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Similar to the MDL criterion, this criterion is used after the spatial smoothing is applied to the array.
Depending upon the properties of the covariance matrix and the preprocessing scheme applied, the penalty function is given by
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Also, the maximum number of signals that can be estimated is given in a way similar to MDL. For a ULA with M elements and N snapshots, the maximum number of the signals determined by this criterion is given by
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For a URA with Mx × My elements and N snapshots, the maximum number of the signals is given by
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The AIC criterion is simulated in two experiments with a 10-ele- ment ULA under SNR of 0 dB. Results are averaged over 50 trials with 250 snapshots per trial. In the first experiment, one signal was impinging on the array. In the second experiment, three signals were impinging on the array. Figure 4.5 shows the results. As seen, the AIC function achieves minimum right at the numbers of the signals impinging. In other words, the AIC method successfully estimates the numbers of the signals impinging.
4.4 Conclusion
In this chapter, techniques to deal with correlated signals and to determine the number of signals are described with literature references provided for further detailed presentations of each technique. Simulations were run and results are presented to give a better understanding of the techniques. As observed, these techniques play a vital role in DOA
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Figure 4.5 Histogram of the AIC for computing the number of signals. (a) Two signals impinging and (b) three signals impinging.
estimation and thus form a mandatory requirement before any DOA algorithms are run. We assume hereafter that while discussing the DOA estimation algorithms, we have had the number of signal sources and have resolved the issue of the correlation among the signals, by applying the techniques described in this chapter. In the following chapters, we will focus on the one of the most popular subspace-based DOA estimation scheme, ESPRIT, with these assumptions.
References
[1]Pillai, S. U., and B. H. Kwon, “Forward/Backward Spatial Smoothing Techniques for Coherent Signal Identification,” IEEE Trans. on Acoustics, Speech and Signal Processing, Vol. 37, No. 1, January 1989, pp. 8–15.
[2]Evans, J. E., J. R. Johnson, and D. F. Sun, High Resolution Angular Spectrum Estimation Techniques for Terrain Scattering Analysis and Angle of Arrival Estimation in ATC Navigation and Surveillance System, M.I.T. Lincoln Lab., Lexington, MA.
[3]M. Haardt, Efficient One-, Two-, and Multidimensional High-Resolution Array Signal Processing, New York: Verlag, 1997.
[4]Bachl, R., “The Forward-Backward Averaging Technique Applied to TLS-ESPRIT Processing,” IEEE Trans. on Signal Processing, Vol. 43, No. 11, November 1995, pp. 2691–2699.
[5]Pillai, S. U., Array Signal Processing, New York: Springer-Verlag, 1989.
[6]Wang, H., and K. J. R. Liu, “2-D Spatial Smoothing for Multipath Coherent Signal Separation,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 34, No. 2, April 1998, pp. 391–405.
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[7]Chandna, R., and A. Mahadar, “2D Beamspace ESPRIT with Spatial Smoothing,”
IEEE Trans. on Acoust., Speech, Signal Processing, 1998.
[8]Liberti, Jr., J., and T. S. Rappaport, Smart Antennas for Wireless Communications: IS-95 and Third Generation CDMA Applications, Upper Saddle River, NJ: Prentice-Hall, 1999.
[9]Rissanen, J., “Modeling by the Shortest Data Description,” Automatica, Vol. 14, 1978, pp. 465–471.
[10]Akaike, H., “Information Theory and Extension of the Maximum Likelihood Principle,” Proc. of 2nd Intl. Symp. on Information Theory, 1973, pp. 267–281.
[11]Wax, M., and T. Kailath, “Detection of Signals by Information Theoretic Criterion,”
IEEE Trans. on Acoust., Speech, Signal Processing, Vol. ASSP-33, No. 2, April 1985, pp. 387–392.
5
DOA Estimations with ESPRIT
Algorithms
5.1 Introduction
Most of the algorithms discussed up to now depend on the precise knowledge of the array steering matrix A(θ). For every θi, the corresponding array response, a(θi), must be known. This is obtained by either direct calibration in the field, or by analytical means using information about the position and the response of each individual element of the array; this is normally an expensive and time-consuming task. Furthermore, errors in the calibration may seriously degrade the estimation accuracy. Also, the spectral-based DOA algorithms involve an exhaustive search through all possible angles or steering vectors to find the locations of the power spectral peaks and to estimate the DOA, which is computationally intensive.
ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) overcomes these problems with a dramatic reduction in computational and storage requirements by exploiting a property called the shift invariance of the array [1]. Unlike most DOA estimation methods such as MUSIC, ESPRIT does not require that the array manifold steering vectors be precisely known, so the array calibration requirements are not stringent. This chapter reviews the basics of algorithms belonging to the family of ESPRIT. Section 5.1 explains the fundamental principle of ESPRIT-based algorithms. Section 5.2 applies this principle to uniform linear arrays. Section 5.3 discusses the computation of signal
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