# Non-circular signal DOA estimation based on coprime array MIMO radar

## 1 Introduction

Multiple-input multiple-output (MIMO) technology was introduced into the radar field by Lincoln Laboratory in the USA in 2003 , and the concept of MIMO radar was proposed. Compared with traditional radars, MIMO radar has greater advantages in azimuth resolution, array freedom, multi-target parameter estimation, and anti-jamming capabilities and has been studied and paid attention by many scholars. Direction of arrival [2,3,4,5,6] (DOA) estimation, as an important research content in array signal processing, has been widely used in sonar, radar, medical, and wireless communication fields [7,8,9]. In recent years, the DOA estimation problem of MIMO radar has been widely concerned and has become a research hotspot.

Notations: Lower-case (upper-case) bold characters are used to represent vectors (matrices). The superscripts (·)T, (·)*, (·)H denote the transpose, conjugate, and conjugate transpose of operation, respectively. E{·} is exploited to represent the expectation of $$\otimes$$ refers Kronecker product, $$\circ$$ refers the Khatri–Rao product. diag{·} denotes the diagonal matrix whose diagonal elements are the elements ·.

## 2 Preliminaries

The geometric structure of the expanded coprime array MIMO radar is shown in Fig. 1. Both the transmitter array and the receiver array are composed of two uniform sparse subarrays, the two subarrays are expanded side by side, and the last element of subarray 1 is used as the first element of subarray 2. The sparse uniform subarray 1 and subarray 2 together form the expanded coprime array. Subarray 1 contains M array elements, and the distance between each adjacent array element is Nd (where N is the number of array elements and d is half the wavelength). The other subarray 2 contains N array elements, and the distance between each adjacent array element is Md (where M is the number of array elements and d is half the wavelength). The two numbers M and N are mutually prime, and the distance between adjacent elements of subarray 1 and subarray 2 is greater than half of the wavelength. The formed coprime array contains M + N − 1 elements, as shown in Fig. 1. The coprime array takes the first element as the reference point, and the position of the element can be expressed as:

$$L_{{\text{s}}} = \left\{ {Mnd|0 \le n \le N - 1} \right\}U\left\{ {Nmd|0 \le m \le M - 1} \right\}$$
(1)

where d = λ/2, λ is the wavelength.

Assume that there are K narrowband far-field uncorrelated signals that are incident on the coprime array at angles [θ1, θ2, …, θk], and then the direction vectors of the Kth target of the transmitter array and the receiver array are:

$${\varvec{a}}_{{\text{t}}} \left( {\theta_{k} } \right) = \left[ {{\kern 1pt} {\varvec{a}}_{{{\text{t}}1}}^{{\text{T}}} \left( {\theta_{k} } \right),{\varvec{a}}_{{{\text{t}}2}}^{{\text{T}}} \left( {\theta_{k} } \right){\kern 1pt} } \right]^{{\text{T}}}$$
(2)
$${\varvec{a}}_{{\text{r}}} \left( {\theta_{k} } \right) = \left[ {{\varvec{a}}_{{{\text{r}}1}}^{{\text{T}}} \left( {\theta_{k} } \right),{\varvec{a}}_{{{\text{r}}2}}^{{\text{T}}} \left( {\theta_{k} } \right)} \right]^{{\text{T}}}$$
(3)

where at1(θk), at2(θk) are the direction vectors of the transmitter array subarray 1 and subarray 2, respectively. ar1(θk), ar2(θk) are the direction vectors of the receiver array subarray 1 and subarray 2, respectively. The expressions of the direction vectors of the transmitter subarray and the receiver subarray are, respectively:

$${\varvec{a}}_{{{\text{t}}1}} \left( {\theta_{k} } \right) = {\varvec{a}}_{{{\text{r}}1}} \left( {\theta_{k} } \right) = \left[ {1,e^{{ - j\frac{{2\pi Nd\sin \left( {\theta_{k} } \right)}}{\lambda }}} , \ldots ,e^{{ - j\frac{{2\pi \left( {M - 1} \right)Nd\sin \left( {\theta_{k} } \right)}}{\lambda }}} } \right]^{{\text{T}}}$$
(4)
$${\varvec{a}}_{{{\text{t}}2}} \left( {\theta_{k} } \right) = {\varvec{a}}_{{{\text{r}}2}} \left( {\theta_{k} } \right) = \left[ {1,e^{{ - j\frac{{2\pi Md\sin \left( {\theta_{k} } \right)}}{\lambda }}} , \ldots ,e^{{ - j\frac{{2\pi \left( {N - 1} \right)Md\sin \left( {\theta_{k} } \right)}}{\lambda }}} } \right]^{{\text{T}}}$$
(5)

