3858 IEEE TRANSACTIONS ONVEHICULAR TECHNOLOGY, VOL. 57, NO. 6, NOVEMBER 2008 Fig. 4. Exact and approximate LCR curves for the EGC and MRC techniques using four i.n.i.d. Nakagami-m fading channels. been investigated, and in all cases, our proposed approximations outperform those of [6]. Figs. 2–4 portray the exact and approximate output envelopes and LCRs, respectively, of multibranch EGC and MRC receivers operating with four Nakagami-m fading channels. Note that, as far as the proposed approximations are concerned, the EGC scenario is always more favorable than that of the MRC, yielding remarkable results. REFERENCES [1] M. Nakagami, “The m-distribution—A general formula of intensity dis- tribution of rapid fading,” in Statistical Methods in Radio Wave Prop. Elmsford, NY: Pergamon, 1960. [2] N. C. Beaulieu and A. A. Abu-Dayya, “Analysis of equal gain diversity on Nakagami fading channels,” IEEE Trans. Commun., vol. 39, no. 2, pp. 225–234, Feb. 1991. [3] N. C. Beaulieu, “An infinite series for the computation of the com- plementary probability distribution function of a sum of independent random variables and its application to the sum of Rayleigh random variables,” IEEE Trans. Commun., vol. 38, no. 9, pp. 1463–1474, Sep. 1990. [4] G. K. Karagiannidis, “A closed-form solution for the distribution of the sum of Nakagami-m random phase vectors,” IEEE Commun. Lett., vol. 10, no. 12, pp. 828–830, Dec. 2006. [5] P. Dharmawansa, N. Rajatheva, and K. Ahmed, “On the distribution of the sum of Nakagami-m random variables,” IEEE Trans. Commun., vol. 55, no. 7, pp. 1407–1416, Jul. 2007. [6] J. C. S. Santos Filho and M. D. Yacoub, “Nakagami-m approxima- tion to the sum of M non-identical independent Nakagami-m variates,” Electron. Lett., vol. 40, no. 15, pp. 951–952, Jul. 2004. [7] T. Piboongungon, V. A. Aalo, and C.-D. Iskander, “Average error rate of linear diversity reception schemes over generalized Gamma fading channels,” in Proc. IEEE Southeastcon, Fort Lauderdale, FL, Apr. 2005, pp. 265–270. [8] N. C. Sagias, G. K. Karagiannidis, P. T. Mathiopoulus, and T. A. Tsiftsis, “On the performance analysis of equal-gain diversity receivers over gen- eralized Gamma fading channels,” IEEE Trans. Wireless Commun., vol. 5, no. 10, pp. 2967–2975, Oct. 2006. [9] V. A. Aalo, T. Piboongungon, and C.-D. Iskander, “Bit-error rate of bi- nary digital modulation schemes in generalized Gamma fading channels,” IEEE Commun. Lett., vol. 9, no. 2, pp. 139–141, Feb. 2005. [10] G. K. Karagiannidis, T. A. Tsiftsis, and N. C. Sagias, “A closed- form upper-bound for the distribution of the weighted sum of Rayleigh variates,” IEEE Commun. Lett., vol. 9, no. 7, pp. 589–591, Jul. 2005. [11] A. J. Coulson, A. G. Williamson, and R. G. Vaughan, “Improved fad- ing distribution for mobile radio,” Proc. Inst. Electr. Eng.—Commun., vol. 145, no. 3, pp. 197–202, Jun. 1998. [12] G. Fraidenraich, M. D. Yacoub, and J. C. S. Santos Filho, “Second-order statistics of maximal-ratio and equal-gain combining in Weibull fading,” IEEE Commun. Lett., vol. 9, no. 6, pp. 499–501, Jun. 2005. [13] G. Fraidenraich, J. C. S. Santos Filho, and M. D. Yacoub, “Second-order statistics of maximal-ratio and equal-gain combining in Hoyt fading,” IEEE Commun. Lett., vol. 9, no. 1, pp. 19–21, Jan. 2005. [14] M. D. Yacoub, C. R. C. M. da Silva, and J. E. V. Bautista, “Second- order statistics for diversity-combining techniques in Nakagami-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 1464–1470, Nov. 2001. [15] D. B. da Costa et al., “Second-order statistics of equal-gain and maximal- ratio combining for the α–μ (generalized gamma) fading distribution,” in Proc. IEEE Int. Symp. Spread Spectr. Tech. Appl., Manaus, Brazil, Aug. 2006, pp. 342–346. [16] S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J., vol. 23, pp. 282–332, Jul. 1944. [17] M. D. Yacoub, “The α–μ distribution: A physical fading model for the Stacy distribution,” IEEE Trans. Veh. Technol., vol. 56, no. 1, pp. 27–34, Jan. 2007. [18] M. D. Yacoub, “The κ–μ distribution and the η–μ distribution,” IEEE Antennas Propag. Mag., vol. 49, no. 1, pp. 68–81, Feb. 2007. [19] D. G. Brennan, “Linear diversity combining techniques,” in Proc. IRE, Jun. 1959, vol. 47, pp. 1075–1102. Chaos-Based Radars for Automotive Applications: Theoretical Issues and Numerical Simulation Ennio Gambi, Franco Chiaraluce, Member, IEEE, and Susanna Spinsante, Member, IEEE Abstract—This paper focuses on a possible application of chaotic sig- nals as an alternative to more conventional spreading schemes in direct- sequence spread spectrum (DS-SS) automotive radars, the latter being a key component for future road safety systems. Due to their very good cor- relation properties, chaotic sequences are potentially able to outperform previous options, like Gold codes, with regard to the detection probability and the number of available sequences. Numerical examples are given, in some typical scenarios and under severe operation conditions, due to the presence of interfering radars. Index Terms—Chaotic signals, long-range radars (LRRs), road safety, spread spectrum radars. I. I NTRODUCTION Improving road safety has become one of the most important priori- ties for government policymakers all over the world. Many research programs have been developed, in cooperation with academic and industrial partners, both in Europe (see e-Safety [1]) and elsewhere (see [2]). In the United States, safety is an important issue for the Wireless Access in Vehicular Environments system [3], whose stan- dardization is in progress. In these programs, the most important ob- jective is crash prevention, which can be reached by designing systems that allow accurate knowledge of the location, speed, acceleration, or deceleration of nearby vehicles. A key step is the development Manuscript received May 21, 2007; revised January 17, 2008, March 13, 2008, and March 17, 2008. First published March 31, 2008; current version published November 12, 2008. The review of this paper was coordinated by Prof. T. Lok. The authors are with the Dipartimento di Elettronica, Intelligenza Artificiale e Telecomunicazioni, Università Politecnica delle Marche, 60131 Ancona, Italy (e-mail: e.gambi@univpm.it; f.chiaraluce@univpm.it; s.spinsante@univpm.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2008.921632 0018-9545/$25.00 © 2008 IEEE