International Journal of Antennas and Propagation

Volume 2019, Article ID 2837315, 10 pages

https://doi.org/10.1155/2019/2837315

## DOA Estimation for Highly Correlated and Coherent Multipath Signals with Ultralow SNRs

^{1}Hubei Engineering Research Center of Video Image and HD Projection, School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan 430205, China^{2}School of Information Engineering, Nanchang Institute of Technology, Nanchang 330099, China

Correspondence should be addressed to Yang Li; moc.qq@88255158

Received 3 April 2019; Accepted 4 September 2019; Published 13 October 2019

Academic Editor: Ana Alejos

Copyright © 2019 Li Cheng 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

In a typical multipath propagation environment, there exists a strong direct path signal accompanying with several weak multipath signals. Due to the strong direct path interference and other masking effects, the Direction-of-Arrival (DOA) of a weak multipath signal is hard to be estimated. In this paper, a novel method is proposed to estimate the DOA of multipath signals with ultralow signal-to-noise ratio (SNR). The main idea is to increase the SNR and signal-to-interference ratio (SIR) of the desired multipath signal in time-delay domain before DOA estimation processing. Firstly, the cross-correlation functions of the direct path signal and the received array signal are calculated. Then, they are combined and constructed to an enhanced array signal. Under certain conditions, the SNR and SIR of the desired signal can be significantly increased. Finally, the DOAs of multipath signals can be estimated by conventional technologies, and the associated time delays can be measured on the DOA-time-shift map. The SNR and SIR gains of the desired signal are analyzed theoretically, and theoretical analysis also indicates that the Cramer–Rao bound can be reduced. Simulation examples are presented to verify the advantages of the proposed method.

#### 1. Introduction

Direction-of-Arrival (DOA) plays an important role in wireless communication, radar, sonar, and other areas. In the past few decades, many methods, such as subspace-based approaches, maximum likelihood methods, and sparse-representation-based methods [1–5] have been studied extensively on this topic.

In some multipath environments, such as ionospheric sounding [6], global navigation satellite system [7], and airborne radar [8], some of the incident signals are highly correlated with each other, and some of them can be coherent. In this scenario, the performance of conventional DOA estimation methods is affected dramatically. For example, the popular subspace-based methods MUltiple SIgnal Classification (MUSIC) fails to span the signal subspace in the case of correlated waves. MUSIC can be extended to be applicable by adopting decorrelation preprocessing, such as the spatial smoothing [9] and Toeplitz approximation techniques [10]. Also, the maximum likelihood methods [11] and sparse-representation-based methods [12] can be applicable to the coherent scenario. However, they do not consider the case that the signal-to-noise ratio (SNR) of the desired multipath signal whose DOA needs to be estimated can be very low, and there exists strong direct path which can mask the adjacent weak path. For example, in passive bistatic radar system, the power difference between the echo signal and direct path signal can be larger than 100 dB, which indicates the signal-to-interference ratio (SIR) of the desired echo signal is very low [13], thus estimating the DOA of echo signal is a challenging problem [14].

In low SNR case, the number of sources is hard to estimate correctly, and the signal subspace may swap with the noise subspace [15]. Some research studies have been conducted: a MUSIC-like DOA estimation method without estimating the number of sources is proposed in [16], its performance holds stable when SNR is low; a sparse representation of array covariance vectors in an overcomplete basis is proposed in [17], which is statistically robust in low SNR cases; a two-stage DOA estimation method for low SNR signals is proposed in [18], which enhances the angular resolution and estimation accuracy, particularly for the case when the array antenna elements received the low SNR signals; a linear prediction orthogonal propagator method is proposed in [19], which is more robust to the noise, especially in low SNR scenarios. Although these methods improved the performance of DOA estimation in low SNR scenarios, the performance is constrained by the Cramer–Rao bound (CRB). The CRB relates to many factors, such as the number of snapshots, the array aperture, angular separations, and correlations of the signals, no doubt that SNR is one of the most important factors among them.

This work mainly focuses on the DOA estimation for partly correlated or coherent signals with ultralow SNR in multipath environment, and the precondition that a strong direct path signal is included in the multipath is required, or the reference signal is available in advance. To overcome the SNR limitation, we provide a novel method that can enhance the desired multipath signal before the DOA estimation procedure. We realize this goal through four steps: firstly, the direct path signal is obtained by the conventional DOA estimation and beamforming technologies with the received original array signal (OAS); secondly, the cross-correlation of the direct path signal and OAS are calculated, and then they are combined to a single snapshot array signal; thirdly, the single snapshot is extended to multisnapshots-enhanced array signal (EAS), under certain conditions, the EAS with particular time shift can be enhanced. At last, by using the time-shift scanning and conventional DOA estimation approaches, the DOAs of multipath signals can be estimated and the associated time delays can be measured on a two-dimensional DOA-time-shift map.

