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

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 [3–7] 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,…,𝑡_𝑢] represents a diagonal matrix with diagonal entries 𝑡₁,…,𝑡_𝑢.

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=𝑏₁𝑏₂⋯𝑏_𝑚^𝑇,(1) where 𝐬=[𝑠₁𝑠₂…𝑠_𝑞]^𝑇 is a 𝑞×1 signal vector, 𝐧 is an 𝑚×1 noise process vector, and 𝐀(𝜃)=[𝐚(𝜃₁),𝐚(𝜃₂),…𝐚(𝜃_𝑞)] represents a 𝑚×𝑞 matrix of steering vectors. The 𝑖th column 𝐚(𝜃_𝑖)=[1,𝑒^{𝑗2𝜋𝑑/𝜆𝑖sin𝜃},…,𝑒^{𝑗2𝜋𝑑/𝜆𝑖(𝑚−1)sin𝜃}]^𝑇 is the steering vector 𝑠_𝑖 coming from 𝜃_𝑖, 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 as𝚷^⟂𝜃_𝑛=𝐈−𝐚^′𝜃_𝑛𝐚^′𝜃_𝑛^𝐻,𝑛=1,2,…,180Δ,(2) where 𝜃_𝑛=𝑛Δ∈[−90^∘,90^∘] denotes each search angle, Δ is search angle step, and 𝐚^(𝜃_𝑛) is the normalized version of 𝐚(𝜃_𝑛), determined as 𝐚^(𝜃_𝑛)=𝐚(𝜃_𝑛)/‖𝐚(𝜃_𝑛)‖. Then, use (2) to form the following projected power spectrum:𝐏_Π𝜃_𝑛=1‖‖𝚷^⟂𝜃_𝑛𝐛‖‖.(3) When 𝜃_𝑛≠𝜃_𝑖 for 𝑖=1,2,…𝑞, the projected signal is given by𝐛^⟂=𝚷^⟂𝜃_𝑛𝐛=^𝑞𝑖=1𝚷^⟂𝜃_𝑛𝑠_𝑖𝐚𝜃_𝑖+𝚷^⟂𝜃_𝑛𝐧.(4)

Then, (4) is composed of all 𝑞 incident signals with noise added. However, when 𝜃_𝑛=𝜃_𝑖 for 𝑖=1,2,…𝑞, 𝐛^⟂=𝚷^⟂𝜃_𝑖𝐛=^{𝑞−1𝑗=1∶𝑗≠𝑖}𝚷^⟂𝜃_𝑖𝑠_𝑗𝐚𝜃_𝑗+𝚷^⟂𝜃_𝑖𝐧.(5)