Therefore, the array flow matrix of the transmitter array and the receiver array are At and Ar, respectively. It can be expressed as:

$${\varvec{A}}_{{\text{t}}} \left( \theta \right) = \left[ {\begin{array}{*{20}c} {{\varvec{a}}_{{\text{t}}} \left( {\theta_{1} } \right),{\varvec{a}}_{{\text{t}}} \left( {\theta_{2} } \right), \ldots ,{\varvec{a}}_{{\text{t}}} \left( {\theta_{K} } \right)} \\ \end{array} } \right]$$
(6)
$${\varvec{A}}_{{\text{r}}} \left( \theta \right) = \left[ {\begin{array}{*{20}c} {{\varvec{a}}_{{\text{r}}} \left( {\theta_{1} } \right),{\varvec{a}}_{{\text{r}}} \left( {\theta_{2} } \right), \ldots ,{\varvec{a}}_{{\text{r}}} \left( {\theta_{K} } \right)} \\ \end{array} } \right]$$
(7)

From the formula (6) and (7), the flow pattern A of the entire virtual array can be obtained as:

\begin{aligned} {\varvec{A}} & = {\varvec{A}}_{{\text{r}}} \circ {\varvec{A}}_{{\text{t}}} = \left[ {\begin{array}{*{20}c} {{\varvec{a}}\left( {\theta_{1} } \right),{\varvec{a}}\left( {\theta_{2} } \right), \ldots ,{\varvec{a}}\left( {\theta_{K} } \right)} \\ \end{array} } \right] \\ & = \left[ {{\varvec{a}}_{{\text{r}}} \left( {\theta_{1} } \right) \otimes {\varvec{a}}_{{\text{t}}} \left( {\theta_{1} } \right),{\varvec{a}}_{{\text{r}}} \left( {\theta_{2} } \right) \otimes {\varvec{a}}_{{\text{t}}} \left( {\theta_{2} } \right), \ldots ,{\varvec{a}}_{{\text{r}}} \left( {\theta_{K} } \right) \otimes {\varvec{a}}_{{\text{t}}} \left( {\theta_{K} } \right)} \right] \\ \end{aligned}
(8)

where $${\varvec{A}}_{{\text{r}}} \circ {\varvec{A}}_{{\text{t}}}$$ is the Khatri–Rao operator, $${\varvec{a}}\left( {\theta_{k} } \right){ = }{\varvec{a}}_{{\text{r}}} \left( {\theta_{K} } \right) \otimes {\varvec{a}}_{{\text{t}}} \left( {\theta_{K} } \right)$$, $$\otimes$$ represents the Kronecker product. The received signal of the coprime array can be expressed as :

$${\varvec{x}}(t) = {\varvec{As}}(t) + {\varvec{n}}(t)$$
(9)

where s(t) is the source signal vector (non-circular signal vector in this article), n(t) is the noise vector, and A is the array flow matrix.

For the non-circular signal s(t), according to its definition, the non-circular signal is relative to the circular signal. If the signal has the characteristics of rotation invariance, the signal s(t) is called the circular signal. That is, when E{s(t)} = 0, $$E\{{\varvec{s}}(t){\varvec{s}}^{{\text{H}}} (t)\} \ne {\mathbf{0}}$$ and $$E\{{\varvec{s}}(t){\varvec{s}}^{{\text{T}}} (t)\} = {\mathbf{0}}$$ are established at the same time, s(t) is the circular signal. Conversely, if the signal does not have the characteristics of rotation invariance, then the signal s(t) is called a non-circular signal. That is, when E{s(t)} = 0, $$E\{{\varvec{s}}(t){\varvec{s}}^{{\text{H}}} (t)\} \ne {\mathbf{0}}$$ and $$E\{{\varvec{s}}(t){\varvec{s}}^{{\text{T}}} (t)\} = {\mathbf{0}}$$ are established at the same time, s(t) is the non-circular signal. The non-circular signal s(t) can be expressed as [34, 35]:

$${\varvec{s}}(t) = \user2{\psi s}_{0} (t)$$
(10)

where $${\varvec{\psi}} = {\text{diag}}\{e_{1}^{{ - j\phi_{1} }} ,e^{{ - j\phi_{2} }} , \ldots ,e^{{ - j\phi_{K} }} \}$$, ϕK is the Kth non-circular phase of the non-circular signal, $${\varvec{s}}_{0} (t) \in {\mathbb{R}}^{K \times 1}$$.

