Abstract
In this paper, we propose a vectorized noncircular MUSIC (VNCM) algorithm based on the concept of the coarray, which can construct the difference and sum (diff–sum) coarray, for direction finding of the noncircular (NC) quasi-stationary sources. Utilizing both the NC property and the concept of the Khatri–Rao product, the proposed method can be applied to not only the ULA but also sparse arrays. In addition, we utilize the quasi-stationary characteristic instead of the spatial smoothing method to solve the coherent issue generated by the Khatri–Rao product operation so that the available degree of freedom (DOF) of the constructed virtual array will not be reduced by half. Compared with the traditional NC virtual array obtained in the NC MUSIC method, the diff–sum coarray achieves a higher number of DOFs as it comprises both the difference set and the sum set. Due to the complementarity between the difference set and the sum set for the coprime array, we choose the coprime array with multiperiod subarrays (CAMpS) as the array model and summarize the properties of the corresponding diff–sum coarray. Furthermore, we develop a diff–sum coprime array with multiperiod subarrays (DsCAMpS) whose diff–sum coarray has a higher DOF. Simulation results validate the effectiveness of the proposed method and the high DOF of the diff–sum coarray.
Highlights
Noncircular (NC) signals, such as amplitude-modulated (AM) signals and binary phase-shift keying (BPSK)-modulated signals, have been widely applied in various communication systems [1,2,3,4,5,6,7,8].Different from circular signals, which can only use the information in the covariance matrix for direction finding, NC signals can use the information in both the covariance matrix and the elliptic covariance matrix for direction finding
The virtual array generated in this kind of method should be able to achieve the same number of degree of freedom (DOF) as that generated in the vectorized NC MUSIC (VNCM) method
The configurations utilized are respectively the coprime array with multiperiod subarrays (CAMpS) with ( M, N, P1, P2 ) = (4, 3, 2, 3), the diff–sum coprime array with multiperiod subarrays (DsCAMpS) with ( M, N, P2 ) = (4, 3, 3) and the uniform linear array (ULA)= {0, 1 . . . , 14}
Summary
Noncircular (NC) signals, such as amplitude-modulated (AM) signals and binary phase-shift keying (BPSK)-modulated signals, have been widely applied in various communication systems [1,2,3,4,5,6,7,8]. A lot of DOA estimation algorithms for NC sources have been developed, such as the NC MUSIC method [9], NC Root-MUSIC method [1], NC ESPRIT method [10] and NC Unitary ESPRIT method [11] These traditional NC DOA estimation algorithms utilize the complex conjugate counterpart of the received signals to obtain the NC covariance matrix, which corresponds to a virtual array consisting of the physical array and its flipped array [12]. Constructing a novel virtual array with larger array aperture than the difference coarray is another useful way to increase the DOFs. In [16], we utilized both the temporal information and the spatial information of the received signals to propose the vectorized conjugate augmented MUSIC (VCAM) algorithm, which can construct the difference and sum (diff–sum) coarray. E [] is used to denote the expectation operation and vec(.) represents the vectorizing operation. ⊗ and respectively denote the left Kronecker product and the Khatri–Rao product
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