International Journal of Antennas and Propagation

Volume 2015, Article ID 135753, 10 pages

http://dx.doi.org/10.1155/2015/135753

## Direction of Arrival Estimation Accuracy Enhancement via Using Displacement Invariance Technique

^{1}College of Electronic and Information Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China^{2}College of Astronautics, Nanjing University of Aeronautics & Astronautics, Nanjing 210016, China^{3}Faculty of Engineering, Alexandria University, Alexandria 21615, Egypt

Received 22 April 2015; Revised 2 August 2015; Accepted 5 August 2015

Academic Editor: Giuseppe Castaldi

Copyright © 2015 Youssef Fayad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new algorithm for improving Direction of Arrival Estimation (DOAE) accuracy has been carried out. Two contributions are introduced. First, Doppler frequency shift that resulted from the target movement is estimated using the displacement invariance technique (DIT). Second, the effect of Doppler frequency is modeled and incorporated into ESPRIT algorithm in order to increase the estimation accuracy. It is worth mentioning that the subspace approach has been employed into ESPRIT and DIT methods to reduce the computational complexity and the model’s nonlinearity effect. The DOAE accuracy has been verified by closed-form Cramér-Rao bound (CRB). The simulation results of the proposed algorithm are better than those of the previous estimation techniques leading to the estimator performance enhancement.

#### 1. Introduction

Direction of arrival Estimation (DOAE) for an antenna array is an important process because DOA is the creator of the tracking gate dimensions (the azimuth and the elevation). Accurate DOAE for the moving target leads to the reduction of the angle glint error which affects the accuracy of the tracking radars.

Doppler frequency shift is a significant phenomenon induced by the target movement. Lack of attention to the influence of Doppler shift on the ESPRIT method leads to DOAE process with poor accuracy.

The ESPRIT and its extracts have been widely studied in one-dimensional (1D) DOAE for uniform linear array (ULA) [1–4], nonuniform linear array (NULA) for spatial multiresolution [5–8], or temporal multiresolution [9–11] and also extended to two-dimensional (2D) DOAE [12–18]. All of these ESPRIT methods have been developed to upgrade the accuracy of DOAE with low calculation costs. However, these works did not pay attention to the effect of the Doppler frequency of the moving target on the DOAE accuracy. On the other hand, some other great efforts have been done to estimate Doppler frequency via using the fast Fourier transform (FFT), but it has a high order computation time for a large number of samples [19]. Another method applying ESPRIT technique to compute Doppler shift via employing the rotational factor resulted from time delay sampling [7, 8, 20]. However, these methods require intensive matrix computations or iterative optimization techniques.

This paper presents firstly a novel method to detect the target movement via estimating its Doppler frequency by comparing the displacement invariances of transmitter and receiver arrays, which enables the measurement of the sensor array displacement invariance fluctuation that resulted from wavelength change induced by the target movement. Secondly, for target moves with uniform acceleration, Doppler frequency correction has been embedded into the ESPRIT algorithm to refine the DOA estimated value. Subspace concept has been applied to reduce the model’s nonlinearity effect and to realize parallel processing which leads to the enhancement of the estimation accuracy with low computational load [5, 6].

The paper is organized as follows. In Section 2, the two-dimensions time multiresolution ESPRIT (2D T-ESPRIT) DOAE technique has been introduced, the time multiresolution displacement invariance technique (T-DIT), and the Doppler effect of the moving target have been incorporated into the estimation algorithm. In Section 3, the simulation results are presented, and Section 4 is the conclusion.

#### 2. Proposed Algorithm

##### 2.1. The Measurement Model

In this model, the transmission medium is assumed to be isotropic and nondispersive, so that the radiation propagates in straight lines, and the sources are assumed to be as a far field away the array. Consequently, the radiation impinging on the array is a summation of the plane waves. The signals are assumed to be narrow-band processes, and they can be considered to be sample functions of a stationary stochastic process or deterministic functions of time [3]. Considering that there are narrow-band signals and the center frequency is assumed to have the same , the th signal can be written aswhere is the signal of the th emitting source at time instant , is the carrier phase angles that are assumed to be random variables, each uniformly distributed on and all statistically independent of each other, and is the incident electric field that can be written as components form. As a general expression, we omit the subscript, and thenwhere and are the horizontal and the vertical components of the field, respectively.

Define as the auxiliary polarization angle and as the polarization phase difference, and then,The incident field can be also expressed in Cartesian coordinate system

Figure 1 shows that a planar antenna array has elements indexed as , along and directions, respectively. For any pair (), its coordinate is , where , , and and are reference displacements between neighbor elements along - and -axes. The array elements are oriented in plane, and the space phase factors along and directions are expressed aswhere denote the th source elevation angle and azimuth angle, respectively, and is the wavelength of the th signal. The measurement vector can be expressed aswherestands for the additive white Gaussian noise (AWGN).