COHERENT M-ary quadrature amplitude modulation

1994 IEEE TRANSACTIONS ON WIRELESS COUNICATIONS, VOL. 4, NO. 5, SEPTEBER 2005 Invertible Bounds for -QA in Rayleigh Fading Andrea Conti, ember, IEEE, oe Z. Win, Fellow, IEEE, and arco Chiani, Senior ember, IEEE Abstract In this letter, we derive tight invertible bounds on the bit-error probability BEP for the coherent detection of -ary quadrature amplitude modulation with Gray code bit mapping in Rayleigh fading channels. These bounds enable us to easily obtain tight lower and upper bounds on the bit-error outage BEO, i.e., BEP-based outage probability, in a log-normal shadowing environment. As examples of applications, these bounds are used to investigate the BEO and mean spectral efficiency for slow adaptive modulation. Index Terms Adaptive modulation, bit-error outage BEO, bit-error probability BEP evaluation, fading channel, -ary quadrature amplitude modulation -QA, shadowing. I. INTRODUCTION COHERENT -ary quadrature amplitude modulation -QA schemes have been a popular choice for numerous practical applications due to its bandwidth efficiency. Notable examples include digital video broadcasting DVB, wireless local area networks, and metropolitan area networks. It has also been considered recently for high-speed data transmission via the use of adaptive modulation/coding over fading channels [1] [4]. For a fixed quality of service QoS, adaptive techniques are typically used to maximize the throughput, which depends strongly upon the bit error probability BEP. The exact analysis of the BEP for the coherent detection of -QA, even in an additive-white-gaussian-noise AWGN channel is often complicated, and usually requires the evaluation of the complementary error function, or equivalently, the Gaussian Q-function. The exact BEP expression for an AWGN channel for cases of =4, 16, and 64, can be found in [5] and [6]. Although early work on -QA dates back to the early 1960s [7] [10], the evaluation of BEP for arbitrary is still of current interest [11] [13]. 1 A widely used methodology anuscript received July 27, 2003; revised June 9, 2004 and October 20, 2004; accepted October 27, 2004. The editor coordinating the review of this paper and approving it for publication is N. andayam. This work was supported in part by the Charles Stark Draper Laboratory Robust Distributed Sensor Networks Program, in part by the Office of Naval Research Young Investigator Award N00014-03-1-0489, in part by the National Science Foundation under Grant ANI-0335256, in part by the inistero dell Istruzione, Università e della Ricerca Scientifica IUR within the Project VICom, and in part by the Institute of Advanced Study Natural Science and Technology Fellowship. This paper was presented in part at the Conference on Information Sciences and Systems, Princeton, NJ, arch 2002. A. Conti and. Chiani are with the Istituto di Elettronica e di Ingegneria dell Informazione e delle Telecomunicazioni IEIIT Bologna BO/Consiglio Nazionale delle Ricerche CNR, Dipartimento di Elettronica, Informatica e Sistemistica DEIS, University of Bologna, 40136 Bologna, Italy e-mail: aconti@deis.unibo.it; mchiani@deis.unibo.it.. Z. Win is with the Laboratory for Information and Decision Systems LIDS, assachusetts Institute of Technology, Cambridge, A 02139 USA e-mail: moewin@mit.edu. Digital Object Identifier 10.1109/TWC.2005.853914 1 For a brief history of -QA, see [14]. to estimate BEP is to first obtain the symbol-error probability SEP, and then dividing it by the number of bits per symbol [5], [15]. This implies that, if Gray-code bit mapping is used, only the error event of exchanging the correct symbol with an adjacent one is accounted for, and other error events are neglected. However, the accuracy of this approximation may be not sufficient, especially for low and moderate signal-to-noise ratios SNRs. Another approach is to fit the simulated BEP with a parameterized exponential function [1], [3], [4], [16]. Using this technique, good approximations may be obtained for certain ranges of SNR. To improve the accuracy for all values of SNR, new approaches have been developed [11], [12]. In particular, a methodology based on signal-space concepts was proposed to closely approximate the BEP over an AWGN channel [11]. A recursive algorithm that uses the relationship among different constellation sizes was