The main contributions of the work are as follows: (a) based on the OAS, a new array signal EAS is constructed, in which the desired multipath signal can be enhanced; (b) the SNR gain and SIR gain in the EAS are derived in theory; (c) the DOA and the associated time delay of the multipath signal with ultralow SNR can be estimated or measured with the EAS; (d) the CRB for DOA estimation can be reduced with the proposed method starting with the OAS.

The rest of the paper is organized as follows: The basic signal model and array signal processing are provided in Section 2. The proposed method and its implementation are developed in Section 3. Some simulation examples are presented in Section 4. A brief conclusion appears in Section 5. In the paper, , , , , , , and superscript denote the expectation, transpose, Hermitian transpose, inverse, absolute value, Hadamard product, and complex conjugate, respectively. Boldfaced variable denotes matrix or vector.

#### 2. Problem Formulation

##### 2.1. Array Signal Model

Consider a multipath propagation environment, assuming that there is one direct path signal with its SNR significantly larger than other multipath signals, and there is no strong interference with the same frequency. Multipath signals incident on a uniform linear array (ULA) with *M* omnidirectional antennas, and the source is in the far field of the array. The received array signal can be expressed as [9]where denotes the snapshot point, the number of multipath signals is , the *i*th multipath signal is with steering vector , denotes amplitude attenuation, and denotes time delay. Let represents the direct path; we can make and for reference. Array signal , steering vector , and array noise are dimensional complex vector. The signals are stationary Gaussian random process, and the additive noise is a spatially white Gaussian process. The power of direct path signal and noise is and , respectively, and the power of *i*th multipath signal is . is called OAS in the following.

The data covariance matrix can be expressed as . Since the actual covariance matrix is unknown in practice, it is often replaced by snapshots sample data covariance matrix . The data covariance matrix can be further expressed in different forms which depend on whether the signals are coherent or not [20].

The eigen decomposition of iswhere and are the eigenvalues and the corresponding eigenvectors of . The eigenvalues are sorted in descending order, , where *d* is the number of the distinguishable signals. The eigenvectors corresponding to the largest *d* eigenvalues span the signal subspace , and the other eigenvectors corresponding to the smaller eigenvalues span the noise subspace .

##### 2.2. DOA Estimation

There are numerous methods that can be used to estimate the DOA of directional signal with OAS. One of the classic methods is Capon spatial spectrum, which is calculated bywhere is the DOA of the incident signal. The peaks of the spatial spectrum correspond to the DOAs of distinguishable signals. Although the Capon spatial spectrum is implemented easily, its resolution is limited, and the performance is decreased in low SNR scenarios.

The MUSIC is one of the very popular methods for super-resolution DOA estimation [21], and it is given by

Resolution and accuracy of an eigen structure method like MUSIC depend on the number of snapshots, array aperture, angular separations, SNRs, and correlations of the signals, etc. The CRB provides an algorithm-independent benchmark against which various algorithms can be compared. The stochastic CRB of DOA estimation with OAS is computed as [22]where , , , and , . When other parameters are fixed, the snapshots number *K* and the SNR (relate to and ) decide the CRB of OAS. In the following, we will develop a novel method to reduce the CRB with the same OAS.

#### 3. The Proposed Method

##### 3.1. Direct Path Signal Beamforming

Firstly, the direct path signal is required. In some applications, the reference signal such as the pilot signal in communication and the transmitted signal in active radar are priori known, and the direct path signal can be obtained from the reference signal. However, the reference signal is unavailable in many applications. The beamforming technology which is used to filter the directional signal in space can be used to obtain the direct path signal: , where is the beamformer’s weight vector with its main beam pointing to the DOA of direct path signal. In order to suppress the interference adaptively as well as prevent desired signal self-nulling, the robust adaptive beamforming technologies are suggested to be used. The robust Capon beamformer [23], the robust adaptive beamformer [24], or their improved technologies [25, 26] can be used. After the adaptive beamforming, we assume the obtained direct path signal is pure enough, which means .

##### 3.2. Array Cross-Correlation

The self-correlation function of is calculated by

When the time shift , . The self-correlation function has a property that the response is maximized when the time shift equals to 0:

The cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. The discrete form of cross-correlation function of the *m*th antenna’s received signal and direct path signal is defined as

The peaks of cross-correlation function correspond to the multipath signals that are matched in .

Taking a combination of as , it can be decomposed as

The matrix is dimensional, and its form is similar with OAS: the signal component is , and the noise component is .

Assuming that the path-1 signal with is the desired signal whose DOA needs to be estimated. If , (9) can be decomposed as

Through the Cauchy–Schwarz inequality, we have

The SIR and the SNR of the path-1 signal in and are listed in Table 1. Results show that the SIR and SNR of the path-1 is enhanced in .