Improved ESPRIT Method for Joint Direction-of-Arrival and Frequency Estimation Using Multiple-Delay Output

Abstract

An automatic pairing joint direction-of-arrival (DOA) and frequency estimation is presented to overcome the unsatisfactory performances of estimation of signal parameter via rotational invariance techniques- (ESPRIT-) like algorithm of Wang (2010), which requires an additional pairing. By using multiple-delay output of a uniform linear antenna arrays (ULA), the proposed algorithm can estimate joint angles and frequencies with an improved ESPRIT. Compared with Wang’s ESPRIT algorithm, the angle estimation performance of the proposed algorithm is greatly improved. The frequency estimation performance of the proposed algorithm is same with that of Wang’s ESPRIT algorithm. Furthermore, the proposed algorithm can obtain automatic pairing DOA and frequency parameters, and it has a comparative computational complexity in contrast to Wang’s ESPRIT algorithm. By the way, this proposed algorithm can also work well for nonuniform linear arrays. The useful behavior of this proposed algorithm is verified by simulations.

1. Introduction

Uniform linear antenna (ULA) arrays have been used in radar, sonar, electron reconnaissance, seismic data processing, and so on [1–4]. The direction-of-arrival (DOA) and frequency estimation of signals impinging on the ULAs are two fundamental problems in array processing [5–10]. The problem of joint DOA and frequency estimation arises in the applications of fields mentioned above. For example, these parameters can be applied to locate the targets for radars and to locate pilot tones in electron reconnaissance systems [11]. Furthermore, a precise estimation of these parameters is helpful to attain a better pulse descriptor word (PDW) and thus enhances the system performance. Optimal techniques based on maximum likelihood [12] are often applicable but might be computationally prohibitive. Some ESPRIT-based joint angle and frequency estimation methods have been proposed in [13–17]. Mathews [13] discuss this problem in the context of radar applications. Pro-ESPRIT is proposed to estimate angle and frequency. Haardt and Nossek [14] discuss the problem in the context of mobile communications for space division multiple access applications. Their method is based on unitary-ESPRIT, which involves a certain transformation of the data to real valued matrices. Multiresolution ESPRIT is used for joint angle frequency estimation in [15]. ESPRIT method is used for frequency and angle estimation under uniform circular array in [16, 17]. These ESPRIT-based joint angle and frequency estimation methods can give satisfactory estimation performances, but require additional pairing.

The authors in [18] proposed a joint angle and frequency estimation using multiple-delay output based on ESPRIT, which outperform the conventional ESPRIT algorithm, but it requires an additional pairing. The additional pairing fails to work for low signal-to-noise ratio (SNR), which can be shown from the simulation results in Figure 1. Furthermore, the angle estimation of Wang’s ESPRIT algorithm can be improved further.

Figure 1 

Angle-frequency scatter of the Wang’s ESPRIT algorithm at SNR = 0 dB.

An improved ESPRIT for joint direction-of-arrival (DOA) and frequency estimation is presented in this paper. Compared with Wang’s ESPRIT algorithm, the algorithm has the improved performance of DOA estimation. Moreover, the proposed algorithm can obtain automatically paired DOA and frequency estimation, while Wang’s ESPRIT algorithm requires the additional pairing. The proposed algorithm has a comparative computational complexity in contrast to Wang’s ESPRIT algorithm. Moreover, this proposed algorithm can also work well for nonuniform linear arrays.

Notation 1. (⋅)^𝑇, (⋅)^𝐻, (⋅)⁻¹, (⋅)⁺, and ⊙ denote transpose, conjugate transpose, pseudoinverse operations, and Hadamard product, respectively. Angle(⋅) refers to the phase of a complex number. diag(𝐯) stands for diagonal matrix, the diagonal of which is a vector 𝐯.

2. The Data Model

We consider a uniform linear array with spacing 𝑑. There is an array of 𝑀 sensors on which 𝐾 incident waves impinge (𝑀>𝐾), if the signals are all within the assumed narrow (with respect to the center frequency) band of the receiver, the signal received at the mth antenna is 𝑦_𝑚(𝑡)=^𝐾_𝑘=1𝑒𝑚=0,1,2,…^{𝑗2𝜋𝑚𝑑𝑓𝑘sin(𝜃𝑘)/𝑐},𝑠𝑀_𝑘(𝑡−)+1,𝑛_𝑚(𝑡),(1) where 𝑠_𝑘(𝑡) is the narrow-band signal of the kth source. 𝑐 is velocity of light. 𝜃_𝑘,𝑓_𝑘 are the DOA and the frequency of the kth signal, respectively. 𝑛_𝑚(𝑡) represents an additive noise term which is assumed to be zero mean and stationary.

