By solving equation (22), the optimization results can be obtained, including the total number of physical sensors, the specific situation of the optimal array element configuration, and maximum DOF with the sum-difference coarray of generalized extended twolevel nested MIMO radar, as shown in Table 2.
To illustrate the distribution characteristics of the virtual array elements given by the proposition clearly, an example is shown in Figure 3, where M 1 � N 1 � 2, M 2 � N 2 � 3, α � 2M 2 (M 1 + 1) + 1 � 19, and only the positive part is demonstrated due to the symmetry of SDCA. Moreover, nested MIMO [25] and Yang nested MIMO [27] are also given for comparison. It can be found that the consecutive lags of GENA-TR are [0, 180] in this example, which is higher than nested MIMO and Yang nested MIMO, and SDCA does not have holes. Table 3 shows the consecutive lags and DOF of different MIMO geometries with a given total number of physical elements. It can be clearly seen that the GENA-TR not only retains the original advantages of the existing sparse array MIMO radar, whose sum-difference coarray is a hole-free ULA, but also significantly enhances DOF. Next, the redundant virtual array elements formed by sum-difference coarray can be averaged [30] and combined with the MUSIC algorithm based on Toeplitz matrix reconstruction for DOA estimation.

Simulation Results
In this section, several numerical simulations are presented to verify the rationality and superiority of the proposed nested MIMO radar (GENA-TR) and compare with other sparse array MIMO radars, including nested MIMO [25], NA-TR [26], Yang nested MIMO [27], Zheng nested MIMO [28], and ENA-TR. e total number of physical sensors is set as G � 10.

DOF Comparison.
In this numerical simulation, we compare the maximum DOF of GENA-TR with other nested MIMO geometries, where we change the total number of physical sensors from 10 to 20. It can be seen from Figure 4 that the maximum DOF of various nested MIMO radars increase with the number of physical array elements, and the proposed nested MIMO radar has the most obvious growth trend. In addition, it should be noted that when the number of physical elements is the same, the DOF of ENA-TR is only 4 higher than that of NA-TR, whereas GENA-TR can obtain a higher DOF than other MIMO radars.

Spatial Spectrum.
In this numerical simulation, the Toeplitz matrix reconstruction based on the MUSIC algorithm is adopted to validate the spatial spectrum performance of GENA-TR and Yang nested MIMO radar [27], as shown in Figures 5 and 6. Suppose that there are K � 101 far-field uncorrelated narrowband targets uniformly distributed from −60°to 60°at an angular interval of 1.2°, where the signal-to-noise ratio (SNR) is equal to 10 dB, the number of snapshots L � 1000 and the search angel range is [−90°: 0.01°: 90°]. It is obvious that GENA-TR can obtain better peaks and accurately estimate the DOA of all targets, whereas Yang nested MIMO radar has false peaks. Moreover, it is worth noting that the reason why NA-TR, ENA-TR, nested MIMO, and Zheng nested MIMO cannot estimate 101 targets is that MUSIC algorithm based on Toeplitz matrix reconstruction causes their DOF to be reduced by half, becoming 17, 19, 41, and 81, respectively.

Root Mean Square Error.
In this numerical simulation, the root mean square error (RMSE) of DOA estimation for GENA-TR and other nested MIMO radars is compared via Monte Carlo experiments. It is assumed that there are K � 13 far-field uncorrelated narrowband targets evenly distributed in [−60°: 10°: 60°]. Figure 7 depicts the RMSE of different nested MIMO radars versus SNR, where the number of snapshots is L � 500. Figure 8 shows the RMSE of different nested MIMO radars versus the number of snapshots, where SNR � 0 dB. e RMSE of DOA estimation can be calculated as where T � 200 represents the number of total Monte Carlo simulations, θ k denotes the true DOA, and θ ⌢i k implies the estimated DOA of the i-th trials.
It can be clearly seen from Figures 7 and 8 that as SNR and the number of snapshots increase, the DOA estimation accuracy of each MIMO radar has been improved, and simultaneously the DOA estimation performance of GENA-  TR is better than that of other MIMO radars. is is because the proposed GENA-TR can obtain a lager virtual element aperture when the number of physical sensors is the same.  Figure 9 shows the PD of different nested MIMO radars versus SNR, where the number of snapshots is L � 200. Figure 10 shows the PD of different nested MIMO radars versus the number of snapshots, where SNR � −10 dB. Here, PD is defined as the ratio of the number of numerical simulations with the DOA estimation error within ±0.01°in the total experiments.
As shown in Figures 9 and 10, the PD of each nested MIMO radar has been improved as SNR and the number of snapshots increase. In addition, under the same SNR or the same number of snapshots, GENA-TR has a higher PD. Especially under the conditions of low SNR and low snapshots, the PD of the proposed nested MIMO radar is significantly better than that of other nested MIMO radars.

Resolution Performance.
In this numerical simulation, the resolution performance of DOA estimation with GENA-TR and other nested MIMO radars is compared. Here,   resolution is defined as both the deviation between the true DOA and the estimated DOA of two closely targets which are less than one-half of the difference between the true angles of them [28]. Assume that there are K � 2 far-field uncorrelated narrowband targets located at [ 5°, 5.5°]. It can be clearly seen from Figure 11 that NA-TR and ENA-TR cannot distinguish the above two targets, while other nested MIMO radars can distinguish the above two targets. Furthermore, GENA-TR has higher resolution and sharper peaks due to its higher degrees of freedom.

Conclusions
In this paper, a generalized extended two-level nested MIMO radar array configuration using extended two-level nested array is proposed, which can significantly improve the virtual array apertures and degrees of freedom, and the sum-difference coarray is a ULA without holes. In addition, the closed-form expressions of the maximum DOF are derived for a given number of physical sensors. At last, some numerical simulations are conducted to illustrate the advantages of the proposed GENA-TR in DOF, the estimation accuracy and the angle resolution.
Data Availability e data supporting the conclusion of the article are shown in the research paper.   Mathematical Problems in Engineering