Sparse Billboard and T-Shaped Arrays for Two-Dimensional Direction of Arrival Estimation

In two-dimensional direction of arrival (2D-DOA) estimation, planar arrays can estimate the elevation and azimuth angles simultaneously. However, many planar array topologies such as billboard, L-shaped, T-shaped, and 2D nested arrays suffer from mutual coupling that results from the small separation between the physical sensors (antennas), which limits the estimation capability of the sensor array. In an attempt to reduce mutual coupling between sensors, this article proposes sparse billboard and T-shaped arrays in which the number of closely separated sensors is significantly reduced. In addition to extending the CRB for fourth order coarray, this article also derives closed-form expressions for the sensor locations and the number of consecutive lags or the uniform degrees of freedom (uDOF), in the fourth-order difference coarray (FODC). Simulation results demonstrate the robustness of the proposed sparse arrays against mutual coupling.

There are several metrics to compare 2D sparse arrays, including but not limited to: number of virtual lag locations, required aperture size, resolution, and mutual coupling. Researchers focused on proposing configurations that have closed-form expressions for antenna locations and the achieved degrees of freedom (DOF), which is defined as a measure of the maximum number of sources that can be concurrently estimated. The DOF in 1D arrays is upper bounded by the number of virtual lags in the second order difference coarray (SODC) [12].
In this article, the L-shaped array is modified by adding a third leg either at 45 • between the other two legs or along one of the legs, where three similar 1D sparse arrays are used. The candidate sparse arrays are the conventional coprime [35], rotated conventional nested [36], and super nested arrays [37]. The rotated T-shaped structure can be considered as a modified version of [21]. It has been shown that large DOF can be realized by exploiting the fourth-order difference coarray (FODC). In addition to extending the CRB for fourth order coarray, closed-form expressions for sensor locations and number of consecutive lags, or uniform-DOF (uDOF), in the FODC are derived for all arrays. The maximum DOF is achieved when the coprime pairs N 1 and N 2 are selected as close as possible. Additionally, the maximum DOF of nested and super nested based structures is achieved following the approach in [36]. All proposed configurations have better uDOF compared with hourglass array (HA) [25] and 2D nested planar array (2DNA) [23]. The proposed T-shaped nested array has comparable uDOF with the L-shaped nested array (ALNA) [20] which requires very large aperture size.
The weight functions are also derived and investigated. The most significant weights have constant values irrespective of number of sensors, except for nested based structure. The HA has a hole-free difference coarray. However, the weight values are larger compared with the proposed configurations. The proposed arrays have promising performance for 2D-DOA estimation in the presence of mutual coupling compared with state of art.
The rest of the article is organized as follows: Section II introduces the structure of the proposed arrays. Section III presents the model for 2D-DOA estimation. Section IV explains the performance metrics used to evaluate the proposed arrays. Section V presents the derived Cramér-Rao bound (CRB). Section VI presents the results and discussions about the weight function, number of consecutive lags, and estimation accuracy. Finally, Section VII concludes the article.

II. PROPOSED ARRAY AND 2D-DOA MODELS
The developed arrays are derived from the prototype billboard and the T-shaped (rotated-T) arrays shown in Fig. 1(a) and (b), respectively. All ULAs are replaced with 1D sparse arrays at a time, including coprime array [35], nested array [36], and the super nested array [37].

