Complex elliptically symmetric random variables - Generation, characterization, and circularity tests

IEEE Transactions on Signal Processing

Volumn 59, Issue 1, 2011, Pages 58-69

  Ollila, Esa ^a   Eriksson, Jan ^a   Koivunen, Visa ^a  

^a AALTO UNIVERSITY   (Finland)

Author keywords

Asymptotic distribution; circularity quotient; complex kurtosis; complex random variable (r.va.); elliptical distributions

Indexed keywords

ASYMPTOTIC DISTRIBUTIONS; CIRCULARITY QUOTIENT; COMPLEX KURTOSIS; COMPLEX RANDOM VARIABLES; ELLIPTICAL DISTRIBUTIONS;

ASYMPTOTIC ANALYSIS; HIGHER ORDER STATISTICS; SIGNAL PROCESSING; STATISTICAL TESTS;

RANDOM VARIABLES;

EID: 79955767863     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2010.2083655     Document Type: Article

References (46)

1. R. A. Wooding, "The multivariate distribution of complex normal vari- ables," Biometrika, vol. 43, pp. 212-215, 1956.
2. A. van den Bos, "The multivariate complex normal distribution-A generalization," IEEE Trans. Inf. Theory, vol. 41, no. 2, pp. 537-539, 1995.
3. B. Picinbono, "Second order complex random vectors and normal distributions," IEEE Trans. Signal Process., vol. 44, no. 10, pp. 2637-2640, 1996.
4. E. Ollila and V. Koivunen, "Generalized complex elliptical distribu- tions," in Proc. 3rd IEEE Sens. Array Multichannel Signal Process. Workshop(SAM'04),Barcelona,Spain,Jun.18-21,2004,pp.460-464.
5. P. R. Krishnaiah and J. Lin, "Complex elliptically symmetric distribu- tions," Comm. Statist.-Theory Meth., vol. 15, pp. 3693-3718, 1986.
6. E. Ollila and V. Koivunen, "Robust antenna array processing using M-estimators of pseudo-covariance," in Proc. 14th IEEE Int. Symp. Pers., Indoor and Mobile Radio Commun. (PIMRC'03), Beijing, China, Sep. 7-10, 2003, pp. 2659-2663.
7. M. Novey, T. Adali, and A. Roy, "Circularity and Gaussianity detec- tionusingthecomplexgeneralizedGaussiandistribution," IEEESignal Process. Lett., vol. 16, no. 11, pp. 993-996, 2009.
8. M. Novey, T. Adali, and A. Roy, "A complex generalized Gaussian distribution-characterization, generation, and estimation," IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1427-1433, 2010.
9. P. Tsakalides and C. L. Nikias, "The robust covariation based MUSIC (roc-MUSIC) algorithm for bearing estimation in impulsive noise envi- ronments," IEEE Trans. Signal Process., vol. 44, no. 7, pp. 1623-1633, 1995.
10. P. T. Tsakalides, "Array signal processing with -stable distributions," Ph.D. dissertation, Univ. Southern Calif., Los Angeles, 1995.
11. B. Picinbono, "On circularity," IEEE Trans. Signal Process., vol. 42, no. 12, pp. 3473-3482, 1994.
12. J. Eriksson, E. Ollila, and V. Koivunen, "Essential statistics and tools for complex random variables," IEEE Trans. Signal Process., vol. 58, no. 10, pp. 5400-5408, 2010.
13. J. Eriksson, E. Ollila, and V. Koivunen, "Statistics for complex random variables revisited," in Proc. IEEE Int. Conf. Acoust., Speech Signal Process. (ICASSP'09), Taipei, Taiwan, 2009, pp. 3565-3568.
14. B. Picinbono andP. Chevalier,"Widely linear estimation with complex data,"IEEETrans.SignalProcess.,vol.43,no.8,pp.2030-2033,1995.
15. F.D. NeeserandJ.L. Massey,"Propercomplex randomprocesseswith applications to information theory," IEEE Trans. Inf. Theory, vol. 39, no. 4, pp. 1293-1302, 1993.
16. P. J. Schreier and L. L. Scharf, "Second-order analysis of improper complex random vectors and processes," IEEE Trans. Signal Process., vol. 51, no. 3, pp. 714-725, 2003.
17. M. Haardt and F. Römer, "Enhancements of unitary ESPRIT for non- circular sources," in Proc. Int. Conf. Acoust., Speech Signal Process. (ICASSP'04), Montreal, Canada, May 2004.
18. P. Chargé, Y. Wang, and J. Saillard, "A non-circular sources direction nding methods using polynomial rooting," Signal Process., vol. 81, pp. 1765-1770, 2001.
19. H. Abeida and J.-P. Delmas, "Statistical performance of music-like algorithms in resolving noncircular sources," IEEE Trans. Signal Process., pp. 4317-4329, 2008.
20. J. Eriksson and V. Koivunen, "Complex-valued ICA using second order statistics," in Proc. IEEE Workshop on Mach. Learn. Signal Process. (MLSP'04), Sao Luis, Brazil, 2004.
