Mathematical Analysis and Improvement of the Maximum Spatial Eigenfilter for Direction of Arrival Estimation




Direction of arrival, Maximum likelihood estimation, Noise reduction, Spatial filtering


Maximum spatial eigenfiltering improves the accuracy of maximum likelihood direction-of-arrival estimators for closely-spaced signal sources but may interchangeably attenuate widely-spaced signal sources, producing a severe performance degradation. Although this behavior has been observed experimentally, it still lacks a mathematical explanation. In our previous work, we overcame these limitations using a differential spectrum-based spatial filter but this still caused a small degradation in the DOA estimate. In this paper, we develop a mathematical analysis of how the signal source separation and the Karhunen-Loève expansion affect the passbands of the maximum spatial eigenfilter. The farther the sources, the less significant is the maximum eigenvalue of the spatial correlation matrix and its corresponding eigenvector. Then, the magnitude response of the maximum spatial eigenfilter no longer approximates the spatial power spectrum and is not guaranteed to place multiple passbands around the signal sources. Consequently, we propose a spatial filter built from the eigenvectors of the entire signal subspace. This filter showed an overall runtime smaller than that of our previous work. It also provides a significant reduction in the threshold signal-to-noise ratio for closely-spaced signal sources and does not hamper the estimation for widely-spaced signal sources.


A. Lopes, I. S. Bonatti, P. L. D. Peres, C. A. Alves, “Improving the MODEX algorithm for direction estimation,” Signal Processing, vol. 83, no. 9, pp. 2047–2051, Sep. 2003.

R. Krummenauer, M. Cazarotto, A. Lopes, P. Larzabal, P. Forster, “Improving the threshold performance of maximum likelihood estimation of direction of arrival,” Signal Processing, vol. 90, no. 11, pp. 1582–1590, Nov. 2010.

A. B. Gershman, P. Stoica, “New MODE-based techniques for direction finding with an improved threshold performance,” Signal Processing, vol. 76, pp. 221–235, 1999.

P. Forster, G. Vezzosi, “Application of spheroidal sequences to array processing,” in IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ‘87), Dallas, USA, 1987, pp. 2268–2271.

S. Haykin, Adaptive Filter Theory, 4th ed., Englewood Cliffs, NJ, USA: Prentice-Hall, 2001.

R. P. Lemos, H. V. L. e Silva, E. L. Flôres, J. A. Kunzler, D. F. B. Beltrán, “Spatial filtering based on differential spectrum for improving ML DOA estimation performance,” IEEE Signal Processing Letters, vol. 23, no. 12, pp. 1811–1815, Aug. 2016.

Y. R. Ferreira, R. P. Lemos, “A new DOA estimation algorithm based on angle search through the difference between the principal singular values,” in Proceedings of the International Microwave and Optoelectronics Conference, Brasília, Brazil, 2005, pp. 283–286.

Y. R. Ferreira, R. P. Lemos, “A new DOA estimation algorithm based on differential spectrum,” in Proceedings of the Eighth International Symposium on Signal Processing and Its Applications, Sydney, Australia, 2005, pp. 303–307.

H. V. L. Silva, R. P. Lemos, Y. R. Ferreira, L. G. R. Guedes, “A branch-and-bound inspired technique to improve the computational efficiency of DOA estimation,” Signal Processing, vol. 93, no. 4, pp. 947–956, Apr. 2013.

S. M. Kay, Modern Spectral Estimation - Theory and Application, Englewood Cliffs, NJ, USA: Prentice Hall, 1988.

P. Stoica, A. Nehorai, “Performance study of conditional and unconditional direction-of-arrival estimation,” IEEE Transactions on Acoustic, Speech and Signal Processing, vol. 38, no. 10, pp. 1783–1795, Oct. 1990.

F. Li, R. J. Vaccaro, “Unified analysis for DOA estimation algorithms in array signal processing,” Signal Processing, vol. 25, pp. 147–169, 1991.

M. G. Bellanger, Adaptive Digital Filters, 2nd ed., New York, NY, USA: Marcel Dekker, 2001.

J. K. Thomas, L. L. Scharf, D. W. Tufts, “The probability of a subspace swap in the SVD,” IEEE Transactions on Signal Processing, vol. 43, no. 3, pp. 730–736, Mar. 1995.

M. Hawkes, A. Nehorai, P. Stoica, “Performance breakdown of subspace-based methods: Prediction and cure,” in Proceedings of 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ‘01), Salt Lake City, USA, 2001, pp. 4005–4008.

M. Shaghaghi, S. A. Vorobyov, “Subspace leakage analysis and improved DOA estimation with small sample size,” IEEE Transactions on Signal Processing, vol. 63, no. 12, pp. 3251–3265, Jun. 2015.

H. L. V. Trees, Optimum Array Processing. Part IV of Detection, Estimation and Modulation Theory, New York, NY, USA: John Wiley and Sons, 2002.




How to Cite

Pinto Lemos, R., Leao e Silva, H. V., Flôres, E. L., & Kunzler, J. A. (2021). Mathematical Analysis and Improvement of the Maximum Spatial Eigenfilter for Direction of Arrival Estimation. Journal of Microwaves, Optoelectronics and Electromagnetic Applications (JMOe), 20(1), 76–91.



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