BER and SER Analyses for M-Ary Modulation Schemes Under Symmetric Alpha-Stable Noise

Home , Atmospheric noise

Globecom 2014 - Wireless Communications Symposium 1 BER and SER Analyses for M-ary Modulation Schemes Under Symmetric Alpha-Stable Noise

Fan Yang† and Xi Zhang‡§ †School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China ‡Networking and Information Systems Laboratory, Department of Electrical and Computer Engineering Texas A&M University, College Station, TX 77843, USA §National Mobile Communications Research Laboratory, Southeast University, Nanjing, P. R. China E-mail: {yangf [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 characteristic 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 − α| |α indicator for evaluating channel qualities. ϕU (ω) = exp ( γs ω ). (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 ∫ ∞ 1 − α| |α signaling under SαS noise can be derived from its Gaussian f(u; α) = exp ( γs ω ) cos(uω)dω. (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 |U|] ( 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 { [ ( ) ]}  πα SCHEMES UNDER SαSNOISE  exp −γα |ω|α 1−jβsign(ω) tan +jµω ,  s 2  Notice that only Cauchy distribution has closed-form pdf  α ≠ 1; { [ for ] } in algebraic-tailed distributions when 0 < α < 2. We ﬁrst ϕ (ω)= U  2 α = 2  exp −γ |ω| 1+jβsign(ω) ln |ω| +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 ω 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 ∈ (0, 2], β is the skew the entire range of values for α ∈ (0, 2]. Thus we can parameter with β ∈ [−1, +1], γs is a scale parameter of generate a mapping mechanism from standardized normal distribution that controls the spread satisfying γs ∈ (0, +∞), distribution to an arbitrary SαS distribution. Subsequently, the and µ is the location parameter of the distribution such that bit error probability of BPSK signaling is derived. Based on µ ∈ (−∞, +∞). the framework of Gaussian counterpart, we can further extend If U ∼ S(α, 0, γs, 0) with β and µ are equal to zero, it to obtain the analytical expressions of M-ary modulation follows symmetric alpha-stable (SαS) distribution. Thus the system under SαS noise.

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A. Performance Analysis for BPSK Modulation can be expressed as ∫ ∞ ( ) A random variable Y follows general normal (or Gaussian) + 1 z2 Q(x) = √ exp − dz distribution with mean µ and variance σ2 can be represented as 2 x 2π ( ) ∼ 2 ∫ ∞ Y N(µ, σ ). The probability density function of Gaussian + 1 v2 = √ exp − dv distribution is given by √ 2 π 4 ( ) ∫ 2x − 2 +∞ √1 −(y µ) g(y) = exp 2 . (8) = √ f0(v; α = 2)dv. (17) σ 2π 2σ 2x We can set µ = 0 and σ2 = 1 in a general normal In Gaussian case, the relation between the Q-function and the tail probability function is shown below distribution, deriving a standard normal distribution. The stan- √ dardized process for a general normal distributed variable Y Q(x) = Qs( 2x; α = 2). (18) is implemented by changing Y to Z = (Y − µ)/σ, i.e., Z ∼ N(0, 1), yielding Now, we consider the bit error probability of BPSK modulation in the general Cauchy distribution (α = 1) which is the

