- •Contents
- •Preface
- •1.1 Smart Antenna Architecture
- •1.2 Overview of This Book
- •1.3 Notations
- •2.1 Single Transmit Antenna
- •2.1.1 Directivity and Gain
- •2.1.2 Radiation Pattern
- •2.1.3 Equivalent Resonant Circuits and Bandwidth
- •2.2 Single Receive Antenna
- •2.3 Antenna Array
- •2.4 Conclusion
- •Reference
- •3.1 Introduction
- •3.2 Data Model
- •3.2.1 Uniform Linear Array (ULA)
- •3.3 Centro-Symmetric Sensor Arrays
- •3.3.1 Uniform Linear Array
- •3.3.2 Uniform Rectangular Array (URA)
- •3.3.3 Covariance Matrices
- •3.4 Beamforming Techniques
- •3.4.1 Conventional Beamformer
- •3.4.2 Capon’s Beamformer
- •3.4.3 Linear Prediction
- •3.5 Maximum Likelihood Techniques
- •3.6 Subspace-Based Techniques
- •3.6.1 Concept of Subspaces
- •3.6.2 MUSIC
- •3.6.3 Minimum Norm
- •3.6.4 ESPRIT
- •3.7 Conclusion
- •References
- •4.1 Introduction
- •4.2 Preprocessing Schemes
- •4.2.2 Spatial Smoothing
- •4.3 Model Order Estimators
- •4.3.1 Classical Technique
- •4.3.2 Minimum Descriptive Length Criterion
- •4.3.3 Akaike Information Theoretic Criterion
- •4.4 Conclusion
- •References
- •5.1 Introduction
- •5.2 Basic Principle
- •5.2.1 Signal and Data Model
- •5.2.2 Signal Subspace Estimation
- •5.2.3 Estimation of the Subspace Rotating Operator
- •5.3 Standard ESPRIT
- •5.3.1 Signal Subspace Estimation
- •5.3.2 Solution of Invariance Equation
- •5.3.3 Spatial Frequency and DOA Estimation
- •5.4 Real-Valued Transformation
- •5.5 Unitary ESPRIT in Element Space
- •5.6 Beamspace Transformation
- •5.6.1 DFT Beamspace Invariance Structure
- •5.6.2 DFT Beamspace in a Reduced Dimension
- •5.7 Unitary ESPRIT in DFT Beamspace
- •5.8 Conclusion
- •References
- •6.1 Introduction
- •6.2 Performance Analysis
- •6.2.1 Standard ESPRIT
- •6.3 Comparative Analysis
- •6.4 Discussions
- •6.5 Conclusion
- •References
- •7.1 Summary
- •7.2 Advanced Topics on DOA Estimations
- •References
- •Appendix
- •A.1 Kronecker Product
- •A.2 Special Vectors and Matrix Notations
- •A.3 FLOPS
- •List of Abbreviations
- •About the Authors
- •Index
Overview of Basic DOA Estimation Algorithms |
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3.7 Conclusion
This chapter discussed the signal and data model that we are using for this book. Centro-symmetric arrays that are demanded by many DOA estimation algorithms are reviewed. A few well-known DOA estimation techniques of simple to complicated algorithms are described; they are all based on the power spectrum that is inversely proportional to covariance of received signals. Preliminary computations were performed to illustrate the performance of these DOA estimation schemes.
References
[1]Haardt, M., Efficient One-, Two-, and Multidimensional High-Resolution Array Signal Processing, New York: Shaker Verlag, 1997.
[2]Krim, H., and M. Viberg, “Two Decades of Array Signal Processing Research,” IEEE Signal Processing Magazine, Vol. 13, No. 4, July 1996, pp. 67–94.
[3]Balanis, C. A., Antenna Theory: Analysis and Design, 3rd ed., New York: Wiley, 2005.
[4]Xu, G., R. H. Roy, and T. Kailath, “Detection of Number of Sources Via Exploitation of Centro-Symmetry Property,” IEEE Trans. on Signal Processing, Vol. 42, No. 1, January 1994, pp. 102–112.
[5]Lee, A., “Centrohermitian and Skew-Centrohermitian Matrices,” Linear Algebra and Its Applications, Vol. 29, 1980, pp. 205–210.
[6]Haardt, M., et al., “2D Unitary ESPRIT for Efficient 2D Parameter Estimation,”
Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, Vol. 3, May 9–12, 1995, pp. 2096–2099.
[7]Liberti, Jr., J. C., and T. S. Rappaport, Smart Antennas for Wireless Communications: IS-95 and Third Generation CDMA Applications, Upper Saddle River, NJ: Prentice-Hall, 1999.
[8]Litva, J., and T. K. Y. Lo, Digital Beamforming in Wireless Communications, Norwood, MA: Artech House, 1996.
[9]Capon, J., “High-Resolution Frequency-Wavenumber Spectrum Analysis,” Proc. IEEE, Vol. 57, No. 8, August 1969, pp. 2408–2418.
[10]Makhoul, J., “Linear Prediction: A Tutorial Review,” Proc. IEEE, Vol. 63, No. 4, April 1975, pp. 561–580.
[11]Ziskind, I., and M. Wax, “Maximum Likelihood Localization of Multiple Sources by Alternating Projection,” IEEE Trans. Acoust., Speech, Signal Processing, Vol. 36, No. 10, October 1998, pp. 1553–1560.
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Introduction to Direction-of-Arrival Estimation |
[12]vander Veen, A. J., E. F. Deprettere, and A. L. Swindlehurst, “Subspace Based Signal Analysis Using Singular Value Decomposition,” Proc. IEEE, Vol. 81, No. 9, September 1993, pp. 1277–1308.
[13]Schmidt, R. O., “Multiple Emitter Location and Signal Parameter Estimation,” IEEE Trans. on Antennas Propagation, Vol. 34, No. 3, March 1986, pp. 276–280.
[14]Wax, M., “Detection and estimation of superimposed signals,” Ph.D. dissertation, Stanford University, Stanford, CA, 1985.
[15]Ronhovde, A., et al., “High-Resolution Beamforming for Multibeam Echo Sounders Using Raw EM3000 Data,” http://www.ifi.uio.no/sverre/papers/Oceans.pdf.
[16]Roy, R., and T. Kailath, “ESPRIT-Estimation of Signal Parameters Via Rotational Invariance Techniques,” IEEE Trans. on Acoust., Speech, Signal Processing, Vol. 37, No. 7, July 1989, pp. 984–995.
4
Preprocessing Schemes and Model
Order Estimation
4.1 Introduction
The DOA estimation algorithms described in the previous chapter assume noncoherent impinging signals. However, if the impinging signals are highly correlated or coherent to each other, most of the algorithms will fail to give reliable DOA estimates because the data covariance matrix Rxx received by an array becomes singular or ill-conditioned. This situation is not uncommon in multipath scenarios. In a rich multipath environment, the signals impinging on the array are often delayed and scaled versions of each other and hence are highly correlated or coherent. To circumvent this problem, the data covariance matrix is processed before “feeding” them to the DOA estimation algorithms. These techniques are called preprocessing techniques and they play a vital, if not a mandatory, role in estimating a DOA. Two well-known and established preprocessing schemes, namely spatial smoothing and forward backward averaging, will be discussed in this chapter.
In addition to the signal coherence issue, the number of signals impinging on the array is assumed to be known so far. In reality, however, this number is not known and has to be estimated from the data received. The DOA estimation algorithms depend completely on the assumption of knowledge of number of impinging signals. Hence, estimation of this number plays another key role in a DOA estimation process and invites
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