developed [12]. These approximations are reasonably close to simulation results or to the exact expression in an AWGN channel for the BEP of interest i.e., less than 10 2. Recently, the exact evaluation of the BEP in an AWGN channel, assuming Gray-code bit mapping, was generalized to arbitrary [13]. When considering BEP in fast-fading channels, previously discussed BEP expressions in an AWGN channel referred to as instantaneous BEP can be viewed as a conditional BEP conditioned on the instantaneous SNR. 2 In general, BEP in fastfading channels can be obtained by averaging the instantaneous BEP. Unless otherwise stated, the terms SNR and BEP will be used in the following to denote the mean SNR and the mean BEP, where the average is performed over the fast fading. In addition to BEP, explicit expression for the inverse BEP i.e., SNR as a function of the BEP is required in many important problems related to wireless mobile communication [17], [18]. One noticeable example is provided by the outage probability OP defined in terms of BEP, that is, the probability that the BEP exceeds a maximum tolerable level. We shall refer to this QoS measure as bit error outage BEO. This definition of OP is appropriate for digital communication systems, and the derivation of BEO requires such inversion. However, inverting BEP is not a straightforward procedure and requires a numerical root evaluation due to the fact that, in general, the inverse BEP function does not exist, although closed-form BEP expressions can be found in some cases. In fact, all the available BEP expressions are not useful in obtaining the inverse BEP. To make problems of this nature analytically tractable, we propose to replace the exact BEP by bounds that are invertible and tight for BEP values of interest. In this paper, we derive invertible bounds on the BEP for coherent detection of -QA in Rayleigh fading channels. We 2 The term fading channels is used for channels corrupted by both fading and AWGN. 1536-1276/$20.00 2005 IEEE

IEEE TRANSACTIONS ON WIRELESS COUNICATIONS, VOL. 4, NO. 5, SEPTEBER 2005 1995 obtain both upper and lower bounds that are valid for arbitrary constellation size. These bounds are useful because they are not only tight for BEP values of interests, but also invertible. As example of applications, we first consider the BEO in a lognormal shadowing environment, and analyze its dependence on the channel parameters. As a second example, we consider the slow adaptive modulation SA technique, where modulation parameters are adapted to the mean bit-error rate BER. II. BEP ANALYSIS In the following, we will express the performance by means of the complementary error function that is related to the Gaussian Q-function by Qx =1/2 erfc x/ 2. The BEP P b γ as a function of bit SNR γ can be obtained by averaging the conditional BEP P b e γ conditioned to the instantaneous SNR γ over the channel ensemble as 3 P b γ =E γ {P b e γ} = + 0 P b e ξp γ ξ,γdξ 1 where, due to Rayleigh fading, the probability density function pdf of γ is p γ ξ,γ =1/γexp ξ/γ for ξ 0, and 0 otherwise. To evaluate the BEP, since the conditional BEP expression involves the complementary error function, it is useful to recall the following integral [15]: Iγ = + 0 erfc γ ξ p γ ξ,γdξ =1 1+γ. 2 In the following, we first present the exact BEP, and then derive tight invertible bounds on the BEP that are valid for arbitrary constellation size. A. Exact Solution The exact instantaneous BEP expressions for =4, 16, and 64 are known for many years see, e.g., [5] and [6]. Despite the fact that the -QA technique has been extensively studied, a generalization to arbitrary has been developed only recently in [13] as P b e γ = 1 log2 2 [ erfc 2i +1 log 2 k=1 i 2 + 1 2 1 2 k 1 i=0 ] 3 log 2 γ 2 1 1 i 2 where x denotes the largest integer less than or equal to x commonly referred to as floor operation. Note that the 3 E x{ } denotes the expectation operator with respect to x. 3 instantaneous BEP is a linear combination of erfc functions and can be rewritten as P b e γ = A,j erfc B,j γ. 4 j By using the linearity property of the expectation operator together with 2, the instantaneous BEP in 4 can be averaged over Rayleigh fading as P b γ = j Hence, the exact BEP is given by P b γ = log 1 2 log2 2 A,j IB,j γ. 5 k=1 i 2 + 1 2 1 2 k 1 i=0 1 i 2 32i +1 2 log I 2 γ. 6 2 1 We now define the