228 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001 BER Computation of 4=M -QAM Hierarchical Constellations Pavan K. Vitthaladevuni, Student Member, IEEE and Mohamed-Slim Alouini, Member, IEEE Abstract—Hierarchical constellations offer a different degree of protection to the transmitted messages according to their relative importance. As such they found interesting application in digital video broadcasting systems as well as wireless multimedia services. Although a great deal of attention has been devoted in the recent literature to the study of the bit error rate (BER) performance of uniform quadrature amplitude modulation (QAM) constellations, very few results were published on the BER performance of hierarchical QAM constellations. Indeed the only available expressions are “leading-term” approximate BER expressions for 4/16-QAM and 4/64-QAM. In this paper, we obtain exact and generic expresfor the BER of the 4 -QAM (square and rectansions in gular) constellations over additive white Gaussian noise (AWGN) and fading channels. For the AWGN case, these expressions are in the form of a weighted sum of complementary error functions and are solely dependent on the constellation size , the carrier-to-noise ratio, and a constellation parameter which controls the relative message importance. Because of their generic nature, these new expressions readily allow numerical evaluation for various cases of practical interest. In particular numerical results show that the leading-term approximation gives significantly optimistic BER values at low carrier-to-noise ratio (CNR) in particular over Rayleigh fading channels but is quite accurate in the high CNR region. Index Terms—BER computation, digital broadcasting, embedded modulation, fading channels, gray mapping, hierarchical modulations, multiresolution transmission, QAM constellations. I. INTRODUCTION I N HIS STUDY of broadcast channels, Cover [1] showed about three decades ago that one strategy to guarantee basic communication in all conditions is to divide the broadcasted messages into two or more classes and to give every class a different degree of protection according to its importance. The goal is that the most important information (known as basic or coarse data) must be recovered by all receivers while the less important information (known as refinement, detail, or enhancement data) can only be recovered by the “fortunate” receivers which benefit either from better propagation conditions (e.g., closer to the transmitter and/or with a direct line-of-sight path) or from better RF devices (e.g., lower noise amplifiers or higher antenna gains). Motivated by this information–theoretic study, Manuscript received August 13, 2001. This work is supported in part by the National Science Foundation grant CCR-9983462. This is an expanded version of work which is going to be presented at the 12th IEEE International Symposium on Personal, Indoor and Mobile Communications (PIMRC 2001), San Diego, CA, September 2001. The authors are with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: {pavan; alouini}@ece.umn.edu). Publisher Item Identifier S 0018-9316(01)10594-9. many researchers have shown since then that one practical way of achieving this goal relies on the idea of hierarchical modulations (known also as embedded or multiresolution modulations) which are constituted of constellations with nonuniformly spaced signal points [2]–[4]. This concept was studied further in the early nineties for digital video broadcasting systems [3], [5] and has gained more recently new actuality with i) the demand to support multimedia services by simultaneous transmission of different types of traffic, each with its own quality requirement [6]–[8], and ii) a possible application in the DVB-T standard [9] in which hierarchical modulations can be used on OFDM subcarriers. Yet the exact bit error rate (BER) evaluation performance of these types of modulations in additive white Gaussian channel (AWGN) or fading channels has not been investigated. The only available expressions are approximate BER expressions for 4/16-QAM and 4/64-QAM [5], [6] and for multicast -PSK [7]. In this paper, we focus on the BER computation -QA5M family of modulation wherein a 4-QAM of the -QAM constellation constellation is embedded into an (see Fig. 1). For this family of hierarchical constellation refinement information bits are sent along two basic information bits per channel access. Recent work related to this topic is described in what follows. Generic (in ) BER approximate expressions for uniform -QAM has been developed in [10] and [11] based on signal–space concepts and a recursive algorithm, respectively. Exact expressions for the BER of 16-QAM and 64-QAM were derived in [12]. More recently, Yoon et al. [13] obtained the exact and generic (in ) expression for the BER of uniform square QAM. Motivated by this elegant and general result we set out to see if a similar generic BER expression could be -QAM constellations. Starting with found for hierarchical some relatively simple examples (4/16-QAM, 