21. J. Eriksson and V. Koivunen, "Complex random vectors and ICA models: Identiability, uniqueness and seperability," IEEE Trans. Inf. Theory, vol. 52, no. 3, pp. 1017-1029, 2006.
22. H. Li and T. Adali, "A class of complex ICA algorithms based on the kurtosis cost function," IEEE Trans. Neural Netw., vol. 19, no. 3, pp. 408-420, 2008.
23. S. C. Douglas, "Fixed-point algorithms for the blind separation of arbi- trarycomplex-valuednon-Gaussiansignalmixtures,"EURASIPJ.Adv. Signal Process., vol. 2007, no. 1, p. 83-83, 2007.
24. E. Ollila and V. Koivunen, "Complex ICA using generalized uncorre- lating transform," Signal Process., vol. 89, no. 5, pp. 365-377, 2009.
25. A. K. Jagannatham and B. D. Rao, "Cramér-Rao lower bound for con- strained complex parameters," IEEE Signal Process. Lett., vol. 11, no. 11, pp. 875-878, 2004.
26. E. Ollila, V. Koivunen, and J. Eriksson, "On the Cramér-Rao bound for the constrained and unconstrained complex parameters," in Proc. IEEE Sens. Array Multichannel Signal Process. Workshop (SAM'08), Darmstadt, Germany, Jul. 21-23, 2008, pp. 414-418.
27. P. J. Schreier, L. L. Scharf, and A. Hanssen, "A generalized likelihood ratio test for impropriety of complex signals," IEEE Signal Process. Lett., vol. 13, no. 7, pp. 433-436, 2006.
28. E. Ollila, "On the circularity of a complex random variable," IEEE Signal Process. Lett., pp. 841-844, 2008.
29. A. T. Walden and P. Rubin-Delanchy, "On testing for impropriety of complex-valued Gaussian vectors," IEEE Trans. Signal Process., vol. 57, no. 3, pp. 835-842, 2009.
30. E. Ollila and V. Koivunen, "Adjusting the generalized likelihood ratio test of circularity robust to non-normality," in Proc. 19th IEEE Work- shop on Signal Process. Adv. Wireless Commun. (SPAWC'09), Perugia, Italy, Jun. 21-24, 2009, pp. 558-562.
31. D. H. Brandwood, "A complex gradient operator and its applications in adaptive array theory," Inst. Elect. Eng. Proc. F and H, vol. 1, pp. 11-16, 1983.
32. H. Li and T. Adal, "Complex-valued adaptive signal processing using nonlinear functions," EURASIP J. Adv. Signal Process., 2008, Article ID 765615, 9 pp.
33. K. Kreutz-Delgado, "The complex gradient operator and the CR-cal- culus," Lecture Notes Supplement [online] 2007.
34. M. Bilodeau and D. Brenner, Theory of Multivariate Statistics. New York: Springer-Verlag, 1999.
35. J.-P. Delmas and H. Abeida, "On the degree of second-order non-cir- cularity of complex random variables," in Proc. Int. Conf. Acoust., Speech, Signal Process. (ICASSP'08), 2008.
36. J.-P. Delmas and H. Abeida, "Asymptotic distribution of circularity coefcients estimate of complex random variables," Signal Process., vol. 89, no. 12, pp. 2670-2675, 2009.
37. K.-T. Fang, S. Kotz, and K. W. Ng, Symmetric Multivariate and Re- lated Distributions. London: Chapman and Hall, 1990.
38. P. Amblard, M. Gaeta, and L. Lacoume, "Statistics for complex vari- ables and signals-Part I: Variables," Signal Process., vol. 53, no. 1, pp. 1-13, 1996.
39. J. M. Chambers, C. L. Mallows, and B. W. Stuck, "A method for sim- ulating stable random variables," J. Amer. Stat. Assoc., vol. 71, pp. 340-344, 1976.
40. J. H. McCulloch, Stable Random Number Generator-stabrnd.m (Matlab) [Online]. Available: http://www.econ.ohio-state.edu/jhm/jhm.html
41. T.-H. Liu and J. M. Mendel, "A subspace-based direction nding al- gorithm using fractional lower order statistics," IEEE Trans. Signal Process., vol. 49, pp. 1605-1613, 2001.
42. S. Cambanis, S. Huang, and G. Simons, "On the theory of elliptically contoured distributions," J. Mult. Anal., vol. 11, no. 3, pp. 368-385, 1981.
43. E. Ollila, CES Matlab Software [Online]. Available: http://cc.oulu./ ~esollila/CES
44. C. R. Rao, Linear Statistical Inference and its Applications. New York: Wiley, 1973.
45. G. Frahm, "Asymptotic distributions of robust shape matrices and scales," J. Mult. Anal., vol. 100, pp. 1329-1337, 2009.
46. J. R. Magnus and H. Neudecker, Matrix Differential Calculus With Ap- plications in Statistics and Econometrics, 2nd ed. Chichester, U.K.: Wiley, 1999, 422 pp.