## 3 Methods

### 3.1 Expanded coprime array MIMO radar non-circular signal dimensionality reduction DOA estimation method

From the formula (9) and (10), the received signal of the coprime array can be expressed as:

$${\varvec{x}}(t) = \user2{A\psi s}_{0} (t) + {\varvec{n}}(t)$$
(11)

Using the non-circular characteristic of the signal s(t), the array flow matrix can be reconstructed, and the received signal can be reconstructed as:

\begin{aligned} {\varvec{y}}(t) & = \left[ {\begin{array}{*{20}l} {{\varvec{x}}(t)} \hfill \\ {{\varvec{x}}^{*} (t)} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\user2{A\psi }} \\ {{\varvec{A}}^{*} {\varvec{\psi}}^{*} } \\ \end{array} } \right]{\varvec{s}}_{0} (t) + \left[ {\begin{array}{*{20}c} {{\varvec{n}}(t)} \\ {{\varvec{n}}^{*} (t)} \\ \end{array} } \right] \\ & = {\varvec{Bs}}_{0} (t) + {\varvec{n}}_{0} (t) \\ \end{aligned}
(12)

where $${\varvec{n}}_{0} (t) = \left[ {\begin{array}{*{20}c} {{\varvec{n}}(t)} \\ {{\varvec{n}}^{ * } (t)} \\ \end{array} } \right]$$, $${\varvec{B}} = \left[ {\begin{array}{*{20}c} {{\varvec{A\varPsi}}} \\ {{\varvec{A}}^{ * }{\varvec{\varPsi}}^{ * } } \\ \end{array} } \right]{ = }\left[ {{\varvec{b}}\left( {\theta_{1} ,\phi_{1} } \right),{\varvec{b}}\left( {\theta_{2} ,\phi_{2} } \right), \ldots ,{\varvec{b}}\left( {\theta_{K} ,\phi_{K} } \right)} \right]^{{\text{T}}}$$, where

$${\varvec{b}}\left( {\theta_{K} ,\phi_{K} } \right){ = }\left[ {\begin{array}{*{20}l} {{\varvec{a}}\left( {\theta_{k} } \right)e^{{ - j\phi_{k} }} } \hfill \\ {{\varvec{a}}^{*} \left( {\theta_{k} } \right)e^{{j\phi_{k} }} } \hfill \\ \end{array} } \right]$$
(13)

Then the covariance matrix $${\varvec{R}} = E[{\varvec{y}}(t){\varvec{y}}^{{\text{H}}} (t)]$$ of the received signal can be obtained from L snapshots. That is

$$\hat{\user2{R}} = \frac{1}{L}\sum\limits_{l = 1}^{L} {{\varvec{y}}\left( {t_{l} } \right){\varvec{y}}^{{\text{H}}} \left( {t_{l} } \right)}$$
(14)

Perform eigen decomposition on the covariance matrix A, we can get

$$\hat{\user2{R}}{ = }{\varvec{E}}_{{\text{s}}} {\varvec{D}}_{{\text{s}}} {\varvec{E}}_{{\text{s}}}^{{\text{H}}} + {\varvec{E}}_{{\text{n}}} {\varvec{D}}_{{\text{n}}} {\varvec{E}}_{{\text{n}}}^{{\text{H}}}$$
(15)

where Ds is K × K diagonal matrix, whose diagonal elements are composed of K larger eigenvalues of the covariance matrix. Es is the signal subspace, which is the space formed by the eigenvectors corresponding to the K larger eigenvalues of the covariance matrix. Dn is composed of 2(M + N − 1)2 − K smaller eigenvalues with smaller diagonal elements. En is the noise subspace, which is the space formed by the eigenvectors corresponding to the 2(M + N − 1)2 − K smaller eigenvalues of the covariance matrix $$\hat{\user2{R}}$$. According to the orthogonality between the noise subspace and the direction vector, the following spatial spectrum function is constructed as :

$$P(\theta ,\phi ) = \frac{1}{{{\varvec{b}}^{{\text{H}}} (\theta ,\phi ){\varvec{E}}_{{\text{n}}} {\varvec{E}}_{{\text{n}}}^{{\text{H}}} {\varvec{b}}(\theta ,\phi )}}$$
(16)