4/64-QAM, and 4/256-QAM) this paper shows that it is actually possible to find for a pattern and to obtain an exact and generic expression in the BER of these constellations first for square QAMs then for nonsquare QAMs. The remainder of this paper is organized as follows. Section II presents the system model and parameters as well as the method to compute the average BER of the base and refinement information bits in terms of the BER of the in-phase and quadrature-phase bits. Section III reviews the “leading-term” approximation approach developed in [5], [6] to find the BER of such constellations. Section IV tracks the pattern in the BER of the in-phase and quadrature-phase bits and obtains the resulting new generic BER expressions. Section V presents some numerical 00018–9316/01$10.00 © 2001 IEEE VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF -QAM HIERARCHICAL CONSTELLATIONS 229 Fig. 1. 4=M -QAM square and rectangular (nonsquare) hierarchical constellations. results as well as their interpretation. Finally, Section VI concludes the paper by summarizing the main contributions. II. SYSTEM MODEL AND PARAMETERS A. System Model -QAM constellation with Karnaugh map style For a Gray mapping (see Fig. 2 for the 4/16-QAM example). We assume that we have two incoming streams of data. One carrying the basic information and the other carrying the refinement refinement information. For every channel access bits are sent along with 2 basic bits, as shown in Fig. 3. The two basic bits are assigned the most significant bit (MSB) position in the in-phase (I) and quadrature-phase (Q). In the square -QAM case , refinement bits are assigned to the in-phase and the rest refinement bits are assigned to the quadrature-phase. On the -QAM case , other hand, for the rectangular refinement bits are assigned to the in-phase refinement bits are assigned and the rest to the quadrature-phase. In this fashion the base bits Fig. 2. Embedded 4/16-QAM constellation with constellation points Gray coded in i q i q fashion. 230 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001 M ) 0 2 refinement bits are sent along with 2 basic bits. Fig. 3. System Model: log ( can be viewed as being transmitted by selecting one of the four fictitious symbols, whereas the refinement bits can be viewed as symbols surrounding the being transmitted via one of the selected fictitious symbol. For a -QAM rectangular constellation, it can be shown is given by that the average energy per symbol B. System Parameters represents the dis1) Distances: As shown in Fig. 2, represents tance between the two fictitious symbols whereas the distance between two neighboring symbols within one quadwhich is the minimum rant. Another important distance is distance between two symbols in adjacent quadrants. Note that -QAM case, can be related to and by in the square (1) as the constellation size parameter and We define as the constellation priority parameter which controls the we have a uniform relative message priorities. When corresponds to a uniform 16-QAM. When 4-QAM, while the basic bits are given a higher priority, while these . bits are given less protection for -QAM square constellation, it can 2) Energies: For a be shown that the average energy per symbol 1 is given by (5) In this case the first two terms represent the energy in the base bits whereas the last term represents the energy of the refinement bits. Hence the ratio of the energy in the fictitious 4-QAM constellation to the total average symbol energy is given in the rectangular constellation case by (6) 3) BER Parameterization: As we will see in Section III -QAM over AWGN the following section, the BER of channels is a weighted sum of complementary error functions whose arguments are in the following form: (7) (2) or equivalently (3) The first term in the above expression (3) represents the average symbol energy of a 4-QAM system with constellation points . On the other hand, the second term separated by a distance -QAM in (3) represents the average symbol energy of an . We define with constellation points separated by a distance the ratio of the energy in the fictitious 4-QAM constellation to the total average symbol energy: (4) s 1Throughout the paper the superscript will refer to square QAM and refer to rectangular (nonsquare) QAM. r will where is nonnegative integer, is a strictly positive integer is the two-sided power spectral density of number, and the AWGN. These arguments can be expressed solely in terms as of , , and the carrier-to-noise ratio (CNR) (8) where for the square -QAM case (9) -QAM case (see equation (10) whereas for the rectangular at the bottom of the page). Note that alternatively the overall BER can be expressed solely in terms of , , and . (10) VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF -QAM HIERARCHICAL CONSTELLATIONS 231 C. BER for the Base and Refinement Bits -QAM: The BER of the base information 1) Square is given by (11) and denote the BER of the th where -QAM constellain-phase and quadrature phase bits of a tion, respectively. On the other hand, the BER of