Table of Contents
ISRN Signal Processing
Volume 2011 (2011), Article ID 751670, 8 pages
http://dx.doi.org/10.5402/2011/751670
Research Article

Fast DOA Estimation Algorithm Based on a Combination of an Orthogonal Projection and Noise Pseudo-Eigenvector Approach

Department of Electrical Engineering, Kun-Shan University, No. 949, Dawan Rd., Yongkang City, Tainan County 710, Taiwan

Received 3 November 2010; Accepted 5 December 2010

Academic Editors: C.-W. Kok, E. D. Übeyli, and Y. Xiang

Copyright © 2011 Ching Jer Hung. 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

This paper presents a new fast direction of arrival (DOA) estimation technique, using both the projection spectrum and the eigenspectrum. First, the rough DOA range is selected using the projection spectrum; then, a linear matrix equation is used to acquire a noise pseudo-eigenvector. Finally, the fine DOA estimation is obtained from an eigenspectrum approach based on the noise pseudo-eigenvector. Without the need to form the covariance matrix from a block of the array data and without a prior knowledge of the number of incoming signals, reduced complexity is achieved, in contrast to conventional subspace-based algorithms. Simulation results show that the proposed algorithm has a good resolution performance and deals well with both uncorrelated and correlated signals. Since the new approach can reduce computational complexity while maintaining better or similar resolution capability, it may provide wider application prospects in real-time DOA estimation when contrasted to other comparable methods.

1. Introduction

The direction of arrival (DOA) has been applied in many fields, including speech, radio, telecommunication, and medical signal processing. In the past few decades, a large number of accurate high-resolution DOA estimation techniques have been proposed. One popular approach is the subspace-based method, used in the well-known MUltiple Signal Classification (MUSIC) algorithm [1] and Estimation of Signal Parameters by Rotational Invariance Techniques (ESPRIT) [2]. However, the conventional subspace-based method has some inherent problems that require the assumption of the uncorrelated incident signals, prior knowledge of the number of incoming signals, a block of data snapshots to form a covariance matrix, and an eigendecomposition computation. We know that the estimations for the covariance matrix and the number of the incoming signals, the computation for eigendecomposition, and the procedure for decorrelating the high correlated signals are computationally intensive and time-consuming. Therefore, the conventional subspace-based will remain severely limited in fast DOA estimation under a nonstationary environment if the above issues are not resolved.

A direct data domain least squares (D3LS) approach [37] has recently been proposed and applied effectively in adaptive array processing. Based on the D3LS approach, methods [8, 9] developed a fast DOA estimation algorithm process, called the pseudo-covariance matrix technique, which estimated fast varying signals in two steps. The rough incidence angle ranges are first obtained using the bearing response; then, exact incidence angles are determined by combing the bearing response and the directional spectrum. The bearing response and the directional spectrum are obtained using a pseudo-covariance matrix, based on a single snapshot [8]. The advantage of the approach given in [8] is that the DOA can be estimated on a snapshot-by-snapshot basis, without the need of prior knowledge of the number of incoming signals and without the need of the covariance matrix from a block of the array data. Kim’s algorithm performs well for both uncorrelated and correlated signals coexisting in nonstationary environments. However, it requires the solution of a nonlinear-generalized-eigenvalue equation of a pseudo-covariance matrix, resulting in a high computational burden, and, using this method, the number of resolved signals is at most half of the number of the sensor array. In order to enhance the resolution of Kim’s method, an improved pseudo-covariance matrix scheme was proposed. It was Wen’s method [9]which is called forward-backward pseudo-covariance matrix. Wen’s method processes more signals and obtains a better resolution than the forward-only matrix. However, it uses singular-value decomposition (SVD), requiring a greater computational burden than does Kim’s method.

In this paper, an effective method is proposed for developing a fast DOA estimation algorithm when uncorrelated and correlated signals are present in highly nonstationary environments. The proposed method consists of two approaches. First, we use a suboptimal projection spectrum approach to select the rough DOA ranges of the incoming signals. This is done forming small spectrum peaks, generated by removing the signal from a given direction, instead of solving a more exact projection operator, as in Kim’s and Wen's methods. The whole process reduces algorithm computational cost. Secondly, the solution of a linear-matrix equation, formed by a forward-backward data matrix, is applied as a noise pseudo-eigenvector, which performs a function similar to a noise eigenvector. Then, a fine DOA estimation can be obtained from an eigenspectrum. This approach allows us to reduce computational cost and obtain a good algorithm resolution. In contrast to Kim’s and Wen's algorithms, the proposed algorithm has better or similar capability of resolution but offers lower computational complexity. Moreover, the proposed algorithm deals well with both uncorrelated and correlated signals and processes only a signal data snapshot without the use of the covariance matrix. Therefore, it may provide wider application prospects in real-time DOA estimation.

