IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005 Efficient Nakagami- 413 m Fading Channel Simulation Norman C. Beaulieu, Fellow, IEEE, and Christine Cheng Abstract—An efficient method for generating correlated Nakagami- fading envelope samples is presented; this method is applicable for arbitrary values of the fading parameter . The new method is compared to other methods used to generate Nakagami- random variates. An accurate approximation to the inverse Nakagami- cumulative distribution function, valid for all values of , is derived. Uncertainties regarding the autocorrelation of the Nakagami- fading process are discussed. The fading envelope autocorrelation is determined by simulation and asymptotic analysis. Index Terms—Fading channels, probability, simulation, stochastic processes. I. INTRODUCTION T HE Nakagami- distribution has gained widespread application in the modeling of physical fading radio channels [1]–[8]. Through the parameter , this distribution can model signal fading conditions that range from severe to moderate, to light or no fading. The primary justification for the use of the Nakagami- fading model is its good fit to empirical fading data [9]–[11]. Much theoretical and numerical analysis of the performances of diverse communication systems operating in Nakagami fading has been reported in the literature [1]–[8], [12]–[15]. Fewer results pertaining to the computer simulation of Nakagami- fading have been reported. Recognizing that the square root of a sum of squares of zero-mean identically distributed Gaussian random variables (RVs) has a Nakagami distribution with [16, pp. 46–48] leads to a “brute force method” for integer and half-integer values of . This method is used [17], but becomes more inefficient computationally as increases. Previous work that deals with the simulation of Nakagami- fading channels includes the following. Yip and Ng [18] proposed a Nakagami- fading channelsimulation method based on implementing the product of a square-root beta process and a complex Gaussian process. This method is restricted to values of . Dersch and Rúegg [19] developed a semi-empirical method, partially simulation based, using a theoretical model proposed in [20] with measured data. This method requires determining the values of certain coefficients from measured data. Furthermore, the generation of 100 dissimilarly distributed Gaussian RVs is required to obtain one random envelope/phase sample pair. In this paper, we give a procedure for simulating both the amplitude and the phase processes of a Nakagami- fading channel, for arbitrary values of the fading parameter . In deManuscript received January 13, 2001; revised June 14, 2004 and October 5, 2004. The review of this paper was coordinated by Prof. E. Sourour. N. C. Beaulieu is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]). C. Cheng is with Nortel Networks, Nepean, ON K1Y 4H7, Canada. Digital Object Identifier 10.1109/TVT.2004.841555 signing our phasor Nakagami-fading simulation procedure, we discuss longstanding uncertainties regarding the autocorrelation of the Nakagami-fading amplitude model appropriate to Nakagami amplitude fading. In addition, we derive a new useful approximation to the inverse cumulative distribution function (cdf) of the Nakagami distribution, valid for arbitrary values of fading parameter . This paper is organized as follows. Section II presents a description of the Nakagami- fading model and a brief discussion of its uncertainties. Previous simulation methods are examined prior to giving our new method, as well as some rationales for our design. In Section III, we derive a useful approximation to the inverse function of the Nakagami cdf needed for our simulator design. Section IV examines some statistical properties of our simulator. Comparisons of the new method with other methods are given in Section V and some conclusions are given in Section VI. II. NAKAGAMI- FADING MODEL In this section, we present the Nakagami- fading model. The properties of this model as well as some uncertainties associated with the model are examined. Also, previous simulation methods are discussed in detail. The bases and design of our new simulation method are given. Nakagami [1] has reported that signal amplitude fading in radio wave propagation can be well modeled by the probability density function (pdf) (1a) where (1b) is the expectation operator and (1c) In (1a), is a parameter that controls the severity, or depth, of the amplitude fading. The value results in the widespread Rayleigh-fading model [1], while values of less than one correspond to fading more severe than Rayleigh fading and greater than one correspond to fading less severe values of than Rayleigh fading. Fig. 1 shows the pdf for some different values of . Note that for integer and half-integer values of , the pdf is that of the amplitude of a sum of squared independent Gaussian RVs appropriately normalized.1 That is (2) 1The Nakagami distribution in these cases is equivalent to the generalized Rayleigh distribution described in [16, pp. 46–48]. 