The outputs of the uniform linear antenna arrays (ULA) is 𝐘_𝟎=𝑦₀(𝑛),𝑦₁(𝑛𝑛),=𝑦₂1(,𝑛2),,3…,…,𝑦,𝑁._𝑀−1(𝑛)^𝑇,(2)

Based on this available samples, the problem is to estimate the angles and the frequencies of all sources from the 𝑁 times snapshots. Suppose the number of signals 𝐾 is assumed known. Denote the state vector, we get 𝐘_𝟎=𝐀𝐒+𝐍_𝟎,(3) where 𝐍_𝟎𝐒=𝑠₁,𝑠₂,…,𝑠_𝐾^𝑇∈𝐂^𝐾×𝑁,𝐀=⎡⎢⎢⎢⎣=𝑒1𝐧₀,𝐧₁,…1𝐧_𝑀−1^𝑇⋯∈𝐂1^𝑀×𝑁_−𝑗𝛼1,𝑒^−𝑗𝛼2⋯𝑒^{−𝑗𝛼𝐾}𝑒⋮⋮⋱⋮_{−𝑗(𝑀−1)𝛼1}𝑒^{−𝑗(𝑀−1)𝛼2}⋯𝑒^{−𝑗(𝑀−1)𝛼𝐾}⎤⎥⎥⎥⎦,(4) where 𝛼_𝑘=2𝜋𝑑𝑓_𝑘sin(𝜃_𝑘)/𝑐, 𝑘=1,2,…,𝐾.

In order to joint estimate DOA and frequency, we add 𝑃−1 delayed outputs for the received signal of array antenna. We suppose that0<(𝑃−1)𝜏<1/max(𝑓_𝑖).

The delayed signal for (1) with delay 𝜏 is 𝑦_𝑚(𝑡−𝜏)=^𝐾_𝑘=1=𝑒^{𝑗2𝜋𝑚𝑑𝑓𝑘sin(𝜃𝑘)/𝑐}𝑠_𝑘(𝑡−𝜏)+𝑛^𝑚(𝑡)_𝐾_𝑘=1𝑒^{𝑗2𝜋𝑚𝑑𝑓𝑘sin(𝜃𝑘)/𝑐}𝑠_𝑘(𝑡)𝑒^{−𝑗2𝜋𝑓𝑘𝜏}+𝑛^𝑚(𝑡).(5)

The delayed signal for (5) with 𝜏 can be denoted as 𝐘₁=𝐀Φ𝐒+𝐍₁,(6) where Φ=diag𝑒^−𝑗𝛽1,𝑒^−𝑗𝛽2,…,𝑒^{−𝑗𝛽𝐾},(7) where 𝛽_𝑘=2𝜋𝑓_𝑘𝜏, 𝑘=1,2,…,𝐾.

The delayed signal for (5) with 𝑝𝜏 can be denoted as 𝐘_𝑝=𝐀Φ^𝑝𝐒+𝐍_𝑝,𝑝=0,1,…,𝑃−1.(8)

According to (3), (6), and (8), we define 𝐘=⎡⎢⎢⎣𝐘^𝟎𝐘₁𝐘⋮_𝑃−1⎤⎥⎥⎦=⎡⎢⎢⎣𝐀𝐀Φ⋮𝐀Φ_𝑃−1⎤⎥⎥⎦𝐒+⎡⎢⎢⎣𝐍𝐍^𝟎₁𝐍⋮_𝑃−1⎤⎥⎥⎦.(9)

3. Joint Angle and Frequency Estimation

We can use received signal to attain the direction matrix 𝐀 and the delay matrix Φ, and then estimate angle and frequency. The covariance matrix of the received signal can be reconstructed via 𝐑_𝑌=𝐘𝐘^𝐻. Using eigenvalue decomposition of 𝐑_𝑌, we can get the signal subspace 𝐄_𝑠. In the free-noise case, 𝐄_𝑠 can be denoted as 𝐄_𝑠=⎡⎢⎢⎣𝐀𝐀Φ⋮𝐀Φ_𝑃−1⎤⎥⎥⎦𝐓,(10) where 𝐓 is a 𝐾×𝐾full-rank matrix.