This paper is organized as follows. In Section 2, we briefly introduce the problem statement and signal model. In Section 3, we present the proposed DOA estimation algorithm. In Section 4, some simulation results and a comparison of the proposed algorithm to MUSIC, ESPRIT, Kim’s, and Wen’s methods are presented. Finally, conclusions are presented in Section 5.

2. Problem Statement and Signal Model

In the section, we prepare mathematical models and representations of signals. In what follows, the operators []𝑇, []𝐻, [], and denote transpose, conjugate transpose, conjugate, and vector norm, respectively. diag[𝑡1,,𝑡𝑢] represents a diagonal matrix with diagonal entries 𝑡1,,𝑡𝑢.

Our objective is to estimate the DOA from 𝑞 incident signals coming from a received single snapshot. Let a single data snapshot vector 𝐛 received from a uniform linear array with 𝑚-omnidirectional sensors, representing a linear combination of 𝑞(𝑚>𝑞) narrow-band signals added with zero mean white Gaussian noise process, be given by 𝐛=𝐀(𝜃)𝐬+𝐧=𝑏1𝑏2𝑏𝑚𝑇,(1) where 𝐬=[𝑠1𝑠2𝑠𝑞]𝑇 is a 𝑞×1 signal vector, 𝐧 is an 𝑚×1 noise process vector, and 𝐀(𝜃)=[𝐚(𝜃1),𝐚(𝜃2),𝐚(𝜃𝑞)] represents a 𝑚×𝑞 matrix of steering vectors. The 𝑖th column 𝐚(𝜃𝑖)=[1,𝑒𝑗2𝜋𝑑/𝜆𝑖sin𝜃,,𝑒𝑗2𝜋𝑑/𝜆𝑖(𝑚1)sin𝜃]𝑇 is the steering vector 𝑠𝑖 coming from 𝜃𝑖, 𝑖=1,2,,𝑞. Parameter 𝑑 is the distance between the sensors and 𝜆 is the wavelength of the carrier.

3. Fast DOA Estimation of Proposed Method

In this section, the proposed DOA estimation is derived, determining the rough and fine DOA estimation algorithms, presented in Sections 3.1 and 3.2, respectively.

3.1. Rough DOA Estimation Algorithm

In Kim’s and Wen’s methods, the rough incidence angle ranges are estimated by using the bearing response, requiring the use of nonlinear-generalized-eigenvalue or SVD algorithm in the different look directions. However, such processing may bring high computation complexity against real-time requirement. In the subsection, we present a simple novel DOA estimation algorithm, called the orthogonal projection technique, to estimate the rough DOA ranges from incoming signals using a single snapshot for achieving fast tracking varying signals under highly nonstationary environments. Let us define an orthogonal projection matrix as𝚷𝜃𝑛=𝐈𝐚𝜃𝑛𝐚𝜃𝑛𝐻,𝑛=1,2,,180Δ,(2) where 𝜃𝑛=𝑛Δ[90,90] denotes each search angle, Δ is search angle step, and 𝐚(𝜃𝑛) is the normalized version of 𝐚(𝜃𝑛), determined as 𝐚(𝜃𝑛)=𝐚(𝜃𝑛)/𝐚(𝜃𝑛). Then, use (2) to form the following projected power spectrum:𝐏Π𝜃𝑛=1𝚷𝜃𝑛𝐛.(3) When 𝜃𝑛𝜃𝑖 for 𝑖=1,2,𝑞, the projected signal is given by𝐛=𝚷𝜃𝑛𝐛=𝑞𝑖=1𝚷𝜃𝑛𝑠𝑖𝐚𝜃𝑖+𝚷𝜃𝑛𝐧.(4)

Then, (4) is composed of all 𝑞 incident signals with noise added. However, when 𝜃𝑛=𝜃𝑖 for 𝑖=1,2,𝑞, 𝐛=𝚷𝜃𝑖𝐛=𝑞1𝑗=1𝑗𝑖𝚷𝜃𝑖𝑠𝑗𝐚𝜃𝑗+𝚷𝜃𝑖𝐧.(5)