After reconstructing the receiving matrix from non-circular signals, the spatial spectrum function is a two-dimensional spectral peak search, which is highly complex, and the following dimensionality reduction processing is performed. Firstly, reconstruct the formula (16), then:

$${\varvec{b}}(\theta ,\phi ) = \left[ {\begin{array}{*{20}l} {{\varvec{a}}\left( \theta \right)e^{ - j\phi } } \hfill \\ {{\varvec{a}}^{*} \left( \theta \right)e^{j\phi } } \hfill \\ \end{array} } \right]{ = }\left[ {\begin{array}{*{20}c} {{\varvec{a}}(\theta )} & {{\mathbf{0}}_{{M_{1} \times 1}} } \\ {{\mathbf{0}}_{{M_{1} \times 1}} } & {{\varvec{a}}^{ * } (\theta )} \\ \end{array} } \right] \times \left[ {\begin{array}{*{20}c} {e^{ - j\phi } } \\ {e^{j\phi } } \\ \end{array} } \right] = {\varvec{P}}(\theta ){\varvec{e}}_{0} (\phi )$$
(17)

where

$${\varvec{a}}\left( \theta \right){ = }{\varvec{a}}_{{\text{r}}} \left( \theta \right) \otimes {\varvec{a}}_{{\text{t}}} \left( \theta \right)$$
(18)
$${\varvec{P}}(\theta ) = \left[ {\begin{array}{*{20}c} {{\varvec{a}}(\theta )} & {{\mathbf{0}}_{{M_{1} \times 1}} } \\ {{\mathbf{0}}_{{M_{1} \times 1}} } & {{\varvec{a}}^{ * } (\theta )} \\ \end{array} } \right]$$
(19)
$${\varvec{e}}_{0} (\phi ){ = }\left[ {\begin{array}{*{20}c} {e^{ - j\phi } } \\ {e^{j\phi } } \\ \end{array} } \right]$$
(20)

Define function V(θ,ϕ):

$$V(\theta ,\phi ) = \frac{1}{P(\theta ,\phi )}{ = }{\varvec{b}}^{{\text{H}}} (\theta ,\phi ){\varvec{E}}_{{\text{n}}} {\varvec{E}}_{{\text{n}}}^{{\text{H}}} {\varvec{b}}(\theta ,\phi )$$
(21)

substituting formula (17) into the formula (21), we can get:

\begin{aligned} V(\theta ,\phi ) & = {\varvec{e}}_{0} (\phi )^{{\text{H}}} {\varvec{P}}(\theta )^{{\text{H}}} {\varvec{E}}_{{\text{n}}} {\varvec{E}}_{{\text{n}}}^{{\text{H}}} {\varvec{P}}(\theta )e_{0} (\phi ) \\ & = {\text{e}}^{ - j\phi } {\varvec{e}}_{0} (\phi )^{{\text{H}}} {\varvec{P}}(\theta )^{{\text{H}}} {\varvec{E}}_{{\text{n}}} {\varvec{E}}_{{\text{n}}}^{{\text{H}}} {\varvec{P}}(\theta ){\varvec{e}}_{0} (\phi ){\text{e}}^{j\phi } \\ \end{aligned}
(22)

where let $${\varvec{q}}(\phi ) = {\varvec{e}}_{0} (\phi ){\text{e}}^{j\phi }$$, $${\varvec{Q}}(\theta ){\mathbf{ = }}{\varvec{P}}(\theta )^{{\text{H}}} {\varvec{E}}_{{\text{n}}} {\varvec{E}}_{{\text{n}}}^{H} {\varvec{P}}(\theta )$$. Then V(θ,ϕ) can be expressed as:

$$V(\theta ,\phi ) = {\varvec{q}}(\phi )^{{\text{H}}} {\varvec{Q}}(\theta ){\varvec{q}}(\phi )$$
(23)

where the formula (22) is about the problem of quadratic optimization, to find its optimal solution (θ,ϕ). Firstly, increase the constraint of eHq(ϕ) = 1 to eliminate the solution of q(ϕ) = 0, so as to obtain the optimal solution (θ,ϕ) of V(θ,ϕ). Use constraints to reconstruct the secondary optimization problem and seek the optimal solution, that is:

$$\min {\varvec{q}}(\phi )^{{\text{H}}} {\varvec{Q}}(\theta ){\varvec{q}}(\phi )\quad {\text{s.t.}}\quad {\varvec{e}}^{{\text{H}}} {\varvec{q}}(\phi ) = 1$$
(24)