g(y)dy = g0(z)dz, (9) only case that has closed-form pdf in algebraic-tailed distributions. The probability density function of Cauchy distribution where g0(z) is the pdf of standard normal distribution given can be represented as by γs 1 ( ) f(u; α = 1) = 2 2 . (19) 1 z2 π γs + u g0(z) = √ exp − . (10) 2π 2 We assume that the bipolar bits has equal transmission probability. Then the bit error probability of Cauchy case can According to [14], the analytical bit error probability of BPSK be calculated as ∫ modulation under AWGN channel can be expressed as the +∞ P BPSK { } form of Q-function b,α = Pr U > A = f(u; α = 1)du ( ) ∫ A ∫ √ +∞ +∞ 2E γs 1 1 1 P BPSK b = du = dv. (20) b = Q , (11) 2 2 2 N A π γs + u A π 1 + v 0 γs Then we have where Q(·) is the Q-function defined as: ( ) ∫ A +∞ ∫ ( ) ∫ P BPSK +∞ 2 +∞ b,α = Qs ; α = 1 = f0(v; α = 1)dv, (21) 1 z γs A Q(x) , √ exp − dz = g0(z)dz. (12) γs 2π x 2 x where f0(v; α = 1) is the pdf of standard Cauchy distribution By substituting Eq. (11) into Eq. (12), we have given by (√ ) ∫ 1 1 +∞ 2E f0(v; α = 1) = . (22) P BPSK b π 1 + v2 b = Q = √ g0(z)dz. (13) N0 2Eb N0 We recall that the bit error probability of BPSK modulation in Gaussian case can be written as According to [12], if random variable U ∼ S(α, 0, γ , 0), ( ) ∫ s A +∞ P BPSK then V = U/γs follows the so-called standard SαS distri- b = Qs ; α = 2 = √ f0(v; α = 2)dv. (23) ∼ γs 2 Es bution, i.e., V S(α, 0, 1, 0). The standardized process for N0 general SαS distributed variable yields According to Eq. (7), we obtain √ A ( 2 −1) E f(u; α)du = f0(v; α)dv, (14) = 4C α s . (24) γ g N s 0 √ where f0(v; α) is the pdf of standard SαS distribution by We can see that the lower limit of integration 2 Es/N0 setting γs = 1 in Eq. (4), i.e., in Eq. (23) is equal to A/γ in Eq. (21) when α = 2 in ∫ s 1 +∞ Eq. (24). Because the geometric power is defined consistently f (v; α) = exp (−|ω|α) cos(vω)dω, (15) ∈ 0 π continuous for α (0, 2], we can generate the following 0 performance mapping mechanism of BPSK modulation system where the integral formula for f0(v; α) can be implemented from the standard Gaussian to general SαS distribution as by [15] effectively. Referring to Eq. (12), we can define the follows (√ ) (√ ) ( √ ) tail probability function for SαS distribution as 2E 2E E ∫ P BPSK b s s +∞ b =Q =Q =Qs 2 ; α=2 N0 N0 N0 Qs(x; α) , f0(v; α)dv. (16) ( ) (√ ) x 2 − CM A ( α 1) Es BPSK = Q ; α =Q 4Cg ; α =P , 2 2 s γ s N b,α Notice that we have σ = 2√γs in Gaussian case (α = 2). By s 0 making substitution for v = 2z, the Q-function of Eq. (12) (25)

3985 Globecom 2014 - Wireless Communications Symposium 4 where the operation CM= denotes consistently mapping (CM), approximated by using the ﬁrst two dominant term, i.e.,   which is continuous for the whole entire of α, i.e., α ∈ (0, 2]. √ √ M − 1 3 Es P MQAM ≈ √ √  N0  b erfc − M log2 M 2(M 1) B. Extension to MPSK Modulation √  √  M − 2 3 Es First we consider bit error rate (BER) and symbol error rate + √ √ erfc 3 N0  2(M − 1) (SER) performance of the MPSK under AWGN channel. The M log2 M √   √  relation between Es/N0 and Eb/N0 is given by 3 Eb k 3 Eb k  N0   N0  = η1Q + η2Q 3 , Es Eb − − = k , (26) M 1 M 1 N N 0 0 (31) where k = Rc log2 M represents the information bits per where  ( ) symbol, Rc is the code rate of the system and Rc = 1 in  4 1  η1 = 1 − √ ; our case for uncoded system. k M ( ) (32) As [14] indicated that the MPSK has comparable perfor-   4 2 η2 = 1 − √ . mance as MQAM when M = 4, however, the latter is superior k M to the former when M ≥ 4. Thus, we only consider the and erfc(·) is the complementary error-function deﬁned as: representative QPSK with an initial phase π/4 instead of ∫ MPSK modulation, and show its SER performance as +∞ ( ) , √2 − 2 ( ) ( ) erfc(x) exp z dz, (33) √ √ π x E E P QPSK = 2Q s − Q2 s and the complementary error-function can be further expressed M N N ( 0 ) ( 0 ) in terms of the Q-function as √ √ ( ) 2E 2E √ = 2Q b − Q2 b . (27) erfc (x) = 2Q 2x . (34) N0 N0 The symbol error probability of square MQAM modulation The BER of QPSK modulation is equivalent to that of the is shown below [17] BPSK case shown in Eq. (13). √  √  3 Es 3 Es Based on the mapping mechanism of BPSK signaling under P MQAM = ξ Q  N0  − ξ Q2  N0  general SαS noise, α ∈ (0, 2], in Eq. (25), we drive M 1 M − 1 2 M − 1 (√ ) (√ ) √  √  2 Eb Eb 2Es CM ( −1) Es 3 k 3 k α  N0  2  N0  Q = Qs 4Cg ; α . (28) = ξ1Q − ξ2Q (35) N0 N0 M − 1 M − 1