Then, (5) is composed of 𝑞−1 incident signals with noise added, where the steering vector 𝐚(𝜃_𝑖) 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 𝐏_Π(𝜃_𝑖) when the steering vector 𝐚(𝜃_𝑛) coincides with the source direction, that is, 𝜃_𝑛=𝜃_𝑖, 𝑖=1,2,…𝑞. By using the projection matrix 𝚷^⟂(𝜃_𝑛) for each steering vector 𝐚(𝜃_𝑛), the rough DOA incident signals ranges are estimated by searching local peaks from 𝐏_Π(𝜃_𝑛), 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 𝑃_Π(𝜃_𝐶min) has a minimum value at angle 𝜃_𝑛=𝜃_𝐶min, the steering vector 𝐚(𝜃_𝐶min) should be guaranteed to be too far from the signal directions. In order to acquire a noise pseudo-eigenvector, the steering vector 𝐚(𝜃_𝐶min) 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 𝐁_𝟏 and the sensors {2,3,…,𝑀+1} form the second subarray of 𝐁_𝟏 and so on; the sensors {𝑀+2𝑁−1,𝑀+2𝑁−2,…,𝑀+𝑁} form the first subarray of 𝐁_𝟐 and the sensors {𝑀+2𝑁−2,𝑀+2𝑁−3,…,𝑀+𝑁−1} form the second subarray of 𝐁_𝟐 and so on, that is, we rearrange the received single snapshot 𝐛=[𝑏₁𝑏₂…𝑏_𝑚]^𝑇 to form the following two data matrices:𝐁₁=⎡⎢⎢⎢⎣𝑏¹𝑏₂…𝑏_𝑀+𝑁𝑏₂𝑏₃…𝑏_𝑀+𝑁+1𝑏⋮⋮⋱⋮_𝑀𝑏_𝑀+1…𝑏_{2𝑀+𝑁−1}𝐁⎤⎥⎥⎥⎦,(6)₂=⎡⎢⎢⎢⎢⎣𝑏^{∗𝑀+2𝑁−1}𝑏_{∗𝑀+2𝑁−2}…𝑏_𝑁∗𝑏_{∗𝑀+2𝑁−2}𝑏_{∗𝑀+2𝑁−3}…𝑏_∗𝑁−1𝑏⋮⋮⋱⋮_∗𝑀+𝑁𝑏_{∗𝑀+𝑁−1}…𝑏_1∗⎤⎥⎥⎥⎥⎦,(7) where 𝑀 is equal to ⌈2𝑚/3⌉, 𝑁(𝑁>𝑞) is equal to ⌈(𝑚−1)/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==⎡⎢⎢⎢⎢⎣𝐁𝑏_1𝑇1𝐁_2𝑇𝑏₂…𝑏_𝑀𝑏_{∗𝑀+2𝑁−1}𝑏_{∗𝑀+2𝑁−2}…𝑏_∗𝑀+𝑁𝑏₂𝑏₃…𝑏_𝑀+1𝑏_{∗𝑀+2𝑁−2}𝑏_{∗𝑀+2𝑁−3}…𝑏_{∗𝑀+𝑁−1}𝑏⋮⋮⋱⋮⋮⋮⋱⋮_𝑀+𝑁𝑏_𝑀+𝑁+1…𝑏_{2𝑀+𝑁−1}𝑏_𝑁∗𝑏_∗𝑁−1…𝑏_1∗⎤⎥⎥⎥⎥⎦.