A. BILLBOARD ARRAYS
The first sparse billboard design is achieved by replacing each subarray in Fig. 1(a) by the conventional coprime array [35]. Coprime arrays consist of two uniform linear subarrays having N 1 and N 2 elements, where N 1 and N 2 are two coprime integers, and N 2 > N 1 . The elements of the subarray that has N 1 elements are spaced by N 2 d, while the elements of the subarray that has N 2 elements are spaced by N 1 d, with d being the minimum separation between any two elements which is set as half the wavelength λ 2 . The sensor locations are given as the union of the two sets, Coprime array has a total of N c = N 1 + N 2 − 1 sensors (one sensor is shared between the subarrays). Fig. 1(c) shows an example of coprime array with N 1 = 4 and N 2 = 5. The set of the elements of the billboard array is where and the set g = P c describes the linear array used to construct the 2D array. The total number of elements is To generate the billboard nested array, the coprime array is replaced by the nested array introduced in [36]. The conventional nested array consists of two collinearly placed subarrays with different interelement spacing. Assume that subarray1 has N 1 elements with interelement spacing of d. Subarray2 has N 2 elements, but with interelement spacing of (N 1 + 1)d. Sensor locations are given as follows Nested array has a total of N n = N 1 + N 2 sensors. Fig. 1(d) shows an example of nested array with N 1 = 4 and N 2 = 4. The 2D billboard nested array is constructed in the same way using (2)-(5) by setting g = P rn , where P rn = (N 2 (N 1 + 1) − 1)d − P n . Note that the nested array is rotated by swapping the positions of the dense subarray and the sparse subarray to improve the DOF and reduce the mutual coupling. The total number of elements is The billboard super nested array is constructed using the super nested array introduced in [37]. Super nested array is a modified version of the nested array that significantly reduces mutual coupling by relocating some of the elements of the nested array. The super nested array is used to define the billboard super nested array as in (2)-(5) with g = P sn . Super nested array with N s elements can be constructed with N 1 ≥ 4 and N 2 ≥ 3 [37]. Note that the definition of the set P sn is eliminated for brevity, and interested readers are referred to [37]. Similar to the nested case, the total number of elements is N = 3N s − 2 = 3(N 1 + N 2 ) − 2, where N s is the total number of sensors in the super nested array.

B. T-SHAPED ARRAYS
The T-shaped arrays are constructed using three sparse arrays similar to the billboard case, but the third subarray is located along the negative side of the x-axis, see Fig. 1(b). The set of elements in the T-shaped 2D array is given as where G x and G y are as defined in (3) and (4), respectively, and: To construct the T-shaped coprime, nested, and super nested arrays, both (7) and (8) are used with the set g being equal to P c , P rn , and P sn , respectively. The total number of elements is the same as their counterparts using billboard structures. 1

III. 2D-DOA ESTIMATION
Assume that K uncorrelated signal sources located in the far-field of the sensor array generate narrowband signals that impinge on a 2D array. The kth source has an azimuth angle θ k ∈ [0, π] and an elevation angle φ k ∈ [0, 2π ]. The received signal at the output of the array over T samples or snapshots can be expressed as where y(t ) = [y 1 (t ), y 2 (t ), . . . , y N (t )] T , n(t ) = [n 1 (t ), n 2 (t ), . . . , n N (t )] T is white Gaussian noise with zero mean and uncorrelated with the transmitted signal, is the manifold matrix of size N × K, with a k (θ k ,φ k ) being a steering vector that has an element (n x , n y ) ∈ S B or S T given by e j2π (θ k n x +φ k n y ) , whereθ k = d λ sin θ k cos φ k and φ i = d λ sin θ k sin φ k are the normalized DOAs. The 2D-DOA estimation is based on the fourth-order cumulant. The fourth-order cumulant matrix is given as [38]: where [.] * and [.] H represent the complex conjugate and Hermitian operators, respectively, A = A A * with ⊗ and being the Kronecker product and Khatri-Rao product, a(θ k ,φ k ) = a(θ k ,φ k ) ⊗ a * (θ k ,φ k ), A and Y = [y 1 (t 1 ), y 2 (t 2 ), . . . , y N (t T )] are matrices of size N 2 × K and N × T , respectively, and C s = diag[g 1 , g 2 , . . . , g K ] is a diagonal matrix with g k = (s k (t ), s * k (t ), s k (t ), s * k (t )) being the kurtosis of the kth source signal and (.) denotes the cumulants operator. Vectorizing C 4 yields [18]: where vec(.) is the vectorization operator which turns a matrix into a column vector, The new extended steering matrix,Ā, is a function of the FODC, which has a total of l u unique lags (the number of unique entries in each column ofĀ). Though, the number of consecutive lags generated by the FODC is l c < l u . The measurements associated with consecutive lags are extracted and sorted to form a new vector as: where B is a matrix of size l c × K, with b i (θ i ,φ i ) being a steering vector that has an element (n x , n y ) ∈ U 4 given by e j2π (θ i n x +φ i n y ) , U 4 is the largest URA segment in the FODC [20], [39]. By considering the consecutive segment of virtual lags (largest symmetric URA around the origin [20], [39]), 2D-DOA estimation can be performed finally based on r using 2D unitary ESPRIT algorithm [40]. In the presence of mutual coupling, sensors are influenced by their neighboring elements, and (9) becomes Y = CA(θ,φ)S + N, where C is the mutual coupling matrix modeled as in [25], [37]. This matrix can be approximated by a B-banded symmetric Toeplitz matrix depending on the separation between the elements as [41]: where n 1 , n 2 ∈ S, . 2 is the l 2 -norm of a vector, and c 0 , c 1 , . . . , c B are the mutual coupling coefficients with [25], [37].