The method of solving the optimal solution using the Lagrange multiplier method, construct the cost function L(θ,ϕ), that is:

$$L(\theta ,\phi ) = {\varvec{q}}(\phi )^{{\text{H}}} {\varvec{Q}}(\theta ){\varvec{q}}(\phi ) - \lambda \left( {{\varvec{e}}^{{\text{H}}} {\varvec{q}}(\phi ) - 1} \right)$$
(25)

where λ is constant. The partial derivative of formula (25) can be obtained as follows:

$$\frac{\partial L(\theta ,\phi )}{{\partial {\varvec{q}}(\phi )}} = 2{\varvec{Q}}(\theta ){\varvec{q}}(\phi ) + \lambda {\varvec{e}}$$
(26)

where let the formula (26) be equal to 0, then we can get:

$${\varvec{q}}(\phi ){ = } - \frac{\lambda }{2}{\varvec{eQ}}^{ - 1} (\theta )$$
(27)

Because of the constraint of eHq(ϕ) = 1, combined with formula (27), we can get:

$${\varvec{q}}(\phi ) = \frac{{{\varvec{Q}}^{ - 1} (\theta ){\varvec{e}}}}{{{\varvec{e}}^{{\text{H}}} {\varvec{Q}}^{ - 1} (\theta ){\varvec{e}}}}$$
(28)

Combining formulas (24) and (28) we can obtain:

$$\hat{\theta } = {\text{argmin}}\frac{1}{{{\varvec{e}}^{{\text{H}}} {\varvec{Q}}^{ - 1} (\theta ){\varvec{e}}}} = {\text{argmax}}\,{\varvec{e}}^{{\text{H}}} {\varvec{Q}}^{ - 1} (\theta ){\varvec{e}}$$
(29)

Because $${\varvec{Q}}(\theta ){\mathbf{ = }}{\varvec{P}}(\theta )^{{\text{H}}} {\varvec{E}}_{{\text{n}}} {\varvec{E}}_{{\text{n}}}^{{\text{H}}} {\varvec{P}}(\theta )$$, then the one-dimensional spectral peak search function can be obtained:

$$f(\theta ) = {\varvec{e}}^{{\text{H}}} {\varvec{Q}}^{ - 1} (\theta ){\varvec{e}} = {\varvec{e}}^{{\text{H}}} \left( {{\varvec{P}}(\theta )^{{\text{H}}} {\varvec{E}}_{{\text{n}}} {\varvec{E}}_{{\text{n}}}^{{\text{H}}} {\varvec{P}}(\theta )} \right)^{ - 1} {\varvec{e}}$$
(30)

In the process of searching for the above-mentioned spectral peaks, the number of snapshots is limited due to the actual situation, which affects the accuracy of the noise subspace En, and reduces the DOA estimation performance of the algorithm. For this reason, use the power series of the noise feature to modify the corresponding noise subspace En, and the power series of the noise feature:

$${\varvec{C}}_{n} = \left[ {\lambda_{K + 1}^{n} e_{K + 1} ,\lambda_{K + 2}^{n} e_{K + 2} , \ldots ,\lambda_{{(M + N - 1)^{2} }}^{n} e_{{(M + N - 1)^{2} }} } \right]$$
(31)

where using formula (31), the above Q(θ) can be re-expressed as:

$${\varvec{Q}}(\theta ) = \mathop \sum \limits_{i = K + 1}^{{(M + N - 1)^{2} }} \lambda_{i}^{2n} \left( {\left| {{\varvec{P}}(\theta )^{{\text{H}}} {\varvec{e}}_{i} } \right|^{2} } \right)$$
(32)

where substituting formula (32) into the formula (30), the corrected one-dimensional peak search function can be obtained:

$$f(\theta ) = {\varvec{e}}^{{\text{H}}} {\varvec{Q}}^{ - 1} (\theta ){\varvec{e}} = {\varvec{e}}^{{\text{H}}} \left( {\mathop \sum \limits_{i = K + 1}^{{(M + N - 1)^{2} }} \lambda_{i}^{2n} \left( {\left| {{\varvec{P}}(\theta )^{{\text{H}}} {\varvec{e}}_{i} } \right|^{2} } \right)} \right)^{ - 1} {\varvec{e}}$$
(33)

In summary, the implementation steps of the non-circular signal dimensionality reduction DOA estimation method based on the expanded coprime array MIMO radar proposed in this article are shown in Table 1.