relative importance of the jth term in AWGN, F A,j, and in Rayleigh fading, F R,j,as F A,j = A,j erfc B,j γ P b e γ 7 F R,j = A,j I B,j γ. P b γ 8 In Table I, the relative importance of each term in 4 and 5 is reported for =16, 64, 256, and 1024 at SNRs corresponding to instantaneous BEP and to a BEP of 10 2. This table provides insight on how the different terms affect the performance. Specifically, it shows that a larger number of terms are required to accurately estimate the BEP in Rayleigh fading. B. Lower Bound The truncation to the first term in 4 that is given by P b,l e γ =A,1 erfc B,1 γ 9 with A,1 = [2 1]/ log 2 and B,1 = 3 log 2 /[2 1]. By investigating the construction of 4 for each of the in-phase and quadrature signals as in [13], one can arrive at the conclusion that 9 is a lower bound on the instantaneous BEP, that is, P b,l e γ P b e γ. In fact, one can observe that terms for j 2 are decreasing in absolute value and alternated in sign such that their summation is greater than zero. By averaging 9, we obtain the lower bound for the BEP in Rayleigh fading channel as P b,l γ = 1 log2 γ 2 1 3log 2 + γ P b γ. 10

1996 IEEE TRANSACTIONS ON WIRELESS COUNICATIONS, VOL. 4, NO. 5, SEPTEBER 2005 TABLE I RELATIVE IPORTANCE OF EACH jth TER AT INSTANTANEOUS OR EAN BEP 10 2 FOR =16, 64, 256, 1024; F A,j IS FOR AWGN, F R,j IS FOR RAYLEIGH FADING We will show in the numerical results section that this lower bound is sufficiently tight for the BEP of interest. Note that P b,l γ is invertible, which offers analytical tractability for both BEO and SA analysis in Sections III and IV. C. Upper Bound We introduce the function gγ = T /γ, where T = log 2 1 2 [log2 ] 2 2 k=1 i 2 + 1 2 1 2 k 1 i=0 i 2 1 32i +1 2 11 is dependent only on the constellation size. Note that gγ has the same structure as 6, except the term I{[32i + 1 2 log 2 /2 1]γ} that we have substituted by its asymptotic behavior for large SNRs { 1/[32i + 1 2 log 2 ]}1/γ using 1 Bx/1+Bx= 1/2Bx+ Ox 2 for x. We now define { P b,u γ = 1 min 2, T }. 12 γ Since P b,u γ is invertible, it offers analytical tractability for both BEO and SA analysis in Sections III and IV. Typical values of interest are T =0.25, 0.497, 1.113, 2.753, 7.353, and 20.84 for =4, 16, 64, 256, 1024, and 4096, respectively. Let us consider the usual BEP behavior for P b γ and gγ in logarithmic scale as a function of γ in decibel. Note that here P b γ is concave, whereas gγ is linear in γ decibel. It is clear from 6 that P b 0 g0. Furthermore, P b γ and gγ are both decreasing, and the two curves osculate at γ.

IEEE TRANSACTIONS ON WIRELESS COUNICATIONS, VOL. 4, NO. 5, SEPTEBER 2005 1997 Therefore, P b γ gγ. On the other hand, P b γ 1/2, and hence, 12 represents an upper bound for the BEP, i.e. P b γ P b,u γ. 13 In the following, we will compare our bounds with some commonly used approximations. D. Approximations for the BEP A typical approach to estimate the instantaneous BEP is to divide instantaneous SEP by the number of bits per symbol see, e.g., [5] and [15]. By averaging the instantaneous SEP P s e γ =A erfc [ B γ A ] 4 erfc B γ 14 where A = 2[ 1/ ] and B = [3 log 2 /2 1], the approximate BEP in Rayleigh fading becomes we will refer to this approximation as A1 A P b γ log 2 [ 4 π B γ 1+B γ tan 1 B γ 1+B γ + A2 4 log 2 ] 1+B γ 1. 15 B γ Note that the right side of 15 serves as the lower bound, since every symbol error is considered to produce one, and only one, bit error. Tight approximations for the BEP in an AWGN channel have recently appeared in literature [11], [12]. As an example, following the signal-space concept in [11], the instantaneous BEP can be written as III. BEO In digital mobile-radio systems, when a fast process is superimposed on a slow process, BEP alone is not sufficient to describe the link quality. In this case, γ, the SNR averaged over the Rayleigh fading process, also varies in time slower than Rayleigh fading due, for example, to a combination of mobility, shadowing, and power control. In such environments, a reasonable performance measure relating to the slow variations of the channel is given by the OP. Here, we consider the BEO, the OP based on the BEP, as P o P b =P {P b γ P b } 19 where Pb is the maximum tolerable BEP [17], [18]. The BEO defined here is an appropriate QoS measure for digital mobile radio. This can be evaluated as P o = p γ ξ dξ 20 A