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International Journal of Antennas and Propagation
Volume 2012, Article ID 309269, 9 pages
http://dx.doi.org/10.1155/2012/309269
Research Article

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

College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 8 June 2012; Revised 5 July 2012; Accepted 18 July 2012

Academic Editor: Stefano Selleri

Copyright © 2012 Wang Xudong 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

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 [14]. The direction-of-arrival (DOA) and frequency estimation of signals impinging on the ULAs are two fundamental problems in array processing [510]. 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 [1317]. 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.

309269.fig.001
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 𝑦𝑚(𝑡)=𝐾𝑘=1𝑒𝑗2𝜋𝑚𝑑𝑓𝑘sin(𝜃𝑘)/𝑐𝑠𝑘(𝑡)+𝑛𝑚(𝑡),𝑚=0,1,2,,𝑀1,(1) where 𝑠𝑘(𝑡) is the narrow-band signal of the kth source. 𝑐 is velocity of light. 𝜃𝑘,𝑓𝑘 are the DOA and the frequency of the kth signal, respectively. 𝑛𝑚(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 𝐘𝟎=𝑦0(𝑛),𝑦1(𝑛),𝑦2(𝑛),,𝑦𝑀1(𝑛)𝑇,𝑛=1,2,3,,𝑁.(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 𝐘𝟎=𝐀𝐒+𝐍𝟎,(3) where 𝑠𝐒=1,𝑠2,,𝑠𝐾𝑇𝐂𝐾×𝑁,𝐍𝟎=𝐧0,𝐧1,𝐧𝑀1𝑇𝐂𝑀×𝑁,𝑒𝐀=111𝑗𝛼1𝑒𝑗𝛼2𝑒𝑗𝛼𝐾𝑒𝑗(𝑀1)𝛼1𝑒𝑗(𝑀1)𝛼2𝑒𝑗(𝑀1)𝛼𝐾,(4) where 𝛼𝑘=2𝜋𝑑𝑓𝑘sin(𝜃𝑘)/𝑐, 𝑘=1,2,…,𝐾.

In order to joint estimate DOA and frequency, we add 𝑃1 delayed outputs for the received signal of array antenna. We suppose that0<(𝑃1)𝜏<1/max(𝑓𝑖).

The delayed signal for (1) with delay 𝜏 is 𝑦𝑚(𝑡𝜏)=𝐾𝑘=1𝑒𝑗2𝜋𝑚𝑑𝑓𝑘sin(𝜃𝑘)/𝑐𝑠𝑘(𝑡𝜏)+𝑛𝑚=(𝑡)𝐾𝑘=1𝑒𝑗2𝜋𝑚𝑑𝑓𝑘sin(𝜃𝑘)/𝑐𝑠𝑘(𝑡)𝑒𝑗2𝜋𝑓𝑘𝜏+𝑛𝑚(𝑡).(5)

The delayed signal for (5) with 𝜏 can be denoted as 𝐘1=𝐀Φ𝐒+𝐍1,(6) where 𝑒Φ=diag𝑗𝛽1,𝑒𝑗𝛽2,,𝑒𝑗𝛽𝐾,(7) where 𝛽𝑘=2𝜋𝑓𝑘𝜏, 𝑘=1,2,…,𝐾.

The delayed signal for (5) with 𝑝𝜏 can be denoted as 𝐘𝑝=𝐀Φ𝑝𝐒+𝐍𝑝,𝑝=0,1,,𝑃1.(8)

According to (3), (6), and (8), we define 𝐘𝐘=𝟎𝐘1𝐘𝑃1=𝐀𝐀Φ𝐀Φ𝑃1𝐍𝐒+𝟎𝐍1𝐍𝑃1.(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 𝐑𝑌=𝐘𝐘𝐻. Using eigenvalue decomposition of 𝐑𝑌, we can get the signal subspace 𝐄𝑠. In the free-noise case, 𝐄𝑠 can be denoted as 𝐄𝑠=𝐀𝐀Φ𝐀Φ𝑃1𝐓,(10) where 𝐓 is a 𝐾×𝐾full-rank matrix.

