1. Introduction
DOA estimation is a critical problem in many applications, such as radar, sonar, and wireless communication systems. Conformal arrays, which are composed of sensors placed on a curved surface, have recently received significant attention due to their good aerodynamic performance [
1]. Common conformal array shapes are cylindrical conformal arrays [
2], conical conformal arrays [
3,
4], and spherical conformal arrays [
5]. These circular array based conformal arrays are the basic structure of most aircraft and are widely used. However, conformal arrays suffer from directional element pattern [
6], shadow effect [
2], and mutual coupling [
7,
8,
9,
10]. Array manifold construction is complicated by these effects, which will corrupt the performance of DOA estimation algorithms. Therefore, current DOA estimation methods cannot be used on CCAs straightforwardly.
To overcome these limitations, several modified DOA estimation algorithms for conformal arrays are suggested. For directive elements, ref. [
11] decomposes the radiation pattern by Fourier series and proposes an effective pattern model. Ref. [
12] proposed the UCA-RARE algorithm which can successively perform 2-D angle estimation. Ref. [
13] optimizes the directivity of the radiation pattern by using CRB. It proved that using directional elements with proper directionality gives better performance than isotropic elements. In order to solve the shadowing effect problem in conformal arrays, ref. [
2] proposed the subarray segmentation method, which divides the cylindrical conformal array into several subarrays. For each subarray, the non-isotropic sector array is transformed into a virtual uniform linear array via interpolation method. In order to compensate for the decrease in the array DOF and aperture caused by the shadow effect, ref. [
14,
15] convert cylindrical conformal arrays to virtual nested arrays [
16]. The accuracy of these interpolation algorithms depends on the size of interpolation step, which may introduce additional errors.
Uniform circular arrays (UCAs) are the basis for solving conformal arrays with circular carriers. The phase mode method [
17] synthesizes the circular array elements into beamspace composed of several beams. Thus, the steering vector is converted into a product of Bessel function and the exponent of azimuth. The manifold separation technique [
18] uses the wavefield modeling [
19] to equate the received signal to spatial signal sampling by arbitrary array. The array received signal with arbitrary structure are broken down to the sampling matrix and the Vandermonde structure vector containing angle information. In recent years, corray methods have received extensive attention [
16,
20,
21]. An array contains
N element can provide
$O\left({N}^{2}\right)$ coarray elements by vectorizing the covariance matrix of the received signal. Namely, the array DOF is expanded to
$O\left({N}^{2}\right)$. Ref. [
22] uses UCA to estimate quasi stationary signals larger than the number of elements by K-R product. Ref. [
23] extend the difference coarray concept to arbitrary arrays and perform azimuth estimation of underdetermined signal sources.
Conical conformal arrays can estimate the elevation angle without suffering from shadow effect when the elevation angle is smaller than the cone angle. When the elevation angle is larger than the cone angle, only half of the elements will detect the signal, and the array is suitable for 1-D DOA estimation in this case. CCA is different from general directional UCA and cylindrical conformal array. Its array element pattern not only needs the transformation of azimuth angle, but also the transformation of elevation direction. We establish the local coordinate system radiation pattern of the CCA using 2-D coordinate transformation and propose the CCA-MST method. However, due to the shadow effect, the circular array degenerates into a sector array. In order to recover the array DOF from the sector array, we introduce the difference coarray method and propose the CCA-Coarray algorithm. When the elevation angle is smaller than the cone angle, all array elements can receive signals. The CCA can therefore perform 2-D DOA estimation. Previous beamspace methods generally use eigensubspace methods [
24,
25]. The propagator method (PM) [
26,
27] circumvents the EVD step by using steering vectors instead of signalsubspaces. However, when the traditional PM is applied to beamspace, its robustness is poor. This is because the beamspace manifold is regulated by Bessel function. In order to develop the detection performance, we restructure the propagator matrix for the CCA-PM algorithm.
The paper is arranged as follows. First in
Section 2, we establish the CCA model, including directional element pattern model and CCA shadow effect model.
Section 3 introduces the proposed 1-D CCA-MST method and CCA-Coarray method.
Section 4 introduces the proposed 2-D CCA-PM algorithm. Then in
Section 5, we analyze the computational complexity of each algorithm, and validate the practice of the various algorithms by simulation. In
Section 6 we conduct experimental measurements.
Section 7 concludes the paper.
