In this section, the array model and signal model of this article are mainly given.

### 2.1 Array model

As shown in Fig. 1, MIMO targets 4 transmit arrays and M receive arrays, that is, the total number of arrays is *M* + 4. Due to the nature of the MIMO array model, the array can be virtualized so that the number of virtualized arrays is 4*M*. The array arrangement of the transmitting array is shown in Fig. 1, and the receiving array is *M* uniform linear arrays with a spacing of 2*Md* along the *x*-axis direction, where \(d = \lambda /2\) and \(\lambda\) is wavelength.

Due to the nature of the MIMO radar, the virtual array is shown in Fig. 2.

Due to the need to construct a coprime array, the last element of sub-array 2 is discarded to form a coprime array, as shown in Fig. 3.

As shown in Fig. 3, the coprime array consists of three sparse linear uniform arrays. Sub-array 1 has *2 M* array elements, and its array element spacing is *Md*; and sub-arrays 2 and 3 have *M-1* and M array elements, and its array element spacing is. The array element spacing *d* is \(\lambda /2\), where \(\lambda\) is the wavelength of the corresponding carrier frequency. By choosing \(M \in {\mathbb{N}}^{ + }\) and \(2M \in {\mathbb{N}}^{ + }\) to be relatively prime (where \({\mathbb{N}}^{ + }\) is expressed as a set of positive integers), the minimum cell spacing along the y-axis is \(\lambda /2\). This article assumes that the array sensor is located at

$$\left\{ {\left( {x,y} \right)|\left( {0,M2md} \right) \cup (d,2Mm_{1} d) \cup \left( {d + Ld,2M^{2} d + 2Mm_{2} d} \right)} \right\}$$

(1)

where \(2m \in [0,2M - 1]\), \(m_{1} \in [1,M - 1]\), \(m_{2} \in [0,M - 1]\), \(n,m_{1} ,m_{2} \in {\mathbb{N}}^{ + }\), where (*x*,*y*) represents the coordinates in the *x − y* plane. To make the distinction simple, let 2*M* = *N*. Then the array sensor is located at

$$\left\{ {\left( {x,y} \right)|\left( {0,M2md} \right) \cup (d,Nm_{1} d) \cup \left( {d + Ld,MNd + Nm_{2} d} \right)} \right\}$$

(2)

### 2.2 Signal model

For the estimation of the one-dimensional wave arrival angle direction, compared with the traditional coprime array, the main difference is that these sub-arrays are no longer collinear, and are placed in parallel at distances *d* and *Ld*, \(\left( {L \in {\mathbb{N}}^{ + } } \right)\), that is, the minimum unit spacing *d* along the *x*-axis. As *L* increases, the aperture of the array also increases, and the resolution also increases. But the larger the aperture, the signal will be correlated, so the value of *L* should not be very large.

The output is

$$\begin{aligned} x(l) & = \left[ {a_{t} \left( {\theta_{1} ,\varphi_{1} } \right) \otimes a_{r} \left( {\theta_{1} ,\varphi_{1} } \right),a_{t} \left( {\theta_{2} ,\varphi_{2} } \right) \otimes a_{r} \left( {\theta_{2} ,\varphi_{2} } \right), \ldots } \right. \\ & \quad \left. {a_{t} \left( {\theta_{k} ,\varphi_{k} } \right) \otimes a_{r} \left( {\theta_{k} ,\varphi_{k} } \right)} \right]S(l) + n(l) \\ \end{aligned}$$

(3)

where \(\theta_{k}\) and \(\varphi_{k}\) in Eq. (3) are the azimuth and elevation angles of the *k*th source, respectively. \(\otimes\) is expressed as the Kronecker product. \(n(l)\) is another noise vector whose elements are independently and evenly distributed in \((i,i,d)\) and obey the Gaussian distribution \(CN\left( {0,\sigma_{n}^{2} I_{{N_{t}^{i} }} } \right)\), where \(i = 1,2,3\). \(a_{t} (\theta_{k} ,\varphi_{k} ) = a_{ty} (\theta_{k} ,\varphi_{k} ) \otimes a_{tx} (\theta_{k} ,\varphi_{k} )\), \(a_{r} (\theta_{k} ,\varphi_{k} ) = a_{ry} (\theta_{k} ,\varphi_{k} ) \otimes a_{rx} (\theta_{k} ,\varphi_{k} )\), where \(a_{ty} (\theta_{k} ,\varphi_{k} )\) and \(a_{tx} (\theta_{k} ,\varphi_{k} )\) are the steering vectors of the transmitting array. At the same time, \(a_{t} (\theta_{k} ,\varphi_{k} ) \otimes a_{r} (\theta_{k} ,\varphi_{k} )\) corresponds to the Kronecker product of the receiving direction vector and the sending direction vector of the *k*th target. \(a_{ty} (\theta_{k} ,\varphi_{k} ) \otimes a_{tx} (\theta_{k} ,\varphi_{k} )\) and \(a_{ry} (\theta_{k} ,\varphi_{k} ) \otimes a_{rx} (\theta_{k} ,\varphi_{k} )\) are the same. Let \(a_{ti} (\theta_{q} ,\varphi_{q} ) \otimes a_{ri} (\theta_{q} ,\varphi_{q} ) = a_{i} (\theta_{q} ,\phi_{q} )\), suppose the relationship after the virtual is:

$$x_{i} (t) = \sum\nolimits_{q = 1}^{{\text{Q}}} {a\left( {\theta_{q} ,\phi_{q} } \right)} e^{{j2\pi \frac{{x_{i} }}{\lambda }\sin (\theta_{q} )\cos (\phi_{q} )}} s_{q} (t) + n_{i} (t)$$

(4)

where

$$a_{i} \left( {\theta_{q} ,\phi_{q} } \right) = \left[ {{{e}}^{{j2\pi \frac{{y_{1}^{i} }}{\lambda }\sin (\theta_{q} )\cos (\phi_{q} )}} , \ldots ,{{e}}^{{j2\pi \frac{{y_{{t_{t}^{i} }}^{i} }}{\lambda }\sin (\theta_{q} )\cos (\phi_{q} )}} } \right]^{T}$$

(5)

Equation (5) represents \((\theta_{q} ,\phi_{q} )\) corresponding to the steering vector of the *i*th sub-array, where \(q = 1, \ldots ,Q\), \(i = 1,2,3\). \(y_{j}^{i} ,1 \le j \le N_{t}^{i}\) is the y coordinate of the *i*th sensor. Where \(N_{t}^{i}\) is the total number of sensors in the *i*th sub-array, that is, \(N_{t}^{1} = 2M\), \(N_{t}^{2} = M - 1\), \(N_{t}^{3} = M\).Similarly, \(x_{i}\) represents the position of the *i*th sub-array along the x-axis, and the noise vector element is in the *i*th sub-array \(n_{i} (t)\), where \((i,i,d)\) is independently and uniformly distributed and obeys the Gaussian distribution \(CN\left( {0,\sigma_{n}^{2} I_{{N_{t}^{i} }} } \right)\), where \(i = 1,2,3\). In order to transform the two-dimensional DOA estimation problem into two independent one-dimensional problems, as shown in Fig. 4, \(\alpha_{q} ,\beta_{q} \in [0^{ \circ } ,180^{ \circ } ]\), where *q* = 1, …, *Q,* respectively, are expressed as the angle between the incident direction and the *y*-axis and *x*-axis The relationship between \(\alpha_{q} ,\beta_{q}\) and \(\theta_{q} ,\phi_{q}\) is

$$\cos \left( {\alpha_{q} } \right) = \sin \left( {\theta_{q} } \right)\sin \left( {\phi_{q} } \right)$$

(6)

$$\cos \left( {\beta_{q} } \right) = \sin \left( {\theta_{q} } \right)\cos \left( {\phi_{q} } \right)$$

(7)

Therefore, the data vector received in Eq. (4) is

$$x_{i} (t) = \sum\nolimits_{q = 1}^{Q} {a_{i} \left( {\alpha_{q} } \right)} {{e}}^{{j2\pi \frac{{x_{i} }}{\lambda }\cos (\beta_{q} )}} s_{q} (t) + n_{i} (t)$$

(8)

The corresponding steering vector is

$$a_{i} (\alpha_{q} ) = \left[ {{{e}}^{{j2\pi \frac{{y_{i} }}{\lambda }\cos (\alpha_{q} )}} , \ldots ,{{e}}^{{j2\pi \frac{{y_{{N_{t} }}^{i} }}{\lambda }\cos (\alpha_{q} )}} } \right]^{T}$$

(9)

Set \(s(t) = [s_{1} (t), \ldots ,s_{Q} (t)]^{T}\) to the signal vector, \(A_{i} = [a_{i} (\alpha_{1} ), \ldots ,a_{i} (\alpha_{Q} )]\) is the array manifold corresponding to the *i*th sub-array, where \(i = 1,2,3\), the data vector of the receiving channel can be written as

$$x_{i} (t) = A_{i} B_{i} s(t) + n_{i} (t)$$

(10)

The diagonal matrix is expressed as

$$B_{i} = {\text{diag}}\left( {\left[ {{{e}}^{{j2\pi \frac{{x_{i} }}{\lambda }\cos (\beta_{1} )}} , \ldots ,{{e}}^{{j2\pi \frac{{x_{i} }}{\lambda }\cos (\beta_{Q} )}} } \right]} \right)$$

(11)

Although traditional methods can achieve high-resolution DOA estimation, the \(Q < N_{t} Q\) conditions must be met to obtain the noise subspace. In application, the problem of detecting information sources with more than the number of array elements has become the focus of research. In this section, an effective method is proposed to achieve the equivalence of differential arrays with a larger number of DOF. In addition, the group sparse array technology is used to improve the estimation accuracy of DOA, and the differential covariance equations of \(x_{i} (t)\) and \(x_{k} (t)\) are constructed.