Then, (5) is composed of 𝑞1 incident signals with noise added, where the steering vector 𝐚(𝜃𝑖) is removed. The norm of (4) will have much greater value than the norm of (5). Thus, it is easy to show that we can obtain a peak in the projected power spectrum 𝐏Π(𝜃𝑖) when the steering vector 𝐚(𝜃𝑛) coincides with the source direction, that is, 𝜃𝑛=𝜃𝑖, 𝑖=1,2,𝑞. By using the projection matrix 𝚷(𝜃𝑛) for each steering vector 𝐚(𝜃𝑛), the rough DOA incident signals ranges are estimated by searching local peaks from 𝐏Π(𝜃𝑛), 𝑛=1,2,,180/Δ.

3.2. Fine DOA Estimation Algorithm

In the conventional subspace-based methods, the covariance matrix is decomposed into its constituent eigenvectors and eigenvalues. Recalling that the eigenvectors of the noise eigenvalues are orthogonal to the signal subspace spanned by the eigenvectors of the signal eigenvalues, then MUSIC algorithm uses the orthogonality of these subspaces efficiently to get a high-resolution DOA estimation. However, such processing is time-consuming against real-time applications and may lose performance for correlated signals. This subsection presents a procedure of finding a noise pseudo-eigenvector based on the solution of a linear-matrix equation, formed by a forward-backward data matrix, for real-time requirement and for the purpose of working well in correlated signals.

