In this section, we introduce the concept of interpolation into the SDCA to further increase the DOF. Based on the concept of array interpolation [23], additional virtual sensors are interpolated into the “holes” of the virtual array to create an interpolated ULA. Moreover, an ANM problem is formulated in order to reconstruct Toeplitz covariance matrix, which can take full advantage of all information to estimate the off-grid DOAs.

### 3.1 Virtual array interpolation for SDCA

For the sake of intuitive understanding, an example is demonstrated in Fig. 2, where \(M=3\) and \(N=7\).

Clearly, we can obtain the nonuniform virtual array derived from the SDCA as shown in Fig. 2a, where the missing elements \(\{\pm 31,\pm 34,\pm 35\}\) in \({\mathbb{S}}_v\) are the holes. In order to generate a virtual ULA without discarding the discontinuous sensors, the additional virtual sensors are filled into the positions of the holes, which are called the interpolated sensors. The interpolated virtual array \({\mathbb{S}}_I\) is depicted in Fig. 2b, where all information received by the virtual sensors in \({\mathbb{S}}_v\) are included. It should be noticed that the interpolated sensors expressed by hollow circles in Fig. 2b are nonfunctional and the corresponding outputs are zero. Accordingly, the received signals from the interpolated sensors are regarded as zero. Thus, the output \({\mathbf{y}}_I\) of the interpolated virtual array \({\mathbb{S}}_I\) can be initialized as

$$\begin{aligned} \langle {\mathbf{y}}_I \rangle _i = {\left\{ \begin{array}{ll} \langle {\mathbf{y}}_v \rangle _i , &{} i\in {\mathbb{S}}_v,\\ 0, &{} i \in {\mathbb{S}}_I \setminus {\mathbb{S}}_v \end{array}\right. } \end{aligned}$$

(16)

where \(\langle \cdot \rangle _i\) represents the virtual sensor at the position *id*. \({\mathbb{S}}_I \setminus {\mathbb{S}}_v\) denotes the elements in \({\mathbb{S}}_I\) but not in \({\mathbb{S}}_v\). In order to successfully use the interpolated virtual array, the unknown virtual signals received by the interpolated sensors should be recovered.

Similar to (15), the ideal received signals of the interpolated virtual array \({\mathbb{S}}_I\) can be expressed as

$$\begin{aligned} {\mathbf{y}}_I ={\mathbf{A}}_I{\mathbf{p}} = \sum _{q=1}^Q{\mathbf{a}}_I(\theta _q)p_q, \end{aligned}$$

(17)

where \({\mathbf{A}}_I = [{\mathbf{a}}_I(\theta _1),{\mathbf{a}}_I(\theta _2),\ldots ,{\mathbf{a}}_I(\theta _Q)]\in {\mathbb{C}}^{|{\mathbb{S}}_I| \times Q}\) represents the steering matrix of the interpolated virtual array. Obviously, \({\mathbf{y}}_I\) behaves like the received signal from a single snapshot. In addition, the rank of its covariance matrix is one and subspace-based DOA estimation techniques cannot be applied to identify multiple sources. To overcome this problem, we divide this interpolated virtual array \({\mathbb{S}}_I\) into \(J=(|{\mathbb{S}}_I|+1)/2\) overlapping sub-arrays, and each sub-array contains *J* elements. Accordingly, \({\mathbf{y}}_I\) can be divided into *J* segments \(\{{\mathbf{y}}^1_I,{\mathbf{y}}^2_I,\ldots ,{\mathbf{y}}^J_I \}\) as shown in Fig. 3, and the equivalent signals of each sub-array \({\mathbf{y}}^j_I\) can be denoted as

$$\begin{aligned} {\mathbf{y}}^j_I ={\mathbf{A}}^j_I{\mathbf{p}} = \sum _{q=1}^Q{\mathbf{a}}^j_I(\theta _q)p_q, \end{aligned}$$

(18)

where \({\mathbf{A}}^j_I = [{\mathbf{a}}^j_I(\theta _1),{\mathbf{a}}^j_I(\theta _2),\ldots ,{\mathbf{a}}^j_I(\theta _Q)]\in {\mathbb{C}}^{J \times Q}(j=1,2,\ldots ,J)\), and \({\mathbf{a}}^j_I(\theta _q)=[e^{-j\pi d_{J-j+1}\sin (\theta _q)},e^{-j\pi d_{J-j+2}\sin (\theta _q)},\ldots ,e^{-j\pi d_{2J-j}\sin (\theta _q)}]^{\text{T}}\) represents the steering vector of the *q*th source corresponding to the *j*th virtual sub-array, where \(d_i\) represents the *i*th virtual sensor position in \({\mathbb{S}}_I\).