0018-9545/$20.00 © 2005 IEEE 414 Fig. 1. Nakagami- IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005 m pdf for various values of fading parameter m. where are independent and identically distributed (i.i.d.) Gaussian RVs each with zero-mean and variance . The RV in (2) has a Nakagami- pdf with parameter and second moment . If the second motends to infinity, ment is held constant, then in the limit as the Nakagami- pdf tends to an impulse as seen in Fig. 1; this is a consequence of the normalization chosen by Nakagami and permits obtaining a static signal amplitude as a limiting case ). (obtained as While (1) gives the pdf of the amplitude of the modeled fading signal, it is not clear what the associated phase pdf should be [1], [18]. A clear and plausible physical or intuitive basis for Nakagami- amplitude fading is not known and this fact has been a basis for some criticism of the use of this distribution to model fading [21]. Nonetheless, the distribution has been widely used owing to empirical justifications [9]–[11], of its quality in matching measured amplitude fading data. We are left with the open question of what phase pdf to associate with Nakagami amplitude fading if our simulation method is to provide phasor (amplitude and phase) fading gains. We postpone discussing how we choose to deal with this uncertainty to discuss a second uncertainty associated with the Nakagami-fading model. It is, perhaps, even less clear what the second- or higher-order distributions should be. The fading makes samples of the signal correlated. Gans [22] has derived the fading spectrum associated with two-dimensional (2-D) isotropic scattering and an omnidirectional receiving antenna, for the case of Rayleigh amplitude fading. The corresponding normalized autocorrelation of the complex Gaussian quadrature components is given by the inverse Fourier transform of the fading spectrum as [23, pp. 39–42] (3) where is the zero-order Bessel function of the first kind, is the maximum Doppler frequency, and represents time. It can be shown that the normalized autocorrelation function of the received signal envelope is then given by [23, p. 50] (4) is the hypergeometric function defined in where [24, p. 555]. To the best of the authors’ knowledge, it has neither been established by physical argument nor by empirical evidence what the autocorrelation of the quadrature signal components or the autocorrelation of the envelope should be for Nakagami fading [18]. We do not rigorously answer the fundamental uncertainties regarding the phase and the signal autocorrelations that should be associated with Nakagami fading, noting that little explicit discussion of these problems can be found in the literature. However, the following plausible artifice permits us to assign an autocorrelation to the faded Nakagami signal. It is based on considering the cdf defined by (5) where is a dummy variable of integration and is the pdf in (1). Consider the fading environments illustrated in Fig. 2. We assume that the signal experiences a multipath transmission environment such that it undergoes slow Rayleigh fading. We assume further that the Rayleigh-faded signal experiences a subsequent transmission environment such that the resultant received signal undergoes slow Nakagami fading. This model puts the Nakagami-fading signal into a correspondence with the Rayleigh-fading signal through the cdfs of each. Thus, if we specify that the Rayleigh-faded signal has a particular autocorrelation function, the autocorrelation function of the BEAULIEU AND CHENG: EFFICIENT NAKAGAMI- FADING CHANNEL SIMULATION Fig. 2. A cdf correspondence between Nakagami- m fading and Rayleigh fading. Fig. 3. Block diagram of the phasor Nakagami- 415 m fading channel simulator. Nakagami-faded signal is determined through this cdf transformation that maps Rayleigh fading into Nakagami fading. We are left with needing to assign a phase to the Nakagami fading to have a phasor Nakagami simulator. Somewhat arbitrarily, we assign a uniform phase to the faded Nakagami signal. This assignment is motivated (but not justified) by the fact that a Rayleigh-faded signal has a uniform phase distribution. This assumption is most common in the literature. Our phasor Nakagami simulation method is illustrated in Fig. 3. An existing Rayleigh-fading