3.1. Frequency Estimation

According to (10), we define 𝐄1 and 𝐄2𝐄1=𝐀𝐀Φ𝐀Φ𝑃2𝐄𝐓,2=𝐀Φ𝐀Φ2𝐀Φ𝑃1𝐓.(11) According to (11), 𝐄2=𝐀Φ𝐀Φ2𝐀Φ𝑃1𝐀𝐓=𝐀Φ𝐀Φ𝑃2𝐓𝐓1Φ𝐓=𝐄1𝐓1Φ𝐓.(12)

Let Ψ=𝐓1Φ𝐓, so Ψ=𝐄1+𝐄2. We use eigenvalue decomposition of Ψ to get the estimate of Φ. Also we obtain an estimation of the matrix 𝐓1 via the eigenvectors of Φ. In the no-noise case, 𝐓1=𝐓1𝚷(13)Φ=𝚷𝑇Φ𝚷,(14) where 𝚷 is a permutation matrix, and 𝚷1=𝚷𝑇.

Because Ψ has the same eigenvalues as Φ, we can get 𝛽𝑘, 𝑘=1,2,…,𝐾 from the eigenvalues 𝜆𝑘, 𝑘=1,2,…,𝐾 of matrix Ψ. And then estimate frequency 𝑓𝑘, 𝑘=1,2,…,𝐾𝑓𝑘=1𝜆2𝜋𝜏angle𝑘.(15)

3.2. Angle Estimation

There exists a transformation matrix 𝐇 corresponding to the finite number of row interchanged operations for subspace 𝐄𝑠 such that 𝐄𝑐=𝐇𝐄𝑠𝐓1=𝐇𝐄𝑠𝐓1𝚷=𝐅𝚷,(16) where 𝐅=𝐇𝐄𝑠𝐓1𝐀=𝐇𝐀Φ𝐀Φ𝑃1=𝐆𝐆Φ𝐆Φ𝑀1,(17) where 𝑒𝐆=111𝑗𝛽1𝑒𝑗𝛽2𝑒𝑗𝛽𝐾𝑒𝑗(𝑃1)𝛽1𝑒𝑗(𝑃1)𝛽2𝑒𝑗(𝑃1)𝛽𝐾,𝑒Φ=diag𝑗𝛼1,𝑒𝑗𝛼2,,𝑒𝑗𝛼𝐾𝐂𝐾×𝐾,(18) where 𝛼𝑘=2𝜋𝑑𝑓𝑘𝜃sin𝑘/𝑐,𝑘=1,2,,𝐾.(19) According to (16) and (17), take the first (𝑀1)𝑃 rows of 𝐄𝑐, we define matrix 𝐄𝑐1 as follows: 𝐄𝑐1=𝐆𝐆Φ𝐆Φ𝑀2𝚷.(20) Taking the last (𝑀1)𝑃 rows of 𝐄𝑐, we define matrix 𝐄𝑐2𝐄𝑐2=𝐆Φ𝐆Φ2𝐆Φ𝑀1𝚷.(21) We define 𝐂=𝐄+𝑐1𝐄𝑐2.(22) According to (20) and (21), in free-noise case, 𝐂 can be denoted as 𝐂=𝚷𝑇Φ𝚷.(23) Then, the DOA estimation is obtained via ̂𝜃𝑘𝑐=arcsin2𝜋𝑓𝑘𝑑𝜉angle𝑘,𝑘=1,2,,𝐾,(24) where 𝜉𝑘 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 𝐑𝑌=𝐘𝐘𝐻.(2)Get the signal subspace 𝐄𝑠 by using EVD of 𝐑𝑌. Construct the matrices 𝐄1 and 𝐄2 from 𝐄𝑠 via (11) and (12).(3)Employ EVD on 𝐄1+𝐄2 to get 𝐓1 and Φ. Estimate frequency via (15).(4)Compute 𝐄𝑐 by (16) and construct matrices 𝐄𝑐1 and 𝐄𝑐2 via (20) and (21).(5)Compute the matrix 𝐂=𝐄+𝑐1𝐄𝑐2 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 𝐑𝑌 needs 𝑂(𝑀3𝑃3); the computation of 𝐄1+𝐄2 requires 𝑂(2𝐾2𝑀(𝑃1)+2𝐾3); eigen decomposition of 𝐄1+𝐄2 needs 𝑂(𝐾3); compute the matrix 𝐂=𝐄+𝑐1𝐄𝑐2 needs 𝑂(2𝐾2(𝑀1)𝑃+2𝐾3). The major computational complexity of this algorithm is 𝑂𝑀2𝑃2𝑁+𝑀3𝑃3+2𝐾2𝑀(𝑃1)+2𝐾3+𝐾3+2𝐾2(𝑀1)𝑃+2𝐾3,(25) while Wang’s ESPRIT algorithm requires 𝑂𝑀2𝑃2𝑁+𝑀3𝑃3+2𝐾2𝑀(𝑃1)+2𝐾3+𝐾3+2𝐾2(𝑀1)+2𝐾3+3𝐾3.(26) According to [19], we derive the CRB for the frequency estimation and angle estimation as 𝜎CRB=2𝐃2𝑁Re𝐻𝚷𝐀𝐏𝐃𝑠𝑇,1(27) where 𝐃=[(𝜕𝐚1/𝜕𝑓1), (𝜕𝐚2/𝜕𝑓2)(𝜕𝐚𝐾/𝜕𝑓𝐾), (𝜕𝐚1/𝜕𝜃1), (𝜕𝐚2/𝜕𝜃2)×(𝜕𝐚𝐾/𝜕𝜃𝐾)],   with 𝐚𝑘 being the kth column of 𝐀𝐏𝑠=1𝑁𝑁𝑡=1𝐬𝑡𝐬𝐻𝑡,Π𝐀=𝐈𝑃𝑀𝐀𝐀𝐻𝐀1𝐀𝐻.(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 𝑑𝑚(𝑚=0,,𝑀1), where 𝑑0=0.