According to Eq. (27) and Eq. (11), then the symbol error where,  ( ) probability and the equivalent bit error probability of QPSK  1  ξ1 = 4 1 − √ ; α α ∈ (0, 2] signaling under general S S noise, , can be repre- ( M )  2 (36) sented as  1  ξ2 = 4 1 − √ . P QPSK M M,α ( ) ( ) √ √ Based on the consistently mapping in Eq. (28), we have 2 −1 2 −1 ( α ) Es 2 ( α ) Es (√ ) (√ ) = 2Qs 2Cg ; α − Q 2Cg ; α s 2 − N0 N0 Es CM ( α 1) Es (√ ) (√ ) Q = Qs 2Cg ; α . (37) N0 N0 2 −1 E 2 −1 E ( α ) b − 2 ( α ) b = 2Qs 4Cg ; α Qs 4Cg ; α N0 N0 Similarly, according to Eq. (35) and Eq. (31), after some (29) algebraic manipulation, the symbol error probability and the bit error probability of MQAM signaling under general SαS and noise, α ∈ (0, 2], are described as v  (√ ) u 2 u 2 −1 ( −1) Eb ( α ) Eb P QPSK α t6Cg k  = Qs 4Cg ; α . (30) QPSK  N0  b,α N P = ξ Q ; α 0 M,α 1 s  M − 1  v  u u 2 −1 C. Extension to MQAM Modulation ( α ) Eb t6Cg k  − ξ Q2  N0 ; α (38) According to [16], the bit error probability of MQAM 2 s  M − 1  signaling (M ≥ 4) with square constellation can be well

3986 Globecom 2014 - Wireless Communications Symposium 5 and v  IV. NUMERICAL SIMULATION RESULTS u u 2 −1 We derived the analytical expressions of BPSK, QPSK and ( α ) Eb t6Cg k  MQAM  N0  MQAM modulation for uncoded systems under SαS noise. We P ≈ η1Qs  ; α b,α M − 1 employ Gray-coded constellation in each modulation scheme. ∈  v  The simulations are based on the SαS framework α (0, 2], u taking 107 independent trials for each E /N . For a given u 2 −1 b 0 ( α ) Eb  t6Cg k  characteristic exponent α, a group of curves including the + η Q 3 N0 ; α. (39) 2 s  M − 1  asymptotic expression, the simulation, and our analytical expression are used for comparison.

D. Asymptotic Performance of M-ary Modulation In order to evaluate the asymptotic performance, the tail -1 probability in a limited case is given by [13] 10 → Cα D 0.2 lim Qs(v; α) , (40) -2 v→∞ vα 10 D 1 where, ( ) D 1.7 -3 1 πα 10 Cα = Γ(α) sin , (41) π 2 BER

