Jensen-Cotes upper and lower bounds on the Gaussian Q-function and related functions

IEEE Transactions on Communications

Volumn 57, Issue 11, 2009, Pages 3328-3338

  De Abreu, Giuseppe Thadeu Freitas ^a  

^a UNIVERSITY OF OULU   (Finland)

Author keywords

Communication; Functional analysis; Statistics

Indexed keywords

BASIC IDEA; COOPERATIVE RELAYING; DECODE-AND-FORWARD; DIFFERENTIAL PHASE-SHIFT KEYING; ELEMENTARY TRANSCENDENTAL FUNCTIONS; FUNCTIONAL ARGUMENTS; GAUSSIAN Q-FUNCTION; JENSEN INEQUALITY; LOWER AND UPPER BOUNDS; LOWER BOUNDS; MARCUM Q-FUNCTION; NEWTON-COTES; NUMERICAL INTEGRATIONS; Q-FUNCTIONS; QUADRATURE RULES; RELATED FUNCTIONS; UPPER AND LOWER BOUNDS;

COMMUNICATION SYSTEMS; DIFFERENTIATION (CALCULUS); FUNCTIONAL ANALYSIS; GAUSSIAN DISTRIBUTION; RELAY CONTROL SYSTEMS;

FUNCTION EVALUATION;

EID: 78649359468     PISSN: 00906778     EISSN: None     Source Type: Journal    
DOI: 10.1109/TCOMM.2009.11.080479     Document Type: Article

References (34)