simulator generates a Rayleigh-faded envelope sample sequence having the specified denote autocorrelation and phase properties. Let the RV an envelope sample and the RV denote a phase sample corresponding to . Then, the transformation TABLE I INVERSE NAKAGAMI CDF APPROXIMATION COEFFICIENTS (6) where is the second moment of the random variable (i.e., ) transforms the RV into a uniform RV on . is the cdf of a Rayleigh This follows from the fact that is the RV [16, p. 45]. The subsequent transformation inverse function of the Nakagami- cdf defined implicitly by (7) by It is well known that transforming a uniform RV results in a transformed RV the inverse cdf function [25, pp. 101–102]. Thus having cdf given by (8) is an RV having Nakagami- distribution. Observe that our simulator design generates a uniform random variable from a Rayleigh random variable using the transformation (6), rather than starting directly from a uniform random variable. This design builds in appropriate correlation properties and appropriate higher order statistical properties into both the generated uniform random variable at the intermediate stage and into the generated Nakagami- random variable at the final (output) stage. There is no closed-form expression for except for . In the next section, we derive a the special case of useful and accurate approximation to the Nakagami- inverse cdf function. 416 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005 Fig. 4. Inverse normalized Nakagami cdf approximation for m = 2. Fig. 5. Inverse normalized Nakagami cdf approximation for m = 10. III. INVERSE CDF APPROXIMATION In order to implement the direct method to generate phasor Nakagami- distributed random variates described in Section II, the Nakagami- inverse cdf function is needed. Since this is not available in closed-form and since approximations to it that are useful for the problem at hand have not appeared in the literature, we derive an accurate useful approximation in this section. This approximation will be useful in other work, for example, in outage probability calculations [14]. We consider a . normalized Nakagami cdf, defined by Hastings [26, pp. 192–193], [27] has derived approximations to the inverse complementary cdf of the Gaussian distribution that are widely used [24, pp. 931–933]. Following the approach of Hastings, we have obtained [28] the approximation (9) where is an ancillary variable defined as (10) , and are coefficients chosen to minimize the . Unfortunately, approximation error and BEAULIEU AND CHENG: EFFICIENT NAKAGAMI- FADING CHANNEL SIMULATION Fig. 6. Relative approximation error of inverse Nakagami cdf approximation for m = 2. Fig. 7. m = 10. Relative approximation error of inverse Nakagami cdf approximation for it is not possible to find one set of coefficients that results in an accurate approximation for all values of the fading [28]. Table I gives coefficients for values of parameter , and . These coefficients were determined according to a minimum–maximum error criterion implemented by numerical than the values in search. Coefficients for other values of Table I can be determined by interpolation or by replicating our approach. , the approximation function , Note that for which gives the exact inverse Rayleigh cdf. (Thus, values of , and are not given in Table I for .) The ac- 417 curacy of the approximation for other values of is illustrated in Figs. 4 and 5, which show the exact inverse cdf (computed numerically) and the approximation obtained using the coeffiand , respectively. Also cients in Table I for shown is the ancillary variable . In these figures, the abssica of each plot has been scaled (nonlinearly) such that the true cdf plots as a straight line; in this way, the error of the approximation is immediately obvious. The relative error defined as Approximation-Function (11) Function is shown for and in Figs. 6 and 7, respectively. In deriving these approximations, we have sought accuracy 418 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005 Fig. 8. Comparison of theoretical and empirical Nakagami- m cdfs for three values of m for small values of cdf argument. Fig. 9. Comparison of theoretical and empirical Nakagami- m cdfs for three values of m for large values of cdf argument. in both “tails” of the cdf approximation, i.e., the region and the region . The former region is important in outage problems where the goal is to minimize the probability that the signal amplitude falls below a specified minimum level. The latter region is important in co-channel interference (CCI) problems where the probability that the CCI exceeds a specified level is important. The approximations for , and and the corresponding relative