By using the orthogonal projection method, we state that if 𝑃Π(𝜃𝐶min) has a minimum value at angle 𝜃𝑛=𝜃𝐶min, the steering vector 𝐚(𝜃𝐶min) should be guaranteed to be too far from the signal directions. In order to acquire a noise pseudo-eigenvector, the steering vector 𝐚(𝜃𝐶min) may be used. First, we form two data matrices, 𝐁1, and 𝐁2, by dividing the uniform linear array into 𝑀+𝑁 high overlapping subarrays of size 𝑀 and 𝑁. Thus, the sensors {1,2,,𝑀} form the first subarray of 𝐁𝟏 and the sensors {2,3,,𝑀+1} form the second subarray of 𝐁𝟏 and so on; the sensors {𝑀+2𝑁1,𝑀+2𝑁2,,𝑀+𝑁} form the first subarray of 𝐁𝟐 and the sensors {𝑀+2𝑁2,𝑀+2𝑁3,,𝑀+𝑁1} form the second subarray of 𝐁𝟐 and so on, that is, we rearrange the received single snapshot 𝐛=[𝑏1𝑏2𝑏𝑚]𝑇 to form the following two data matrices:𝐁1=𝑏1𝑏2𝑏𝑀+𝑁𝑏2𝑏3𝑏𝑀+𝑁+1𝑏𝑀𝑏𝑀+1𝑏2𝑀+𝑁1𝐁,(6)2=𝑏𝑀+2𝑁1𝑏𝑀+2𝑁2𝑏𝑁𝑏𝑀+2𝑁2𝑏𝑀+2𝑁3𝑏𝑁1𝑏𝑀+𝑁𝑏𝑀+𝑁1𝑏1,(7) where 𝑀 is equal to 2𝑚/3, 𝑁(𝑁>𝑞) is equal to (𝑚1)/3, and 𝑦(𝑦<𝑦) is the closest integer to 𝑦. Finally, we combine the forward data matrix (6) and the backward data matrix (7) to form an (𝑀+𝑁)×(𝑀+𝑁) forward-backward data matrix as𝐁0=𝐁1𝑇𝐁2𝑇=𝑏1𝑏2𝑏𝑀𝑏𝑀+2𝑁1𝑏𝑀+2𝑁2𝑏𝑀+𝑁𝑏2𝑏3𝑏𝑀+1𝑏𝑀+2𝑁2𝑏𝑀+2𝑁3𝑏𝑀+𝑁1𝑏𝑀+𝑁𝑏𝑀+𝑁+1𝑏2𝑀+𝑁1𝑏𝑁𝑏𝑁1𝑏1.(8)We know that the resolution capability of the DOA estimation is relative to the number of sensors or to the degree of freedom (DOF) of the solved data matrix [7, 9]. For both the forward-only and backward-only methods, the maximum number of the DOF we can consider is given by 𝑚/2. In the proposed algorithm, we combine the forward data matrix 𝐁1 and backward data matrix 𝐁2 to double the received data and thereby increase the DOF significantly over that of either the forward-only or backward-only method alone. The maximum number of the DOF in the forward-backward method can be increased significantly to 2𝑚/3 without increasing the number of sensors; thus, this is the reason that we set 𝑀=𝑚/3 and 𝑁=(𝑚1)/3. Then, the proposed algorithm with about 2𝑚/3 DOF can provide a better resolution performance than Kim’s forward-only method with about 𝑚/2 DOF, and a resolution performance similar to Wen’s forward-backward method with about 2𝑚/3 DOF. Let 𝐛k denote the 𝑘th column vector of the data matrix (8). Then, we can write𝐛𝑘=𝐀(𝜃)𝐅1𝐬+𝐧k,(𝑘,)=(1,1),(2,2),,(𝑀,𝑀),(𝑀+1,𝑀2𝑁+3),,(𝑀+𝑁,𝑀𝑁+2),(9) where 𝐀(𝜃) and 𝐅 can be expressed as̃𝐚𝐀(𝜃)=𝜃1,̃𝐚𝜃2̃𝐚,,𝜃𝑞,̃𝐚𝜃𝑖=1,𝑒𝜑𝑖,,𝑒(𝑀+𝑁1)𝜑𝑖𝑇,𝑒𝐅=diag𝜑1,,𝑒𝜑𝑞,(10) where 𝜑𝑖=𝑗2𝜋𝑑/𝜆sin𝜃𝑖, 𝑖=1,2,𝑞. Suppose that the first two signals are highly correlated or coherent among the incident signals, that is, 𝑠2=𝜌𝑠1, where 𝜌 denotes a complex coefficient describing the phase and gain relationship between the two coherent signals. Under this assumption, we can rewrite the first column vector of the data matrix (8) as𝐛1=𝐀(𝜃)𝐅𝐬+𝐧1=̃𝐚𝜃1,̃𝐚𝜃2̃𝐚,,𝜃𝑞𝑠1𝜌𝑠1𝑠𝑞+𝐧1=𝑠1̃𝐚𝜃1+𝜌𝑠1̃𝐚𝜃2+𝑠𝑞̃𝐚𝜃𝑞+𝐧1,(11) then𝐛1𝑠1̃𝐚𝜃1=𝜌𝑠1̃𝐚𝜃2++𝑠𝑞̃𝐚𝜃𝑞+𝐧𝟏.(12) Based on (12), [𝐛1𝑠1̃𝐚(𝜃1)] excludes the desired signal 𝑠1̃𝐚(𝜃1), that is, it only includes other 𝑞1 desired signals 𝜌𝑠1̃𝐚(𝜃2),,𝑠𝑞̃𝐚(𝜃𝑞) and noise vector 𝐧1. If we can find a weight vector 𝐰(𝜃1)=[𝑤1(𝜃1),𝑤2(𝜃1),,𝑤𝑀+𝑁(𝜃1)]𝑇 to satisfy [𝜌1𝑠1̃𝐚(𝜃2)++𝑠𝑞̃𝐚(𝜃𝑞)+𝐧1]𝑇𝐰(𝜃1)=0, then (12) can be rewritten as𝐛1𝑠1̃𝐚𝜃1𝑇𝐰𝜃1=0.(13) Therefore,𝐛𝑇1𝐰𝜃1=𝑠1̃𝐚𝜃1𝑇𝐰𝜃1.(12-1) By the same process, we can then get𝐛𝑇2𝐰𝜃1=𝑠1𝑒𝜑𝑖̃𝐚𝜃1𝑇𝐰𝜃1,𝐛𝑇3𝐰𝜃1=𝑠1𝑒2𝜑1̃𝐚𝜃1𝑇𝐰𝜃1,𝐛𝑇𝑀𝐰𝜃1=𝑠1𝑒(𝑀1)𝜑1̃𝐚𝜃1𝑇𝐰𝜃1,𝐛𝑇𝑀+1𝐰𝜃1=𝑠1𝑒(𝑀2𝑁+2)𝜑1̃𝐚𝜃1𝑇𝐰𝜃1,𝐛𝑇𝑀+2𝐰𝜃1=𝑠1𝑒(𝑀2𝑁+3)𝜑1̃𝐚𝜃1𝑇𝐰𝜃1,𝐛𝑇𝑀+𝑁𝐰𝜃1=𝑠1𝑒(𝑀𝑁+1)𝜑1̃𝐚𝜃1𝑇𝐰𝜃1.(12-2)