→ -4 the operation ∫ represents asymptotic, and the gamma func- 10 D 1.99 ∞ x−1 −t AWGN Simulation tion is Γ(x) = 0 t e dt. → AWGN Analytical 1). According to Eq. (25) and and Eq. (30), as (Eb/N0) -5 D 2 SDS Simulation 10 ∞, the asymptotic bit error provability of both BPSK and SDS Analytical QPSK signaling under SαS noise has the form SDS Asymptotic -6 SDS Empirical (√ )−α 10 0 5 10 15 20 25 30 35 40 2 −1 E P BPSK → ( α ) b Eb/No(dB) b,α Cα 4Cg . (42) N0 Fig. 1. The BER of BPSK signaling under SαS noise corresponding to α 2). According to Eq. (29) , as (Eb/N0) → ∞, the asymp- equals 2, 1.99, 1.7, and 0.2, respectively. totic symbol error probability of QPSK signaling under SαS noise has the form We show the BER of BPSK signaling in Fig. 1. In order (√ )−α 2 −1 E to have a deep insight, we add the empirical approximation P QPSK → ( α ) b M,α 2Cα 4Cg obtained in [5] for further comparison. The BER performance N0 (√ )−2α of BPSK signaling under Gaussian noise are used as bench- 2 −1 E mark. We can see from Fig. 1 that the simulation results are − 2 ( α ) b Cα 4Cg . (43) N0 well matched with our analytical BER expressions one-by- one even for small values of α. For a given α, three curves → ∞ 3). According to Eq. (38) and Eq. (39), as (Eb/N0) , (the asymptotic expression, the simulation, and our analytical ≥ the asymptotic performance of MQAM signaling (M 4) expression) are almost overlapped in large SNR region. The under SαS noise has the form v  analytical BER curve under SαS noise is close to the one u −α u 2 −1 under Gaussian case in low SNR region when α approaching ( α ) Eb t6Cg k  P MQAM →  N0  to 2, while it deviated when α gets smaller. M,α ξ1Cα   M − 1 On( √ the other) hand, empirical approximation using Qs 2 Es/N0 in [5] is very close to the exact BER perfor- v − u 2α mance when the value of α approaching to 2, such as α = 1.99 u 2 −1 ( α ) Eb t6Cg k  and α = 1.7. However, this approximation deviates from − ξ C2  N0  , (44) 2 α  M − 1  the three curves (the asymptotic expression, the simulation, and our analytical expression) when α gets smaller.√ It can 2 E /N and be see more clearly√ from Eq. (30) that the value s 0 − v −α [(2/α) 1] u is approximate to 4Cg (Es/N0) when α is close to u 2 −1 ( α ) Eb t6Cg k  2. We can deduce that the approximation used in [5] is not P MQAM → η C  N0  b,α 1 α  M − 1  accurate for small values of α and higher-order modulation types.  v  u −α The BER and SER results for M-ary signaling under SαS u 2 −1 ( α ) Eb noise (α = 1.9) are shown in Fig. 2 and Fig. 3, respectively.  t6Cg k   N0  Also, we employ the Gaussian case as benchmark. The trend + η2Cα 3  . (45) M − 1 for the BER and SER curves under SαS noise corresponding to different order of modulation is similar to the Gaussian

