Bit Error Probability of M-QAM under Impulsive Noise and Fading Modeled by Markov Chains

RADIOENGINEERING, VOL. 27, NO. 4, DECEMBER 2018 1183 Bit Error Probability of M-QAM Under Impulsive Noise and Fading Modeled by Using Markov Chains Hugerles S. SILVA 1, Marcelo S. de ALENCAR 2, Wamberto J. L. de QUEIROZ 1, Rodrigo de A. COELHO 1, Francisco MADEIRO 3 1Dept. of Electrical Engineering, Federal University of Campina Grande, Campina Grande, Paraíba, Brazil 2Dept. of Electrical Engineering, Federal University of Bahia, Salvador, Bahia, Brazil 3University of Pernambuco, Recife, Pernambuco, Brazil {hugerles.silva, rodrigo.almeida}@ee.ufcg.edu.br, {wamberto, malencar}@dee.ufcg.edu.br, [email protected] Submitted February 13, 2018 / Accepted August 12, 2018 Abstract. This article presents new exact expressions, written in terms of elementary transcendental functions, for calculating the bit error probability of M-ary Quadrature Amplitude Modulation (M-QAM) scheme considering the wireless communication channel modeled by a Markov chain with Nstates. For the numerical evaluation of the expressions obtained, a particular case of a Markov chain with two states is considered, with each state representing distinct scenarios. In the first scenario it is considered the presence of Gated Additive White Gaussian Noise (GAWGN) and fading η-µor κ-µ, while the second scenario considers the presence of the Double Gated Additive White Gaussian Noise (G2AWGN) and fading η-µor κ-µ. Bit error probability curves as a function of the signal-to-permanent-noise ratio for different values of the signal-to-impulsive-noise ratio, fading parameters and modulation order Mare also presented. Keywords Markov chains, η-µfading, κ-µfading, bit error probability, impulsive noise 1. Introduction The signal transmitted in industrial environments and indoor environments, such as shopping malls, is subjected to noise from multiple sources and interference in the spectrum band intended for industrial applications in 2.4GHz [1]. Beyond the noise usually modeled as an additive white Gaussian process, these environments also present noise of impulsive nature that arise mainly due to the presence of numerous thyristorized equipment whose switching can generate disturbances in the wireless links. Electric motors, frequency inverters, welding equipment, triggering of lamps, among others, are examples of sources of impulsive noise [2]. Another challenge commonly encountered in industrial environments is fading, which imposes random variations of intensity on the transmitted signal [1]. This phenomenon arisesduetothe largenumberofobjects andmachinespresent in these environments, which are usually constructed with metallic materials and provide appropriate conditions to multipath propagation [3]. The combination of phenomena such as impulsive noise and fading can seriously compromise the quality of the communication link. In such wireless environments, the effects of fading and impulsive noise on the observed signal at the output of a matched filter in the receiver may be modeled by a twoterm composition, one representing the information signal affected by fading and the other term representing the impulsive noise [2]. In non selective frequency channel models, fading is usually considered flat and characterized by a single probability distribution over at least onesymbol time [4]. However, in environments such as shopping malls and factories, a single probability distribution may not be able to appropriately model the intensity variations imposed on the transmitted signal