3.1. Frequency Estimation

According to (10), we define 𝐄₁ and 𝐄₂𝐄₁=⎡⎢⎢⎣𝐀𝐀Φ⋮𝐀Φ_𝑃−2𝐄⎤⎥⎥⎦𝐓,₂=⎡⎢⎢⎢⎣𝐀𝐀𝐀Φ⋮ΦΦ²_𝑃−1⎤⎥⎥⎥⎦𝐓.(11) According to (11), 𝐄₂=⎡⎢⎢⎢⎣𝐀𝐀𝐀Φ⋮ΦΦ²_𝑃−1⎤⎥⎥⎥⎦𝐓=⎡⎢⎢⎣𝐀𝐀Φ⋮𝐀Φ_𝑃−2⎤⎥⎥⎦𝐓𝐓₋₁Φ𝐓=𝐄₁𝐓⁻¹Φ𝐓.(12)

Let Ψ=𝐓⁻¹Φ𝐓, so Ψ=𝐄₁₊𝐄₂. We use eigenvalue decomposition of Ψ to get the estimate of Φ. Also we obtain an estimation of the matrix 𝐓⁻¹ via the eigenvectors of Φ. In the no-noise case, Φ𝐓=⁻¹=𝚷_𝑇𝐓⁻¹𝚷(13)Φ𝚷,(14) where 𝚷 is a permutation matrix, and 𝚷⁻¹=𝚷^𝑇.

Because Ψ has the same eigenvalues as Φ, we can get 𝛽_𝑘, 𝑘=1,2,…,𝐾 from the eigenvalues 𝜆_𝑘, 𝑘=1,2,…,𝐾 of matrix Ψ. And then estimate frequency 𝑓_𝑘, 𝑘=1,2,…,𝐾𝑓_𝑘=12𝜋𝜏angle𝜆^𝑘.(15)

3.2. Angle Estimation

There exists a transformation matrix 𝐇 corresponding to the finite number of row interchanged operations for subspace 𝐄_𝑠 such that 𝐄_𝑐=𝐇𝐄_𝑠𝐓⁻¹=𝐇𝐄_𝑠𝐓⁻¹𝚷=𝐅𝚷,(16) where 𝐅=𝐇𝐄_𝑠𝐓⁻¹=𝐇⎡⎢⎢⎣𝐀𝐀Φ⋮𝐀Φ_𝑃−1⎤⎥⎥⎦=⎡⎢⎢⎣𝐆Φ𝐆⋮𝐆Φ_𝑀−1⎤⎥⎥⎦,(17) where 𝐆=⎡⎢⎢⎢⎣𝑒11⋯1^−𝑗𝛽1𝑒^−𝑗𝛽2⋯𝑒^{−𝑗𝛽𝐾}𝑒⋮⋮⋱⋮_{−𝑗(𝑃−1)𝛽1}𝑒^{−𝑗(𝑃−1)𝛽2}⋯𝑒^{−𝑗(𝑃−1)𝛽𝐾}Φ=diag𝑒_−𝑗𝛼1⎤⎥⎥⎥⎦,,𝑒^−𝑗𝛼2,…,𝑒^{−𝑗𝛼𝐾}∈𝐂^𝐾×𝐾,(18) where 𝛼_𝑘=2𝜋𝑑𝑓_𝑘sin𝜃_𝑘/𝑐,𝑘=1,2,…,𝐾.(19) According to (16) and (17), take the first (𝑀−1)𝑃 rows of 𝐄_𝑐, we define matrix 𝐄_𝑐1 as follows: 𝐄_𝑐1=⎡⎢⎢⎣𝐆Φ𝐆⋮𝐆Φ_𝑀−2⎤⎥⎥⎦𝚷.(20) Taking the last (𝑀−1)𝑃 rows of 𝐄_𝑐, we define matrix 𝐄_𝑐2𝐄_𝑐2=⎡⎢⎢⎢⎣𝐆𝐆Φ𝐆⋮ΦΦ²_𝑀−1⎤⎥⎥⎥⎦𝚷.(21) We define 𝐂=𝐄^+𝑐1𝐄_𝑐2.(22) According to (20) and (21), in free-noise case, 𝐂 can be denoted as 𝐂=𝚷^𝑇Φ𝚷.(23) Then, the DOA estimation is obtained via 𝜃̂_𝑘=arcsin𝑐2𝜋𝑓_𝑘𝑑angle𝜉^𝑘,𝑘=1,2,…,𝐾,(24) where 𝜉_𝑘 is the kth diagonal element of matrix 𝐂.