### 3.2 No phase ambiguity proof

Since the element spacing of the sparse array is greater than half a wavelength, there is a problem of angular ambiguity. But the method in this article adopts the expanded coprime array, which can effectively suppress the phase ambiguity problem. The proof is as follows:

Suppose there is phase ambiguity, that is, there is an ambiguity angle θm, which satisfies:

$${\varvec{a}}(\hat{\theta }_{k} ) = {\varvec{a}}(\theta_{m} )$$
(34)

where $$\hat{\theta }_{k}$$ is the estimated value of the direction of arrival, and θm is the value of the ambiguity angle. From formula (34), we can get:

$${\varvec{a}}_{{\text{r}}} \left( {\hat{\theta }_{k} } \right) \otimes {\varvec{a}}_{{\text{t}}} \left( {\hat{\theta }_{k} } \right){ = }{\varvec{a}}_{{\text{r}}} \left( {\theta_{m} } \right) \otimes {\varvec{a}}_{{\text{t}}} \left( {\theta_{m} } \right)$$
(35)

Substituting formulas (2) and (3) into formula (35), we can get:

$$\left[ {\begin{array}{*{20}c} {{\varvec{a}}_{{{\text{r}}1}} \left( {\hat{\theta }_{k} } \right)} \\ {{\varvec{a}}_{{{\text{r}}2}} \left( {\hat{\theta }_{k} } \right)} \\ \end{array} } \right] \otimes \left[ {\begin{array}{*{20}c} {{\varvec{a}}_{{{\text{t}}1}} \left( {\hat{\theta }_{k} } \right)} \\ {{\varvec{a}}_{{{\text{t}}2}} \left( {\hat{\theta }_{k} } \right)} \\ \end{array} } \right]{ = }\left[ {\begin{array}{*{20}c} {{\varvec{a}}_{{{\text{r}}1}} \left( {\theta_{m} } \right)} \\ {{\varvec{a}}_{{{\text{r}}2}} \left( {\theta_{m} } \right)} \\ \end{array} } \right] \otimes \left[ {\begin{array}{*{20}c} {{\varvec{a}}_{{{\text{t}}1}} \left( {\theta_{m} } \right)} \\ {{\varvec{a}}_{{{\text{t}}2}} \left( {\theta_{m} } \right)} \\ \end{array} } \right]$$
(36)

Expand the formula (36), we can get:

$$\frac{{2\pi d_{l} \sin \hat{\theta }_{k} }}{\lambda } - \frac{{2\pi d_{l} \sin \theta_{m} }}{\lambda } = 2k\pi$$
(37)

where dl is the element spacing. From the formula (37), we can get:

$$\left\{ {\begin{array}{*{20}l} {\sin \hat{\theta }_{k} - \sin \theta_{m} = 2k_{1} {/}M} \hfill \\ {\sin \hat{\theta }_{k} - \sin \theta_{m} = 2k_{2} {/}N} \hfill \\ \end{array} } \right.$$
(38)

where $$k_{1} \in ( - N,N)$$, $$k_{2} \in ( - M,M)$$. In the coprime array, M and N are integers that are mutually prime numbers. According to theorem 1 of the literature , it can be known that there is no M and N in formula (36), that is, there is no blur angle θm that holds true in Eq. (34). That is, it proves that there is no phase ambiguity problem.

## 4 Results and discussion

$${\text{RMSE}} = \frac{1}{k}\mathop \sum \limits_{1}^{k} \sqrt {\frac{1}{100}\mathop \sum \limits_{J = 1}^{100} \left( {\hat{\theta }_{k,J} - \theta_{k} } \right)^{2} }$$
(39)

where J represents the number of Monte Carlo experiments, $$\hat{\theta }_{k,J}$$ represents the estimated DOA value of θk in the Jth experiment, and θk is the true value of the angle.