as the receiver moves and may experience different environment conditions for multipath propagation. In [2] industrial environments are considered and more than one probability distribution is used to characterize the effects of fading. A possible way to model sudden changes in the statistical behavior of the transmitted signal intensity, as the receiver moves through the communication environment, is by means of Markov chains [5–7]. One advantage of considering Markov chains, compared to the non selective frequency model, is that they allow to better describe the statistical changes of fading over time [8]. These variations can be caused by changes in the scenarios in which the transmissions are performed and can affect both up and downlinks on reduced-diameter cells as well as horizontal device-to-device links [9]. DOI: 10.13164/re.2018.1183 SYSTEMS 1184 H. SILVA, M. ALENCAR, W. QUEIROZ, ET. AL., BIT ERROR PROBABILITY OF M-QAM UNDER IMPULSIVE NOISE . .. Other authors have also used Markov chains to describe statistical behavior change in the channel, such as Liu et al. [10] and Altinel and Kurt [11]. Lutz et al. [12], for example, defined a channel model with two states, one classified as good and the other classified as bad, depending on the conditions and level of attenuation in each scenario. Another important work is that of Vucetic and Du [13], which characterized the wireless channel of a particular geographical area of Australia by a Markov chain of four states, combining four different types of conditions. In this work, exact expressions for the computation of the average bit error probability (BEP), Pe, of M-ary Quadrature Amplitude Modulation (M-QAM) scheme are determined considering that the alternations in the fading statistics along the channel are described by the state transitions of a Markov chain with Nstates. The expressions obtained are novel and written in terms of elementary transcendental functions. For the numerical evaluation of the expressions obtained, a particular case of a Markov chain with two states is considered, with each state representing distinct scenarios. In the first state of the chain, the presence of the Gated Additive White Gaussian Noise (GAWGN) and fading η-µor κ-µis considered, whereas in the second state the presence of the Double Gated Additive White Gaussian Noise (G2AWGN) and fading η-µor κ-µis considered. In the mathematical characterization of the impulsive noise, we have considered the models GAWGN and G2AWGN because they are able to characterize, respectively, the occurrence of noisy pulses and bursts of noisy pulses. By its turn, the fading is characterized by the distributions η-µand κ-µ, which are generalist models of fading and encompass, as special cases, the distributions of Rayleigh, Rice and Nakagami, among others, and may be used to characterize fading on a small and large scale, with or without line-of-sight [14]. The methodology used to determine Peis to weight the BEP of the scheme M-QAM under the composite noise models GAWGN and G2AWGN by the probability density function (PDF) of the fading envelope. In this approach, the representation of Craig [15] is used for the function Q(·), expressing Pefor an integral defined in the range of 0 to π/2, in terms of the Moment Generating Function (MGF) of the distributions η-µand κ-µ. 2. Noise Model The mathematical model of noise η(t)is given by [16] η(t)=ηg(t)+C(t)ηi(t),(1) in which ηi(t)represents a zero-mean complex white Gaussian random process with variance σ2 i,C(t)is a signal which models the occurrence of noise