The following summarizes the major steps of this proposed algorithm.(1)Compute the covariance matrix from the received signal via 𝐑_𝑌=𝐘𝐘^𝐻.(2)Get the signal subspace 𝐄_𝑠 by using EVD of 𝐑_𝑌. Construct the matrices 𝐄₁ and 𝐄₂ from 𝐄_𝑠 via (11) and (12).(3)Employ EVD on 𝐄₁₊𝐄₂ to get 𝐓⁻¹ and Φ. Estimate frequency via (15).(4)Compute 𝐄_𝑐 by (16) and construct matrices 𝐄_𝑐1 and 𝐄_𝑐2 via (20) and (21).(5)Compute the matrix 𝐂=𝐄^+𝑐1𝐄_𝑐2 and estimate DOA by (24).

Remark 1. The proposed algorithm can obtain automatically parameter estimation. From (14) and (23), DOA and frequency estimations suffer from the same permutation ambiguity, so we can get automatically paired DOA and frequency.

3.3. Complexity Analysis

Note that M, K, N and P are the number of the antennas, the sources, the snapshots, and delays, respectively. Computational complexity of Wang’s ESPRIT algorithm and this proposed algorithm is a function of M, K, N, and P.

In contrast to Wang’s ESPRIT algorithm, the proposed algorithm has a comparative computational load. For the proposed algorithm, the covariance matrix estimation costs 𝑂(𝑀²𝑃²𝑁); eigen decomposition of 𝐑_𝑌 needs 𝑂(𝑀³𝑃³); the computation of 𝐄₁₊𝐄₂ requires 𝑂(2𝐾²𝑀(𝑃−1)+2𝐾³); eigen decomposition of 𝐄₁₊𝐄₂ needs 𝑂(𝐾³); compute the matrix 𝐂=𝐄^+𝑐1𝐄_𝑐2 needs 𝑂(2𝐾²(𝑀−1)𝑃+2𝐾³). The major computational complexity of this algorithm is 𝑂+𝑀²2𝑃𝐾²₂𝑁+𝑀³𝑃³+2𝐾²𝑀(𝑃−1)+2𝐾³+𝐾³(𝑀−1)𝑃+2𝐾³,(25) while Wang’s ESPRIT algorithm requires 𝑂+𝑀²2𝑃𝐾²₂𝑁+𝑀³𝑃³+2𝐾²𝑀(𝑃−1)+2𝐾³+𝐾³(𝑀−1)+2𝐾³+3𝐾³.(26) According to [19], we derive the CRB for the frequency estimation and angle estimation as CRB=𝜎²2𝑁Re𝐃^𝐻𝚷^⟂𝐀𝐃⊙𝐏_𝑠𝑇,⁻¹(27) where 𝐃=[(𝜕_𝐚1/𝜕_𝑓1), (𝜕_𝐚2/𝜕_𝑓2)⋯(𝜕_𝐚𝐾/𝜕_𝑓𝐾), (𝜕_𝐚1/𝜕_𝜃1), (𝜕_𝐚2/𝜕_𝜃2)×⋯(𝜕_𝐚𝐾/𝜕_𝜃𝐾)],   with 𝐚_𝑘 being the kth column of 𝐀𝐏_𝑠=1𝑁^𝑁_𝑡=1Π_⟂𝐀𝐬_𝑡𝐬^𝐻𝑡,=𝐈_𝑃𝑀−𝐀𝐀^𝐻𝐀⁻¹𝐀^𝐻.(28)

3.4. Discussion

The performance of this proposed algorithm with nonuniform linear array will be discussed in this section. The distance between the mth antenna and the left reference element is 𝑑_𝑚(𝑚=0,…,𝑀−1), where 𝑑₀=0.

There is an array of 𝑀 sensors on which 𝐾 incident waves impinge (𝑀>𝐾). The direction matrix 𝐀^ is 𝐀^=⎡⎢⎢⎢⎣𝑒11⋯1^{−𝑗𝑑1𝛾1}𝑒^{−𝑗𝑑1𝛾2}⋯𝑒^{−𝑗𝑑1𝛾𝐾}𝑒⋮⋮⋱⋮_{−𝑗𝑑𝑀−1𝛾1}𝑒^{−𝑗𝑑𝑀−1𝛾2}⋯𝑒^{−𝑗𝑑𝑀−1𝛾𝐾}⎤⎥⎥⎥⎦,(29) where 𝛾_𝑘=2𝜋𝑓_𝑘sin(𝜃_𝑘)/𝑐, 𝑘=1,2,…,𝐾.