errors are shown graphand is ically in [28]. The relative error is typically no more than for these values of for the region to . These inverse Nakagami cdf approximations are incorporated into the Nakagami- fading simulator given in Fig. 3. In the BEAULIEU AND CHENG: EFFICIENT NAKAGAMI- FADING CHANNEL SIMULATION 419 Fig. 10. Empirical and theoretical phase pdfs of the simulator output. next section, we present some discussion of the statistical properties of the simulator output. IV. SIMULATOR STATISTICS In this section, we examine some statistical properties of the simulator output sequences. The distribution, the amplitude autocorrelation, the squared-amplitude autocorrelation, the quadrature component autocorrelation, and the phase autocorrelation are discussed. Figs. 8 and 9 highlight the cdf and complementary cdf, respectively, of the simulator output sequences. In both figures, the desired theoretical values are plotted as well as the empir, and . It ical simulator values for is seen in Fig. 8 that the small argument “tail” of the simulator distribution is in excellent agreement with the theoretical model. The theoretical and empirical values are graphically indistinin the range . guishable for values of the cdf Fig. 9 gives empirical and theoretical values of the complemen, in the range . Again, tary cdf the agreement between empirical and theoretical values is exin the region cellent. (There is a small difference for to . This difference is expected from the results given for the relative error of the approximation in [28].) Fig. 10 shows the phase pdf of the simulator output determined empirically and the theoretical phase pdf. Note that the generated phase is uniformly distributed in the interval and the empirical and theoretical pdfs are in excellent agreement. Fig. 11 shows a typical amplitude sequence and its corresponding phase sequence. The samples used in this figure have . Observe that, typically, when the a fading parameter signal has deep amplitude fades, it also has marked changes in phase. This behavior is characteristic of real world channels [29] and it is important that a simulator replicate it. The autocorrelation of the amplitude of the envelope is widely used as an indicator of the quality of a fading channel simulator [18], [23, ch. 2], [30, ch. 1]. The Nakagami random variates are obtained by a nonlinear transformation of the Rayleigh (or the underlying complex Gaussian) random variates. The analytical determination of the envelope amplitude autocorrelation is complicated and cumbersome. We, therefore, determine the amplitude autocorrelation function empirically, as discussed further in the sequel. Another relevant statistical measure of the fading process is provided by the squared-envelope autocorrelation function [23, ch. 2], [31]. Here, we derive the squared-envelope autocorrelation function for whole- and half-integer values of the fading parameter . Let and denote absolute time and a time increment, respectively. Then, using (2), the squared-envelope autocorrelation function is given by (12) where denotes the autocorrelation function of the denotes the autocorrelation function squared-envelope, 420 Fig. 11. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005 (a) Typical amplitude sequence and (b) its corresponding phase sequence. of the squared quadrature component, and the linearity property of expectation has been used as well as the independence of and for . Furthermore, one has that [25, pp. 305–306] The autocorrelation function in (14) is not normalized such that . For consistency and clarity, we normalize it; gives then, using (13) (15) for a normal stationary process with zero mean and autocorre. Combining (13) with (12) and using lation gives (14) Note that equality has not been preserved in going from (14) as given in (15) will be considered to (15); only for values of further in the sequel. Fig. 12 shows , and . Also shown are empirically BEAULIEU AND CHENG: EFFICIENT NAKAGAMI- FADING CHANNEL SIMULATION 421 Fig. 12. Normalized theoretical and empirical squared-envelope autocorrelation functions of Nakagami random sequences for five values of Fig. 13. Normalized empirical envelope autocorrelation functions of generated Nakagami sequences for five values of determined values of ; these are in excellent agreement with the theoretical values. Observe that (15) predicts the asymptotes (16) since (17) m. m. inas expected, since the amplitude becomes constant as creases without limit and the Nakagami- pdf tends to an impulse. Fig. 13, shows the normalized amplitude autocorrelation of the Nakagami simulator output, deterfunction mined empirically. The normalized empirical autocorrelation was computed