ηi(t), characterized by a continuous time and discrete Bernoulli random process, and ηg(t) is the background Gaussian noise with zero-mean and variance σ2 g. The product C(t)ηi(t), referred to as impulsive noise, in (1), characterizes the noise ηi(t)gated by the process C(t). The noise can be simple or double gated. In simple gated noise, the noise is referred to as Gated Additive White Gaussian Noise (GAWGN) and the amplitude of C(t)assumes values zero or one randomly, with probabilities 1−pand p, respectively. The signal C(t)has unit amplitude in the interval −αpT/2≤t≤αpT/2and zero otherwise. The variable αpcan assume values between zero and one. In double gated noise, the noise is referred to as Double Gated Additive White Gaussian Noise (G2AWGN) and the random signal C(t)is characterized by the product of two auxiliary processes, C1(t)and C2(t), that take values in the discrete set {0,1}. In the intervals at which C1(t)assumes value one, several noisy pulses, modeled as C2(t)ηi(t), can occur. One can say that the interval for which C1(t)=1 is a burst of noisy pulses C2(t)ηi(t). The product C1(t)C2(t) models the occurrence of bursts of noisy pulses. The function C1(t)assumes the values one and zero with probabilities p1and 1−p1, respectively. This function has unit amplitude in the interval 0≤t≤βT1and zero otherwise. The signal C2(t)assumes the values one and zero randomly, with probabilities p2and 1−p2, respectively, and has unit amplitude in the interval 0≤t≤αpT2and zero otherwise. The variables βeαpassume values between zero and one. 3. Average Bit Error Probability of M-QAM Under Noise and Fading Modeled by Using a NStates Markov Chain It was shown in [17] that the BEP of M-QAM for a given fading z, denoted by P(e|z), under GAWGN, can be written as P(e|z)=2 √Mlog2√M log2√M Õ k=1 (1−2−k)√M−1 Õ i=0 w(i,k,M) ×(αppQ sa(i,M)z2δgδi δg+δi! +(1−αpp)Qqa(i,M)z2δg,(2) in which w(i,k,M)=(−1)ji·2k−1 √Mk·2k−1−i·2k−1 √M +1 2,(3) a(i,M)=3(2i+1)2 (M−1)log2M,(4) Mis the order of the constellation, δgis the signal-topermanent-noise ratio, defined as the ratio of the signal power to the power of the background Gaussian noise that is always present in the system, and δiis the signal-to-impulsive-noise ratio, defined as the ratio between the power of the signal and the power of the impulsive noise that acts in the system. RADIOENGINEERING, VOL. 27, NO. 4, DECEMBER 2018 1185 Making x=z2(5) in (2), P(e|z)can be written as P(e|z=√x)=2 √Mlog2√M log2√M Õ k=1 (1−2−k)√M−1 Õ i=0 w(i,k,M) ×(αppQ sa(i,M)xδgδi δg+δi! +(1−αpp)Qqa(i,M)xδg.(6) The bit error probability, Pe, can be obtained by taking the average of (6) with respect to the PDF of x, that is, Pe=∫∞ 0 P(e|z=√x)fX(x)dx =2 √Mlog2√M log2√M Õ k=1 (1−2−k)√M−1 Õ i=0 w(i,k,M) ×(αpp∫∞ 0 Q sa(i,M)xδgδi δg+δi!fX(x)dx +(1−αpp)∫∞ 0 Qqa(i,M)xδgfX(x)dx,(7) in which fX(x)is the PDF of the fading. An alternative expression for the function Q(·)is given by [15] Q(x)=1 π∫π 2 0 exp −x2 2sin2θdθ. (8) Thus, Pecan be written as Pe=2 π√Mlog2√M log2√M Õ k=1 (1−2−k)√M−1 Õ i=0 w(i,k,M) ×(αpp∫π 2 0∫∞ 0 exp −a(i,M)x 2sin2θ δgδi δg+δifX(x)dxdθ +(1−αpp)∫π 2 0∫∞ 0 exp −a(i,M)x 2sin2θδgfX(x)dxdθ). (9) Since the fading is constant for at least one symbol interval Ts, it is possible to represent the behavior of the channel by means of a Markov chain of Nstates. In the model considered in this article, whenever the fading changes to a particular state, it is assumed that it will remain in this state for a time Ti=kTsseconds. The parameter Tiis a discrete random variable with distribution pTi(t)and average value ¯ ti. In a given state j, the fading envelope is characterized by a random variable Xjwith probability density function fXj(x). Under these conditions, the probability of X≤xcan be written as P(X≤x)= N Õ i=1 N Õ j=1 P(X≤x,Ti,ej) = N Õ i=1 N Õ j=1 P(X≤x|Ti,ej)P(Ti,ej).