IV. PERFORMANCE MEASURES
This section presents some performance measures used to evaluate and compare the proposed arrays. This includes: the number of unique/consecutive lags in the FODC, the aperture size, and the weight function in general.

A. DEGREES OF FREEDOM AND DIFFERENCE COARRAY
When using 4th-order statistics for estimation, the number of unique elements in the FODC is directly related to the DOF, which is the maximum number of detectable uncorrelated sources. This is significant for algorithms that exploit all the elements in the difference coarray even if it is not hole-free. If the used algorithm requires continuous segment, then the number of consecutive lags is more significant. Definition: Difference Coarray: Let a 2D array be specified by a set S, the SODC, D, is the difference between sensor positions as: The FODC, D 4 , can be calculated by taking the differences again but between the virtual positions generated by the set D as: In other words, p 1 = n 1 − n 2 and p 2 = n 3 − n 4 for any arbitrary sensor locations n 1 , n 2 , n 3 , n 4 ∈ S. The 4th-order difference coarray can be rewritten as: D 4 = p 1 − p 2 = (n 1 − n 2 ) − (n 3 − n 4 ) = (n 1 + n 4 ) − (n 2 + n 3 ), or D 4 = (n 4 − n 2 ) + (n 1 − n 3 ). Therefore, the FODC is also equivalent to the difference coarray of the second order sum coarray or the sum coarray of the SODC. The number of unique lags, l u , of the FODC is equal to the cardinality of D 4 , that is l u = |D 4 |. On the other hand, the number of consecutive lags, l c , is equal to the cardinality of U 4 , that is l c = |U 4 |, where U 4 is the largest URA segment in the FODC [20], [39]. The variables l u and l c are also known as the DOF and the uDOF [39]. An example is shown in Fig. 2 for the FODC for all proposed arrays, where N 1 = 4 and N 2 = 5 for coprime case, and N 1 = N 2 = 4 for both nested and super nested cases. Therefore, the total number of elements is N = 22. The red dots in Fig. 2 represent the virtual lag locations, while the blue triangles represent the physical locations of the elements. 2 Table 1 summarizes the number of virtual lags and the required apertures. The achievable uDOF of these configurations are examined in Section VI-B in terms of 2D-DOA estimation.
The objective is to find closed-form expressions for the maximum achievable uDOF. Let's consider first a 1D coprime. Its SODC has 2(N 1 + N 2 ) − 1 consecutive lags in the range of −(N 1 + N 2 − 1) : (N 1 + N 2 − 1) and > N 1 N 2 unique lags. The holes affect the FODC, though the structure guarantees that the FODC realizes at least −2( In case of nested and super nested arrays, the SODC and the FODC have −(N 1 + 1)N 2 + 1 : Focusing only on one array along any axis, relating the example above to this discussion and considering the FODC, coprime array has 33 consecutive lags, while nested and super nested arrays have 77 consecutive lags. Now if there is another identical array along the negative side of the same axis, as in the T-shaped arrays, this number will be doubled. Finally, this number will be squared, (. 2 ), if this happens across all axes. Actually, larger URA is expected to be generated due to the contribution between the utilized three sparse arrays to form the billboard or the T-shaped configurations.
Extensive analysis was conducted to derive closed-form expressions for the uDOF, l c , of the proposed arrays. Understanding the underlying structures was incorporated to finalize the derivation. The resultant FODCs always have symmetric URA around the origin. Thus, the idea starts by finding the (x, y) coordinate of any virtual lag on one of the four corners within the resultant URA (generated by the FODC). Then an expression, X , is drafted for the coordinates for different cases of N 1 and N 2 . This expression is confirmed by intensive simulation. After that, the expression is doubled and incremented by one to account for the zero axis, that is 2X + 1. Finally, 2 A MATLAB code available in https://github.com/alawsh21/Sparse-Billboard-and-T-Shaped-Arrays-for-Two-Dimensional-Direction-of-Arrival-Estimation.git is provided to construct the proposed arrays and generate the FODC for any arbitrary N. VOLUME 4, 2023 325 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  the result is squared to account for all consecutive lags as: (2X + 1) 2 . Table 2 illustrates all closed-form expressions used to find the maximum achievable l c or uDOF. Apart from super nested array, the formulas are applicable for arbitrary N 1 and N 2 , provided that the greatest common divisor (GCD) is 1, i.e., GCD(N 1 , N 2 ) = 1, and N 2 > N 1 for coprime case. The expressions related to super nested arrays are valid for the optimal selection of N 1 and N 2 . 