the refinement , is obtained by averaging over all the reinformation, maining in-phase and quadrature bits, yielding (12) -QAM: We lose symmetry between 2) Rectangular in-phase and quadrature phase bits. However the in-phase bits -QAM rectangular constellation will behave in the of a -QAM square same manner as the in-phase bits of a constellation. Hence we can write the BER for these in-phase , as bits, (13) -QAM rectOn the other hand the quadrature bits of a angular constellation will behave in the same manner as the -QAM square constellation. Hence quadrature bits of a , as we can write the BER for these quadrature bits, Fig. 4. The I sub-channel for 4/16-QAM case. 2 subchannels, I and Q. In the case of square constellations, the I and Q sub-channels have the same average bit error probability. So, we show the derivation for the I subchannel alone. Consider the I subchannel bits shown in Fig. 4. The in symbol 1000 is given by probability of error for the bit . Similarly, the bit error probability . for bit in symbol 1010 is given by We consider just the leading term, as an approximation for the for bit , i.e., bit error probability (17) Now, consider the second subchannel . The bit error probability for this subchannel in symbol 1000 is given by (18) For symbol 1010, it is (19) (14) Therefore the overall BER of the base information is given by So, the average bit error probability channel is given by (15) whereas the overall BER of the refinement info is given by (see equation (16) at the bottom of the page). Based on all these equations, we can conclude that the BER -QAM computation of the base and refinement bits of a (square and rectangular) constellation reduces to finding an expression for the BER of the in-phase bits only. III. LEADING-TERM APPROXIMATION The BER expressions for -QAM can be approximated to the leading term in error function, as has been done by [5] for the 4/16-QAM and 4/64-QAM constellations. A. 4/16-QAM Constellation This approach yields good results for 4/16-QAM constellations over AWGN channels. The data bits are divided into for the sub- (20) Neglecting the higher order terms, we just have (21) B. 4/64-QAM Constellation Similar to what has been done in the case of 4/16-QAM, we can extend the result to 4/64-QAM. In the case of 64-QAM, we have 6 bits per symbol. That gives 3 bits per subchannel. The base bit has the following protection distances with equal , , , . Considering probability: just the leading term, the probability of error is (22) (16) 232 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001 Fig. 5. The I sub-channel for 4/64-QAM case. Similarly, for bit and , we have the error probability as (23) and, (24) (29) IV. EXACT BER As we will see later Section V, the approximate approach yields good results for the bit error probability at high CNR end. But, at low CNR, it appears under estimating the BER. The performance deteriorates further, if the channel fades. In this section, we derive the exact BER expressions. A. 4/16-QAM Constellation C. 4/256-QAM Constellation The BER expression for this constellation helps us discover the pattern, using which we can write the general expression -QAM constellations. Similar to for the BER for general Section III, we can write Referring to Fig. 2 we can write (25) (26) B. 4/64-QAM Constellation (30) Referring to Fig. 5 we can write (27) (28) (31) VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF Fig. 6. -QAM HIERARCHICAL CONSTELLATIONS 233 Three sets of coefficients have been tracked. The sum of these three coefficients gives the coefficients for the uniform case. (33) D. General Expression for Square in AWGN -QAM Constellations As can be seen in the equations of the Section III, there are three types of terms to be tracked. First, coefficients of the functions whose arguments are of the form alone. Second, coefficients of the functions whose argusuch that . ments are of the form Finally, we track the rest of the coefficients. The behavior of these coefficients is illustrated in Fig. 6. Based on these observations, a pattern can be found for these sets of terms. is different from that for the other This pattern for the bit we can write bits. In particular, for (32) (34) and for , we have the following general expression 234 Fig. 7. IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001 Comparison between the exact and the average BER of 4/256-QAM with R = 0 8 over AWGN channels. : (38) and for where by is the floor function, (35) (36) and (37) Recall that knowledge of the BER of the in-phase bits of square QAMs as given in a generic fashion in (34) and (35), is sufficient for the computation of the overall BER of any square -QAM as discussed in Section II-C. As a or rectangular check for the validity of the proposed expressions (34) and (35), it can be easily verified that they reduce to the corresponding ex(uniform QAM case). pression of [13], [14] when Using the notation of Section II-B, the new generic BER expressions (34) and (35) over AWGN channels can be expressed in terms of the CNR , as (39) where, (40) VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF Fig. 8. -QAM HIERARCHICAL CONSTELLATIONS Effect of Rayleigh fading on the BER performance of 4/256-QAM with R = 0 8. : Fig. 9. Effect of the variation of R on the BER of the base and refinement bits over AWGN channels. 