Combining (12-1) and (12-2) into matrix notation gives𝐛𝑇1𝐛𝑇2𝐛𝑇𝑀𝐛𝑇𝑀+1𝐛𝑇𝑀+2𝐛𝑇𝑀+𝑁𝐰𝜃1=𝑠1̃𝐚𝜃1𝑇𝑒𝜑1̃𝐚𝜃1𝑇𝑒(𝑀1)𝜑1̃𝐚𝜃1𝑇𝑒(𝑀2𝑁+2)𝜑1̃𝐚𝜃1𝑇𝑒(𝑀2𝑁+3)𝜑1̃𝐚𝜃1𝑇𝑒(𝑀𝑁+1)𝜑1̃𝐚𝜃1𝑇𝐰𝜃1=𝑠1̃𝐚𝜃1𝑇𝑠1𝑒𝜑1̃𝐚𝜃1𝑇𝑠1𝑒(𝑀1)𝜑1̃𝐚𝜃1𝑇𝑠1𝑒(𝑀2𝑁+2)𝜑1̃𝐚𝜃1𝑇𝑠1𝑒(𝑀2𝑁+3)𝜑1̃𝐚𝜃1𝑇𝑠1𝑒(𝑀𝑁+1)𝜑1̃𝐚𝜃1𝑇𝑤1𝜃1𝑤2𝜃1𝑤𝑀+𝑁𝜃1=𝑠1𝑤1𝜃1̂̃𝐚𝜃1++𝑠1𝑤𝑀+𝑁𝜃1̂̃𝐚𝜃1̂̃𝐚=𝛼𝜃1,(14) where 𝑠𝛼=1𝑤1𝜃1++𝑠1𝑤𝑀+𝑁𝜃1,̂̃𝐚𝜃1=1,𝑒𝜑1,,𝑒(𝑀1)𝜑1,𝑒(𝑀2𝑁+2)𝜑1,,𝑒(𝑀𝑁+1)𝜑1𝑇,(15) then 𝐛𝑇1𝐛𝑇2𝐛𝑇𝑀𝐛𝑇𝑀+1𝐛𝑇𝑀+2𝐛𝑇𝑀+𝑁𝐰𝜃1̂̃𝐚𝛼𝜃1̂̃𝐚𝜃1.(16) Note that (16) skips the scalar 𝛼 because it does not affect the vector direction of weight vector 𝐰(𝜃1). Weight vector 𝐰(𝜃1) has pattern nulls at the incident angles of the rest of incoming signals, 𝜃2,𝜃3,,𝜃𝑞. Therefore, if the steering vector ̂̃𝐚(𝜃𝐶min) is used in (16), then the corresponding weight vector 𝐰(𝜃𝐶min) will be orthogonal to the signal steering vectors ̃𝐚(𝜃𝑖) of the incoming signals, that is, 𝐰(𝜃𝐶min)𝑇̃𝐚(𝜃𝑖)0, where 𝑖=1,2,,𝑞. In this paper, weight vector 𝐰(𝜃𝐶min) is a noise pseudo-eigenvector because it performs a function similar to that of a noise eigenvector. Then, fine peaks corresponding to the DOA of the incoming signals can be found in eigenspectrum 𝑃𝐸1(𝜃𝑛) using 𝑃𝐸1𝜃𝑛=1|||𝐰𝜃𝐶min𝑇̃𝐚𝜃𝑛|||2.(17) Moreover, since an eigenspectrum provides higher resolution DOA estimation than a projection spectrum, steering vector ̂̃𝐚(𝜃𝐸min) with minimum eigenspectrum 𝑃𝐸1(𝜃𝐸min) via (17) is more likely to be constrained too far from the signal steering vectors ̃𝐚(𝜃𝑖) than steering vector ̂̃𝐚(𝜃𝐶min). The corresponding weight vector 𝐰(𝜃𝐸min) is obtained by using ̂̃𝐚(𝜃𝐸min) in (16), which is more orthogonal to the signal steering vectors ̃𝐚(𝜃𝑖) than the weight vector 𝐰(𝜃𝐶min). Then, we can obtain a more accurate DOA estimation of the incoming signals than (17) by performing an eigenspectrum as follows:𝑃𝐸2𝜃𝑛=1|||𝐰𝜃𝐸min𝑇̃𝐚𝜃𝑛|||2.(18) It is known that a linear matrix equation can be solved very efficiently by applying a conjugate gradient (CG) algorithm. The conjugate gradient method can provide a faster rate of convergence than other gradient methods and can be implemented to operate in real-time utilizing a digital signal processing chip [7, 10]. Therefore, the use of the CG algorithm makes our proposed algorithm suitable for real-time implementation. The procedure of our proposed fast DOA estimation is as follows.(1)Obtain the rough DOA ranges of the incoming signals and a steering vector ̂̃𝐚(𝜃𝐶min) by using the projection method via (3).(2)Use the steering vector ̂̃𝐚(𝜃𝐶min) in (16) to obtain a noise pseudo-eigenvector 𝐰(𝜃𝐶min).(3)Obtain the fine DOA estimation of the incoming signals by searching for local peak values in the eigenspectrum 𝑃𝐸1(𝜃𝑛) via (17) within the rough DOA ranges.(4)Use the steering vector ̂̃𝐚(𝜃𝐸min) in (16) to obtain a noise pseudo-eigenvector 𝐰(𝜃𝐸min).(5)Fine-tune the DOA estimation of the incoming signals in step (3) by searching for local peak values of the eigenspectrum 𝑃𝐸2(𝜃𝑛) via (18).