(10) Given that [18] P(Ti,ej)= P(ej)¯ tiδ[i−j] ÍN u=1ÍN v=1¯ tuP(ev)δ[u−v],(11) where δ[i−j]is the Kronecker delta, which assumes value equal to one if i=jand zero otherwise; ¯ tiis the average duration of the i-th state and P(ej)is the probability that the chain be in the j-th state after ntransitions, it follows that P(X≤x)= N Õ i=1 N Õ j=1 P(X≤x|Ti,ej) ×P(ej)¯ tiδ(i−j) ÍN u=1ÍN v=1¯ tuP(ev)δ[u−v] = N Õ j=1 P(X≤x|Tj,ej)P(ej)¯ tj ÍN v=1¯ tvP(ev) = N Õ j=1 P(Xj≤x)P(ej)¯ tj ÍN v=1¯ tvP(ev).(12) Hence, the probability density function fX(x)is obtained from the derivative of the cumulative distribution function FX(x)=P(X≤x)and can be written as fX(x)= N Õ j=1 fXj(x)P(ej)¯ tj ÍN v=1¯ tvP(ev),(13) in which fXj(x)is the PDF of Xin the j-th state of the Markov chain. A simplification of this model can be performed considering that the chain remains in a certain state during a symbol interval and then performs a transition to another state. This means that all states have the same average duration value, since fading can be considered independent of time for at least one symbol interval Ts. Considering a Markov chain of Nstates, the probability of X≤xcan be written as P(X≤x)= N Õ j=1 P(X≤x,ej) = N Õ j=1 P(X≤x|ej)P(ej)(14) = N Õ j=1 P(Xj≤x)P(ej). 1186 H. SILVA, M. ALENCAR, W. QUEIROZ, ET. AL., BIT ERROR PROBABILITY OF M-QAM UNDER IMPULSIVE NOISE . .. Hence, the PDF fX(x)is given by fX(x)= N Õ j=1 fXj(x)P(ej),(15) in which fXj(x)is the PDF of Xin the j-th state of the Markov chain and P(ej)is the probability that the chain be in the j-th state. Substituting (15) in (9), it follows that Pe=2 π√Mlog2√M log2√M Õ k=1 (1−2−k)√M−1 Õ i=0 N Õ j=1 w(i,k,M) ×P(ej) ×(αpp∫π 2 0∫∞ 0 exp −a(i,M)x 2sin2θ δgδi δg+δifXj(x)dxdθ +(1−αpp)∫π 2 0∫∞ 0 exp −a(i,M)x 2sin2θδgfXj(x)dxdθ). (16) Given that [19] ∫∞ 0 e−sx fX(x)dx=MX(s),(17) in which MX(s)represents the MGF of fX(x), it follows that it is possible to write (16) as Pe=2 π√Mlog2√M log2√M Õ k=1 (1−2−k)√M−1 Õ i=0 N Õ j=1 w(i,k,M) ×P(ej)(αpp∫π 2 0 MXja(i,M) 2sin2θ δgδi δg+δidθ +(1−αpp)∫π 2 0 MXja(i,M)δg 2sin2θdθ).(18) For the model G2AWGN, the bit error probability conditioned to zis given by [17] P(e|z)=2 √Mlog2√M log2√M Õ k=1 (1−2−k)√M−1 Õ i=0 w(i,k,M) ×(αpβp1p2Q sa(i,M)z2δgδi δg+δi! +(1−αpβp1p2)Qqa(i,M)z2δg.(19) Using (5), it follows that P(e|z=√x)is given by P(e|z=√x)=2 √Mlog2√M log2√M Õ k=1 (1−2−k)√M−1 Õ i=0 w(i,k,M) ×(αpβp1p2Q sa(i,M)xδgδi δg+δi! +(1−αpβp1p2)Qqa(i,M)xδg.(20) The average bit error probability for a Markov chain with Nstates for the G2AWGN model is given by Pe=2 π√Mlog2√M log2√M Õ k=1 (1−2−k)√M−1 Õ i=0 N Õ j=1 w(i,k,M) ×P(ej)(αpβp1p2∫π 2 0 MXja(i,M) 2sin2θ δgδi δg+δidθ +(1−αpβp1p2)∫π 2 0 MXja(i,M)δg 2sin2θdθ),(21) in which MX(·)represents the MGF. 4. Moment Generating Function of the Distributions η-µand κ-µ If zin (2) and (19) has distribution η-µor κ-µ, the probability density function of of x=z2, in (6) and (20), is given by [19] fX(x)=1 2√x[fZ(√x)+fZ(−√x)].(22) If zhas distribution η-µ, it follows that fX(x)is given by fXη−µ(x)=2√πµµ+0.5hµxµ−0.5 Γ(µ)Hµ−0.5Ωµ+0.5exp −2µhx Ω ×Iµ−0.52µHx Ωu(x),(23) in which u(·) represents the unit step function, Ωif the average power of the signal, Γ(·) represents the Gamma function and the parameters Hand hare given by H=η−1−η 4and h=2+η−1+η 4(24) for the first format and η,0< η < ∞, is the ratio of the power of the phase and quadrature components. The generalized moment of that distribution is given by [20] MXη−µ(s)=4µ2h (2(h−H)µ+sΩ)(2(h+H)µ+sΩ)µ .