Globecom 2014 - Wireless Communications Symposium 1 BER and SER Analyses for M-ary Modulation Schemes Under Symmetric Alpha-Stable Noise Fan Yangy and Xi Zhangzx ySchool of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China zNetworking and Information Systems Laboratory, Department of Electrical and Computer Engineering Texas A&M University, College Station, TX 77843, USA xNational Mobile Communications Research Laboratory, Southeast University, Nanjing, P. R. China E-mail: fyangf [email protected], [email protected] Abstract—Symmetric alpha-stable (SαS) distribution has been generally does not exist, except for the scenarios of Gaussian widely used to model undesirable impulsive noise disturbance and Cauchy distributions with characteristic exponent α = 2 in many scenarios. Due to lack of probability density function and α = 1, respectively. The performance of an uncoded (pdf) of SαS distribution (except Cauchy and Gaussian cases), the general closed-form expression of the bit error probability or system especially for higher-order modulation under arbitrary symbol error probability for M-ary modulation schemes has not SαS noise (0 < α ≤ 2) remain unexplored, presenting the been derived yet, preventing the derivation of the exact coding exact coding gain from being derived. gain from being feasible. By employing geometric power involved The use of high-order modulation can achieve higher speed in zero-order statistics, we create a mapping mechanism which communication by allowing more bits per symbol transmitted is consistently continuous along the entire range of characteris- tic values. Then we derive the accurate bit error probability in parallel [9]. The need for high capacity transmission with and symbol error probability of M-ary modulation schemes M-ary (M ≥ 4) modulation such as M-ary quadrature under SαS noise. Our obtained derivations agree well with our amplitude modulation (MQAM) and M-ary phase-shift keying simulations, which provide a unified framework for uncoded (MPSK), is highly demanded in impulsive noise environments. systems using M-ary modulation under additive white Gaussian This trend can be seen more clearly in power line communi- noise (AWGN) and additive white symmetric alpha-stable noise (AWSαSN). Also, it enables the design of capacity approaching cation (PLC) applications. The HomePlug Green PHY (GP) codes especially for higher-order modulation scenarios. supports quadrature phase shift-keying (QPSK) modulation in smart grid applications, while the HomePlug AV provides up Index Terms—M-ary modulation, symmetric alpha-stable noise (SαS), probability density function (pdf), smart grid, non- to 1024-QAM in its physical layer. Moreover, the HomePlug Gaussian, geometric signal-to-noise ratio (GSNR), zero-order AV2 takes much higher orders of modulation 4096-QAM into statistics. consideration. As [10] stated that all complex baseband additive white SαS I. INTRODUCTION noise (AWSαSN) samples are independent of each other for LPHA-STABLE noise processes with infinite-variance any α. If the carrier and bandpass sampling frequencies satisfy A that satisfy generalized central limit theorem (GCLT) a certain condition, the real and imaginary components of the can appear in many practical scenarios. The symmetric alpha- baseband AWSαSN sample can be independent. This feature stable (SαS) process with heavy tailed distributions is shown provides an elegant way to implement M-ary modulation to give a good approximation to real-world impulsive noise, schemes under AWSαSN channel. The authors of [5] proposed such as in power lines noise, underwater acoustic noise, and a p-norm metric for Viterbi decoding under AWSαSN channel, atmospheric noise [1]–[4]. and obtained an approximate bit error probability through ≤ Because communication systems originally designed for empirical experiments for 1 < α 2. The approximation Gaussian noise may perform very poorly in impulsive noise, is efficient when α is close to 2 (the Gaussian case), while several channel coding schemes using binary phase shift- the gap between the approximation and theoretical value keying (BPSK) modulation were proposed for performance increases as the value of α gets smaller. Moreover, the authors improvement [2], [5]–[8]. However, the closed-form expres- of [5] only considered the empirical approximation for BPSK sion of probability density function (pdf) of SαS distribution signaling, but didn’t derive the theoretical results for higher- order constellations. This work was supported in part by the U.S. National Science Foundation Conventionally, the bit error probability and symbol error under Grant CNS-1205726, the U.S. National Science Foundation CAREER Award under Grant ECCS-0348694, the National Natural Science Foundation probability of a communication system can be measured by of China (No. 61301272), the Open Research Fund of National Mobile Com- the ratio of signal energy per bit Eb to the noise power