A Novel Nested Array Structure for DOA Based on CCM of Subarrays

An innovative nested array structure that can notably increase the DOF and the linear arrays’ direction of arrival (DOA) estimated performance is proposed. For the conventional nested array, we vectorize the ACM’s output of the whole array and then remove the repeated rows to get a virtual array’s output. In contrast with classical nested arrays, the superiority of the new nested array is that it utilizes the combination of reverse ordering and conjugation of the inner uniform linear array (ULA) for one of the two subarrays and the sparse outer ULA for another to increase the array aperture. This means the outer sparse ULA’s sensor spacing is approximately the length of the entire inner ULA in the classical nested array structure. Yet, it is about twice the length in the new nested array structure. We generate one lengthy consecutive virtual uniform array excluding any redundant virtual sensors by vectorizing the two subarrays’ cross-correlation matrix (CCM), which is mentioned above. With the export signal of the virtual array and its conjugate form of it, an equivalent covariance matrix of full rank is constructed, which is called the Toeplitz matrix, to compensate for the rank lack of the virtual array’s ACM. For the sake of obtaining the DOA of input signals, the conventional DOA estimation method will be implemented on the ACM. The increase in outer sparse ULA’s sensor spacing ensures an increase in virtual array aperture in principle, which means it is capable of increasing the number of DOA estimations, which is called DOF, and optimizing the DOA estimated performance in contrast with the ULA and classical nested arrays. Results from trials certify the preponderance of the proposed structure in the aspect of DOF and the accuracy of DOA estimation.


INTRODUCTION
A sampling of input signals is performed by antenna arrays in order to detect the source signal or calculate the signal parameters.A major application of the antenna array is getting the direction of arrival (DOA) information, which is essential in respect of sensor signal processing, like radar, SONAR, navigation, satellite communications, and electromagnetic reconnaissance [1].A uniform linear array (ULA) is capable of distinguishing N 1   signal sources with N sensors using conventional methods like MUSIC [2] or ESPRIT [3].Over the past decades, however, it has been mostly constrained to ULAs [4].
Arrays with nonuniform sparse elements obtain growth degrees of freedom (DOF), simultaneously narrowing down the coupling effects between array elements, in contrast with ULAs [5,6] under the same array number [7][8][9].So, it has attracted extensive attention from the academic circle.Recently, it has become increasingly popular to use nested arrays because it is possible to construct sparse arrays systematically and express their numbers of DOF in the closed form [10].The issue of DOA estimation making use of nested arrays has been extensively considered.Nevertheless, as the field of DOA estimation has developed, higher requirements have been imposed.
So, in this paper, a more effective new nested array for direction finding is suggested.Unlike the classical nested array structure, our new nested array structure estimates the source DOA through the cross-correlation matrix (CCM) of sub-arrays instead of the auto-correlation matrix (ACM) of the whole array's output.And the CCM is obtained by multiplying the original export signal from one subarray by the combination of reverse ordering and conjugation of the export signal from the other.Thus, a dramatic increase in DOF can be achieved with the new nested array structure compared with ULA and the classical nested array.And it can distinguish more sources not only from the number of actual sensors but also from the classical nested array.Particularly, under the condition of a large quantity of sensors, the DOF of the new nested array is almost twice as large as that of the classical nested array.Furthermore, with the same RMSE, the new nested array structure is improved by 6 dB and 19 dB under the condition of high SNR, in the aspect of the performance of DOA estimation, compared with that of the classical nested array and ULA, respectively.Results from simulations validate the proposed structure.

 
diag a denotes a diagonal matrix that adopts the elements of a as its diagonal entries. represents the Khatri-Rao product., where

ARRAY STRUCTURE AND SIGNAL MODEL
, 1, , 0,1, , ,2 1, , . p is an integer vector comprising the positional information of sensors in the classical nested array structure.The outer sparse ULA's sensor spacing of the classical nested array is N 1 +1, as is shown in Figure 1, which limits the DOF of the virtual arrays derived by the classical nested arrays.Nevertheless, the outer sparse ULA's sensor spacing of the new nested array is 2N 1 -1, as is shown in Figure 2, which is almost twice the length of that of the classical nested arrays.The sensors' position information of the new nested array structure is ) , which increases the DOF which can be proved in Section 4.1. where indicates the output signal, ( ) t  is the additive white Gaussian noise.