Specifically, the sensor positions of the reference virtual sub-array are \(\{0,d,\ldots ,(J-1)d\}\) as shown in Fig. 3. By setting \(j=1\), we can obtain the steering vector corresponding to the *q*th source

$$\begin{aligned} {\mathbf{a}}^1_I(\theta _q)=[e^{-j\pi \sin (\theta _q)},\ldots ,e^{-j\pi (J-1)\sin (\theta _q)}]^{\text{T}}. \end{aligned}$$

(19)

Accordingly, the received signals of the reference virtual sub-array can be expressed as

$$\begin{aligned} {\mathbf{z}}= {\mathbf{y}}^1_I = \sum _{q=1}^Q{\mathbf{a}}^1_I(\theta _q)p_q. \end{aligned}$$

(20)

Hence, we can construct the following atomic set in continuous domain by using the steering vector of the reference virtual sub-array as

$$\begin{aligned} {\mathcal{A}} = \{{\mathbf{a}}^1_I(\theta )|\theta \in [-90^\circ ,90^\circ ] \}. \end{aligned}$$

(21)

The atomic norm of the output \({\mathbf{z}}\) received from the reference virtual sub-array is defined as the minimum number of atoms in \({\mathcal{A}}\) that can express \({\mathbf{z}}\), i.e.,

$$\begin{aligned} \Vert {\mathbf{z}}\Vert _{{\mathcal{A}}} = \hbox{inf}\{t>0:{\mathbf{z}}\in t \hbox{conv}({\mathcal{A}}) \} = \hbox{inf}\left\{ \sum _{q}p_q:{\mathbf{z}} = \sum _{q}p_q{\mathbf{a}}^1_I(\theta _q), p_q>0 \right\} , \end{aligned}$$

(22)

where \(\hbox{conv}({\mathcal{A}})\) represents the convex hull of the atom set \({\mathcal{A}}\) and inf is the infimum of infinite set.

### 3.2 Toeplitz matrix reconstruction for DOA estimation

Based on (18), we can obtain the multiple measurements as \({\mathbf{Y}}_I = [{\mathbf{y}}^1_I,{\mathbf{y}}^2_I,\ldots ,{\mathbf{y}}^J_I] \in {\mathbb{C}}^{J \times J}\). Thus, we can express the correlation statistic of the equivalent virtual signal \({\mathbf{Y}}_I\) as

$$\begin{aligned} {\mathbf{R}}_{{Y}_I} =\dfrac{1}{J}{\mathbf{Y}}_I{\mathbf{Y}}_I^{\text{H}}= \dfrac{1}{J}\sum _{j=1}^J {\mathbf{y}}^j_I{{\mathbf{y}}^j_I}^{\text{H}}, \end{aligned}$$

(23)

where \({\mathbf{R}}_{{Y}_I}\in {\mathbb{C}}^{J \times J}\) is a full rank covariance matrix. Since the elements located at the holes in \({\mathbb{S}}_I\) are set to be zeros, each element in \({\mathbf{R}}_{{Y}_I}\) suffers a deviation during the summation process in (23). Therefore, \({\mathbf{R}}_{{Y}_I}\) can’t be directly calculated. Encouragingly, according to [26], a new Hermitian Toeplitz matrix \({\mathbf{R}}\) is associated with \({\mathbf{R}}_{{Y}_I}\), and they are related as \({\mathbf{R}}^2=J{\mathbf{R}}_{{Y}_I}\). It is possible to construct the Hermitian Toeplitz covariance matrix \({\mathbf{R}}\) from the second-order statistics \({\mathbf{y}}_I\) by