The proposed algorithm is simulated on MATLAB R2018b software to verify its performance. Set the transmitting array element and the receiving array element to M = 3 and N = 4, respectively. The number of sources is 2, the target angle is 10° and 20°, the number of snapshots is 100, and the number of Monte Carlo experiments is 100. Figure 3 shows the estimation results of the proposed algorithm for all targets when the SNR = 10 dB. It can be seen that the algorithm in this article can accurately estimate the angle of multiple independent targets at the same time.

Figure 4 shows the DOA of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm under the condition of 100 snapshots under different SNB estimated performance. It can be seen from Fig. 4 that with the gradual improvement in the signal-to-noise ratio of the three algorithms, the root mean square error RMSE is all getting smaller, and the DOA estimation performance is improved. In addition, it can be seen from the above figure that the algorithm proposed in this article is better than the other two algorithms, and the DOA estimation performance is better. It can also be seen that the relatively prime array can significantly improve the DOA estimation performance compared to the uniform linear array.

Figure 5 shows the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm under 100 effective computer simulation experiments, the target detection success rate varies with the SNR. Target detection success rate, that is, the ratio of the number of successful DOA estimates to the number of trials. When $$\left| {\hat{\theta }_{K} - \theta_{K} } \right| < 1$$ (where $$\hat{\theta }_{k}$$ is the estimated value of DOA and θk is the actual value), it is deemed to be a successful estimate of DOA. It can be seen from Fig. 5 that the success rate of the three algorithms increases as the SNR increases. When the SNR is equal to 10 dB, the success rates of the three algorithms all reach 100%. But when the SNR is less than 10 dB, the success rate of this algorithm is better than the other two algorithms. In the case of low SNR, it still maintains a high success rate. This shows that the algorithm in this article is still applicable under the condition of low SNR. The algorithm in this paper uses MIMO radar, which greatly expands the array aperture and improves the degree of freedom of the array. The introduction of non-circular signals further expands the receiving array and improves the measurement accuracy of the algorithm. Finally, the noise subspace is further modified to further improve the DOA estimation accuracy. Therefore, the success rate of this algorithm is higher than the other two algorithms.

Figure 6 shows the variation of the RMSE of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm under different snapshots. It can be seen from Fig. 6 that as the number of snapshots increases, the RMSE of the three algorithms gradually decreases, and the estimation performance gradually improves. Among them, the classic MUSIC algorithm has the worst angle estimation performance, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm has the best angle estimation performance and is more stable. The algorithm in this article still has good estimation performance when the number of snapshots is low.

Figure 8 shows the variation of the RMSE of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO radar non-circular signal dimensionality reduction MUSIC algorithm under different numbers of array elements. Set the number of transmitting array elements to 4, and the number of receiving array elements to 3, 5, 7, 9, and 11. The number of sources is 2, the target angle is 10° and 20°, the number of snapshots is 100, and the number of Monte Carlo experiments is 100. The experiment parameters are simulated by MATLAB, and RMSE of the three algorithms decreases with the increase of the number of elements, and the accuracy of angle estimation is improved. Among them, the classic MUSIC algorithm has the lowest angle estimation accuracy, and the algorithm proposed in this article has the highest accuracy. In the case of a small number of transmitted array elements, the algorithm proposed in this article has higher estimation accuracy than the other two algorithms.

Table 5 shows the specific values of the RMSE of the classic MUSIC algorithm, the traditional coprime array MUSIC algorithm, and the expanded coprime array MIMO non-circular signal dimensionality reduction MUSIC algorithm under different numbers of array elements. It can be seen from Table 5 that when the number of array elements is small, the angle estimation accuracy of the classic MUSIC algorithm is the lowest, and the angle estimation accuracy of the expanded coprime array MIMO non-circular signal dimensionality reduction MUSIC algorithm is the highest. When the number of array elements is 3, the RMSE of the proposed algorithm is about 62% lower than that of the classic MUSIC, and the RMSE of the traditional coprime array MUSIC algorithm is about 54% lower than that of the classic MUSIC algorithm. As the number of array elements increases, the accuracy of each algorithm is gradually improved. The angle estimation accuracy of the algorithm proposed in this article is the highest, and the angle estimation accuracy of the classic MUSIC algorithm is the lowest.