where p γ ξ is the pdf of γ, and A is the outage domain defined as A = {γ : P b γ >Pb }. Since the function P bγ is strictly decreasing in its argument, the BEO becomes P o P b = γp b 0 p γ ξ dξ 21 and, hence, the crucial point in evaluating the BEO becomes inverting P b γ, i.e., finding γ Pb 1 =Pe Pb, which, in general, requires numerical root evaluation. The above difficulty is alleviated by replacing γ by the bounds on the inverse BEP derived from the bounds on the BEP 10 and 12. In particular, the required SNR, γ = P 1 b P b, to achieve a target BEP with Pb [0,A,1] can be lower and upper bounded by P b e γ A log 2 /2 i=1 erfc [ 2i 1 ] B γ. 16 where γ L P b γ P b γ U P b 22 By averaging this expression over Rayleigh pdf, we obtain the approximation A2 P b γ A log 2 /2 i=1 2i 1 2 B γ 1+2i 1 2 B γ. 17 In [16], an exponential bound for the instantaneous BEP was derived, and it has been modified for better approximation in [1], [3], and [4]. The average over the Rayleigh fading of this expression is presented in [4], giving the following approximation, A3, of the BEP P b γ 0.2 1 1.6 log 2 γ + 1. 18 The bounds and approximations will be compared together with the exact solution in Section V. γ L Pb = P 1 b,l P b = 2 1 3log 2 log2 1 γ U P b = P 1 b,u P b log2 1 P b P b 2 2 23a = T Pb. 23b Using 22, 23a, and 23b, and the fact that p γ ξ in 21 is nonnegative, we arrive at the lower and upper bounds on the BEO as P o,l P b P o P b P o,u P b 24 where P o,l Pb and P o,upb are obtained by substituting γ Pb in 21 with γ L P b and γ U P b, respectively.

1998 IEEE TRANSACTIONS ON WIRELESS COUNICATIONS, VOL. 4, NO. 5, SEPTEBER 2005 As an example of application, let us derive the BEO in log-normal shadowing environments. It has been shown that shadowing in mobile-radio systems is well modeled by a lognormal distribution [19]. In this case, γ is a log-normal distributed random variable RV with parameters µ db and σdb 2, that is, γ db = 10 log 10 γ is a Gaussian RV with mean µ db and variance σdb 2. Since the logarithm is monotonic, the BEO is lower and upper bounded by P o,l Pb=Q µdb 10 log 10 γ L P b σ db σ db 25a P o,u Pb µdb 10 log =Q 10 γ U P b. 25b IV. SA In the literature, different adaptive-modulation techniques for maximizing the capacity have been proposed, in which the modulation parameters are chosen according to the instantaneous SNR [1] [4]. The instantaneous SNR thresholds for adapting the constellation size can be determined from the inversion of the exact instantaneous BEP expression or, when not possible, from its approximations [1] [4]. With these techniques, the receiver must reliably estimate the channel fast enough to track the fast-fading evolution, and must send reliably and instantaneously the information regarding the constellation size to be used back to the transmitter. To reduce the feedback-channel load, model-based channel-prediction algorithms can be adopted [3]. It is essential to predict channel coefficients several tens to hundreds of symbols ahead in order to realize these methods. oreover, in rapidly changing mobileradio environments, the vehicle speed and scattering geometry change continuously, and thus, the model parameters need to be recomputed frequently [3]. We denote these adaptation techniques based on the knowledge of the fast-fading state as fast adaptive modulation FA. The adaptation of modulation parameters based on the knowledge of the BEP averaged over an interval of a few seconds i.e., the BEP averaged over fast fading appears much more reliable and simple. With this SA technique, the constellation must only follow the slowly varying shadowing level; hence, it is much more simple to implement due to the slower feedback rate. 4 Constellation sizes corresponding to the SNR per bit can be determined from the inverse BEP expressions. Since the exact BEP expression for -QA is not invertible, we use the invertible bounds derived in Section II. So, for a fixed target BEP, from expressions 23a and 23b, we derive the maximum number of bits per symbol per dimension i.e., log 2 as a function of the received SNR. oreover, to quantify the gain of the SA technique, we evaluate the BEO and the mean spectral efficiency [E s = E γ {log 2 } bit/second/hertz] for different values of the target BEP and shadowing parameters. 