# Joint Angle and Frequency Estimation Using Multiple-Delay Output Based on ESPRIT

- Wang Xudong
^{1}Email author

**2010**:358659

https://doi.org/10.1155/2010/358659

© Wang Xudong. 2010

**Received: **29 August 2010

**Accepted: **21 December 2010

**Published: **28 December 2010

## Abstract

This paper presents a novel ESPRIT algorithm-based joint angle and frequency estimation using multiple-delay output (MDJAFE). The algorithm can estimate the joint angles and frequencies, since the use of multiple output makes the estimation accuracy greatly improved when compared with a conventional algorithm. The useful behavior of the proposed algorithm is verified by simulations.

## Keywords

## 1. Introduction

Antenna array has been used in many fields such as radar, sonar, electron reconnaissance and seismic data processing. The direction-of-arrival- (DOA-) estimation of signals impinging on an array of sensors is a fundamental problem in array processing [1–5]. Angle estimation and frequency estimation [6, 7] are two key problems in the signal processing field. The problem of joint DOA and frequency estimation arises in the applications of radar, wireless communications and electron reconnaissance. For example, these parameters can be applied to locate the radars and to locate pilot tones in electron reconnaissance systems [8]. 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 [9] are often applicable but might be computationally prohibitive. Some ESPRIT-based joint angle and frequency estimation methods have been proposed in [10–14]. Zoltowski and Mathew [10] discuss this problem in the context of radar applications. Pro-ESPRIT is proposed to estimate angle and frequency. Haardt and Nossek [11] 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. Multi resolution ESPRIT is used for joint angle frequency estimation in [12]. ESPRIT method is used for frequency and angle estimation under uniform circular array in [13, 14]. References [15, 16] proposed the trilinear decomposition method for joint angle and frequency estimation method. The other joint angle and frequency estimation method is proposed in [17–24].

This paper uses multiple-delay output, so as to achieve the purpose of improving estimation accuracy. This algorithm has the improved performance compared with conventional method. The proposed algorithm is applicable to uniform linear array.

Note 1.

We denote by the matrix transpose, and by the matrix conjugate transpose. The notation refers to the Moore-Penrose inverse (pseudoinverse).

## 2. The Data Model

*i*th source has a carrier frequency . The signal received at the th antenna is

## 3. Joint Angle and Frequency Estimation

### 3.1. Frequency Estimation

### 3.2. Angle Estimation

In contrast to ESPRIT algorithm [13], this algorithm has a high computational load, which is usually dominated by formation of the covariance matrix, matrix inversion and calculation of EVD. The major computational complexity of this algorithm is , while ESPRIT requires , where , , , and are the number of antennas, delays, snapshots, and sources.

## 4. Simulation Results

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

Define , where is the estimated angle/frequency, and is the perfect angle/ frequency.

Simulation 1

Simulation 2

Simulation 3

Simulation 4

Simulation 5

Simulation 6

## 5. Conclusion

This work presents a new ESPRIT algorithm-based joint angle and frequency estimation using multiple-delay output. The advantage of this proposed algorithm using the multiple-delay output over the conventional algorithm is that the estimation accuracy has been greatly improved.

## Declarations

### Acknowledgments

This paper is supported by China NSF Grant (60801052), Aeronautical Science Foundation of China (2009ZC52036), Ph.D. Programs Foundation of China's Ministry of Education (200802871056) and Nanjing University of Aeronautics and Astronautics Research Funding (NS2010109, NS2010114). The authors thank Professor Zhang Xiaofei for his help.