Under the expanded coprime array MIMO radar signal model, set the number of snapshots to 100, the number of sources to 2, the target angle to 10° and 20°, the number of receiving array elements N remains unchanged, and the number of transmitting array elements M to 3, 5, 7. Carry out 100 Monte Carlo experiments, get the RMSE graph of angle measurement accuracy with SNR under different numbers of transmitting array elements. It can be seen from Fig. 9 that in the case of different transmitting array elements, the RMSE of the algorithm proposed in this paper decreases as SNR increases, and the angle measurement estimation accuracy becomes better. The larger the SNR, the smaller the influence of noise on the algorithm, and the higher the accuracy of DOA estimation. When the number of transmitter array elements is 3, the root mean square error is large and the measurement accuracy is low. However, when the number of transmitter array elements is 7, the RMSE becomes smaller and the angle measurement estimation accuracy is improved. It can be seen that under the same circumstances, the more the number of launching array elements, the smaller the root mean square error, and the better the accuracy of the measurement angle.

Table 6 shows the specific values of the root mean square error (RMSE) of the algorithm proposed in this paper with the change of the SNR under different numbers of transmitting array elements. It can be seen from Table 6 that, in the case of the same SNR, the angle estimation accuracy of the transmitting array element number 3 is the lowest, and the angle estimation accuracy of the transmitting array element number 7 is the highest. As the SNR increases, the RMSE becomes smaller and the algorithm accuracy improves. When the SNR is 5 dB, the RMSE of the transmitting array element M = 5 is about 45% lower than the RMSE of M = 3. The RMSE of the transmitting array element M = 7 is about 60% lower than the RMSE of M = 3. Therefore, the number of array elements has a great influence on the DOA estimation accuracy. The more the number of array elements, the higher the DOA estimation accuracy. However, the actual number of array elements is affected by the hardware. The more array elements, the larger the hardware volume. Therefore, the number of array elements is not infinite and needs to be selected according to the actual situation.

## 5 Conclusion

This paper proposes a non-circular signal DOA estimation method based on coprime array MIMO radar, which solves the problems of low degree of freedom, small array aperture, and phase ambiguity of traditional coprime array DOA estimation methods. In this paper, the array model combines a coprime array with MIMO radar, which greatly improves the array aperture, increases the degree of freedom, and improves the accuracy of DOA estimation. Then, the non-circular signal is introduced, and the receiving matrix is effectively expanded by using the non-circular characteristic, and the parameter estimation performance and the estimation accuracy of multiple sources are improved. Then use the idea of dimensionality reduction to reduce the dimensionality of the two-dimensional MUSIC algorithm to reduce the complexity of the algorithm. The power series of noise eigenvalues is used to correct the noise subspace, which further improves the accuracy of the algorithm. In addition, coprime array is used as the receiving array, which eliminates the phase ambiguity problem caused by the distance between the array elements larger than half the wavelength. Finally, the effectiveness of the algorithm is verified by simulation experiments. Compared with the classic MUSIC algorithm and the traditional MUSIC algorithm, the algorithm in this paper can better improve the DOA estimation accuracy and successful resolution. And it still maintains superior performance under low SNR. In the future, the propagator method will be integrated into the method research of this paper to further reduce the complexity of the algorithm, so as to improve the real-time performance of the algorithm in practical applications. The research method will be applied to vehicle radar. And continue to study the processing of multi-source signals and circular signals.

## Availability of data and materials

All data generated or analysed during this study are included in this article.

## Abbreviations

MIMO:

Multiple-input multiple-output

DOA:

Direction of arrival

MUSIC:

Multiple signal classification algorithm

SNR:

Signal-to-noise ratio

RMSE:

Root mean square error

CA:

Coprime array

NRC-MIMO MUSIC:

Non-circular signal dimensionality reduction DOA estimation method based on the expanded coprime array MIMO radar

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## Acknowledgements

The authors would like to thank Aisuo Jin for his academic support, and I can finish the thesis successfully.

## Funding

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61801170 and 61801435, and Jiangsu Overseas Visiting Scholar Program for University Prominent Yong & Middle-aged Teachers and Presidents.

## Author information

Authors

### Contributions

JCT analysed and interpreted the data regarding the NRC-MIMO MUSIC, and was a major contributor in writing the manuscript. ZZJ and YDY analysed the theory of MIMO and related DOA estimation. ZF provided experimental ideas. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Fei Zhang.

## Ethics declarations

Not applicable.

Not applicable.

### Competing interests

The authors declare that they have no competing interests. 