1. M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels: A Unified Approach to Performance Analysis. New York: Wiley, 2000.
2. M. K. Simon, Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers and Scientists, 2nd ed. Springer, Nov. 2006.
3. P. O. Börjesson and C. E. Sundberg, "Simple approximations of the the error function Q (x) for communications applications", IEEE Trans. Commun., vol. 27, pp. 639-643, Mar. 1979.
4. N. C. Beaulieu, "A simple series for personal computer computation of the error function", IEEE Trans. Commun., vol. 37, no. 9, pp. 989-991, Sep. 1989.
5. C. Tellambura and A. Annamalai, "Efficient computation of erfc (x) for large arguments", IEEE Trans. Commun., vol. 48, pp. 529-532, Apr. 2000.
6. G. K. Karagiannidis and A. S. Lioumpas, "An improved approximation for the Gaussian Q-function", IEEE Commun. Lett., vol. 11, no. 8, pp. 644-646, Aug. 2007.
7. J. S. Dyer and S. A. Dyer, "Corrections to, and comments on, "An improved approximation for the Gaussian Q-function"", IEEE Commun. Lett., vol. 12, no. 4, p. 231, Apr. 2008.
8. J. W. Craig, "A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations", in Proc. IEEE Military Commun. Conf. (Milcom'91), vol. 2, McLean, USA, Nov. 1991, pp. 571-575.
9. M. K. Simon, "Single integral representations of certain integer powers of the Gaussian Q-function and their application", IEEE Commun. Lett., vol. 6, no. 12, pp. 532-534, Dec. 2002.
10. -, "A simpler form of the Craig representation for the twodimensional joint Gaussian Q-function", IEEE Commun. Lett., vol. 6, no. 2, pp. 49-51, Feb. 2002.
11. S. Yousefi and B. Holmes, "A simple form for the two-dimensional Q-function suitable for performance evaluation of communication systems", in Proc. IEEE 61st Veh. Technol. Conf. (VTC'05 Spring), vol. 2, Stockholm, Sweden, May 2005, pp. 1091-1095.
12. S. Park and U. J. Choi, "A generic Craig form for the two-dimensional Gaussian Q-function", ETRI J., vol. 29, no. 4, pp. 516-517, Aug. 2007.
13. P. Y. Kam and R. Li, "A new geometric view of the first-order Marcum Q-function and some simple tight erfc-bounds", in Proc. IEEE 63rd Veh. Technol. Conf. (VTC'06 Spring), vol. 5, Melbourne, Australia, May 2006, pp. 2553-2557. (Pubitemid 46509275)
14. R. Li and P. Y. Kam, "Computing and bounding the generalized Marcum Q-function via a geometric approach", in Proc. IEEE International Symp. Inform. Theory (ISIT'06), Seattle, USA, July 2006.
15. M. K. Simon, "The Nuttall Q-function - its relation to the Marcum Q-function and its application in digital communication performance evaluation", IEEE Trans. Commun., vol. 50, no. 11, pp. 1712-1715, Nov. 2002.
16. S. Mihos, V. M. Kapinas, and G. K. Karagiannidis, "Lower and upper bounds for the generalized Marcum and Nuttall Q-functions", in Proc. IEEE International Symp. Wireless Pervasive Computing (ISWPC'08), Santorini, Greece, May 2008.
17. V. M. Kapinas, S. K. Mihos, and G. K. Karagiannidis, "On the monotonicity of the generalized Marcum and Nuttall Q-functions", IEEE Trans. Inf. Theory, to be published. [Online]. Available: http://arxiv.org/abs/ 0712.4103v3.
18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed. Dover Publications, 1965.
19. M. Chiani, D. Dardari, and M. K. Simon, "New exponential bounds and approximations for the computation of error probability in fading channels", IEEE Trans. Commun., vol. 2, no. 4, pp. 840-845, July 2003.
20. B. J. Malesevic, "Application of λ-method on Shafer-Fink's inequality", Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat. 8, 1997.
21. D. S. Mitrinović, Analytic Inequalities. Berlin: Springer-Verlag, 1970.
22. Á. Baricz, "Mills' ratio: monotonicity patterns and functional inequalities", J. Math. Anal. Appl., vol. 340, no. 2, pp. 1362-1370, 2008.
23. W. Gautschi, "Some elementary inequalities relating to the gamma function and incomplete gamma function", J. Math Phys., vol. 38, no. 77-81, 1959.
24. H. Alzer, "On some inequalities for the incomplete gamma function", Math. Comp., vol. 66, no. 218, pp. 771-778, 1997.
25. Á. Elbert and A. Laforgia, "An inequality for the product of two integrals relating to the incomplete gamma function", J. Inequal. Appl., vol. 5, no. 1, pp. 39-51, 2000.
26. M. B. Ruskai and E. Werner, "Study of a class of regularizations of 1/|x| using gaussian integrals", SIAM J. Math. Anal. 32, vol. 32, no. 2, pp. 435-463, 2000.
27. S. Kullback and R. A. Leibler, "On information and sufficiency", Annals Mathematical Statistics, vol. 22, no. 1, pp. 79-86" Mar. 1951.
28. M. K. Simon and D. Divsalar, "Some new twists to problems involving the Gaussian probability integral", IEEE Trans. Commun., vol. 6, no. 2, pp. 200-210, Feb. 1998.
29. 2, f and t distributions with applications", J. Roy. Statist. Soc., vol. B 35, no. 3, pp. 480-492, 1973.
30. Y. Sun and Á. Baricz, "Inequalities for the generalized Marcum qfunction", Appl. Math. Comput., vol. 203, no. 1, pp. 134-141, 2008.
31. th Annual Globecom Conf. (GLOBECOM'07), Washington, DC, USA, Nov. 2007, pp. 1754-1758.
32. D. Chen and N. Laneman, "Modulation and demondulation for cooperative diversity in wireless systems", IEEE Trans. Wireless Commun., vol. 5, no. 7, pp. 1785-1794, July 2006.
33. Q. Zhao and H. Li, "Differential modulation for cooperative wireless systems", IEEE Trans. Signal Process., vol. 55, no. 5, pp. 2273-2283, May 2007. (Pubitemid 46691605)
34. -, "Performance of cooperative relay with binary modulation in Nakagami-m fading channels", IEEE Trans. Veh. Technol., vol. 57, no. 5, pp. 3310-3315, Sep. 2008.