Rapport technique #INRS-EMT Exact Expression for the BER of Rectangular QAM with Arbitrary Constellation Mapping

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1 Rapport technique #INRS-EMT Exact Expression for the BER of Rectangular QAM with Arbitrary Constellation Mapping Leszek Szczeciński, Cristian González, Sonia Aïssa Institut National de la Recherche Scientifique INRS-EMT University of Quebec Montreal, Canada Universidad Técnica Federico Santa María Department of Electronic Engineering Valparaíso, Chile 1 INRS-EMT, June 30, 004

2 Abstract The exact expression for the bit error rate (BER) of rectangular quadrature amplitude modulation (QAM) is given. The presented closed-form formula is independent of the bit mapping in use. It is thus particularly useful in the analysis of modulation schemes employing non-gray mapping, e.g., those designed for iterative (turbo) demapping. Compared to the so-called expurgated bound and the union bound, our expression is shown to accurately predict the BER, which is particularly important in the low signal-to-noise ratio (SNR) range at which strong error-correcting codes ensure satisfactory performance. I. INTRODUCTION Calculation of the bit error rate (BER) is of fundamental interest in digital communications [1]. Based on the assumption of Gray mapping, exact expressions for the BER of different modulation schemes have previously been presented, e.g., in [] [3, Ch. ] [4]. Although Gray mapping is very popular in communications systems, the importance of other mapping strategies [5] [6] has grown, especially thanks to the iterative (turbo) de-mapping [7] [8] that offers significant performance gains over Gray mapping. The general expression for the BER in arbitrary mapping, is based on the union bound, which is known to be inaccurate in the low SNR range [9, Ch. 4.3.] [6]. This is exactly where a precise evaluation of the BER is necessary when strong error-correcting codes, e.g., turbo codes, are used. To partially palliate the deficiency of the union bound, the so-called expurgated bound, which eliminates unnecessary terms from the union bound, was introduced in [6]. In this letter, we develop the exact closed-form expression for the uncoded BER in rectangular QAM with equidistant constellation points. Unlike previous contributions, we do not make any assumption as to the bit-to-symbol mapping in use. Deriving the expression for the BER, we consider additive white Gaussian noise channel (AWGN), 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 pre-calculated complementary error functions. In the following, the system model, basic assumptions and notations are introduced in Section II. The expression for exact BER is derived in Section III, followed by a numerical example 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 the bits, denoted by y, are mapped into symbols x via a memoryless and arbitrary 1 mapper M{ }. As a result, x(n) = M{y(n)} A, where n is the discrete time, y(n) = [y(nb + 1),..., y(nb + B)] B is the modulating codeword, B is the set of all possible codewords and A = {a 1,..., a L } is the modulation constellation with L = B. In the following, we consider rectangular B -ary quadrature amplitude modulation (QAM), i.e., A = A R {ja I }, where j = 1, and is the cartesian product. The sets A R = {a R,1,..., a R,LR } and A I = {a I,1,..., a I,LI } contain, respectively, the real and imaginary parts of the symbols, where L R > 1, L I > 1, and L = L R L I. We assume that A R and A I contain equidistant elements, i.e., a R,k+1 a R,k = a I,k+1 a I,k = d min, and are centered at the origin, which yields a R,LR = a R,1 M R and a I,LI = a I,1 M I. All these assumptions, adopted for simplicity of derivation, may further be relaxed. The channel output is given by r(n) = x(n)+η(n), where η(n) is a zero-mean complex white Gaussian noise with variance N 0, and - when E b = 1 B L l a l is the average bit s energy - E b /N 0 represents the signal-to-noise ratio (SNR). Given the observation r(n), the detector takes decision in favor of the codeword labelling the closest constellation symbol ŷ(n) = b B if r(n) Z b = {r : r M{b} < r M{ b}, b b}, (1) 1 i.e., Gray mapping is a particular case. This corresponds to widely adopted simplified decision scheme, while the exact metrics calculation should consider all the symbols and the noise level N 0 [6].

3 PSfrag replacements Fig. 1. stripe. 16QAM modulation used for simulations and - shaded - examples of decision regions: a square, a quarter-plane and a half-infinite where Z b is the decision region corresponding to the symbol M{b}. III. CLOSED-FORM EXPRESSION FOR THE BER When sending the codeword b, i.e., x(n) = M{b}, the errors occur when r(n) falls into Z b, where b b. The number of bits in error due to this event depends on the Hamming distance d H (b, b) between b and b. Therefore, averaging over all values of b, and over all possible error events weighted by the corresponding Hamming distance gives the following expression for the average BER: BER = 1 d B B H (b, b)pr{r(n) Z b x(n) = M{b} }, () b B b B where Pr{ } denotes conditional probability. In the case of rectangular QAM, the decision regions Z b are squares of size d min d min with the exception of the regions associated with symbols on the border of the constellation which can be halfinfinite stripes or quarter-planes, cf. Fig. 1. Since, the real and imaginary part of r(n) are independent and each decision region can also be written as Z b = Z R,b {jz I,b } (where Z R,b and Z I,b are, respectively, the real and imaginary part of the elements in Z b ), the conditional probability required by () can be calculated as Pr{r(n) Z b x(n) = M{b}} = Pr{R[r(n)] Z R, b x(n) = M{b}} Pr{I[r(n)] Z I, b x(n) = M{b}}, (3) where R[ ] and I[ ] are real and imaginary parts respectively. Because R[r(n)] and I[r(n)] are normally distributed with variance N 0 / and respective means R[M{b}]

