IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
389
Exact Expression for the BER of Rectangular QAM With Arbitrary Constellation Mapping Leszek Szczecin´ski, Cristian González, and Sonia Aïssa
Abstract—The exact closed-form expression for the bit-error rate (BER) of rectangular quadrature amplitude modulation (QAM) is given. The presented formula is independent of the bit mapping and it is thus particularly useful in the design and analysis of modulation schemes employing non-Gray mapping. Compared with the so-called expurgated bound and the union bound, our expression is shown to accurately predict the BER in the low signal-to-noise ratio range where the bounding techniques fail. Index Terms—Bit-error rate (BER), Gray mapping, non-Gray mapping, performance evaluation, quadrature amplitude modulation (QAM).
I. INTRODUCTION
C
ALCULATION of the bit-error rate (BER) is of fundamental interest in digital communications [1]. Exact expressions for the BER of various modulation schemes using Gray mapping have previously been presented, e.g., in [2]–[8, Ch. 2]. Although Gray mapping is very popular in communications systems, other mapping strategies are starting to gain interest and popularity. Indeed, the application of non-Gray mapping was already reported in previous papers; in [9], as a means to improve the BER, taking into account the nonuniform signaling which may appear due to the residual redundancy of source coding, and in [10] and [11], in the context of iterative (turbo) demapping. Given the lack of exact expression, the BER under non-Gray mapping can be approximated using bounding techniques, such as the union bound [12, Ch. 4.3.2]. However, this bound is known to be inaccurate in the low-signal-to-noise ratio (SNR) range [12, Ch. 4.3.2], [13], and even its improved version, the so-called expurgated bound [13], only partially palliates the deficiency of the approximation at low SNR values. The rising interest in non-Gray mapping and the lack of the exact BER expression provide the motivation of this letter, in which we develop the closed-form formula for the uncoded BER of rectangular QAM modulation using non-Gray mapping. The expression we introduce is simple, yet not trivial, and, unlike Paper approved by A. Zanella, the Editor for Wireless Systems of the IEEE Communications Society. Manuscript received February 16, 2005; revised August 10, 2005. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Research Grants 249704-02 and 222907-01, and in part by Le Fond quebecois de la recherche sur la nature et les technologies (FQRNT), QC, Canada, under Research Grant 2003-NC-81788. L. Szczecin´ski and S. Aïssa are with the Institut National de la Recherche Scientifique, INRS-EMT, Montreal, QC H5A 1K6, Canada (e-mail:
[email protected];
[email protected]). C. González is with the Universidad Técnica Federico Santa María, Department of Electronic Engineering, Valparaíso, Chile (e-mail: gonzalez@ emt.inrs.ca). Digital Object Identifier 10.1109/TCOMM.2006.869879
previous contributions [2], [4], it does not require any assumption as to the bit-to-symbol mapping in use. Deriving the expression for the BER, we consider the additive white Gaussian noise (AWGN) channel, divide the complex plane into decision regions related to each of the constellation symbols, and show that the probability of falling into a decision region can be obtained using products of well-known complementary error functions. In the following, the system model, basic assumptions, and notations are introduced in Section II. The exact expression for the BER is derived in Section III, followed by numerical examples, shown in Section IV, to illustrate the relevance of the proposed expression when applied in the low-SNR range. Our conclusions are drawn in Section V. II. SYSTEM MODEL Consider the system where bits, denoted by , are mapped , into symbols via a memoryless and arbitrary1 mapper , where is the discrete time, is the modulating codeword taken is from the set of all possible codewords , is the constellation the modulation constellation, and size. In the following, we consider rectangular -ary quadrature , where amplitude modulation (QAM), i.e., , and is the Cartesian product. The sets and with sizes and , respectively (i.e., ), contain the real and imaginary parts of the symbols. For simplicity of derivation, and contain equidistant elements, i.e., we assume that , and are centered at the origin. These assumptions may further be relaxed if necessary. and Also, by we denote the maximum absolute value of the real and, respectively, imaginary values of the symbols. , where The channel output is given by is a zero-mean complex white Gaussian noise with variance . The average bit’s energy is given by , and it is easy to show that (1) Given the observation , the detector takes decision in favor of the codeword labeling the closest constellation symbol2 if (2) 1Gray
mapping is a particular case.
2This corresponds to a simplified decision scheme, while the exact calculation
should consider all the symbols and the noise level
0090-6778/$20.00 © 2006 IEEE
N
[13].
390
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
where and
is the decision region corresponding to the symbol is the Euclidean distance.
III. CLOSED-FORM EXACT EXPRESSION FOR THE BER , the errors When sending the codeword , i.e., occur when falls into , where . The number of bits in error due to this event depends on the Hamming distance between and . Therefore, averaging over all values of , and over all possible error events weighted by the corresponding Hamming distance gives the following expression for the average BER: BER (3) where denotes conditional probability. are In the case of a rectangular QAM, the decision regions squares of size , with the exception of the regions associated with the symbols on the border of the constellation, which can be half-infinite stripes or quarter planes, cf. Fig. 1. are independent, and Since the real and imaginary parts of each decision region can also be written as (where and are, respectively, the real and imaginary part of the elements in ), the conditional probability required by (3) can be calculated as
Fig. 1. 16-QAM modulation used for simulations. Shaded regions show examples of decision regions: a square, a quarter plane, and a half-infinite stripe.
