Exact Bit Error Probability of M -QAM Modulation Over Flat Rayleigh Fading Channels Waslon T. A. Lopes∗ , Wamberto J. L. Queiroz† , Francisco Madeiro‡ and Marcelo S. Alencar§ ∗ Faculdade
´ AREA1, Salvador, BA, Brazil de Fortaleza, Fortaleza, CE, Brazil ‡ Escola Polit´ecnica de Pernambuco, Universidade de Pernambuco, Recife, PE, Brazil § Universidade Federal de Campina Grande, Campina Grande, PB, Brazil E-mails:
[email protected],
[email protected],
[email protected] and
[email protected] † Universidade
Abstract— In this paper we derive a general and closedform expression for the bit error probability of square M -ary quadrature amplitude modulation (M -QAM) for a Rayleigh fading channel.
derived in [7] an expression for the BEP of square M QAM for an AWGN channel. It is given by Pb =
I. I NTRODUCTION The growing need for improvements in capacity and performance of wireless communications systems has imposed some challenges in the scenario of achieving high transmission rates, suitable to accommodate the ever-increasing multimedia traffic and applications. In this context, spectrally-efficient modulation schemes have gained great attention. M -ary quadrature amplitude modulation (M -QAM) is an attractive technique for achieving high data rate transmission without increasing the bandwidth of wireless communications systems. Although many works (e.g. [1]–[6]) have been devoted to assess the performance of quadrature amplitude modulation in terms of bit error rate, only recently, in a paper by Cho and Yoon [7], a closed-form expression for the bit error probability (BEP) of an arbitrary square M QAM constellation for an additive white Gaussian noise (AWGN) channel has been derived. In this paper, results presented by Cho and Yoon in [7] are used to derive a general and closed-form expression for the BEP of an arbitrary M -QAM for a flat Rayleigh fading channel when Gray code bit mapping is employed. The exact BEP expression derived in this paper provides a convenient way to evaluate the performance of QAM for various cases of practical interest. II. BEP DERIVATION Based on the consistency of the bit mapping of a Gray coded signal constellation [8], Cho and Yoon have
1 √
log2
log2
M
√
XM
Pb (k),
(1)
k=1
with 1 Pb (k) = √ M
√ (1−2−k ) M −1(
X
w(i, k, M )·
i=0
s
erfc (2i + 1)
3 log2 M · γ 2(M − 1)
!)
(2)
,
where i · 2k−1 1 k−1 + w(i, k, M ) = (−1) · 2 − √ , 2 M (3) γ = Eb /N0 denotes the signal-to-noise ratio (SNR) per bit, ⌊x⌋ denotes the largest integer smaller than x, and erfc(·) denotes the complementary error function, given by Z ∞ 2 2 e−t dt. (4) erfc(x) = √ π x k−1
⌊ i·2√M ⌋
The main insight of Cho and Yoon [7] was to express the BEP of square M -QAM for an AWGN channel in terms of a weighted sum of complementary error functions. Those weights incorporate the effect of the position k of the bits in a log2 M -bit symbol on the BEP. Consider now a flat Rayleigh fading channel with fading amplitude α, whose probability density function (pdf) is expressed as 2
pA (α) = 2αe−α u(α),
(5)
where u(·) is the unit step function. Assume also perfect fading estimation and symbol synchronization. The corresponding BEP of square M -QAM for a given α may be obtained from (1)-(3) if one takes into account that after the fading effect the received signal-to-noise ratio will be modified by α2 [3]. So, the probability that the kth bit is in error for a given α can be expressed as 1 Pb|α (k) = √ M
√ (1−2−k ) M−1(
X
erfc (2i + 1)
√ ( (1−2−k ) M −1
X
w(i, k, M )·
i=0
q
1 − q
3(2i+1)2 log2 M ·γ 2(M −1)
3(2i+1)2 log2 M ·γ 2(M −1)
+1
)
(11)
.
.
(6)
III. R ESULTS
i=0
3 log2 M · γ · α2 2(M − 1)
1 PRay (k) = √ M
It is worth mentioning that the expression obtained is this paper is simpler than the one that could be obtained from the results presented by Yoon and Cho in [10]. The computation of the BEP by the results presented in [10] involves a hypergeometric function.
w(i, k, M )·
s
with
!)