3 When N 1 and N 2 are selected as close as possible in case of coprime, the maximum uDOF is achieved, similar to the 1D case [35]. Table 2 illustrates that the billboard coprime array has (4N 4 2 ) uDOF, when N 1 = N 2 − 1, that is 2N 1 > N 2 . On the other hand, if we substitute N 1 = N 2 − 1 for N 1 > 2 (optimal selection) in the corresponding formula of the Tshaped coprime array from Table 2, we end up with l c = (2(N 2 2 + N 2 ) − 3) 2 . Therefore, the array has (4(N 2 2 + N 2 ) 2 ) Arrival-Estimation.git is developed to compare the derived and the simulated expressions for any arbitrary N. uDOF, which is larger than that of the billboard coprime array by 4N 2 2 (2N 2 + 1). The proposed nested and super nested arrays realize the maximum performance when N 1 and N 2 are selected as in the 1D case [36]. That is N 1 = N 2 = N n 2 if N n is even, whereas N 1 = N n −1 2 and N 2 = N n +1 2 if N n is odd [36]. Remember that N 1 and N 2 are either equal or N 2 = N 1 + 1. This is also applicable for super nested array-based structures. Nested structures have one expression for any arbitrary N 1 and N 2 , see Table 2. The billboard and T-shaped nested arrays realize ( 1 4 N 2 n (N n + 4) 2 ) = ( 4 36 2 (N + 2) 2 (N + 14) 2 ) uDOF  and (N 2 n (N n + 2) 2 ) = ( 1 9 2 (N + 2) 2 (N + 8) 2 ) uDOF, respectively.
Considering the FODC, a comparison between the different billboard and T-shaped variants in terms of uDOF (l c ) and l u , is illustrated in Figs. 3 and 4, respectively. The optimal conventional arrays are used for all configurations. Specifically, N 1 and N 2 are selected as close as possible for coprime case [35] and N 1 = N 2 = N n 2 or N 1 = N n −1 2 and N 2 = N n +1 2 if N n is even or odd, respectively, for nested and super nested cases [36]. Super nested array can be constructed for N 1 ≥ 4 and N 2 ≥ 3 [37]. This is why the traces in Figs. 3 and 4 start from N = 3N s − 2 = 3(N 1 + N 2 ) − 2 = 22 elements, where N s is the number of sensors of the underlaying super nested array. In case of coprime (N 1 = 2, N 2 = 3) and nested (N 1 = N 2 = 2) based structures, it starts at N = 10 elements. Fig. 3 shows the uDOF or number of consecutive lags versus the total number of elements, N, for all derived expressions in Table 2. All expressions were verified by simulation.
The T-shaped nested array achieves the largest number of consecutive lags. The billboard coprime and nested structures achieve comparable performance. The T-shaped coprime and super nested structures achieve comparable performance. The billboard super nested array realizes the smallest number of consecutive lags among all configurations. Redistributing the elements of the dense array to reduce mutual coupling is the main reason. The uDOF, l c , of the proposed configurations are compared with the HA [25], 2DNA [23], and ALNA [20] based on the FODC. All proposed configurations have better performance compared with HA and 2DNA, as demonstrated in Fig. 3. The ALNA and the T-shaped nested array have comparable uDOF.
The number of simulated unique lags, l u , versus the total number of elements, N, are shown in Fig. 4. The T-shaped structures enjoy larger l u compared with billboard, except for super nested array. The T-shaped nested array shows a clear superiority compared with others. This structure also enjoys few holes. Note how close are the two traces of nested case in Figs. 3 and 4. Though this is at the expense of mutual coupling and aperture size. The billboard coprime, super nested, and T-shaped coprime arrays achieve almost comparable performance. The billboard super nested array realizes the smallest number of consecutive lags among all configurations.