Replacing matrix 𝐀with 𝐀^ in (3)–(12), we can get the frequency estimation by (15).

For a nonuniform linear array, (20) and (21) can be rewritten as follows: 𝐄^𝑐1=⎡⎢⎢⎣𝐆𝐆Φ⋮𝐆Φ¹_𝑀−2⎤⎥⎥⎦𝚷=⎡⎢⎢⎢⎣𝐄𝐄^{𝑐11𝑐12}𝐄⋮_{𝑐1(𝑀−1)}𝐄_𝑐2⎤⎥⎥⎥⎦,=⎡⎢⎢⎣𝐆Φ¹𝐆Φ₂𝐆Φ⋮_𝑀−1⎤⎥⎥⎦𝚷=⎡⎢⎢⎢⎣𝐄𝐄^{𝑐21𝑐22}𝐄⋮_{𝑐2(𝑀−1)}⎤⎥⎥⎥⎦,(30) where Φ_𝑗=diag𝑒^{−𝑗𝑑𝑗𝛾1},𝑒𝑗=^{−𝑗𝑑𝑗𝛾2}1,,…2,,…𝑒,𝑀^{−𝑗𝑑𝑗𝛾𝐾}−1∈,𝐂^𝐾×𝐾,(31)𝐄^𝑐1𝑗 and 𝐄^𝑐2𝑗 represent the (𝑗−1)𝑃+1 to 𝑗𝑃th rows of matrix 𝐄^𝑐1 and 𝐄^𝑐2, respectively.

We define 𝐂_𝑗=𝐄^+𝑐1𝑗𝐄_𝑐2𝑗,𝑗=1,2,…,𝑀−1.(32) Then, the DOA estimation can be obtained via 𝜃̂_𝑘=1𝑀−1^𝑀−1_𝑗=1arcsin𝑐2𝜋𝑓_𝑘𝑑_𝑗−𝑑_𝑗−1angle𝜉^𝑘𝑗𝑘=1,2,…,𝐾,,(33) where 𝜉_𝑘𝑗 is the kth diagonal element of matrix 𝐂_𝑗.

3.5. The Advantages of the Proposed Algorithm

The proposed algorithm has the following advantages.(1)The proposed algorithm has a better angle estimation performance than Wang’s ESPRIT algorithm.(2)The proposed algorithm can obtain automatically paired parameter estimation, while Wang’s ESPRIT algorithm requires additional pairing.(3)The proposed algorithm has a comparative computational complexity in contrast to Wang’s ESPRIT algorithm.(4)This proposed algorithm also suit for nonuniform linear arrays.

4. Simulation Results

We present Monte Carlo simulations that are used to assess the angle and frequency estimation performance of this algorithm. The number of Monte Carlo trials is 1000. Note that 𝑀 is the number of antennas; 𝑃 is the number of the delays; 𝑁 is the number of snapshots; 𝐾 is the number of the sources.

Define RMSE=1𝐾∗^𝐾_𝑘=1⎷10100¹⁰⁰⁰_𝑚=1𝑎_𝑚𝑘−𝑎_0𝑘²,(34) where 𝑎_𝑚𝑘 is the kth estimated angle/frequency, and 𝑎_0𝑘 is the kth perfect angle/frequency.

Simulation 1
The performance of Wang’s ESPRIT algorithm and this proposed algorithm is investigated. 𝑀=12, 𝐾=3, 𝑃=3, and 𝑁=400 in this simulation. Their DOAs are 10^∘, 20^∘, and 30^∘, and their carrier frequencies are 500kHz, 700kHz, and 900kHz. Figures 1 and 2 show the performance of Wang’s ESPRIT algorithm and this proposed algorithm with SNR=0dB. From Figures 1 and 2, we find the proposed algorithm can obtain automatically paired parameter estimation, while Wang’s ESPRIT algorithm cannot, so it works well.

Figure 2 

Angle-frequency scatter of proposed method, SNR = 0 dB.