Some numerical examples of the closed-form expression of the bit error probability of M -ary QAM for a flat Rayleigh fading channel are presented in Fig. 1, which shows the BEP as a function of the SNR per bit for M = 4, 16, 64 and 256. As shown in Fig. 1, −∞ √ the analytical results, obtained from (10), (11) and (3), Z ∞ (1−2−k ) M−1( X 1 strongly agree with Monte Carlo simulation results. It √ = w(i, k, M )· is observed in Fig. 1, for instance, that 3–4 dB of SNR M 0 si=0 !) have to be invested to transmit an extra bit per quadrature 3 log2 M · γ · α2 2 2αe−α dα.component (two extra bits per symbol), to maintain the erfc (2i + 1) 2(M − 1) average bit error probability of 10−2 . (7) IV. C ONCLUSION Considering that erfc(x) = 1 − erf(x) and using the In this paper, an exact and closed-form expression substitution α2 = u, the previous expression becomes has been derived for the bit error probability of M -ary quadrature amplitude modulation for a channel subject to √ Z ∞" (1−2−k ) M−1( Rayleigh fading. The BEP expression for QAM with an X 1 1− PRay (k) = √ w(i, k, M ) · arbitrary constellation size was shown to be in agreement M 0 with Monte Carlo simulation results. s i=0 !# ) 2 As future works, the authors will investigate the effects 3(2i + 1) log2 M · γ · u erf e−u du . of channel estimation errors as well as multipath scenar2(M − 1) (8) ios on the bit error probability of modulation schemes subject to Rayleigh fading. The corresponding BEP can be obtained by averaging (6) with respect to the pdf of α, that is, Z ∞ Pb|α (k) · pA (α) dα PRay (k) =
Taking into account that [9, Eq. 6.283] Z
∞ 0
√
p 1 ψ √ eβx [1 − erf( ψx)]dx = β ψ−β
R EFERENCES − 1 , (9)
for ℜ[ψ] > 0 and ℜ[β] < ℜ[ψ], where ℜ[x] represents the real part of x. Thus, the overall BEP for square M QAM is finally obtained as
PRay =
1 √
log2
log2
M
√
XM k=1
PRay (k),
(10)
[1] J. Lu, K. B. Letaief, J. C.-I. Chuang and M. L. Liou. “M PSK and M -QAM BER Computation Using Signal-Space Concepts”. IEEE Transactions on Communications, vol. 47, no. 2, pp. 181–184, February 1999. [2] L.-L. Yang and L. Hanzo. “A Recursive Algorithm for the Error Probability Evaluation of M -QAM”. IEEE Communications Letters, vol. 4, no. 10, pp. 304–306, October 2000. [3] P.-M. Fortune, L. Hanzo and R. Steele. “On the Computation of 16-QAM and 64-QAM Performance in Rayleigh-Fading Channels”. IEICE Transactions on Communications, vol. E75B, no. 6, pp. 466–475, June 1992.
Bit Error Probability
100
10−1
10−2
10−3
Fig. 1.
256−QAM (Analytical) 256−QAM (Simulation) 64−QAM (Analytical) 64−QAM (Simulation) 16−QAM (Analytical) 16−QAM (Simulation) 4−QAM (Analytical) 4−QAM (Simulation)
0
5
10
15 Eb/N0 (dB)
20
25
30
Bit eror probability of M -QAM as a function of the signal-to-noise ratio per bit (Eb /N0 ) for a flat Rayleigh fading channel.
[4] M. G. Shayesteh and A. Aghamohammadi. “On the Error Probability of Linearly Modulated Signals on Frequency-Flat Ricean, Rayleigh and AWGN Channels”. IEEE Transactions on Communications, vol. 43, no. 2/3/4, pp. 1454–1466, February/March/April 1995. [5] L. Hanzo, R. Steele and P.-M. Fortune. “A Subband Coding, BCH Coding, and 16-QAM System for Mobile Radio Speech Communications”. IEEE Transactions on Vehicular Technology, vol. 39, no. 4, pp. 327–339, November 1990. [6] P. K. Vitthaladevuni and M.-S. Alouini. “BER Computation of 4/M -QAM Hierarchical Constellations”. IEEE Transactions on Broadcasting, vol. 47, no. 3, pp. 228–239, September 2001. [7] 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, pp. 1074– 1080, July 2002. [8] P. J. Lee. “Computation of the Bit Error Rate of Coherent M ary PSK with Gray Code Bit Mapping”. IEEE Transactions on Communications, vol. 34, no. 5, pp. 488–491, May 1986. [9] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Academic Press, 1979. [10] D. Yoon and K. Cho. “General Bit Error Probability of Rectangular Quadrature Amplitude Modulation”. Electronics Letters, vol. 38, no. 3, pp. 131–133, January 2002.