spectral munications Research Laboratory, Southeast University (No. 2012D12), the density N0 or signal-to-noise ratio (SNR) using second-order Applied Basic Research Foundation of Science and Technology Department of Sichuan Province (No. 2014JY0037), and the Fundamental Research Funds statistics. However, the alpha-stable process has infinite vari- for the Central Universities (No. ZYGX2013J008). ance, for which neither the classical second-order statistics nor U.S. Government work not protected by U.S. copyright 3983 Globecom 2014 - Wireless Communications Symposium 2 the higher-order statistics are well defined. According to the characteristic function of a univariate SαS distribution can be zero-order statistics framework [11], geometric signal-to-noise simplified to power ratio (GSNR) defined by geometric power is a suitable − αj jα indicator for evaluating channel qualities. ϕU (!) = exp ( γs ! ): (3) To overcome the aforementioned problems, in this paper we develop the performance modeling technique/framework When α = 2, the SαS distribution is converted to Gaussian 2 2 under the SαS noise. Motivated by the Cauchy distribution, the distribution with finite variance σ = 2γs . When α < 2, the only case that has closed-form pdf for no-Gaussian distribution SαS distribution is algebraic-tailed with infinite variance, and with 0 < α < 2, we create a mapping mechanism from only Cauchy distribution has closed-form pdf. Smaller value standardized normal distribution to general SαS distribution of α indicates the heavier tail of the density function, resulting by utilizing the closed-form pdf of Cauchy distribution. Then in more impulsive noise. We employ GSNR to measure the the analytical performance expressions for binary phase-shift noise impulsiveness, which is defined based on the zero-order keying (BPSK) modulation can be readily derived. Because the statistics framework. Even there is no close form pdf for SαS geometric power is consistently continuous along the entire distribution, we can represent it using integral method as [13] range of values of α, the analytical expressions of M-ary Z 1 1 − αj jα signaling under SαS noise can be derived from its Gaussian f(u; α) = exp ( γs ! ) cos(u!)d!: (4) counterpart. Using our developed mapping mechanism, one π 0 can derive the coding gains under SαS noise for systems Let U be a logarithmic-order variable. According to [11], employing different types of modulations, including, but not the geometric power of U is defined as: limited to, BPSK, QPSK, MQAM, etc. 1 − The rest of the paper is organized as follows. Section II E[log jUj] ( α 1) S0 = S0(U),e = γsCg ; (5) describes system model and the GSNR involved in zero-order statistics. Section III derives the analytical expressions for M- where E(·) denotes taking expectation operation, Cg is the ary modulations. Section IV evaluates our obtained analytical exponential of Eulers constant and Cg ≈ 1:78. bit error probability and symbol error probability through The GSNR ΥG designed involving the Gaussian case (α = simulations. The paper concludes with Section V. 2) can be expressed as ( )2 II. THE SYSTEM MODEL 1 A A2 Υ = = ; (6) G ( 2 −1) We consider the n-th transmit signal sample s (n) (i = 2Cg S0 α 2 i 2Cg γs 1; 2; :::; M) using M-ary modulation and Gray mapping, cor- rupted by a complex AWSαSN channel. Then the correspond- where A is the root mean square (RMS) amplitude of the ing received signal sample r(n) is given by transmitted signal. For the real or the imaginary of complex baseband SαS r(n) = si(n) + w(n); (1) noise, we have where w(n) is the complex symmetric α-stable (SαS) noise E 1 A2 s = Υ = ; (7) sample, which is assumed to be independent and identically G ( 2 −1) N0 2 4C α γ2 distributed (i.i.d.) for its real and imaginary components. g s Generally, a random alpha-stable variable U is called fol- where Es is the signal energy per symbol. lowing S(α; β; γs; µ) distribution [12] can be represented as U ∼ S(α; β; γs; µ), which has characteristic function III. GENERAL PERFORMANCE ANALYSIS FOR M -ARY n h ( ) io 8 πα SCHEMES UNDER SαSNOISE > exp −γα j!jα 1−jβsign(!) tan +jµω ; > s 2 > Notice that only Cauchy distribution has closed-form pdf < α =6 1; { [ for ] } in algebraic-tailed distributions when 0 < α < 2. We first ϕ (!)= U > 2 α = 2 > exp −γ j!j 1+jβsign(!) ln j!j +jµω ; consider the bit error probability of the Gaussian cases ( ) > s π :> and the Cauchy case (α = 1), respectively, where both of the for α = 1; two distributions have close-form standard pdf in alpha-stable (2) framework. Then we indicate that the lower limit of integration ! is the frequency-domain variable, α is usually called the in derivation of the bit error probability is closely related to characteristic exponent and determines the heaviness of the the geometric power, which is consistently continuous along tails for the distribution such that α 2 (0; 2], β is the skew the entire range of values for α 2 (0; 2].