(25) RADIOENGINEERING, VOL. 27, NO. 4, DECEMBER 2018 1187 If zhas distribution κ-µ, it follows that fX(x)is given by fXκ−µ(x)=µ(1+κ)µ+1 2xµ−1 2 κµ−1 2exp(κµ)Ωµ+1 2 exp −µ(1+κ)x Ω ×Iµ−1 2µrκ(1+κ)x Ω!u(x),(26) in which the parameter κis the ratio of the total power of the dominant components and the power of the scattered waves, and Ωis the average power of the signal. The generalized moment for the distribution κ-µis given by [20] MXκ−µ(s)=µ(1+κ) µ(1+κ)+sΩµ exp µ2κ(1+κ) µ(1+κ)+sΩ−µκ. (27) 5. Results Simulations were carried out considering two channel scenarios. The first scenario is characterized by the occurrence of GAWGN noise and fading that can alternate between the models η-µand κ-µ, and the second scenario is characterized by the presence of noise G2AWGN and fading η-µ or κ-µ. The values adopted for the Markov chain parameters are based on the work of Sanchez-Salas and Cuevas-Ruiz [8]. The values used for the parameters of the impulse noise η(t) and the modulation schemes are chosen in accordance with impulsive interference studies described in [21] and with the conditions of simulations performed in [16]. In the present work, simulations were carried out by Monte Carlo method, considering 5 ×106transmitted bits. In the simulations, the matrix of transition probabilities between states is given by [8] P=0.7 0.3 0.1 0.9,(28) initially the states are equiprobable and the steady state is obtained with a high number of transitions between states The absence or presence of line of sight between transmitter and receiver is characterized by η-µor κ-µdistribution, resp. The average bit error probability curves of 64-QAM, for a Markov chain with two states, under the effects of noise GAWGN and η-µor κ-µfading are presented in Fig. 1. The curves are plotted as a function of the signal-to-permanentnoise ratio, δg, for different values of δi, with p=0.3,µ=2.0, Ω=1.0,κ=2.0,η=0.3, and αp=0.1. For low values of signal-to-impulsive-noise ratio, with δi= 1 dB and δi= 10dB, it is observed that the average bit error probability, Pe, changes little with the increase of δg, for values of signal-to-permanent-noise ratio in the range 20–40 dB. For the four values of δiunder consideration, it is observed, for δg in the range 0 to 12 dB, that the average bit error probability curves overlap. Figure1alsopresents thetheoretical Pecurve with the Markov chain subjected to AWGN and fading η-µ or κ-µ. It is observed that the curves with the presence of impulsive noise have a higher BEP value when compared to the theoretical curve of Pewith the communication channel under the effect of AWGN. It is important to observe that the AWGN theoretical curve can be seen as a lower bound of the system performance. It is also observed in Fig. 1 that the average BEP curves tendto become irreducible from agiven signal-to-permanentnoise ratio. For low values δi, such as 1 dB, one has, for δg= 20dB, that the power of the impulsive noise is approximately 80 times greater than the power of the permanent noise, contributing so that the average BEP does not decrease from a given value of δg. For high values of signalto-impulsive-noise ratio, such as 20dB, both permanent and impulsive noise have low power. This explains why the respective average BEP curve presents a greater approximation (when compared to the curves referring to the low signal-toimpulsive-noise ratio) to the curve determined by the AWGN noise and