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Two-dimensional direction-of-arrival (DOA) estimation in coprime planar array involves problems that the complexity of spectral peak search is huge and the noncircular feature of signals is not considered. Considering that unitary estimating signal parameters via rotational invariance techniques (Unitary-ESPRIT) is a low complexity subspace algorithm, an approach to estimate DOA fast for multiple signals is proposed in this paper. We first apply Unitary-ESPRIT to solve one possible value of each signal. Given the relationship between the ambiguous values and real values, we then have all possible values belonging to each subarray. Through finding the common values of two subarrays, we finally obtain the highly precise true DOAs. Moreover, when the signals are noncircular, we present an improved method using noncircular Unitary-ESPRIT, which is favorable in terms of accuracy and degree of freedom. Simulation results demonstrate the effectiveness of our proposed methods.

Direction-of-arrival (DOA) estimation of multiple narrowband signals is a fundamental task in many applications such as radar [

Furthermore, the existing algorithms have just taken the independent narrowband signals into consideration. But sometimes the features of impinging signals can influence the performance. Considering that the general subspace methods cannot be applied to coherent signals, we have proposed an algorithm in [

In order to avoid the high complexity cost by the spectral searching and utilize the signals features, in this paper, a fast DOA estimation method in coprime array for multiple signals is proposed. We first obtain the received signal data and calculate the covariance matrix. Then, when signals are circular, we can apply unitary estimating signal parameters via rotational invariance techniques (Unitary-ESPRIT) [

The paper is organized as follows. We first present a brief signal model in Section

Considering the coprime planar array [

Geometry of coprime planar array [

We then can calculate the covariance matrix of received signals given by

Unitary-ESPRIT is an efficient method, which solves the closed-form solutions of estimated values and reduces much complexity compared with the spectral peak search method, such as 2D-MUSIC. Unfortunately, Unitary-ESPRIT is commonly used in nonsparse arrays and it cannot estimate all ambiguous and real values.

Assume that there is a signal from the direction

We can use Unitary-ESPRIT to obtain one possible estimated value of each signal. First, we define

The last subsection presents the algorithm for circular signals, which can also be used for general independent signals. When there are noncircular signals, the ellipse covariance matrix of them can be nonzero; thus, we need add some additional procession. We assume that the signals are strictly noncircular. At first, we can “double” the available sensors by

A detailed flowchart of this algorithm is shown in Figure

Flowchart of the proposed algorithm.

Calculate the covariance matrix via (

Apply Unitary-ESPRIT and obtain the estimation

If the signals are noncircular, we transform the received signal data as (

Calculate all real values and ambiguous values through (

Resolve the DOA estimation

We first analyze the computational complexity of the proposed method using Unitary-ESPRIT algorithm and compare it with the spectral peak search method based on 2D-MUSIC algorithm and PSS. And then analyze the complexity of proposed method using NC-Unitary-ESPRIT when signals are noncircular.

The complexity of the proposed method using Unitary-ESPRIT mainly concludes three parts. The complexities of calculating the covariance matrix, eigenvalue decomposition, and Unitary-ESPRIT are

Computational complexity comparison.

Algorithm | Complexity |
---|---|

2D-MUSIC | |

PSS | |

Unitary-ESPRIT | |

NC-Unitary-ESPRIT | |

Complexity comparison: (a) versus the number of sensors when

As shown in Figure

The DOF determines the maximum number of signals that we can estimate directly. 2D-MUSIC, PSS and the proposed algorithms are all common values finding method to resolve the DOAs, where the DOFs of those algorithms are limited by the number of subarray sensors. The DOFs of 2D-MUSIC, PSS, and proposed algorithm using Unitary-ESPRIT are equal as

DOF comparison versus the number of sensors.

This section performs the results of simulation experiments comparing the two proposed methods with PSS. To measure the accuracy of the algorithms, define the root mean square error (RMSE) as

The detailed values of each step.

Value | Subarray 1 | Subarray 2 |
---|---|---|

| -0.1465 | -0.3131 |

| -0.1465 | -0.3131 |

| -2.1465, -1.6465, -1.1465, -0.6465, -0.1465, | -2.3131, -1.6464, -0.9798, -0.3131, |

| -2.1465, -1.6465, -1.1465, -0.6465, -0.1465, 0.3535, 0.8535, 1.3535, 1.8535 | -2.3131, -1.6464, -0.9798, -0.3131, 0.3536, 1.0202, 1.6869 |

Distribution of azimuth and elevation estimation (a) Unitary-ESPRIT (b) NC-Unitary-ESPRIT.

RMSE comparison under different SNRs (a) elevation (b) azimuth.

In addition, another advantage of NC-Unitary-ESPRIT is that it can estimate more signals than PSS and Unitary-ESPRIT. Hence, we set four situations where the number of noncircular signals is

RMSE of NC-Unitary-ESPRIT under different number of noncircular signals (a) elevation (b) azimuth.

RMSE comparison under different number of snapshots (a) elevation (b) azimuth.

The paper has proposed a fast DOA estimation approach in a coprime planar array for multiple signals, where circular and noncircular signals are concerned. The paper has described the model and the associated algorithms and analyzed the computational complexity as well as DOF of the proposed methods in comparison with that of existing algorithms. Through the theoretical analysis and simulation experiments, we demonstrate that proposed methods can detect multiple signals and realize the estimation of DOAs. Unitary-ESPRIT can obtain a close accuracy as the spectral peak search methods. Although the spectral peak search methods can obtain a higher accuracy with a small searching step, it costs much larger complexity than Unitary-ESPRIT. Moreover, when we estimate the DOAs of strictly noncircular signals, NC-Unitary-ESPRIT is favorable in both accuracy and DOF compared with existing algorithms and Unitary-ESPRIT. In practice, there can be the coexistence of both circular and noncircular signals. Hence, how to separate the two kinds of signals and resolve their DOAs with the consideration of signals features deserves the further research.

The data, which are produced by simulations, used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The work is supported by the National Natural Science Foundation of China (Grant no. 61401513).