3987 Globecom 2014 - Wireless Communications Symposium 6 case, i.e., the BER and SER degrade as the modulation order V. CONCLUSIONS M increases. By employing zero-order statistics, we created a consistently It can be observed that the QPSK and QAM signaling continuous mapping mechanism acting as a linkage between have almost the same performance either in Gaussian or in the Gaussian and the general SαS framework. Using our impulsive noise channels. Consider impulsive noise channel, developed mapping mechanism, we can simply derive the ana- the SER of the QAM in Eq. (38) (M = 4) is equal to that of lytical BER and SER expressions of M-ary signaling including the QPSK in Eq. (29), and the BER of the QAM in Eq. (39) BPSK, QPSK, and MQAM under SαS noise. We deduced that dominated by the first term is approximate to that of the QPSK the analytical expressions for other conventional modulated in Eq. (30). Moreover, we can see that the asymptotic curves signals under SαS noise can be also derived by using this can only approximate to the exact performance for large SNR mapping mechanism based on its Gaussian counterpart. Our if higher-order modulation is implemented. For example, as analytical expressions have been verified through simulations, shown in Fig. 3, the Eb/N0 is needed above 10dB to get a and provide a valuable benchmark for deriving the exact better SER approximation for QAM signaling, however, it is coding gain under AWSαSN channels. required up to 30dB for 1024-QAM signaling. This can be seen more clearly in Eq.√ (44) that we can get better asymp- REFERENCES [(2/α)−1] totic performance for 6kCg (Eb/N0)/(M − 1) by [1] J. Lin, M. Nassar, and B. Evans, “Non-Parametric Impulsive Noise increasing Eb/N0 in the numerator when a lager M is used Mitigation in OFDM Systems Using Sparse Bayesian Learning,” in IEEE Global Telecommunications Conference (GLOBECOM 2011), in the denominator of the square root. Dec. 2011, pp. 1–5. [2] M. Souryal, E. Larsson, B. Peric, and B. Vojcic, “Soft-Decision Metrics for Coded Orthogonal Signaling in Symmetric Alpha-Stable Noise,” D =1.9 IEEE Transactions on Signal Processing, vol. 56, no. 1, pp. 266–273, AWGN Simulation Jan. 2008. -1 AWGN Analytical 10 [3] M. Chitre, S. Kuselan, and V. Pallayil, “Ambient noise imaging in warm SDS Simulation shallow waters; robust statistical algorithms and range estimation,” The SDS Analytical Journal of the Acoustical Society of America, vol. 132, no. 2, pp. 838– -2 S S Asymptotic 10 D 847, 2012. [4] D. Zha and T. Qiu, “Underwater sources location in non-Gaussian 1024-QAM impulsive noise environments,” Digital Signal Processing, vol. 16, no. 2, -3 10 256-QAM pp. 149–163, 2006. [5] M. Chitre, J. Potter, and S. Ong, “Viterbi Decoding of Convolutional BER 64-QAM Codes in Symmetric Alpha-Stable Noise,” IEEE Transactions on Com- -4 10 munications, vol. 55, no. 12, pp. 2230–2233, Dec. 2007. [6] T. Saleh, I. Marsland, and M. El-Tanany, “Simplified LLR-based Viterbi decoder for convolutional codes in symmetric alpha-stable noise,” -5 10 in IEEE Canadian Conference on Electrical Computer Engineering (CCECE), April 2012, pp. 1–4. 16-QAM [7] T. Shehata, I. Marsland, and M. El-Tanany, “Near Optimal Viterbi -6 QPSK, QAM 10 Decoders for Convolutional Codes in Symmetric Alpha-Stable Noise,” 0 5 10 15 20 25 30 35 40 in IEEE 72nd Vehicular Technology Conference Fall (VTC 2010-Fall), Eb/No(dB) Sept. 2010, pp. 1–5. [8] S. Mohammad, L. Heng-Siong, and C. Teong-Chee, “Decoding of Turbo Codes in Symmetric Alpha-Stable Noise,” ISRN Signal Processing, vol. Fig. 2. The BER of M-ary signaling under SαS noise with α = 1.9. 2011, 2011. [9] J. Forney, G.D. and G. Ungerboeck, “Modulation and coding for linear Gaussian channels,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2384–2415, Oct. 1998. =1.9 0 D [10] A. Mahmood, M. Chitre, and M. Armand, “PSK Communication with 10 AWGN Simulation Passband Additive Symmetric alpha-Stable Noise,” IEEE Transactions AWGN Analytical on Communications, vol. 60, no. 10, pp. 2990–3000, October 2012. -1 10 SDS Simulation [11] J. Gonzalez, J. Paredes, and G. Arce, “Zero-Order Statistics: A Math- SDS Analytical ematical Framework for the Processing and Characterization of Very SDS Asymptotic Impulsive Signals,” IEEE Transactions on Signal Processing, vol. 54, -2 10 1024-QAM no. 10, pp. 3839–3851, Oct. 2006. 256-QAM [12] J. Nolan, Stable distributions: models for heavy-tailed data. Birkhauser, 64-QAM 2003. -3 10 [13] G. Sureka and K. Kiasaleh, “Sub-Optimum Receiver Architecture for SER AWGN Channel with Symmetric Alpha-Stable Interference,” IEEE Transactions on Communications, vol. 61, no. 5, pp. 1926–1935, May -4 10 2013. [14] J. G. Proakis, Spread spectrum signals for digital communications.

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