B. APERTURE SIZE
Due to the use of sparse arrays instead of ULAs in each leg, the proposed structures require larger aperture size than the already existing billboard and T-shaped arrays. This gives higher estimation accuracy but increases at the physical size and space requirements. Expressions for the aperture size are summarized in Table 3. Note that the aperture size of the Tshaped array is twice its counterpart using billboard structure. Nested and super nested structures have equal aperture.
The achieved uDOF is directly proportional to the aperture of the array. Large aperture size is considered a disadvantage when small form-factor is required. Fig. 5 shows the required aperture size for the considered 2D array configurations. Due to the introduced shift between the two nested subarrays, the ALNA, which has the largest uDOF, requires the largest aperture. The proposed T-shaped-based structures require larger aperture than billboard-based structures. HA and 2DNA have comparable apertures.

C. 2D WEIGHT FUNCTION
The weight function is a popular measure used to quantify the performance of antenna arrays for DOA estimation in the presence of mutual coupling. It is known that the closer the sensors are to each other, the more significant the effect of mutual coupling is. The definition of the weight function of the difference coarray for 2D arrays is given as Definition: Weight Function of the Difference Coarray: Let a 2D array be specified by the set S, and let its SODC be D. The weight function of the difference coarray describes how many pairs of elements in S generate each element in D. In other words, how many sensor pairs in S are separated by m x and m y in x and y directions, respectively, [25]: w (m) = |{(n 1 , n 2 ) ∈ S 2 |n 1 − n 2 = m}| (16) where |.| denotes the cardinality operation and m = (m x , m y ) is a vector of two components. The most significant weights that affect mutual coupling are the smallest ones. Particularly, w(0, 1), w(1, 0), w(1, 1), and w(1, −1) are the most important [25]. Further discussion of the obtained values is available in the next section. In addition to increasing the distance between consecutive elements to reduce mutual coupling, the mutual impedance, which depends on the type of antennas, must be properly calibrated and computed [41], [42], [43], [44], [45].

V. THE CRAMÉR-RAO BOUND FOR FOURTH ORDER COARRAY
Few researchers derived the CRB for second order coarray model [46], [47], [48]. In this section, we derive the Cramér-Rao bound (CRB) for 2D-DOA for the constructed 2D arrays based on the fourth order coarray. The parameter vector in (9) and (10) is defined as: The (m, n)-th element of the Fisher information matrix, FIM, is given by: with ∂c ∂η being the derivative of c with respect to η given as: As a matrix, the FIM can be further expressed as: The derivatives in (20) can be calculated based on (11) as: [46], [47]: Here  [46], [47]: Therefore, the FIM becomes FIM = T M H M. The CRB matrix for 2D-DOA is calculated by block-wise inversion as:

VI. RESULTS AND DISCUSSION
The performance results related to the weight function, 2D-DOA estimation, and CRB are discussed in this section.

A. THE WEIGHT FUNCTION
From mutual coupling perspective, the most significant weights in the 2D weight function are w(0, 1), w(1, 0), w(1, 1), and w(1, −1). Apart from nested-based structure, the proposed array structures offer significantly low values, see Table 4. Details on coprime based arrays and nested based arrays are presented, and then extended to super nested configurations.