DOA ESTIMATION ALGORITHM FOR THE NEW NESTED ARRAY STRUCTURE
Instead of vectorizing the auto-correlation matrix (ACM) of the whole array's output and then removing the repeated rows to get a virtual array's output, that is what we did to the classical nested array, we take the nested arrays in two subarrays, as shown in Figure 2. Subarray 1 is a ULA comprising N 1 sensors with spacing 2 d   , and subarray 2 is also a ULA containing 2 1  N  sensors with spacing 0 . The N 1 -th sensor is shared by two subarrays.The outputs of subarrays 1 and 2 are written as: Besides, mean the manifold matrices.The  -th column of 1 ( ) 2)c o s ( ) , , , , The reverse ordering and conjugation of the received signal from subarray 1 are written as: where , and * ( ) ( ) where .
We obtain a cross-correlation matrix xzc R between the signal vectors 1 ( ) where and 2 k  is the power of the  -th source.
We vectorize xzc R to get the following vector: where , , , ( ) ( ), ( ), , ( )  is the noise variance, and e denotes a column vector of     rows with all zeros except two ones at the 1st and (2N 1 -1)-th position.
According to the Equation ( 10), x r represents the equivalent export signal, ( ) x A  is its manifold matrix and the input signal is p .
, , , , , , , , , , and whose  -th column of the manifold matrix is . As a result, the DOF dramatically increases, as it is like a ULA containing N N  physical sensors.The equivalent input signal p is coherent resulting in a rank lack in the ACM.Thus, the traditional DOA estimation algorithm ESPRIT is not capable of being directly implemented for it.As the ACM is a Toeplitz matrix, the ACM X R can be established using X.X R can be expressed as: ( where 1 ( ), , ( ) Applying the ESPRIT-like method after constructing a Toeplitz matrix, we can calculate the  .By operating like the above, we can get the ACM of the virtual array without any redundant elements compared with the classical nested array, which reduces the computational complexity.

DOF
According to Sections 2 and 3, in which the inner array has N 1 sensors, and the outer array has N 2 sensors, the DOF of the classical nested array is The RMSE of ULA, the classical and new nested array versus the number of snapshots under 500 trials is displayed in Figure 5(b), in which the SNR is 10 dB, and the quantity of snapshots modifies from 200 to 2000 with the spacing of 200.As we can see from Figure 5(b), with the increasing quantity of snapshots, for any array structure, the corresponding RMSE decreases gradually.Still, the new nested array apparently achieves the lowest RMSE.The decrease of RMSE is no longer significant when the quantity of snapshots is more than 1000 for all three array structures, which indicates there is no obvious need to increase the number of snapshots when it is more than 1000.

CONCLUSION
This paper offers a novel structure of nested arrays with a sparser outer ULA compared with the classical nested array.Unlike the structure of previously nested arrays, whose virtual arrays are got by vectorizing the ACM of the whole array's output, a novel virtual array generated by the new nested array structure is attained by vectorizing the reassembled CCM of different subarrays excluding any redundant elements.At the back of acquiring the virtual arrays, the full-rank equivalent ACM is erected with the output of the virtual array and its conjugate form.An ESPRIT-like method is done to acquire the DOA information.Numerical results reveal that the performance of DOA estimation is improved by 6 dB and 19 dB compared with that of classical nested arrays and ULAs, respectively.The new nested array has


stand for the complex conjugate, transpose and conjugate transpose, respectively.  E  means the statistical expectation,   vec  denotes the vectorization operator and It is assumed that the nested arrays have N scalar sensors.Being made up of two continuous ULAs, the inner ULA's sensor spacing is 2 d   with N 1 sensors, where  is the carrier wavelength, and the outer ULA's spacing is sensors.It is also assumed that the sensors' positions are p d  

Figure 1 .
Figure 1.The classical nested array structure

Figure 2 ..
Figure 2. The new nested array structure

Figure 5 .
Figure 5. (a) RMSE against the SNR; (b) RMSE against the snapshot number The RMSE of ULA, the classical and new nested array versus SNR from -10 dB to 20 dB with the spacing of 2 dB, under the condition of 500 Monte-Carlo trials, 1000 snapshots, is shown in Figure 5(a).The RMSE of the new nested array shows the optimum performance over the whole SNR scope, followed by the classical nested array and finally ULA.The RMSE of the new and classical nested array maintains a steady decline when SNR > -8 dB, whereas the ULA shares the same condition when SNR > 0 dB.Moreover, the new nested array increases the DOA estimation performance by 6 dB in contrast with the classical nested array and 19 dB compared with the ULA under high SNR conditions.The RMSE of ULA, the classical and new nested array versus the number of snapshots under 500 trials is displayed in Figure5(b), in which the SNR is 10 dB, and the quantity of snapshots modifies from 200 to 2000 with the spacing of 200.As we can see from Figure5(b), with the increasing quantity of snapshots, for any array structure, the corresponding RMSE decreases gradually.Still, the new nested array apparently achieves the lowest RMSE.The decrease of RMSE is no longer significant when the quantity of snapshots is more than 1000 for all three array structures, which indicates there is no obvious need to increase the number of snapshots when it is more than 1000.