4 4 and I[M{b}], the product terms in the right hand side of (3) can be written as Pr{R[r(n)] Z R, b x(n) = M{b}} = 1 πn0 R[M{ b}]+dmin / R[M{ b}] d min / Pr{I[r(n)] Z I, b x(n) = M{b}} = 1 πn0 I[M{ b}]+dmin / I[M{ b}] d min / exp( (t R[M{b}]) /N 0 )dt, (4) exp( (t I[M{b}]) /N 0 )dt. (5) Exceptionally, if the symbol M{ b} is at the constellation border with respect to its real part (i.e., R[M{ b}] = M R ), or imaginary part (i.e., I[M{ b}] = M I ), the upper integration limit in (4) and/or (5) should be replaced by, and this, to reflect the fact that the decision region extends to infinity. Equations (4) and (5) having identical form, we can write (3) as Pr{r(n) Z b x(n) = M{b}} = I ( R[M{b}], R[M{ b}] ) I ( ) I[M{b}], I[M{ b}]. (6) Further, after a simple change of variables in (4) and (5), the integrals denoted by I(, ) in (6) may be expressed using the familiar pre-calculated function Q(t) = 1 π exp( τ /)dτ, so that t ( ) v ṽ Q d min if ṽ = M R/I N0 / I(v, ṽ) = ( ) ( ), (7) Q Q otherwise v ṽ d min N0 / v ṽ + d min N0 / where v and ṽ are generic arguments of I(, ) [i.e., real and/or imaginary parts of M{b} or M{ b} in (6)], and the condition ṽ = M R/I verifies if a symbol M{ b} is located at the constellation border, with respect to its real (M R/I M R ) and/or imaginary part (M R/I M I ). Finally, expression (7) applied to (6) and further used in () results in the proposed closed-form expression for the average BER. Note that the derivation presented here may be straightforwardly extended to rectangular constellations with no equidistant points [4], 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. IV. NUMERICAL EXAMPLE To illustrate the advantage offered by the proposed analytical expression, we compare it to the results of numerical simulations and to the expressions known from the literature, namely, the union bound [9, Ch. 4.3.], and the expurgated bound [6]. The results presented in Fig. were obtained generating 10 6 random bits, which were mapped into the 16-QAM constellation with non-gray mapping taken from from [8] and shown here in Fig. 1. As expected, for high values of SNR E b /N 0, the union and expurgated bounds give results matching well the simulations. The significant discrepancy appears for low SNR values for which the union bound results in a BER even higher than 1. On the other hand, the expression presented in this letter accurately predicts the results of simulations. 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 - is not restricted to Gray mapping. Mappings which are not Gray are particularly important when using the iterative (turbo) de-mapping. 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 where strong error-correcting codes ensure performance gains.

5 Union Bound Expurgated Bound 10 0 Proposed Expression Simulation 10 1 BER PSfrag replacements E b /N 0 Fig.. Fig. 1. Comparison between simulation results and analytical expressions for the raw BER obtained for 16QAM constellation shown in REFERENCES [1] J. G. Proakis, Digital Communications, 4th ed. McGraw-Hill, 000. [] K. Cho and D. Yoon, On the general BER expression of one- and two-dimensional amplitude modulations, IEEE Transactions on Communications, vol. 50, no. 7, July 00. [3] L. Hanzo, C. Wong, and M. S. Yee, Adaptive Wireless Tranceivers. Wiley and Sons, 00. [4] P. K. Vitthaladevuni and M. Alouini, A closed-form expression for the exact BER of generalized PAM and QAM constellations, IEEE Transactions on Communications, vol. 5, no. 5, May 004. [5] E. Zehavi, 8-PSK trellis codes for a rayleigh channel, IEEE Transactions on Communications, vol. 40, no. 3, May 199. [6] G.Caire, G.Taricco, and E. Biglieri, Bit interleaved coded modulation, IEEE Transactions on Information Theory, vol. 44, no. 3, May [7] X. Li and J. Ritcey, Bit-interleaved coded modulation with iterative decoding using soft feedback, Electronic Letters, vol. 34, no. 10, May [8] F. Schreckenbach, N. Görtz, J. Hagenauer, and G. Bauch, Optimization of symbol mappings for bit-interleaved coded modulation with iterative decoding, IEEE Communications Letters, vol. 07, no. 1, Dec [9] S. Benedetto and E. Biglieri, Principles of Digital Transmission with Wireless Applications. Kluwer Academic, 1999.