## Authors’ Affiliations

## References

- Widrow B, Stearns SD:
*Adaptive Signal Processing*. Prentice-Hall, Englewood Cliffs, NJ, USA; 1985.MATHGoogle Scholar - Zhang X, Xu D: Improved coherent DOA estimation algorithm for uniform linear arrays.
*International Journal of Electronics*2009, 96(2):213-222. 10.1080/00207210802526810View ArticleGoogle Scholar - Zhang X, Gao X, Xu D: Multi-invariance ESPRIT-based blind DOA estimation for MC-CDMA with an antenna array.
*IEEE Transactions on Vehicular Technology*2009, 58(8):4686-4690.View ArticleGoogle Scholar - Zhang X, Gao X, Chen W: Improved blind 2D-direction of arrival estimation with L-shaped array using shift invariance property.
*Journal of Electromagnetic Waves and Applications*2009, 23(5):593-606. 10.1163/156939309788019859View ArticleGoogle Scholar - Zhang X, Yu J, Feng G, Xu D: Blind direction of arrival estimation of coherent sources using multi-invariance property.
*Progress In Electromagnetics Research*2008, 88: 181-195.View ArticleGoogle Scholar - Rosnes E, Vahlin A: Frequency estimation of a single complex sinusoid using a generalized Kay estimator.
*IEEE Transactions on Communications*2006, 54(3):407-415.View ArticleGoogle Scholar - Zhang XF, Xu DZ: Novel joint time delay and frequency estimation method.
*IET Radar, Sonar and Navigation*2009, 3(2):186-194. 10.1049/iet-rsn:20080032View ArticleGoogle Scholar - Tsui J:
*Digital Techniques for Wideband Receivers*. 2nd edition. Artech House, Norwood, Mass, USA; 2001.Google Scholar - Djeddou M, Belouchrani A, Aouada S: Maximum likelihood angle-frequency estimation in partially known correlated noise for low-elevation targets.
*IEEE Transactions on Signal Processing*2005, 53(8):3057-3064.MathSciNetView ArticleGoogle Scholar - Zoltowski MD, Mathews CP: Real-time frequency and 2-D angle estimation with sub-Nyquistspatio-temporal sampling.
*IEEE Transactions on Signal Processing*1994, 42(10):2781-2794. 10.1109/78.324743View ArticleGoogle Scholar - Haardt M, Nossek JA: 3-D unitary ESPRIT for joint 2-D angle and carrier estimation.
*Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '97), April 1997, Munich, Germany*255-258.Google Scholar - Lemma AN, van der Veen AJ, Deprettere EF: Joint angle-frequency estimation using multi-resolution ESPRIT.
*Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '98), May 1998, Seattle, Wash, USA*4: 1957-1960.Google Scholar - Lemma AN, van der Veen AJ, Deprettere EDF: Analysis of joint angle-frequency estimation using ESPRIT.
*IEEE Transactions on Signal Processing*2003, 51(5):1264-1283. 10.1109/TSP.2003.810306MathSciNetView ArticleGoogle Scholar - Wang S, Zhou X: Direction-of-arrival and frequency estimation in array signal processing.
*Journal of Shanghai Jiaotong University*1999, 33(1):40-42.Google Scholar - Xiaofei Z, Gaopeng F, Jun Y, Dazhuan X: Angle-frequency estimation using trilinear decomposition of the oversampled output.
*Wireless Personal Communications*2009, 51(2):365-373. 10.1007/s11277-008-9652-5View ArticleGoogle Scholar - Zhang X, Wang D, Xu D: Novel blind joint direction of arrival and frequency estimation for uniform linear array.
*Progress in Electromagnetics Research*2008, 86: 199-215.MathSciNetView ArticleGoogle Scholar - Chen H, Wang Y, Wu Z: Frequency and 2-D angle estimation based on uniform circular array.
*Proceedings of IEEE International Symposium on Phased Array Systems and Technology, 2003*547-552.Google Scholar - Fu T, Jin S, Gao X: Joint 2-D angle and frequency estimation for uniform circular array.
*Proceedings of International Conference on Communications, Circuits and Systems (ICCCAS '06), June 2006*1: 230-233.Google Scholar - Jia W, Yao M, Song J: Joint frequency, two dimensional arrival angles estimations via marked signal subspace.
*Proceedings of the 8th International Conference on Signal Processing (ICSP '06), November 2006*1: 16-20.Google Scholar - Lin CH, Fang WH, Wu KH, Lin JD: Fast algorithm for joint azimuth and elevation angles, and frequency estimation via hierarchical space-time decomposition.
*Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '07), April 2007*2: 1061-1064.Google Scholar - Hasan MA: DOA and frequency estimation using fast subspace algorithms.
*Signal Processing*1999, 77(1):49-62. 10.1016/S0165-1684(99)00022-5MathSciNetView ArticleMATHGoogle Scholar - Jiacai H, Yaowu S, Jianwu T: Joint estimation of DOA, frequency, and polarization based on cumulants and UCA.
*Journal of Systems Engineering and Electronics*2007, 18(4):704-709. 10.1016/S1004-4132(08)60007-9View ArticleMATHGoogle Scholar - Liang J, Zeng X, Ji B, Zhang J, Zhao F: A computationally efficient algorithm for joint range-DOA-frequency estimation of near-field sources.
*Digital Signal Processing*2009, 19(4):596-611. 10.1016/j.dsp.2008.06.006View ArticleGoogle Scholar - Amin M, Zhang Y: Spatial time-frequency distributions and DOA estimation.
*Classical and Modern Direction-of-Arrival Estimation*2009, 185-217.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.