2. CCA Signal Model
The CCA geometry is depicted in
Figure 1 with the height of the cone
h and the cone angle
$\alpha \in \left(0,\pi /2\right)$. The base of the cone lies on the
$xoy$ plane with the radius
r. A spherical coordinate system is established with the origin placed at the center of the cone base. The
N elements are distributed equidistantly over the circumference of the base.The spherical coordinate system is established by taking the center of cone base as the origin.
N elements are arranged identically along the base circumference.
Assume that
P narrow band signals with directions
$({\theta}_{p},{\phi}_{p}),(p=1,2,...,P)$ incident on the array. The ideal received signal data is then given by
where
${\mathbf{A}}_{e}=[{\mathbf{a}}_{e}({\theta}_{1},{\phi}_{1}),{\mathbf{a}}_{e}({\theta}_{2},{\phi}_{2}),...,{\mathbf{a}}_{e}({\theta}_{P},{\phi}_{P})]$ is the
$N\times P$ ideal array manifold,
$\mathbf{s}\left(t\right)$ is the
$P\times 1$ signal matrix and
$\mathbf{n}\left(t\right)$ denote
$N\times 1$ noise vectors. The ideal steering vector
${\mathbf{a}}_{e}({\theta}_{p},{\phi}_{p})$ are expressed as
where
${\zeta}_{p}=\kappa rsin{\theta}_{p},$ $\kappa =2\pi /\lambda $ is the wave number and element angle
${\varphi}_{n}=2\pi (n-1)/N$,
$n=1,2,...,N$.
Each element pattern of the CCA is directional. The antenna element pattern gain is the function of the local coordinate system with elements as the reference point, as shown in
Figure 2. For an arbitrary signal, the radiation pattern gains of diverse array elements are different. The directional array elements of circular arrays usually assume that the elevation angle of the maximum directivity of the pattern is
$\theta ={90}^{\circ}$. However, in CCAs, due to the deflection of the elevation angle, the element pattern cannot be obtained by the rotation of the azimuth angle directly. The incident signal angles in the local coordinate system are obtained by the following coordinate transformation:
Assume that all array elements have a
$1+cos\varphi $ pattern response, the radiation pattern of the array antenna is represented as
The CCA is also affected by the carrier structure, and its received signal has shadow effect. In
Figure 3, when the elevation angle
$\theta $ of a received signal is greater than the cone angle
$\alpha $, only half of the array elements can receive the signal. The rest of the array elements are in the dark side of the carrier. It can be seen from the geometric properties that only when the incident angle in the local coordinate system of the
nth element satisfies
$\left[-\pi /2,\pi /2\right]$, the
nth array element can receive the echo signal of the
pth target. The array that responds to the incident signal is actually a sector array. For analyzing the impact of the shadow effect on CCA target angle estimation, we define the following function
WThrough the above analysis, the received signal of the CCA is expressed as
where
$\mathbf{W}$ is the
$N\times P$ sign matrix,
$\overline{\mathbf{F}}$ is the
$N\times P$ radiation pattern matrix.
$\mathbf{A}=\mathbf{W}\circ \overline{\mathbf{F}}\circ {\mathbf{A}}_{e}$ is the
$N\times P$ CCA manifold matrix, ∘ stands for the Hadamard product. The array response vector (ARV) is given by
where
$F(\theta ,\phi -{\varphi}_{n})=W(\theta ,\phi -{\varphi}_{n})\overline{F}(\theta ,\phi -{\varphi}_{n})$. The covariance matrix is
where
${\mathbf{R}}_{S}=E\left[\mathbf{S}\left(t\right){\mathbf{S}}^{H}\left(t\right)\right]$ is the signal covariance matrix.
${\sigma}^{2}$ is the variance of the noise. Due to the limited length of the received data, the actual covariance matrix estimation is
where
Q is the number of snapshots.
7. Conclusions
In summary, 1-D and 2-D DOA estimation of a conical conformal array is studied. First, the CCA geometric model and signal model are established to solve the issues of directional pattern and shadow effect. Then, the CCA-MST, CCA-Coarray and CCA-PM algorithms are proposed, which are suitable for 1-D and 2-D scenarios, respectively. The results show that in presence of 2 targets, the estimation error of the CCA-Coarray method is less than $0.{3}^{\circ}$ at 4 dB with 1000 snapshots. The estimation error of the CCA-PM method is less than ${1}^{\circ}$ at 6 dB under 1000 snapshots. The method is also more efficient than traditional subspace methods. In the future, the CCA-based mutual coupling compensation algorithm remains to be explored.