$$\begin{aligned} {\mathbf{R}} = \begin{bmatrix} \langle {\mathbf{y}}_I \rangle _J &{}\quad \langle {\mathbf{y}}_I \rangle ^*_{J+1} &{}\quad \cdots &{}\quad \langle {\mathbf{y}}_I \rangle ^*_{2J-1}\\ \langle {\mathbf{y}}_I \rangle _{J+1} &{}\quad \langle {\mathbf{y}}_I \rangle _J &{}\quad \cdots &{}\quad \langle {\mathbf{y}}_I \rangle ^*_{2J-2}\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \langle {\mathbf{y}}_I \rangle _{2J-1} &{}\quad \langle {\mathbf{y}}_I \rangle _{2J-2} &{}\quad \cdots &{}\quad \langle {\mathbf{y}}_I \rangle _J \end{bmatrix}. \end{aligned}$$

(24)

Notice that the interpolated virtual array \({\mathbb{S}}_I\) is symmetric about zero point. The complex-valued signals of the symmetrical pair in \({\mathbf{y}}_I\) are conjugate. Thus, the matrix \({\mathbf{R}}\) is directly derived from the signal statistics \({\mathbf{y}}_I\), which includes all information received by the interpolated virtual array. However, since there are several zero elements filled in \({\mathbf{y}}_I\) which denote the holes, the diagonals in \({\mathbf{R}}\) corresponding positions are zeros. Thus, a binary vector \({\mathbf{b}} \in {\mathbb{R}}^J\) is defined to describe the presence of virtual sensors indexed in \({\mathbb{S}}_I\), where the element of 0 in \({\mathbf{b}}\) stands for the sensor positions to be interpolated and 1 otherwise.

Therefore, according to (22), the Toeplitz covariance matrix reconstruction in (24) can be represented as the following ANM problem

$$\begin{aligned}&\min _{{\mathbf{z}} \in {\mathbb{C}}^J} \quad \Vert {\mathbf{z}}\Vert _{{\mathcal{A}}} \\&\text{ s.t. }\quad {\left\{ \begin{array}{ll} \Vert {\mathcal{T}}({\mathbf{z}})\circ {\mathbf{B}} - {\mathbf{R}} \Vert ^2_F\le \delta , \\ {\mathcal{T}}({\mathbf{z}}) \succeq 0,&{} \end{array}\right. } \end{aligned}$$

(25)

where \({\mathcal{T}}({\mathbf{z}})\) denotes a Hermitian PSD Toeplitz matrix with the optimization variable \({\mathbf{z}}\) as the first column. Therefore, \({\mathcal{T}}({\mathbf{z}})\) can be seen as the covariance matrix of signals corresponding to the reference virtual sub-array. Furthermore, notice that it contains all information of the interpolated virtual array [23]. Here, \({\mathbf{B}} = {\mathbf{b}}{\mathbf{b}}^{\text{T}} \in {\mathbb{R}}^{J \times J}\) is a binary matrix distinguishing the zero (unknown) elements and the nonzero (known) elements in \({\mathbf{R}}\). This makes the nonzero elements in \({\mathbf{R}}\) consist of the elements in the reconstructed covariance matrix \({\mathcal{T}}({\mathbf{z}})\). And \(\Vert {\cdot }\Vert _F\) represents the Frobenius norm, \(\delta\) is a threshold to restrict the discrepancies between the nonzero elements in \({\mathbf{R}}\) and the corresponding elements in \({\mathcal{T}}({\mathbf{z}})\). Moreover, \({\mathcal{T}}({\mathbf{z}}) \succeq 0\) guarantees the Hermitian positive semi-definite Toeplitz structure of the optimum solution. Then, the optimization problem (25) can be written as

$$\begin{aligned}&\min _{{\mathbf{z}} \in {\mathbb{C}}^J} \quad \dfrac{1}{2}\Vert {\mathcal{T}}({\mathbf{z}})\circ {\mathbf{B}} - {\mathbf{R}} \Vert ^2_F+\mu \Vert {\mathbf{z}}\Vert _{{\mathcal{A}}} \\&\text{ s.t. }\quad {\mathcal{T}}({\mathbf{z}}) \succeq 0, \end{aligned}$$

(26)

where \(\mu\) is a regularization parameter.