4 Recall that BEP and SNR denote the mean BEP and the mean SNR averaged over fast fading, respectively. Fig. 1. Exact BEP, approximations, and bounds as a function of bit SNR decibel for coherent detection of 1024-QA, in the presence of AWGN and Rayleigh fading. The SA performance is made in terms of achievable BEO and mean spectral efficiency. With a given received power, as increases, the BEO increases; at the same time, the system becomes more efficient in spectrum utilization, that is, E s increases. In fact, the system falls into outage when even the smallest constellation size does not guarantee the target BEP. Therefore E s ranges from 0, when the system is in outage all the time, to log 2 max bit/second/hertz, when the system never experiences outage. In Section V, we demonstrate that by adopting SA techniques with modulation levels in { min,..., max }, it is possible to obtain the same outage of nonadaptive modulation using = min, as well as mean spectral efficiency better than for nonadaptive modulation using = max. V. N UERICAL RESULTS In this section, we present the numerical results related to our analysis on the bounds for direct and inverse BEP for coherent detection of -QA and their application to BEO and SA analyses. We first present the lower and upper bounds on the BEP in Rayleigh fading channels. In order to investigate the tightness of our bounds, we compare them with the exact and approximate BEP expressions. For brevity, the exact BEP, our bounds, and the approximations are plotted only for = 1024 in Fig. 1. It is observed that our bounds are practically indistinguishable from the exact solution for the BEP of interest less than 10 2. 5 Next, we evaluate the lower and upper bounds for the BEO, given, respectively, by 25a and 25b, in the presence of AWGN, Rayleigh fading, and log-normal shadowing. Fig. 2 shows the BEO versus the target BEP for coherent detection of -QA with =4, 16, 64, 256, 1024, and 4096 for 5 The bounds and approximations improve for smaller constellation size.

IEEE TRANSACTIONS ON WIRELESS COUNICATIONS, VOL. 4, NO. 5, SEPTEBER 2005 1999 Fig. 2. Lower and upper bounds on BEO as a function of target BEP for coherent detection of -QA with =4, 16, 64, 256, 1024, and 4096 in the presence of AWGN, Rayleigh fading, and log-normal shadowing with µ db =50and σ db =8. Fig. 4. Achievable bits per symbol per dimension with SA versus the bit SNR decibel at Pb =10 2 and Pb =10 3 ; LB is related to γ L and UB is related to γ U. Fig. 3. Lower and upper bounds on BEO as a function of µ db at P b =10 3 for coherent detection of -QA with =4, 16, 64, 256, 1024, and 4096 in the presence of AWGN, Rayleigh fading, and log-normal shadowing with σ db =8. µ db =50 and σ db =8. Fig. 3 shows the lower and upper bounds on BEO as a function of µ db at P b =10 3 for coherent detection of -QA with σ db =8and =4, 16, 64, 256, 1024, and 4096. Note that the lower and upper bounds on the BEO are tight, regardless of, P b, and µ db. For a given BEO, one can obtain the lower and upper bounds on the requirement on the parameter µ db corresponding to the median value of the shadowing level. This is useful for the design of digital mobile-radio systems employing -QA with various constellation size. For example, the maximum distance of the radio link can be estimated when the path-loss low is known. As far as the application of bounds 23a and 23b to SA is concerned, in Fig. 4, the achievable number of bits per Fig. 5. Comparison between SA and the fixed-modulation scheme in terms of mean spectral efficiency bit/second/hertz lower bound and BEO upper bound for Pb =10 2,andσ db =8. In the case of SA, stands for max. symbol per dimension as a function of the lower and upper bound on the bit SNR in decibel LB and UB, respectively is shown Pb =10 2 and Pb =10 3. As an example, from Fig. 4, at Pb =10 3, it is possible to use 1024-QA for γ 38.7 db, whereas if γ [34.4, 38.7] db, a constellation size must not exceed = 256, in order to fulfill the requirement on the BEP. In Fig. 5, SA schemes are compared to the nonadaptive modulation scheme by onte Carlo simulations in terms of mean spectral efficiency and BEO at Pb =10 2, for σ db =8, and different values of µ db. As an example, for µ db =35and BEO equal to 5%, the nonadaptive scheme achieves E s up to about 5.8 bit/s/hz using =64. On the other hand, SA schemes achieve about 5.9, 7.8, 9.2,

