COPRIME ARRAY PARAMETERS OPTIMIZATION FOR DOA ESTIMATION

AbstractSparse array such as the coprime array is one of the most preferable sparse arrays for the direction of arrival estimation due to its properties, like simplicity, the capability of resolving more sources than the number of elements and resistance to mutual coupling issue. In this paper, a new coprime array model is proposed to increase the degree of freedom (DOF) and improve the performance of the coprime array. The newly designed array can avoid mutual coupling by minimizing the lag redundancy and expand the central lags in the virtual difference co-array. Thus, the proposed structure can resolve more sources than the prototype coprime array using the same number of elements with the improved direction of arrival estimation. Simulation results demonstrate that the proposed array model is more efficient than the other coprime array model.


I. INTRODUCTION
In array signal processing, an array of elements gathered the received data for estimating the sources signal parameters [1]. The traditional uniform linear array (ULA) consists of N elements with uniform spacing between the elements that can resolve up to N-1 source or degree of freedom (DOF). To identify more sources, extra elements are added to increase the number of DOF that may lead to increased mutual coupling due to the inter-element spacing which is half the wavelength and complexity [2]. To cope with this situation, different types of the sparse array with the different co-array concept are implemented to resolve up to O(N 2 ) source using N elements. As an example, MRA [3], MHA [4], nested array [5], and coprime array [6] are sparse arrays. So as the size of the difference co-array is larger than the number of the physical elements, more uncorrelated sources can be recovered using the DOA estimator. MRA and MHA sparse arrays expanded the size of the difference co-array with a minimum number of holes (empty positions) and reduced the mutual coupling, but they lack the formal expression for the array configuration besides the long processing time to build the physical array [7]. The nested array (NA) and coprime array (CA) focus on these issues and provide large virtual arrays that have a formal expression for the physical array geometry. NA proposed by [5] can generate a hole-free virtual array using two ULAs, one is a dense array with small spacing between the elements and the other is a sparse array. The resulted virtual array suffers from remarkable mutual coupling that declined the DOA estimation performance. One of the most interested sparse array models is the CA that attracts many researchers over the last few. CA consists of two ULA (M,N) with small inter-element spacing between element pairs that eliminate the mutual coupling effect. Lately, different works have been suggested to enhance the CA configuration by maximizing the obtainable DOFs and reducing the mutual coupling affects.
In [6], the extended coprime array (ECA) is proposed to get M N + N − 1 contiguous lags by extending the M-subarray to 2M. The authors in [8] proposed two CA configurations, the former is CACIS in which one sub-array is compressed • The formal expression that identifies the element positions in an array.
• Large virtual elements in the difference co-array to increase the number of resolvable sources.
• The maximum economic array that means all the elements are essential elements that greatly affect the construction of the difference co-array. MRA, MHA, nested array, and cantor array are examples of maximally economic arrays.
In CA with (M,N) elements pair, the number of DOF be enhanced by eliminating the number of lag frequencies in the difference co-array, specifically the cross difference co-array since the self-difference lag frequencies cannot be prevented [16]. In this paper, a new coprime array model is designed based on the above remarks. The proposed array model construct under the coprime array structure accomplishes a higher DOF with reduce mutual coupling and reduced the lag frequencies using the same number of elements as the prototype coprime array.

II. SIGNAL MODEL
For any sensor array with M elements, the element positioned at P = (0, 1, ...M − 1)d, where d is the spacing between elements and d = λ/2 , λ is the signal wavelength and the zero position is the reference position. Suppose that Q far-field, uncorrelated narrowband signal is impinging on the array from the direction θ 1 , θ 2 , ..., θ Q , and then the received data at time t is expressed as: Where P is the source signal covariance matrix, , σ 2 q is the signal power of the qth source. R n is the noise covariance matrix, R n = σ 2 n I M , I M is the identity matrix. For an estimated covariance matrix with T snapshots, the sample covariance matrix is expressed as:

A. Difference Co-Array Concept
For sparse array, to obtain higher DOFs with less number of physical elements, a virtual array is constructed from the difference co-array. Here, some fundamental definition is introduced first.
Definition 1 (Difference Co-array Suppose an array with an integer number of elements positioned at P, the difference set is obtained as follows [5], [17]: Definition 2(Degree of freedom) For a particular array P, the degree of freedom (DOF) is the number of virtual lags of its difference co-array (D). The uDOF is the uniform DOF that indicates the number of DOF of the central lags of the difference co-array. For a co-array MUSIC, the number of uncorrelated sources that can be resolved is (uDOF − 1)/2 [5], [6].