may be expressed using the familiar precalculated function
(4)
if
and are, respectively, where the real and imaginary parts of . and are normally distributed with Because and respective means and variance , the first product term in the right-hand side (RHS) of (4) can be written as shown in (5) at the bottom of the page, and the second product term in the RHS of (4), related to , may be straightforwardly obtained, the imaginary part of with . Further, after a replacing in (5) the subindex simple change of variables in (5) and applying (1), the integral
if
(6)
where , , and are generic arguments of (i.e., real and/or imaginary parts of or and or ), and is given in (1); the condition verifies if the symbol is located at the constellation border, with respect to its real and/or imaginary part. Finally, (6) must be applied to calculate the product terms in (4), which are further used in (3). This results in the proposed closed-form expression for the average BER.
if
if if
(5)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
391
Note that the derivation presented here may be straightforwardly extended to rectangular constellations with no equidistant points [2], introducing the concept of the boundary decisions. In such a case, the decision regions may have different sizes, depending on the constellation point being considered. The proposed expression may be compared with the results of the so-called union bound [12, Ch. 4.3.2] BER
(7)
is used Comparing (3) and (7), we note that the function once per each pair in (7), and it may be evoked up to four times in (3) (twice for the real, cf. (6), and twice for the imaginary part of the symbol). Therefore, the implementation complexity of (3) is roughly four times higher than that of the union bound. The so-called expurgated bound, developed in [13], is tighter than the union bound, and has a similar form as (7), but the summations are taken over a limited set of codewords and . The cost of defining which elements and should be considered in the expurgated bound is relatively high, but difficult to compare directly with the cost of calculation of terms in (7). IV. NUMERICAL EXAMPLE To illustrate the advantage offered by the proposed analytical expression, we compare it with the union bound (7) and the expurgated bound [13] in the case of 16-QAM [Fig. 2(a)] and 64-QAM [Fig. 2(b)]. For 16-QAM, we use the non-Gray mapping taken from [11] and shown here in Fig. 1; for 64-QAM, the mapping is taken from [9, Fig. 9]. As a double check, we contrast the results yielded by the proposed expression with Monte Carlo simulation results obtained by randomly generating bits. As expected, our closed-form solution exactly matches the simulations. We can appreciate in Fig. 2(a) and 2(b) that for high values of , the union and expurgated bounds also give results SNR that well match the simulations. The discrepancy (0.5 dB at BER for 16-QAM) appears for low SNR values.3 For 64-QAM, the discrepancy grows to 2 dB at BER due to an increased number of terms in the RHS of (3). V. CONCLUSION We presented a simple closed-form formula for the BER of rectangular QAM, which, unlike other expressions known from the literature for this type of modulation, may be applied in nonGray mappings. Using numerical simulations and through comparisons with the union and expurgated bounds, we illustrated the advantage of using the derived expression, especially in the low-SNR range. When compared with the union bound, the implementation complexity of the proposed formulas is roughly 3Such a high value may be interesting, since strong error-correcting codes, e.g., turbo codes [14], are able to correct this level of uncoded BER.
Fig. 2. Comparison between simulation results and analytical expressions for the uncoded BER obtained for (a) 16-QAM constellation shown in Fig. 1, and (b) 64-QAM with labeling shown in [9, Fig. 9].
four times higher. This is a manageable complexity increase, thus, the proposed expression should be preferred over the union bound and expurgated bound. REFERENCES [1] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2000. [2] P. K. Vitthaladevuni and M. S. Alouini, “A closed-form expression for the exact BER of generalized PAM and QAM constellations,” IEEE Trans. Commun., vol. 52, no. 5, pp. 698–700, May 2004. [3] J. Lu, K. B. Letaief, J. Chuang, and M. Liou, “ -PSK and -QAM BER computation using signal-space concepts,” IEEE Trans. Commun., vol. 47, no. 2, pp. 181–184, Feb. 1999. [4] 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.
M
M
392
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 3, MARCH 2006
M
[5] M. I. Irshid and I. S. Salous, “Bit-error probability for coherent -ary PSK systems,” IEEE Trans. Commun., vol. 39, no. 3, pp. 349–352, Mar. 1991. [6] A. Conti, M. Z. Win, M. 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. [7] J. Lassing, E. G. Ström, E. Agrell, and T. Ottoson, “Computation of the exact bit-error rate of coherent -ary PSK with Gray code bit mapping,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1758–1760, Nov. 2003. [8] L. Hanzo, C. Wong, and M. S. Yee, Adaptive Wireless Tranceivers. New York: Wiley, 2002. [9] G. Takahara, F. Alajaji, N. C. Beaulieu, and H. Kuai, “Constellation mappings for two-dimensional signaling of nonuniform sources,” IEEE Trans. Commun., vol. 51, no. 3, pp. 400–408, Mar. 2003.
M
[10] X. Li and J. Ritcey, “Bit-interleaved coded modulation with iterative decoding using soft feedback,” Electron. Lett., vol. 34, no. 10, pp. 942–943, May 1998. [11] F. Schreckenbach, N. Görtz, J. Hagenauer, and G. Bauch, “Optimization of symbol mappings for bit-interleaved coded modulation with iterative decoding,” IEEE Commun. Lett., vol. 7, no. 12, pp. 593–595, Dec. 2003. [12] S. Benedetto and E. Biglieri, Principles of Digital Transmission with Wireless Applications. Norwell, MA: Kluwer, 1999. [13] G. Caire, G. Taricco, and E. Biglieri, “Bit interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 927–946, May 1998. [14] C. Berrou and A. Glavieux, “Near-optimum error-correcting coding and decoding: Turbo codes,” IEEE Trans. Commun., vol. 44, no. 10, pp. 1261–1271, Oct. 1996.