1) BILLBOARD COPRIME ARRAY
For billboard coprime array, the weight w(1, 0) describes the number of elements spaced by 1 in x and 0 in y. The minimum spacing in the cross differences between any two legs is N 1 , which is greater than 1. Therefore, we are only left with the self-differences of the leg that lays on the x-axis. It was proved in [49] that a 1D coprime array has only two pairs of sensors separated by d, i.e., w(1, 0) = 2. The same argument holds true for the case of w(0, 1). However, this time, the only contribution to this weight is from the self-differences generated by the y-axis leg. Therefore, w(0, 1) = 2.
To have a separation of 1 in x and y, w(1, 1), the cross differences are all eliminated because the minimum separation between any two legs is N 1 . Moreover, the self-differences of the legs laying on the x and y axes are eliminated as well, because they have either the same x or y coordinates. The only contribution left is from the self-differences of the third leg that lays on the x = y straight line. The elements in this leg have equal x and y coordinates. Therefore, if we consider each coordinate separately, we can use the fact for the 1D coprime array to show that this leg has 2 pairs of elements separated by d in x and y. Hence, w(1, 1) = 2 and the overall spacing between the two elements is √ 2d. Regarding w(1, −1), consider two elements and assume that the first element has less x coordinate value than the second element. The y coordinate of the first element must be greater than that of the second element to be counted in this weight. However, this is impossible; because as you increase x, y either increases, or stays unchanged, thus, w(1, −1) = 0.

2) T-SHAPED COPRIME ARRAY
To explain the result, consider the weight w (1, 0), the elements that generate this weight must have the same y coordinate. This is not possible except for elements on the x-axis. Therefore, the part of the array that lays on the y-axis does not contribute to this weight. Furthermore, we can eliminate the cross differences generated from the two legs that lay on the x-axis; because the minimum distance between the two legs that lay on the positive and negative x-axis, is N 1 which is greater than 1. What is left now to contribute to this weight are the self-differences generated by each leg in the x-axis. It was proved in [49] that a 1D coprime array has only two pairs of sensors separated by d. Therefore, each leg on the x-axis will have 2 elements separated by (1, 0) and the whole 2D array will have w(1, 0) = 4.
The same argument can be used for w(0, 1). This time, the elements present on the x-axis do not contribute to the weight. Therefore, the only contribution is due to the leg that lays on the y-axis which results in a value of 2 for this weight, w(0, 1) = 2. For w (1,1), all elements in the T-shaped structure are either placed in the x or y axes. Moreover, the minimum distance between any two legs is N 1 which is greater than 1. Therefore, w(1, 1) = 0. Regarding w(1, −1), a single leg cannot generate this weight because all legs lay on either x-axis or y-axis. Furthermore, the minimum distance between any two legs is N 1 > 1. Therefore, w(1, −1) = 0.

3) BILLBOARD NESTED ARRAY
For the billboard nested array, the important values for the weight function are summarized in Table 4. Only the selfdifferences generated from the elements within the dense subarray along the x-axis (subarray 1) contribute to w (1, 0), because the elements should have equal y coordinates. So w(1, 0) = N 1 . Similarly, w(0, 1) = N 1 . The self-differences generated form subarray 3 contribute to w(1, 1) = N 1 . All self-differences generated from the three subarrays don't contribute to w (1, −1). This is because when the elements are separated by 1 in the x coordinates, they will be separated by 0 or 1 when subarray 1 or subarray 3 are considered, respectively. In addition, separation by 1 in y coordinates implies 0 separation in x coordinates in case of subarray 2, so w(1, −1) = 0. The cross-differences don't contribute to any of these weights because the minimum spacing between the closest two elements in the three subarrays is (N 1 + 1) > 1.

