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180 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 54, NO. 1, JANUARY 2006 For higher values of the SNR, the bias of the estimator decreases, as expected. Fig. 2. RMSE of the DOA versus the number of antenna elements. VI. CONCLUSION In this paper, it has been shown that the computational complexity can be reduced for the MP method for DOA estimation. A unitary transformation applied to the MP method has been successfully formulated and utilized to convert the complex data matrix to a real matrix, hence reducing the computational complexity significantly. It is seen that when the SNR of the data is greater than 10 dB, then both the MP and the new UMP method can be used to model a given data set by a sum of complex exponentials and the UMP can be implemented on a DSP chip using only real arithmetic. The surprising part is that the real computations come at a cost, particularly for low values of SNR where the variance of the estimates due to the UMP method is larger than that for the MP method. ACKNOWLEDGMENT Grateful acknowledgment is made to the reviewers for suggesting ways to improve the readability of the paper. Fig. 3. Bias of the estimator versus SNR. pointed out in [9]. This causes some degradation for low SNR case. That explains the performance difference between the MP and the UMP methods for the low SNR case. After 8 dB SNR, the eigenvalues are all real. Since the results for are similar to we are giving the results for only . The optimum value for the pencil length, , is chosen to be 3 for efficient noise filtering which is explained in [7]. The root mean square error (RMSE) of the DOA versus the number of antenna elements for the UMP is shown in Fig. 2. As the number of antenna elements increase, the RMSE decreases as expected. This simulation is based on the value of an SNR of 20 dB. The bias for the estimate of the DOA has also been studied. The bias is computed from (45) where denotes the expected value. The bias of the estimator versus SNR is shown in Fig. 3. REFERENCES [1] J. C. Liberti and T. S. Rappaport, Smart Antennas for Wireless Communications. Upper Saddle River, NJ: Prentice Hall, 1999. [2] J. Capon, “Maximum likelihood spectral estimation,” Nonlinear Methods of Spectral Analysis, pp. 155–179, 1979. [3] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag., vol. 34, no. 3, pp. 276–280, Mar. 1986. [4] A. J. Barabell, “Improving the resolution performance of eigenstructurebased direction finding algorithm,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, 1983, pp. 336–339. [5] K. Takao and N. Jijuma, “An adaptive array utilizing an adaptive spatial averaging technique for multi-path environments,” IEEE Trans. Antennas Propag., vol. 35, no. 12, pp. 1389–1396, 1987. [6] Y. Hua and T. K. Sarkar, “Generalized pencil-of-function method for extracting poles of an EM system from its transient response,” IEEE Trans. Antennas Propag., vol. 37, no. 2, pp. 229–234, 1989. [7] T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” IEEE Antennas Propag. Mag., vol. 37, no. 1, pp. 48–54, 1994. [8] K. C. Huang and C. C. Yeh, “Unitary transformation method for angle of arrival estimation,” IEEE Trans. Signal Processing, vol. 39, no. 4, pp. 975–977, 1991. [9] M. Haardt and J. A. Nossek, “Unitary ESPRIT: how to obtain increased estimation accuracy with a reduced computational burden,” IEEE Trans. Signal Processing, vol. 43, no. 5, pp. 1232–1242, 1995. [10] A. Lee, “Centrohermitian and skew-centrohermitian matrices,” Linear Algebra and Its Applications, vol. 29, pp. 205–210, 1980. [11] L. Datta and D. M. Salvatore, “Some results on matrix symmetries and a pattern recognition application,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, no. 4, pp. 992–993, Aug. 1986. [12] R. Bachl, “The forward and backward averaging technique applied to TLS-ESPRIT processing,” IEEE Trans. Signal Processing, vol. 43, no. 11, pp. 2691–2699, Nov. 1995. [13] G. Xu, R. H. Roy, and T. Kailath, “Detection of number of sources via exploitation of centro-symmetric property,” IEEE Trans. Signal Processing, vol. 42, no. 1, pp. 102–111, Jan. 1994. [14] H. V. Trees, Optimum Array Processing, Part IV of Detection, Estimation, and Modulation Theory. New York: Wiley, 2002. [15] Y. Hua, “On Techniques for Estimating the Parameters of Exponentially Damped/Undamped Sinusoids in Noise,” PhD dissertation, Syracuse Univ., Syracuse, New York, Aug. 1988. [16] L. C. Godara, Smart Antennas. Boca Raton, FL: CRC Press, 2004.