Since \({\mathcal{T}}({\mathbf{z}})\) is a PSD Hermitian Toeplitz matrix, if \(k = \hbox{rank}({\mathcal{T}}({\mathbf{z}})) < J\), according to Vandermonde decomposition, \({\mathcal{T}}({\mathbf{z}})\) can be decomposed as [27]

$$\begin{aligned} {\mathcal{T}}({\mathbf{z}}) = \sum _{q=1}^K{\mathbf{v}}_1(\theta _q)p_q{\mathbf{v}}^{\text{H}}_1(\theta _q). \end{aligned}$$

(27)

Then, we can obtain the trace of \({\mathcal{T}}({\mathbf{z}})\) as

$$\begin{aligned} \hbox{Tr}({\mathcal{T}}({\mathbf{z}}))= J\sum _{q=1}^Kp_q, \end{aligned}$$

(28)

where \(\hbox{Tr}(\cdot )\) represents the trace operator. Observing the trace of \({\mathcal{T}}({\mathbf{z}})\) in (28) along with the definition of the atomic norm of \({\mathbf{z}}\) in (22), we can get

$$\begin{aligned} \Vert {\mathbf{z}}\Vert _{{\mathcal{A}}} = \dfrac{1}{J}\hbox{Tr}({\mathcal{T}}({\mathbf{z}})). \end{aligned}$$

(29)

Hence, the ANM problem in (26) can be equivalently transformed as,

$$\begin{aligned}&\min _{{\mathbf{z}} \in {\mathbb{C}}^J} \quad \dfrac{1}{2}\Vert {\mathcal{T}}({\mathbf{z}})\circ {\mathbf{B}} - {\mathbf{R}} \Vert ^2_F+\eta \hbox{Tr}({\mathcal{T}}({\mathbf{z}})) \\&\text{ s.t. }\quad {\mathcal{T}}({\mathbf{z}}) \succeq 0, \end{aligned}$$

(30)

where \(\eta = \dfrac{1}{J}\mu\). Note that the optimization problem (30) is convex and easy to be solved by the CVX software [28]. After the optimal solution \(\hat{{\mathbf{z}}}\) is obtained, the covariance matrix \({\mathcal{T}}(\hat{{\mathbf{z}}})\) of the interpolated ULA can be successfully reconstructed. At the same time, the virtual signals derived from the interpolated sensors which are initialized as zero can also be recovered. Since the signal covariance matrix \({\mathcal{T}}(\hat{{\mathbf{z}}})\) which corresponds to the interpolated virtual ULA is reconstructed, the subspace methods such as the MUSIC based [6, 18] and the ESPRIT based [7] can be employed for DOA estimation. Here, we calculate the MUSIC spatial spectrum by the following formula

$$\begin{aligned} f_{\mathrm{MUSIC}}(\theta ) = \dfrac{1}{{\mathbf{v}}^{\text{H}}_1(\theta ){\mathbf{U}}_{{\mathbf{N}}}{\mathbf{U}}^{\text{H}}_{{\mathbf{N}}}{\mathbf{v}}_1(\theta )}, \end{aligned}$$

(31)

where \({\mathbf{U}}_{{\mathbf{N}}}\) is the noise subspace of \({\mathcal{T}}(\hat{{\mathbf{z}}})\), which is acquired by selecting the eigenvectors corresponding to the \(J-Q\) smallest eigenvalues of \({\mathcal{T}}(\hat{{\mathbf{z}}})\). Finally, the DOA estimation can be acquired by searching for the locations corresponding to the *Q* largest peaks of the spectrum in \(f_{\mathrm{MUSIC}}(\theta )\). Accordingly, the number of targets that can be identified are up to \(J-1\).

The advantages of the proposed algorithm are summarized as follows. Firstly, the discontiguous virtual array is interpolated into a virtual ULA, and all of the information in the virtual array can be effectively utilized. Secondly, the interpolated virtual array of SDCA provides a larger coarray aperture and higher DOF. Furthermore, the atomic norm minimization problem can be performed to reconstruct the covariance matrix of the interpolated virtual array in a gridless manner, which avoids the basis mismatch problem.