2000 IEEE TRANSACTIONS ON WIRELESS COUNICATIONS, VOL. 4, NO. 5, SEPTEBER 2005 and 10.5 bit/s/hz for max =64, 256, 1024, and 4096, respectively, with BEO less than 0.5%, which corresponds to a nonadaptive case with =4. These results in terms of mean spectral efficiency that are quite interesting, because they are close to the maximum obtainable in the case of full service without outage, that is, 6, 8, 10, and 12 bit/s/hz at max = 64, 256, 1024, and 4096, respectively. VI. CONCLUSION In this paper, we analyzed coherent detection of -QA in Rayleigh fading channels. We first derived invertible bounds on the mean BEP that are tight for arbitrary constellation size, and we compared them with the exact solution, as well as various approximations that are currently used in the literature. As an example of applications to digital mobile-radio systems, the BEO in a log-normal shadowing environment is evaluated, and its dependence on the channel parameters and constellation size is analyzed. Finally, the bounds are applied to analyze the SA technique, which adapts the constellation size to the mean SNR. It is shown that for an outage of 5% and a maximum BEP of 10 2, the average spectral efficiency is increased from about 5.8 to about 10.5 bit/s/hz. ACKNOWLEDGENT The authors wish to thank O. Andrisano and K. B. Letaief for helpful discussions, L. Hanzo for providing us with historical perspective on the subject, and J. L. Craig for the careful reading of the manuscript. REFERENCES [1] X. Qiu and K. Chawla, On the performance of adaptive modulation in cellular systems, IEEE Trans. Commun., vol. 47, no. 6, pp. 884 895, Jun. 1999. [2] D. L. Goeckel, Adaptive coding for time-varying channels using outdated fading estimates, IEEE Trans. Commun., vol. 47, no. 6, pp. 844 855, Jun. 1999. [3] A. Duel-Hallen, S. Hu, and H. Hallen, Long-range prediction of fading signals Enabling adapting transmission for mobile radio channels, IEEE Signal Process. ag., vol. 17, no. 3, pp. 62 75, ay 2000. [4] S. T. Chung and A. J. Goldsmith, Degree of freedom in adaptive modulation: A unified view, IEEE Trans. Commun., vol. 49, no. 9, pp. 1561 1571, Sep. 2001. [5]. K. Simon, S.. Hinedi, and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection, 1st ed. Englewood Cliffs, NJ: Prentice-Hall, 1995. [6] W. Webb and L. Hanzo, odern Quadrature Amplitude odulation. Piscataway, NJ: IEEE Press, 1998. [7] C. R. Cahn, Combined digital phase and amplitude modulation communication systems, IEEE Trans. Commun., vol. CO-8, no. 3, pp. 150 155, Sep. 1960. [8] J. C. Hancock and R. W. Lucky, Performance of combined amplitude and phase-modulated communication systems, IEEE Trans. Commun., vol. CO-8, no. 4, pp. 232 237, Dec. 1960. [9] C. N. Campopiano and B. C. Glazer, A coherent digital amplitude and phase modulation scheme, IEEE Trans. Commun., vol. CO-10, no. 1, pp. 90 95, ar. 1962. [10] J. C. Hancock and R. W. Lucky, On the optimum performance of N-ary systems having two degrees of freedom, IEEE Trans. Commun., vol. CO-10, no. 2, pp. 185 192, Jun. 1962. [11] J. Lu, K. B. Letaief, J. C.-I. Chuang, and. L. Liou, -PSK and -QA BER computation using signal-space concepts, IEEE Trans. Commun., vol. 47, no. 2, pp. 181 184, Feb. 1999. [12] L.-L. Yang and L. Hanzo, A recursive algorithm for the error probability evaluation of -QA, IEEE Commun. Lett.,vol.4,no.10,pp.304 306, Oct. 2000. [13] K. Cho and D. Yoon, On the general BER expression of one- and twodimensional amplitude modulations, IEEE Trans. Commun., vol. 50, no. 7, pp. 1074 1080, Jul. 2002. [14] L. Hanzo, W. Webb, and T. Keller, Single- and ulti-carrier Quadrature Amplitude odulation: Principles and Applications for Personal Communications, WLANs and Broadcasting, 1st ed. Piscataway, NJ: IEEE Press, 2000. [15] J. G. Proakis, Digital Communications, 4th ed. New York: cgraw- Hill, 2001. [16] G. J. Foschini and J. Salz, Digital communications over fading radio channels, Bell Syst. Tech. J., vol. 62, no. 2, pp. 429 456, Feb. 1983. [17] A. Conti,. Z. Win,. Chiani, and J. H. Winters, Bit error outage for diversity reception in shadowing environment, IEEE Commun. Lett., vol. 7, no. 1, pp. 15 17, Jan. 2003. [18] A. Conti,. Z. Win, and. Chiani, On the inverse symbol error probability for diversity reception, IEEE Trans. Commun.,vol.51,no.5, pp. 753 756, ay 2003. [19] icrowave obile Communications, W. C. Jakes, Ed. Piscataway, NJ: IEEE Press, 1995.