(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=⌈(𝑚−1)/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 𝐛_k denote the 𝑘th column vector of the data matrix (8). Then, we can write(𝑘,ℎ)=((𝑀1,1+)𝑁,(𝐛2,_𝑘−,2=𝑀),𝐀…−(𝜃𝑁,)(𝐅𝑀+,2𝑀^ℎ−1)𝐬,+),𝐧(𝑀_k,+1,−𝑀−2𝑁+3),…,(9) where 𝐀(𝜃) and 𝐅 can be expressed as𝐀𝐚(̃𝜃)=𝐚̃𝜃₁,𝐚̃𝜃₂,…,𝐚̃𝜃_𝑞,𝜃_𝑖𝐅==d1i,a𝑒g^𝜑𝑖,𝑒…_𝜑1,𝑒^{(𝑀+𝑁−1)𝜑𝑖}^𝑇,,…,𝑒^𝜑𝑞,(10) where 𝜑_𝑖=𝑗^2𝜋𝑑/𝜆^sin𝜃𝑖, 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)𝐅^∘𝐬+𝐧₁=𝐚̃𝜃₁,𝐚̃𝜃₂,…,𝐚̃𝜃_𝑞⎡⎢⎢⎢⎢⎣𝑠¹𝜌𝑠₁𝑠⋮_𝑞⎤⎥⎥⎥⎥⎦+𝐧₁=𝑠₁𝐚̃𝜃₁+𝜌𝑠₁𝐚̃𝜃₂⋯+𝑠_𝑞𝐚̃𝜃_𝑞+𝐧₁,(11) then𝐛₁−𝑠₁𝐚̃𝜃₁=𝜌𝑠₁𝐚̃𝜃₂+⋯+𝑠_𝑞𝐚̃𝜃_𝑞+𝐧_𝟏.(12) Based on (12), [𝐛₁−𝑠₁𝐚̃(𝜃₁)] excludes the desired signal 𝑠₁𝐚̃(𝜃₁), that is, it only includes other 𝑞−1 desired signals 𝜌𝑠₁𝐚̃(𝜃₂),…,𝑠_𝑞𝐚̃(𝜃_𝑞) and noise vector 𝐧₁. If we can find a weight vector 𝐰(𝜃₁)=[𝑤₁(𝜃₁),𝑤₂(𝜃₁),…,𝑤_𝑀+𝑁(𝜃₁)]^𝑇 to satisfy [𝜌₁𝑠₁𝐚̃(𝜃₂)+⋯+𝑠_𝑞𝐚̃(𝜃_𝑞)+𝐧₁]^𝑇𝐰(𝜃₁)=0, then (12) can be rewritten as𝐛₁−𝑠₁𝐚̃𝜃₁^𝑇𝐰𝜃₁=0.(13) Therefore,𝐛^𝑇1𝐰𝜃₁=𝑠₁𝐚̃𝜃₁^𝑇𝐰𝜃₁.(12-1) By the same process, we can then get𝐛𝐛^𝑇2_𝑇3𝐰𝜃₁=𝑠₁𝑒^𝜑𝑖𝐚̃𝜃₁^𝑇𝐰𝜃₁,𝐛_𝑇𝑀𝐰𝜃₁=⋮𝑠₁𝑒^2𝜑1𝐚̃𝜃₁^𝑇𝐰𝜃₁,𝐛_𝑇𝑀+1𝐰𝜃₁=𝑠₁𝑒^{(𝑀−1)𝜑1}𝐚̃𝜃₁^𝑇𝐰𝜃₁,𝐛_𝑇𝑀+2𝐰𝜃₁=𝑠₁𝑒^{(−𝑀−2𝑁+2)𝜑1}𝐚̃𝜃₁^𝑇𝐰𝜃₁,𝐛_{𝑇𝑀+𝑁}𝐰𝜃₁=⋮𝑠₁𝑒^{(−𝑀−2𝑁+3)𝜑1}𝐚̃𝜃₁^𝑇𝐰𝜃₁,𝐰𝜃₁=𝑠₁𝑒^{(−𝑀−𝑁+1)𝜑1}𝐚̃𝜃₁^𝑇𝐰𝜃₁.(12-2)