YILMAZER et al.: UTILIZATION OF A UNITARY TRANSFORM FOR EFFICIENT COMPUTATION 181 Nuri Yilmazer was born in Silifke, Turkey. He received the B.S. degree from Cukurova University, Adana, Turkey, in 1996, and the M.Sc. degree from the University of Florida, Gainesville, in 2000. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering at Syracuse University, Syracuse, New York. His current research interests include digital signal processing and adaptive antenna problems. Jinhwan Koh was born in Taegu, Korea. He received the B.S. degree in electronics from Inha University, Korea, and the M.S. and Ph.D. degrees in electrical engineering from Syracuse University, Syracuse, New York, in 1997 and 1999, respectively. He was formerly with Goldstar Electron Semiconductor Company, Korea. He is now a Professor in the Department of Electronics and Electrical Engineering, Kyungpook National University, Taegu, Korea. His current research interests include digital signal processing related to adaptive antenna problems and data restoration. Tapan K. Sarkar (S’69–M’76–SM’81–F’92) received the B.Tech. degree from the Indian Institute of Technology, Kharagpur, India, in 1969, the M.Sc.E. degree from the University of New Brunswick, Fredericton, NB, Canada, in 1971, and the M.S. and Ph.D. degrees from Syracuse University, Syracuse, NY, in 1975. From 1975 to 1976, he was with the TACO Division of General Instruments Corporation. He was with the Rochester Institute of Technology, Rochester, NY, from 1976 to 1985. He was a Research Fellow at the Gordon McKay Laboratory, Harvard University, Cambridge, MA, from 1977 to 1978. He is now a Professor in the Department of Electrical Engineering and Computer Science, Syracuse University. He has authored or coauthored more than 250 journal articles, numerous conference papers, chapters in books, and several books including his most recent, Iterative and Self Adaptive Finite-Elements in Electromagnetic Modeling (Boston, MA: Artech House, 1998), Wavelet Applications in Electromagnetics and Signal Processing(Boston, MA: Artech House, 2002), and Smart Antennas (New York: Wiley, 2003). He is on the editorial board of J. of Electromagnetic Waves and Applications and Microwave and Optical Technology Letters. His current research interests deal with numerical solutions of operator equations arising in electromagnetics and signal processing with application to system design. Dr. Sarkar is a Registered Professional Engineer in the State of New York. He is a Member of Sigma Xi and the International Union of Radio Science (URSI) Commissions A and B. He received the College of Engineering Research Award in 1996 and the Chancellor’s Citation for Excellence in Research from Syracuse University in 1998. He obtained one of the “Best Solution” awards at the Rome Air Development Center (RADC) Spectral Estimation Workshop in May 1977, and received the Best Paper Award from the National Radar Conference in 1997 and the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY in 1979. He received the title Docteur Honoris Causa from Universite Blaise Pascal, Clermont Ferrand, France, and from the Politechnic University of Madrid, Madrid, Spain, in 1998 and 2004, respectively. He received the medal of the Friend of the City of Clermont Ferrand, France, in 2000. He was the Chairman of the Inter-commission Working Group of International URSI on Time Domain Metrology from 1990 to 1996. He was an Associate Editor for feature articles of the IEEE Antennas and Propagation Society Newsletter from 1986 to 1988. He was a Distinguished Lecturer for the Antennas and Propagation Society from 2000 to 2003. He is currently a Member of the IEEE Electromagnetics Award Board and an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He is the Vice President of the Applied Computational Electromagnetics Society (ACES) and the technical Chair for the combined IEEE 2005 Wireless Conference along with ACES to be held in Hawaii.
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