4) T-SHAPED NESTED ARRAY
For the T-shaped nested array, only the self-differences generated from the elements within the dense subarrays along the x-axis (subarray 1 & 3) contribute to w(1, 0), because the elements should have equal y coordinates. As a result, w(1, 0) = 2N 1 , and similarly w(0, 1) = N 1 . The other two weights w(1, 1) = w(1, −1) = 0 can be proved based on w(1, 0) or w(0, 1). If two elements are separated by 1 in x coordinates, then their separation in y coordinates become 0, not 1 or −1, and vice-versa. The cross-differences don't contribute to any of them because the minimum spacing between the closest two elements in the three subarrays is (N 1 + 1) > 1.
The discussion can be extended to the arrays based on the 1D super nested array. The 1D super nested array has different weights for odd and even values of N 1 [37], that is for odd N 1 , w(1) = 1, and for even N 1 , w(1) = 2. The obtained values for the coprime, nested, and super nested based arrays are presented in Table 4. Note that the nested array-based structures do not possess small or constant values for the weight functions due to the presence of the dense ULA segment in each leg.
The self-differences of each leg have a direct impact on w(0, 1), w(1, 0), w(1, 1), and w(1, −1), if n 2 − n 1 > 1, where n 1 and n 2 are the locations of the first and second elements in each 1D subarray, respectively. In other words, the separation of the first two elements in each leg is greater than d. Using this, we can write the discussed weights as: The values are compared to those achieved by the hourglass array [25], 2DNA [23], and ALNA [20]. Apart from nested based structures, the proposed arrays always have smaller weights except for w (1,0). Due to the dense subarrays, the 2DNA and ALNA have large weight functions, which increase linearly with the number of elements in the dense subarrays. The HA has larger weights compared with the proposed configurations, see Table 4.

B. PERFORMANCE IN DOA ESTIMATION
This subsection presents 2D-DOA estimation based on the FODC with 2D unitary ESPRIT algorithm [40]. To carry out the estimation, only the central URA, U 4 , contiguous part of D 4 is utilized by ESPRIT. The number of snapshots is T = 500 and the SNR = 0dB. A total of K = 4 uncorrelated sources are assumed and their normalized direction-cosines are equally-spaced as in [25] but without any rotation. The configurations presented in Section IV-A are assumed where the total number of elements of each array is N = 22, with N 1 = 4 and N 2 = 5 for coprime, and N 1 = N 2 = 4 for nested and super nested based structures. The mutual coupling parameters are c 1 = 0.3, B = 5, and c l = c 1 l e jπ (l−1)/4 [25]. The root-mean-squared error (RMSE) is used to assess the performance, which combines both azimuth and elevation as: where θ k (i) and φ k ( j) are the estimate ofθ k andφ k , respectively, at the i th Monte Carlo trial, i = 1, 2, . . . , I, and K is the number of sources to be localized. A total of I = 100 Monte-Carlo trials are used. All these parameters are fixed unless otherwise stated.
To examine the achievable DOF, a total of K = 64 sources are assumed in noise free environment and in the absence of mutual coupling. Fig. 6 shows the actual and the estimated cosine-directions marked in circles and dots, respectively. Due to the large uDOF (see Table 1), all arrays resolve all sources correctly. Note that the number of sources is larger than the number of sensors, K > N. Fig. 7 shows the RMSE of the estimated DOAs for the proposed arrays ignoring the effect of mutual coupling when K = 4 sources. The RMSE is calculated when varying the SNR and keeping the number of snapshots as 500 in (a) and varying the number of snapshots and keeping the SNR as 0dB in (b). Hourglass array [25] is simulated using N = 22 elements, with N x = N y = 8. This array has a hole-free SODC and has excellent performance in the presence of mutual coupling. For fair comparison with hourglass array, the FODC is considered which is also a hole-free coarray. The coarray has l u = l c = 841 lags and the array requires 7d × 7d aperture size. It is evident that the performance improves with the increase of SNR and number of samples. Due to their large uDOF, the T-shaped structures realize smaller RMSE at high SNR when mutual coupling is ignored. All proposed arrays attain smaller RMSE compared with hourglass array. Though, the latter requires smaller aperture size.
The effect of mutual coupling is considered in Fig. 8. Arrays based on the coprime and super nested arrays show more robustness against mutual coupling as expected due to their sparseness, except the billboard super nested array. The latter has small uDOF as Table 1 depicts. As per the weights presented in Table 4, the T-shaped super nested array has the smallest RMSE. Although the T-shaped nested array has the largest uDOF (see Table 1), mutual coupling deteriorates the performance. Above 0dB and around 300 samples, mutual coupling becomes dominant, and the performance does not improve.
Since K is small, the impact of mutual coupling is not significant. Increasing the number of sources makes the mutual coupling effect clear and deteriorates the estimation capability of some arrays, despite the large number of consecutive lags. Although 2D sparse arrays may estimate K > N sources (see Fig. 6), the effect of mutual coupling makes the estimation process more challenging. Fig. 9 shows how the RMSE depends on c 1 , the most significant coefficient in mutual coupling model, when SNR = 0 dB, T = 500 samples, and K = 4 sources. It can be concluded that the RMSE is small when c 1 is close to zero. The performance degrades above certain thresholds of c 1 . These thresholds are the approximate values of c 1 at which the arrays can perform well in the presence of mutual coupling. Fig. 9 illustrates that the thresholds for the billboard-based structure are around 0.4 for coprime, 0.35 for nested, and 0.5 for super nested. This indicates that billboard super nested array is more robust to mutual coupling effects than others. Note that this array has also smaller uDOF as illustrated in Table 1. While the thresholds for the T-shaped based structure are around 0.65 for coprime, 0.8 for nested, and 0.75 super nested. Although nested based-structures have large mutual coupling, the large uDOF (see Table 1) contributes more to the performance. Note that the T-shaped based structures realize larger uDOF and have larger aperture size. Compared with state of art, the threshold is around 0.55 for hourglass array. Generally speaking and for the considered arrays, the T-shaped based structure is more robust to mutual coupling effects, where     In the previous scenarios, the total number of elements is the same for all configurations. Configurations with comparable number of consecutive lags are further evaluated. Table 5 demonstrates that the closest billboard super nested array has large uDOF. All arrays require comparable number of elements, except the T-shaped nested array. In the presence of mutual coupling, Fig. 10 shows the RMSE versus SNR with K = 25 sources. Due to the mutual coupling effect, nested structures have the worst RMSE, while the best performance is realized by super nested arrays.
The running time for all configurations is calculated for the scenarios in Fig. 7(a). The execution time over MATLAB was calculated on a PC with an AMD Ryzen 7 5700G processor with Radeon Graphics and 16 GB RAM at ∼ 3.8 GHz. The averaged running time in seconds is presented in Table 6.
The T-shaped nested and billboard super nested arrays require the largest and the smallest running time for estimation, respectively. The running time is proportional to the uDOF as presented in Table 1.