Combining (12-1) and (12-2) into matrix notation gives⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐛^𝑇1𝐛^𝑇2𝐛⋮_𝑇𝑀𝐛^𝑇𝑀+1𝐛^𝑇𝑀+2𝐛⋮_{𝑇𝑀+𝑁}⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝐰𝜃₁=𝑠₁⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐚̃𝜃₁^𝑇𝑒^𝜑1𝐚̃𝜃₁^𝑇𝑒⋮_{(𝑀−1)𝜑1}𝐚̃𝜃₁^𝑇𝑒^{(−𝑀−2𝑁+2)𝜑1}𝐚̃𝜃₁^𝑇𝑒^{(−𝑀−2𝑁+3)𝜑1}𝐚̃𝜃₁^𝑇𝑒⋮_{(−𝑀−𝑁+1)𝜑1}𝐚̃𝜃₁^𝑇⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝐰=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑠₁𝜃₁𝐚̃𝜃₁^𝑇𝑠₁𝑒^𝜑1𝐚̃𝜃₁^𝑇𝑠₁⋮𝑒^{(𝑀−1)𝜑1}𝐚̃𝜃₁^𝑇𝑠₁𝑒^{(−𝑀−2𝑁+2)𝜑1}𝐚̃𝜃₁^𝑇𝑠₁𝑒^{(−𝑀−2𝑁+3)𝜑1}𝐚̃𝜃₁^𝑇𝑠₁⋮𝑒^{(−𝑀−𝑁+1)𝜑1}𝐚̃𝜃₁^𝑇⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑤₁𝑤₂𝜃₁𝑤⋮_𝑀+𝑁𝜃₁=𝑠₁𝜃₁⎤⎥⎥⎥⎥⎦=𝛼𝐚𝑤̂̃₁𝜃₁𝐚̂̃𝜃₁+⋯+𝑠₁𝑤_𝑀+𝑁𝜃₁𝐚̂̃𝜃₁𝜃₁,(14) where 𝐚̂̃𝛼=𝑠₁𝑤₁𝜃₁+⋯+𝑠₁𝑤_𝑀+𝑁𝜃₁,𝜃₁=1,𝑒^𝜑1,…,𝑒^{(𝑀−1)𝜑1},𝑒^{(−𝑀−2𝑁+2)𝜑1},…,𝑒^{(−𝑀−𝑁+1)𝜑1}^𝑇,(15) then ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐛^𝑇1𝐛^𝑇2𝐛⋮_𝑇𝑀𝐛^𝑇𝑀+1𝐛^𝑇𝑀+2𝐛⋮_{𝑇𝑀+𝑁}⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝐰𝜃₁≈𝛼𝐚̂̃𝜃₁≡𝐚̂̃𝜃₁.(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,…,𝜃_𝑞. Therefore, if the steering vector 𝐚̂̃(𝜃_𝐶min) is used in (16), then the corresponding weight vector 𝐰(𝜃_𝐶min) will be orthogonal to the signal steering vectors 𝐚̃(𝜃_𝑖) of the incoming signals, that is, 𝐰(𝜃_𝐶min)^𝑇𝐚̃(𝜃_𝑖)≈0, where 𝑖=1,2,…,𝑞. In this paper, weight vector 𝐰(𝜃_𝐶min) 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 𝑃_𝐸1(𝜃_𝑛) using 𝑃_𝐸1𝜃_𝑛=1|||𝐰𝜃_𝐶min^𝑇𝐚̃𝜃_𝑛|||².(17) Moreover, since an eigenspectrum provides higher resolution DOA estimation than a projection spectrum, steering vector 𝐚̂̃(𝜃_𝐸min) with minimum eigenspectrum 𝑃_𝐸1(𝜃_𝐸min) via (17) is more likely to be constrained too far from the signal steering vectors 𝐚̃(𝜃_𝑖) than steering vector 𝐚̂̃(𝜃_𝐶min). The corresponding weight vector 𝐰(𝜃_𝐸min) is obtained by using 𝐚̂̃(𝜃_𝐸min) in (16), which is more orthogonal to the signal steering vectors 𝐚̃(𝜃_𝑖) than the weight vector 𝐰(𝜃_𝐶min). Then, we can obtain a more accurate DOA estimation of the incoming signals than (17) by performing an eigenspectrum as follows:𝑃_𝐸2𝜃_𝑛=1|||𝐰𝜃_𝐸min^𝑇𝐚̃𝜃_𝑛|||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 𝐚̂̃(𝜃_𝐶min) by using the projection method via (3).(2)Use the steering vector 𝐚̂̃(𝜃_𝐶min) in (16) to obtain a noise pseudo-eigenvector 𝐰(𝜃_𝐶min).(3)Obtain the fine DOA estimation of the incoming signals by searching for local peak values in the eigenspectrum 𝑃_𝐸1(𝜃_𝑛) via (17) within the rough DOA ranges.(4)Use the steering vector 𝐚̂̃(𝜃_𝐸min) in (16) to obtain a noise pseudo-eigenvector 𝐰(𝜃_𝐸min).(5)Fine-tune the DOA estimation of the incoming signals in step (3) by searching for local peak values of the eigenspectrum 𝑃_𝐸2(𝜃_𝑛) 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 𝐿≫𝑚>𝐾≫2 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 𝜃̂₁ and 𝜃̂₂ are such that both |𝜃̂₁−𝜃₁| and |𝜃̂₂−𝜃₂| are less than |𝜃₁−𝜃₂|/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.

(a)

(b)

(c)

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.

(a)

(b)

(c)

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.

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.

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.

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.