according to (19) This asymptotic behavior is evident in Fig. 12. Note further that (18) where is the th sample of the amplitude sequence, is the total number of samples, is the discrete relative time differdenotes an empirical average of . ence, and 422 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005 Fig. 14. Normalized quadrature component autocorrelation functions of generated Nakagami sequences for five values of The averages were computed using random samples. (The empirical squared-envelope autocorrelation was computed similarly.) Observe that, as for the squared-envelope autocorrelation, there are asymptotes for large values of and these asymptotes depend on . Again, the asymptotic, large , autocorrelation approaches the value 1 as increases without limit. The autocorrelation of the quadrature components of the Nakagami-fading process are also of interest. This is shown in Fig. 14 for five finite values of fading parameter . Interestingly, there is little difference in for different values of . This is explained as follows. One has (20) where (21) is the autocorrelation function of the projection of the phase process onto a coordinate axis and where the independence of the fading amplitude process and the fading phase process has been used. Observe now, from Fig. 13, that for the typical values of considered there. Thus, (22) for different values of . Furthermore, as increases, the approximate equality in (22) becomes more accurate. That this is so can be seen from the results in Fig. 13. It also follows from the fact that as increases, the amplitude spread becomes narrower as the amplitude approaches a constant. Fig. 14 also . It is seen in Fig. 14 that (22) is a very good apshows proximation. Thus, the autocorrelation of the quadrature com- m. is seen to be an exponent of the Rayleigh-fading cellent approximation to the autocorrelation of the quadrature component of the Nakagami fading for the conditions considered in Fig. 14. Finally, we point out that the design of our simulator (Fig. 3) implies that the autocorrelation of the Nakagami- fading process phase is identical to the phase autocorrelation of the 2-D, isotropic scattering Rayleigh fading process. This can be found in, for example, [30, Ch. 1]. V. EFFICIENCY In this section, the efficiency of the new method is compared to the efficiency of the “brute-force” method. Both methods were programmed in the C language. The input to each method was an autocorrelated complex Gaussian random sequence generated using the algorithm published by Young and Beaulieu [32]. The comparisons were based on the genNakagami- random variates for values of eration of , and . A comparison to the method of [18] was abandoned due to the complexity of implementing the algorithm of [18] and because it is only . valid for restricted values of The new method requires only two independent autocorrelated Gaussian random variates for the generation of each Nakagami- random variate. In contrast, the “brute-force” method independent autocorrelated Gaussian random requires variates to generate one Nakagami- random variate. Table II shows the generation times of the two methods for selected values of . Observe that the new method has the shorter generation time for all values of . Also, the generation times are essentially independent of the parameter . In contrast, the run times for the brute-force method increase as increases. When , the time taken by the brute-force method is four times the time required by the new method. Importantly, the new method is suited for all values of fading parameter . BEAULIEU AND CHENG: EFFICIENT NAKAGAMI- FADING CHANNEL SIMULATION TABLE II EFFICIENCY COMPARISON BETWEEN THE BRUTE FORCE METHOD AND APPROXIMATION METHOD (GENERATION OF 2 RANDOM SAMPLES) VI. CONCLUSION A new efficient method for generating Nakagami- random samples has been presented. The new method can be used to generate Nakagami- random variates for arbitrary values of , whereas previous generation methods are restricted to values of between 0 and 1 and integer and half-integer values of . The new method required less computer time than the standard method in all cases examined. Longstanding uncertainties associated with the autocorrelation of the Nakagami- fading model have been discussed. The autocorrelation of Nakagami- fading sequences was specified using a transformation mapping Rayleigh sequences into Nakagami- sequences. Simulation was then used to obtain heretofore unknown autocorrelations of Nakagami- fading for different values of . A closed-form expression for the asymptotic (large time lag) value of the envelope autocorrelation was derived for arbitrary values of . It was seen that the