C. CRB FOR FOURTH ORDER COARRAY
The derived CRB of the proposed configurations requires very high computation. The size of the term C T 4 ⊗ C 4 in (25) is N 4 × N 4 . Consequently, this cannot be easily handled. To present the CRB, the total number of elements is reduced from N = 22 to N = 10. The billboard and T-shaped super nested arrays cannot be constructed because the minimum values are N 1 ≥ 4 and N 2 ≥ 3 [37]. Coprime and nested arrays with N 1 = 2, N 2 = 3 and N 1 = 2, N 2 = 2 are assumed. Fig. 11 shows the RMSE with mutual coupling versus SNR when T = 500 samples, K = 4 sources, and N = 10 elements. The selected parameters lead to equal weight values for the considered configurations. Due to their large aperture and uDOF, nested based structures have small RMSEs. Billboard coprime array has poor performance due to the small uDOF. The CRBs for the four considered arrays, included based on (26), are comparable. Even when configurations have equal FODC, their CRBs are not the same because the CRB depends on the sensor locations [50].

VII. CONCLUSION
In this article, we examined six 2D arrays derived from the billboard and T-shaped arrays. Each 2D array is constructed using three identical 1D sparse arrays; namely, coprime, nested, and super nested arrays. The proposed arrays achieve large DOF when the FODC is exploited. They also enjoy closed-form sensor locations, and have closed formulas for the number of consecutive lags. The T-shaped structures result in a higher DOF compared with the billboard structure. Among the six arrays, the four arrays based on coprime and super nested arrays offer significantly reduced values for the smallest weights in the 2D weight function of the difference coarray. Simulation results confirmed the robustness of the proposed arrays in the presence of mutual coupling.