235 236 IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001 Fig. 10. Effect of the variation of R on the BER of the base and refinement bits over Rayleigh fading channels. Fig. 11. Effect of the variation of M on the BER of the base and refinement bits over AWGN channels. VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF Fig. 12. Effect of the variation of -QAM HIERARCHICAL CONSTELLATIONS 237 M on the base and refinement bits over Rayleigh fading channels. E. Extension to Flat Fading Channels Since the BER expressions over AWGN are in the form of weighted sum of complementary error functions, these results are readily extendable to flat fading channels with and without diversity reception. Therefore, the AWGN expressions (38) and (39) are valid for flat fading channels provided the definition function is changed accordingly. For example, for of Rayleigh channels where is the Marcum -function, is the 0th-order modified Bessel function of the first kind, and (44) where is the Rician factor. (41) V. NUMERICAL RESULTS where is the average CNR. For Nakagami- channel channels (42) is the Gauss hypergeometric function and where is the Nakagami fading parameter. For the Rician fading channels (43) We note that the analytical expressions derived in this paper, and the corresponding numerical results presented in this section have been verified extensively by Monte Carlo simulations for the various constellations under consideration. Thus, the reader can be totally confident in the correctness of the newly derived closed form expressions and the accuracy of the numerical results illustrated below. Figs. 7 and 8 compare the exact and the approximate BER expressions for 4/256-QAM over AWGN and Rayleigh fading channels, respectively. These numerical results show that the leading-term approximation gives significantly optimistic BER values at low CNR in particular over Rayleigh fading channels. In the case of Rayleigh fading channels, even at a high average received CNR, the leading term approximation is significantly optimistic. Figs. 9 and 10 illustrate the effect of the variation of on the BER of the base and refinement bits over AWGN and Rayleigh 238 Fig. 13. IEEE TRANSACTIONS ON BROADCASTING, VOL. 47, NO. 3, SEPTEMBER 2001 Generalized hierarchical QAM constellations. fading channels, respectively. When (uniform 256QAM), we find that the base bits are less protected than the , we find that base bits are refinement bits. When relatively well protected, at the cost of the BER of the refinement bits. on Figs. 11 and 12 illustrate the effect of the variation of the base and refinement bits over AWGN and Rayleigh fading channels, respectively. As increases, the BER of base bits increases by a negligible amount, but the BER of refinement bits increases significantly. This is because, the fictitious 4-QAM symbols move by a negligible amount, but the refinement symbols around these fictitious 4-QAM symbols get crowded. VI. CONCLUSION We derived a generic (in ) closed form expression for the -QAM constellations. For the AWGN case, these BER of expressions are in the form of a weighted sum of complementary error functions and are solely dependent on the constellation size , the carrier-to-noise ratio, and a constellation parameter which controls the relative message importance. Because of their generic nature, these new expressions readily allow numerical evaluation for various cases of practical interest. Numerical results show that the leading-term approximation underestimates the BER at the low CNR end, in particular over Rayleigh fading channels but is quite accurate at the high CNR end. More recently, we have derived a recursive way of computing the BER of generalized square -QAM constellations as shown in Fig. 13 (see [15]). This recursive algorithm gives an alternative way of computing the exact BER of generalized -QAM conhierarchical -QAM constellations, of which stellations form a special case. ACKNOWLEDGMENT The authors wish to thank Dr. D. Yoon of the Department of Information and Communications Engineering, Taejon University, Taejon, Korea for the pre-print of [14]. REFERENCES [1] T. Cover, “Broadcast channels,” IEEE Trans. on Inform. Theory, vol. IT-18, pp. 2–14, January 1972. [2] C.-E. W. Sundberg, W. C. Wong, and R. Steele, “Logarithmic PCM weighted QAM transmission over Gaussian and Rayleigh fading channels,” IEE Proc., vol. 134, pp. 557–570, October 1987. [3] K. Ramchandran, A. Ortega, K. M. Uz, and M. Vetterli, “Multiresolution broadcast for digital HDTV using joint source/channel coding,” IEEE J. Select. Areas Commun., vol. 11, January 1993. [4] L.