Definition 3 (weight function)
For an array P, the weight function w(m) is the number of the elements pair that construct the virtual elements in the difference co-array D with the index position (m) [18] [9]. It can be described as: The weight function gives an indication of the element allocation in an array. When the weight is larger than one, it refers that there are additional elements pair with a small distance in the physical array that leads to serious mutual coupling [19]. The weight function of the first three elements w(1), w(2) and w (3), which illustrate the number of element pairs can rule the mutual coupling action [20].
By vectoring R in equation (2), we get Where y is the virtual measurement of the received vector for the virtual array with central lags for both the positive and negative parts.
is the virtual steering matrix, represents the Kronecker product, and b is the signal vector. The corresponding received signal can be found from the difference co-array after discharging the repeated rows. Then, the spatial smoothing technique is applied to the central lags only that have consecutive virtual elements [10]. In this paper, for DOA estimation we use both the spatial smoothing technique and interpolation technique. For spatial smoothing, the Toeplitz matrix is implemented on the contiguous lags to build a full rank covariance matrix as follow: : : : Where ρ refers to the uDOF for the positive side only. R can be used to estimate the DOA with less multiplication complexity. The nuclear norm minimization interpolation technique mentioned in ref [21] is applied to manage the holes in the difference co-array and exploit all the virtual elements to increase the number of DOF.  The physical element's position in the PCA can be expressed in the following sets:

B. Prototype Coprime Array Structure
The total number of physical elements isN + M − 1, since the two subarrays are located at collinear and the first element at zero position is the reference elements for the two subarrays. The difference co-array set of the PCA is given by the self-difference of the two subarrays and the cross difference between the two subarrays as follows: The PCA model has some frequent lags in both the self-difference and the cross-difference. The cross difference lags result from the overlapped elements in P 1 − P 2 and P 2 − P 1 . The general structure of the difference PCA is demonstrated in second layer is at 2M + N and 2M + N + 1, the third layer at 3M + N, 3M + N + 1 and , 3M + N + 2 [16]. The length of the central lag in the resulted virtual array can't provide the effective number of resolvable DOF after employing the spatial smoothing [22].
If N is an even, then The proposed PCA model has nil cross-difference redundancy and large difference co-array with more unique lags in the resulted virtual array. The number of uDOF and aperture size of the proposed array model is as listed in Table I.  29] will generate for the positive side, 17 uniform DOF, 22 unique lags, and aperture size is 29 using 8 elements which are more than the traditional PCA, CACIS and the proposed PCA in [16].

IV. SIMULATION RESULTS
The performance of the proposed array model is evaluated and compared with PCA, CACIS (CF=2) and proposed array by [16]. To compare the number of obtainable uDOFs, aperture size and the unique lags using the same number of elements which is M+N-1, Table II [16] 2M A coprime array consists of 8 elements with M=4 and N=5 as shown in Fig. 4. In Fig. 4  the proposed array in [16] and our proposed array can recover the entire sources successively and with higher resolution.
However, the proposed array model can perform better since it presents less RMSE with the generation of higher uDOF and small weight functions that reduce RMSE.
Then, a comparison of the different array configuration is performed regarding the root mean squared error (RMSE). The RMSE is described as :  Fig. 7(a), when the SNR is greater than 5dB, the RMSE is near to zero since it obtains larger contiguous lags of the resulted virtual array. In Fig. 7(b), the RMSE is near zero when the number of snapshots is more than 500 snapshots.