The computational cost of solving a nonlinear generalized eigenvalue algorithm is about o(𝑘3) flops for a 𝑘×𝑘 matrix. The computational cost of solving an SVD algorithm is about o(𝑘𝑟2) flops for a 𝑘×𝑟 matrix. The computational cost of solving the CG algorithm is about o(𝑘2) flops per iteration for a 𝑘×𝑘 matrix. The size of resolved data matrix is approximately 𝑚/2×𝑚/2 in Kim’s method, 2𝑚/3×(2𝑚/3+2) in Wen’s method and 2𝑚/3×2𝑚/3 in our proposed algorithm. The main computational complexity difference between the other methods and the proposed algorithm is the weight vector calculation, which requires about 𝐿o(𝑚3/8) from solving a nonlinear generalized eigenvalue algorithm in Kim’s method, 𝐿o(8𝑚3/27) from solving a SVD algorithm in Wen’s method, and 2𝐾o(4𝑚2/9) in our proposed method, which is based on the CG algorithm. A flop is defined as a complex floating-point multiplication operation, where o() denotes “the order of ”, 𝑚 is the number of the sensors, 𝐿 denotes the total scanning number in the possible DOA range (i.e., the 𝐿 different looking directions), and 𝐾 is the number of iterations for executing the CG algorithm. We can see that the computational complexity of the DOA estimation for fast varying signals for our proposed method is significantly smaller than those that of Kim’s, and Wen’s methods because 𝐿𝑚>𝐾2 in general.

4. Computer Simulation

To compare the performance of the described DOA finding method, MUSIC, ESPRIT, Wen’s, and Kim’s method, in terms of the probability of resolution, the methods were investigated in the narrowband farfield DOA estimations with computer experiments. The probability of resolution is defined as follows. Two signals with DOA 𝜃1 and 𝜃2 are said to be resolved if the respective estimates ̂𝜃1 and ̂𝜃2 are such that both |̂𝜃1𝜃1| and |̂𝜃2𝜃2| are less than |𝜃1𝜃2|/2 [11]. Four different experiments were done with variations in SNR value, the separation between sources, and the number of sensors.

In each experiment, it was assumed that the powers of all the incoming sources were equal. The noise at each array sensor was assumed to be additive white Gaussian noise process with zero mean. We also assumed a uniform linear array with half-wavelength element spacing. We examined the conventional MUSIC and ESPRIT algorithm when the number of snapshots was 100, whereas the proposed algorithm, Kim’s, and Wen’s methods only used a single snapshot. A total of 300 independent test runs were done to get each simulated point, and the DOA scanning was performed over [0°,90°] with a step size of Δ=0.1. In these simulations, the solution of a linear matrix equation was solved using the CG method.