-F. Wei, “Coded modulation with unequal error protection,” IEEE Trans. Commun., vol. COM-41, pp. 1439–1449, October 1993. [5] M. Morimoto, H. Harada, M. Okada, and S. Komaki, “A study on power assignment of hierarchical modulation schemes for digital broadcasting,” IEICE Trans. Commun., vol. E77-B, December 1994. [6] M. Morimoto, M. Okada, and S. Komaki, “A hierarchical image transmission system in a fading channel,” in Proc. IEEE Int. Conf. Univ. Personal Comm. (ICUPC’95), October 1995, pp. 769–772. [7] M. B. Pursley and J. M. Shea, “Nonuniform phase-shift-key modulation for multimedia multicast transmission in mobile wireless networks,” IEEE J.Select. Areas Commun., vol. SAC-17, pp. 774–783, May 1999. [8] , “Adaptive nonuniform phase-shift-key modulation for multimedia traffic in wireless networks,” IEEE J. Select. Areas Commun., vol. SAC-00, pp. 1394–1407, August 2000. [9] “DVB-T standard: ETS 300 744, Digital broadcasting systems for television, sound and data services; framing structure, channel coding and modulation for digital terrestrial television,” ETSI Draft, vol. 1.2.1, no. EN300 744, 1999. [10] J. Lu, K. B. Letaief, J. C.-I. Chuang, and M. L. Liou, “ -PSK and -QAM ber computation using signal–space concepts,” IEEE Trans. Commun., vol. COM-47, pp. 181–184, Februrary 1999. [11] L.-L. Yang and L. Hanzo, “A recursive algorithm for the error probability evaluation of -QAM,” IEEE Commun. Letters, vol. 4, pp. 304–306, October 2000. M M M VITTHALADEVUNI AND ALOUINI: BER COMPUTATION OF -QAM HIERARCHICAL CONSTELLATIONS [12] M. O. Fitz and J. P. Seymour, “On the bit error probability of QAM modulation,” International Journal of Wireless Information Networks, vol. 1, no. 2, pp. 131–139, 1994. [13] D. Yoon, K. Cho, and J. Lee, “Bit error probability of -ary quadrature amplitude modulation,” in Proc. IEEE Veh. Technol. Conf. (VTC’2000Fall), Boston, Massachussets, September 2000, pp. 2422–2427. [14] K. Cho and D. Yoon, “On the general BER expression of -ary square QAM signals,” IEEE Trans. Commun, January 2001. [15] P. K. Vitthaladevuni and M.-S. Alouini, “A recursive algorithm for BER computation of generalized hierarchical -QAM constellations,” in Proc. IEEE global commun. Conf. (GLOBECOM’2001), San Antonio, Texas, November 2001. M M M 239 Mohamed-Slim Alouini (S’94-M’99) was born in Tunis, Tunisia. He received the “Diplôme d’Ingénieur” degree from the Ecole Nationale Supérieure des Télécommunications (TELECOM Paris), Paris, France, and the “Diplôme d’Etudes Approfondies (D.E.A.)” degree in Electronics from the University of Pierre & Marie Curie (Paris VI), Paris, France, both in 1993. He received the M.S.E.E. degree from the Georgia Institute of Technology (Georgia Tech), Atlanta, GA, USA, in 1995, and the Ph.D. degree in Electrical Engineering from the California Institute of Technology (Caltech), Pasadena, CA, USA, in 1998. While completing his D.E.A. thesis, he worked with the optical submarine systems research group of the French national center of telecommunications (CNET-Paris B), on the development of future transatlantic optical links. While -band satellite channel at Georgia Tech, he conducted research in the area of characterization and modeling. From June 1998 to August 1998, he was a PostDoctoral Fellow with the Communications group at Caltech carrying out research on adaptive modulation techniques and on CDMA mobile communications. He joined the department of Electrical and Computer Engineering of the University of Minnesota, Minneapolis, in September 1998, where his current research interests include statistical modeling of multipath fading channels, adaptive modulation techniques, diversity systems, and digital communication over fading channels. Dr. Alouini has published several papers on the above subjects and he is co-author of the recent Wiley Interscience textbook Digital Communication over Fading Channels. He is a recipient of a National Semiconductor Graduate Fellowship Award, the Charles Wilts Prize for outstanding independent research leading to a Ph.D. degree in Electrical Engineering at Caltech, and co-recipient of the 1999 Prize Paper Award of the IEEE Vehicular Technology Conference (VTC’99-Fall), Amsterdam, The Netherlands, for his work on the performance evaluation of diversity systems. He was awarded a 1999 CAREER Award from the National Science Foundation, a McKnight Land-Grant Professorship by the Board of Regents of the University of Minnesota in 2001, and a Top Instructor Award (ECE Department) by the Institute of Technology Student Board at the University of Minnesota in June 2001. He is an editor for the IEEE Transactions on Communications (Modulation & Diversity Systems) and for the Wiley Journal on Wireless Systems and Mobile Computing. K Pavan K. Vitthaladevuni (S’2000) received the Bachelor of Technology (B.Tech) degree in Electrical Engineering from Indian Institute of Technology, Madras, India, in 1999 and the M.S.E.E. degree from University of Minnesota, Twin Cities in 2001. He is currently a graduate research assistant in the Department of Electrical and Computer Engineering at the University of Minnesota, Twin Cities, and is working toward his Ph.D. His research interests include Digital Communications over fading channels and Information Theory.