There is an array of 𝑀 sensors on which 𝐾 incident waves impinge (𝑀>𝐾). The direction matrix 𝐀 is 𝐀=𝑒111𝑗𝑑1𝛾1𝑒𝑗𝑑1𝛾2𝑒𝑗𝑑1𝛾𝐾𝑒𝑗𝑑𝑀1𝛾1𝑒𝑗𝑑𝑀1𝛾2𝑒𝑗𝑑𝑀1𝛾𝐾,(29) where 𝛾𝑘=2𝜋𝑓𝑘sin(𝜃𝑘)/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: 𝐄𝑐1=𝐆𝐆Φ1𝐆Φ𝑀2𝐄𝚷=𝑐11𝐄𝑐12𝐄𝑐1(𝑀1),𝐄𝑐2=𝐆Φ1𝐆Φ2𝐆Φ𝑀1𝐄𝚷=𝑐21𝐄𝑐22𝐄𝑐2(𝑀1),(30) where Φ𝑗𝑒=diag𝑗𝑑𝑗𝛾1,𝑒𝑗𝑑𝑗𝛾2,,𝑒𝑗𝑑𝑗𝛾𝐾𝐂𝐾×𝐾,𝑗=1,2,,𝑀1,(31)𝐄𝑐1𝑗 and 𝐄𝑐2𝑗 represent the (𝑗1)𝑃+1 to 𝑗𝑃th rows of matrix 𝐄𝑐1 and 𝐄𝑐2, respectively.

We define 𝐂𝑗=𝐄+𝑐1𝑗𝐄𝑐2𝑗,𝑗=1,2,,𝑀1.(32) Then, the DOA estimation can be obtained via ̂𝜃𝑘=1𝑀1𝑀1𝑗=1𝑐arcsin2𝜋𝑓𝑘𝑑𝑗𝑑𝑗1𝜉angle𝑘𝑗,𝑘=1,2,,𝐾,(33) where 𝜉𝑘𝑗 is the kth diagonal element of matrix 𝐂𝑗.

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 1RMSE=𝐾𝐾𝑘=1110001000𝑚=1𝑎𝑚𝑘𝑎0𝑘2,(34) where 𝑎𝑚𝑘 is the kth estimated angle/frequency, and 𝑎0𝑘 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.

309269.fig.002
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.

309269.fig.003
Figure 3: Angle estimation performance comparison at M = 12, K = 3, P = 3, and 𝑁=100.
309269.fig.004
Figure 4: Frequency estimation performance comparison at M = 12, K = 3, P = 3, and 𝑁=100.
309269.fig.005
Figure 5: Angle estimation performance comparison at M = 12, K = 4, P = 3, and 𝑁=400.
309269.fig.006
Figure 6: Angle estimation performance comparison at M = 16, K = 3, P = 3, and 𝑁=400.
309269.fig.007
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.

fig8
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.

fig9
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.

fig10
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.

fig11
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.

309269.fig.0012
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).

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