Simulation 2
We compare this proposed algorithm with Wang’s ESPRIT algorithm and CRB. From Figures 3 and 4, we find that this proposed algorithm has better angle estimation performance than Wang’s ESPRIT algorithm and has the same frequency estimation accuracy. Figure 5 to Figure 7 give more angle estimation performance comparison results between these two methods at different M, K, P, and 𝑁. Since the frequency estimation performance of the proposed is same with that of Wang’s ESPRIT, it is not necessary to plot it from Figures 5, 6, and 7.

Figure 3 

Angle estimation performance comparison at M = 12, K = 3, P = 3, and 𝑁=100.

Figure 4 

Frequency estimation performance comparison at M = 12, K = 3, P = 3, and 𝑁=100.

Figure 5 

Angle estimation performance comparison at M = 12, K = 4, P = 3, and 𝑁=400.

Figure 6 

Angle estimation performance comparison at M = 16, K = 3, P = 3, and 𝑁=400.

Figure 7 

Angle estimation performance comparison at M = 12, K = 3, P = 4, and 𝑁=400.

Simulation 3
This proposed algorithm performance under different snapshots 𝑁 is investigated in this simulation. 𝑀=12, 𝐾=3, and 𝑃=3are used in this simulation. Figure 8 shows the angle-frequency estimation performance under different 𝑁. We find that the angle-frequency estimation performance of this algorithm is improved with 𝑁 increasing.

(a)   Angle estimation

(b) Frequency estimation

Figure 8 

Angle-frequency estimation with different snapshot N.

Simulation 4
The performance of this algorithm under different source number 𝐾 is investigated in the simulation. 𝑀=12, 𝑃=3, and 𝑁=400 are used in this simulation. The source number 𝐾 is set to 2, 3, and 4. This proposed algorithm has different performance under different source numbers, as shown in Figure 9. From Figure 9, we find that angle and frequency estimation performance of this algorithm degrades with the increasing of the source number 𝐾.

(a)   Angle estimation

(b) Frequency estimation

Figure 9 

Angle-frequency estimation with different sources.

Simulation 5
The performance of this algorithm under different antenna number 𝑀 is investigated in the simulation. 𝐾=3, 𝑃=3, and 𝑁=400 are used in this simulation. The antenna number 𝑀 is set to 8, 12, and 16. This proposed algorithm has different performance under different antenna numbers, as shown in Figure 10. From Figure 10, we find that angle and frequency estimation performance of this algorithm is improved with 𝑀 increasing.

(a)   Angle estimation

(b) Frequency estimation

Figure 10 

Angle-frequency estimation with different antennas.

Simulation 6
The performance of this algorithm under different delay number 𝑃 is investigated in the simulation. 𝑀=12, 𝐾=3, and 𝑁=400 are used in this simulation. The delay number 𝑃 is set to 2, 3, 4, and 5. This proposed algorithm has different performance under different delay numbers, as shown in Figure 11. From Figure 11, we find that angle and frequency estimation performance of this algorithm is improved with 𝑃 increasing.

(a)   Angle estimation

(b) Frequency estimation

Figure 11 

Angle-frequency estimation with different delay number.

Simulation 7
The performance of this proposed algorithm with nonuniform linear arrays is investigated. 𝑀=8, 𝐾=3, 𝑃=3, and 𝑁=400 are used in this simulation. Their DOAs are 10^∘, 20^∘, and 30^∘, and their carrier frequencies are 500kHz, 700kHz, and 900kHz. Figure 12, shows this proposed algorithm with nonuniform linear arrays with SNR=0dB. From Figure 12, we find the proposed algorithm can also work well in the case of nonuniform linear arrays.

Figure 12 

Angle-frequency scatter for nonuniform linear arrays, SNR = 10 dB.

5. Conclusion

This paper has presented an improved joint angle-frequency estimation method, which has better angle estimation performance than Wang’s ESPRIT algorithm and has the same frequency estimation accuracy. The computational complexity of this proposed algorithm is comparative in contrast to Wang’s ESPRIT algorithm. Since the DOA and frequency estimations suffer from the same permutation ambiguity, this novel method can obtain automatically paired DOA and frequency. This advantage is more obvious when the input SNR is below 0 dB. Furthermore, the proposed algorithm can also work well in the case of nonuniform linear arrays.

Acknowledgments

This paper is supported by China NSF Grant (61201208), Aeronautical Science Foundation of China (2009ZC52036), Nanjing University of Aeronautics and Astronautics Research Funding (NN2012068), and the Fundamental Research Funds for the Central Universities (NZ2012010, kfjj120115, kfjj20110215).