In the first experiment, three signals with the same SNR of 20 dB were impinging on the array of 23 sensors. Figures 1(b) and 1(c) show that we can estimate the three uncorrelated signals from [2.3°,9.7°,17.8°] using the proposed algorithm and MUSIC, respectively. Figure 2(b) shows that we can still estimate the two coherent signals from [2.3°,9.7°] and one uncorrelated signal from [17.8°] using the proposed algorithm but Figure 2(c) shows that MUSIC was unable to estimate the DOA of the two coherent signals. Also, Figures 1(a) and 2(a) show that the angles [2.3°,9.7°,17.8°] of the three incident signals were within the estimated possible ranges.

fig1
Figure 1: DOA estimation using the proposed algorithm and MUSIC for three uncorrelated signals from [2.3°,9.7°,17.8°] under the SNR of 20 dB and the number of sensors 𝑚=23.
fig2
Figure 2: DOA estimation using the proposed algorithm and MUSIC for two coherent signals from [2.3°,9.7°] and one uncorrelated signal from [17.8°] under the SNR of 20 dB and the number of sensors 𝑚=23.

In the second experiment, we considered three uncorrelated signals from [2.3°,9.7°,17.8°], impinging on the array of 23 sensors. Figure 3 shows the comparison result of the probability of resolution versus various SNR values. As seen, ESPRIT has best resolution, the proposed algorithm and Wen’s method have almost identical resolution, and Kim’s method has worst resolution. When an SNR = −5 dB, the resolution probability of the proposed algorithm, Kim’s and Wen’s methods was less than 50% while it is close to 60% for ESPRIT.

751670.fig.003
Figure 3: Comparison of results of the probability of resolution from ESPRIT, the proposed algorithm, Kim’s, and Wen’s methods, with for three uncorrelated signals from [2.3°,9.7°,17.8°] having different SNR, where the number of sensors 𝑚=23.

In the third experiment, three uncorrelated signals from [2.3°,2.3° + δθ,2.3° + 2δθ], impinging on the array of 23 sensors and SNR of 20 dB, were considered where the δθ denote a small angle spacing. Figure 4 shows the comparison result of the probability of resolution versus the angle spacing of the incoming signals. As seen in that figure, ESPRIT achieves a higher rate of successful source separation in comparison with the proposed algorithm, Kim’s, and Wen’s methods. For an angle spacing 𝛿𝜃=2, the resolution probability of the proposed algorithm, Kim’s and Wen’s methods was close to 0%. When an angle spacing 𝛿𝜃<7, the Wen’s method and the proposed algorithm had better performance than Kim’s method.

751670.fig.004
Figure 4: Comparison of results from ESPRIT, the proposed algorithm, Kim’s, and Wen’s methods, for the probability of resolution versus the angle spacing of the incoming signals for three uncorrelated signals from [2.3°,9.7°,17.8°] having the SNR of 20 dB, where the number of sensors 𝑚=23.

In the fourth experiment, we considered three uncorrelated signals from [2.3°,9.7°,17.8°], impinging on the array with SNR of 20 dB. Figure 5 shows the comparison result of the probability of resolution versus the number of sensors for ESPRIT, the proposed algorithm, Kim’s, and Wen’s methods. The simulations showed that the performance could be improved by using more number of sensors for all algorithms, but that more processing time is required.

751670.fig.005
Figure 5: Comparison of results of results of the probability of resolution from ESPRIT, the proposed algorithm, Kim’s, and Wen’s methods, with three uncorrelated signals from [2.3°,9.7°,17.8°] having the different number of sensors, where the SNR of 20 dB.

As expected, from Figure 1 to Figure 5, the simulations showed that the proposed method outperformed Kim’s methods and performed almost the same as Wen’s method with regards to resolution capabilities because the effective array size in both the proposed algorithm and Wen’s method is 15, but in Kim’s method is 11. If the number of signal sources is known exactly and has enough input data snapshots, of course, the MUSIC or ESPRIT provides us with better performance than the proposed algorithm in cases with uncorrelated signals.

5. Conclusion

A new fast direction of arrival estimation technique is presented based on both a simple projection spectrum and the eigenspectrum. The advantages of this method over MUSIC, ESPRIT, Kim’s, and Wen’s methods are as follows: (1) it has a lower computation loads; (2) it allows arbitrary signal statistics, for example, nonstationary and coherent; (3) it is capable of tracking high varying signals. Computer simulations have shown that the proposed algorithm handles the coherent or uncorrelated signals well, whereas MUSIC does not, as well as better resolution as compared with Kim’s method, or similar resolution to Wen's method. Since this new approach improves performance by reducing computational complexity while maintaining sufficient resolution in highly nonstationary environments, it will have a wider range of prospective applications in real-time DOA estimation than other comparable methods.

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