autocorrelation of the envelope tends to 1 as the fading parameter tends to infinity. m REFERENCES [1] M. Nakagami, “The -distribution, a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. G. Hoffman, Ed, Oxford, U.K.: Pergamon, 1960. [2] Q. T. Zhang, “Maximal-ratio combining over Nakagami fading channels with an arbitrary branch covariance matrix,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1141–1150, Jul. 1999. [3] M. S. Alouini and A. J. Goldsmith, “Area spectral efficiency of cellular mobile radio system,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1047–1066, Jul. 1999. [4] M. S. Alouini, “A unified approach for calculating error rates of linearly modulated signals over generalized fading channels,” IEEE Trans. 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VT-21, no. 1, pp. 27–38, Feb. 1972. [23] G. L. Stüber, Principles of Mobile Communication. Boston, MA: Kluwer, 1996. [24] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972. [25] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1995. [26] C. Hastings, Approximations for Digital Computers. Princeton, NJ: Princeton Univ. Press, 1955. [27] , “Rational approximation in high-speed computing,” in Proc. Computation Seminar, International Business Machines Corporation, Dec. 1949, pp. 57–61. [28] C. Cheng, “A Nakagami-m fading channel simulator,” M.Sc. thesis, Dept. Elect. Comp. Eng., Queen’s Univ., Kingston, ON, Canada, 2000. [29] M. Fattouche and H. Zaghloul, “Equalization of =4 offset DQPSK transmitted over flat fading channels,” in Proc. IEEE Int. Conf. Communications (ICC’92), Chicago, IL, June 1992, pp. 296–298. [30] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [31] D. Parsons, The Mobile Radio Propagation Channel. New York: Wiley, 1992. [32] D. J. Young and N. C. Beaulieu, “The generation of correlated Rayleigh random variates by inverse discrete Fourier transform,” IEEE Trans. Commun., vol. 48, no. 7, pp. 1114–1227, Jul. 2000. 424 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 54, NO. 2, MARCH 2005 Norman C. Beaulieu (S’82–M’86–SM’89–F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D. degrees in electrical engineering from the University of British Columbia, Vancouver, Canada, in 1980, 1983, and 1986, respectively. He was a Queen’s National Scholar Assistant Professor from September 1986 to June 1988, an Associate Professor from July 1998 to June 1993, and a Professor from July 1993 to August 2000 with the Department of Electrical Engineering, Queen s University, Kingston, ON, Canada. In September 2000, he became the iCORE Research Chair in Broadband Wireless Communications at the University of Alberta, Edmonton, Canada, and, in January 2001, he became the Canada Research Chair in Broadband Wireless Communications. His current research interests include broadband digital communications systems, fading channel modeling and simulation, interference prediction and cancellation, decision-feedback equalization, and space–time coding. Dr. Beaulieu is a Member of the IEEE Communication Theory Committee and served as its Representative to the Technical Program Committee for the 1991 International Conference on Communication and as Co-Representative to the Technical Program Committee for the 1993 International Conference on Communications and the 1996 International Conference on Communications. He was General Chair of the Sixth Communication Theory Mini-Conference in association with GLOBECOM’97 and Co-Chair of the Canadian Workshop on Information Theory 1999. He has been an Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS for wireless communication theory since January 1992 and has served on the Editorial Board of the PROCEEDINGS OF THE IEEE since November 2000. He served as Editor-in-Chief of the IEEE TRANSACTIONS ON COMMUNICATIONS from January 2000 to December 2003 and as an Associate Editor of the IEEE COMMUNICATIONS LETTERS for wireless communication theory from November 1996 to August 2003. He received the Natural Science and Engineering Research Council of Canada (NSERC) E. W. R. Steacie Memorial Fellowship in 1999. He was awarded the University of British Columbia Special University Prize in Applied Science in 1980 as the highest standing graduate in the Faculty of Applied Science. He is a Fellow of The Royal Society of Canada. Christine Cheng received the B.Sc. degree in radio and electronics from Peking University, Beijing, China, in 1988, and the M.Sc. degree in electrical and computer engineering from Queens University, Kingston, ON, Canada, in 2000. From 1988 to 1997, she was with the Founder System Corporation, Beijing, China. In 1999, she joined Nortel Networks Corporation, Nepean, ON, Canada, as a Member of R&D staff.