The cross-covariance matrix of the data vectors accepted by subarrays \(x_{i} (t)\) and \(x_{k} (t),1 \le i,k \le 3\) can be obtained. The cross-covariance matrix is

$$\begin{aligned} R_{{x_{ik} }} & = E\left[ {x_{i} (t)X_{k}^{H} (t)} \right] \\ & = \sum\nolimits_{q = 1}^{Q} {\sigma_{q}^{2} {{e}}^{{j2\pi \frac{{(x_{i} - x_{k} )}}{\lambda }}} } a_{i} \left( {\alpha_{q} } \right)a_{k}^{H} \left( {\alpha_{q} } \right) + n_{i} (t)n_{k}^{H} (t) \\ & = \left\{ {\begin{array}{*{20}l} {A_{i} R_{ss} D_{ik} A_{k}^{H} } & {i \ne k} \\ {A_{i} R_{ss} A_{i}^{H} + \sigma_{n}^{2} I_{{N_{t}^{i} }} ,} & {i = k} \\ \end{array} } \right. \\ \end{aligned}$$

(12)

where \(R_{s} = E[s(t)s^{H} (t)] = {\text{diag}}([\sigma_{1}^{2} , \ldots ,\sigma_{Q}^{2} ])\) is the covariance matrix of the \(Q \times Q\) dimensional signal, and its diagonal term represents the scattered power of the signal. In addition,

$$D_{ik} = B_{i} B_{k}^{H} = {\text{diag}}\left\{ {\left[ {{\text{e}}^{{j2\pi \frac{{(x_{i} - x_{k} )}}{\lambda }\cos (\beta_{1} )}} , \ldots ,{\text{e}}^{{j2\pi \frac{{(x_{i} - x_{k} )}}{\lambda }\cos (\beta_{Q} )}} } \right]^{T} } \right\}$$

(13)

When *i* = *k*, it becomes the identity matrix.

The matrix \(R_{{x_{ik} }}\) is quantized to obtain the following measurement vector:

$$z_{ik} = {\text{vec}}\left( {R_{{x_{ik} }} } \right) = \left\{ {\begin{array}{*{20}l} {\overline{A}_{ik} b_{ik} ,} & {i \ne k} \\ {\overline{A}_{ik} b_{ik} + \sigma_{n}^{2} i,} & {i = k} \\ \end{array} } \right.$$

(14)

where

$$\overline{A}_{ik} = \left[ {\overline{a}_{ik} \left( {\alpha_{1} } \right), \ldots ,\overline{a}_{ik} \left( {\alpha_{Q} } \right)} \right]$$

(15)

$$b_{ik} = \left[ {\sigma_{1}^{2} {{e}}^{{j2\pi \frac{{(x_{i} - x_{k} )}}{\lambda }\cos (\beta_{1} )}} , \ldots ,\sigma_{1}^{2} {{e}}^{{j2\pi \frac{{(x_{i} - x_{k} )}}{\lambda }\cos (\beta_{Q} )}} } \right]^{T}$$

(16)

where \(\overline{a}_{ik} (\alpha_{q} ) = a_{i} (\alpha_{q} ) \otimes a_{k}^{*} (\alpha_{q} )\), \(1 \le q \le Q\), \(\left( \cdot \right)^{*}\) is denoted as conjugate. \(i = {\text{vec}}(I_{{N_{t}^{i} }} )\), using the van der Monte structure of vectors \(a_{i} (\alpha_{q} )\) and \(a_{k} (\alpha_{q} )\), the entry in \(\overline{a}_{ik} (\alpha_{q} )\) retains the \({\text{e}}^{j\pi (Mn - Nm)} \cos (\alpha_{q} )\) factor. Therefore, \(z_{ik}\) can be regarded as a data vector received from a single snapshot signal vector \(b_{ik}\), and the array manifold \(A_{ik}\) corresponds to a virtual array whose virtual elements are located in the self-hysteresis and cross-lag between different sub-array sets. Due to the relative prime properties of *M* and *N*, there are fewer redundant elements in these virtual arrays. Therefore, the degree of freedom in the common array is greatly increased, so that more sources of \(N_{t}\) can be estimated with fewer array elements.