η-µor κ-µfading. However, even for this case of high δi, the average BEP also tends to become irreducible from a given value of δg, because in this scenario the power of the permanent noise is much larger than the power of the impulsive noise. This contributes to the receiver’s decisionmaking regions becoming more indefinite and, consequently, correct decoding of the transmitted bits is impaired. In Fig. 2 average bit error probability curves are presented, for different values of the modulation order M, for the channel modeled by a Markov chain with two states, under the effects of noise GAWGN and η-µor κ-µfading, considering µ=2.0,αp=0.3,p=0.4,η=0.2,κ=2.0,Ω=1.0 and δi= 20dB. As the parameter Mincreases, the greater is the number of symbols present in the constellation, and consequently the receiver is more likely to make an incorrect decision. Therefore, in this scenario, for fixed values of signal-to-permanent-noise ratio, it follows that greater average bit error probabilities are obtained. For δgranging from 30 to 40 dB, average bit error probability less than 10−5 is obtained with M= 16. An average bit error probability of 10−3is achieved with δg≈15.0 dB for M= 16 and with δg≈26.6 dB for M= 256. It is also observed in Fig. 2 that the presence of impulsive noise causes the BEP curves deviate from the curve obtained when only AWGN noise is present in the system. In Fig. 3 average bit error probability curves of 64-QAM are presented for a Markov chain with two states, under GAWGN and η-µor κ-µfading, with µ=2.0,η=0.3, Ω=1.0,αp=0.1,p=0.2and δi= 20dB. The Pecurves are plotted as a function of the signal-to-permanent-noise ratio, δg, for different values of κ. One observes in Fig. 3 that greater average bit error probability is obtained as κincreases. It is also observed that the BEP curves for the four considered κvalues tend to decrease faster when noise in the channel is the AWGN. In turn, around δg=20 dB, when the power of the impulsive noise is greater than the power of the permanent AWGN noise, the BEP curves tend to decrease 1188 H. SILVA, M. ALENCAR, W. QUEIROZ, ET. AL., BIT ERROR PROBABILITY OF M-QAM UNDER IMPULSIVE NOISE . .. slowly when compared to AWGN curves. It is observed that Pe=10−3is obtained with δg≈17.3 dB for κ= 0.3, while it is obtained with δg≈21.1 dB for κ= 6.3. In Fig. 4, average bit error probability curves of 64-QAM are presented, for a Markov chain with two states, under the effects of noise G2AWGN and η-µor κ-µ fading, with µ=2.0,κ=1.0,Ω=1.0,η=1.5,αp= 0.5, p2= 0.5, β= 0.5 and p1= 0.5, for four different values of signal-to-impulsive-noise ratio, δi. For comparative purposes, the BEP curve is also shown in Fig. 4 when the Markov chain is subjected to AWGN and η-µor κ-µfading. It is observed that the BEP decreases as δgincreases, for fixed values of δi. As the value of δiincreases, it is noted that the BEP curves tend to follow the curve provided for the case where only AWGN noise and fading are present in the channel. In addition, it is observed that values of Peless than 10−4are obtained only with high values of δi, such as 20 dB. Average bit error probability curves for four values of modulation order M, for a channel modeled by a Markov chain with two states, subjected to noise G2AWGN and η-µ or κ-µfading, are presented in Fig. 5, for δi= 20 dB, µ=2.0, κ=1.0,Ω=1.0,η=1.5,αp= 0.5, p2= 0.5, β= 0.5 δg(dB) 0 5 10 15 20 25 30 35 40 Pe 10-6 10-5 10-4 10-3 10-2 10-1 100 101 δi= 1 dB (Theoretical) δi= 10 dB (Theoretical) δi= 15 dB (Theoretical) δi= 20 dB (Theoretical) AWGN δi= 1 dB (Simulated) δi= 10 dB (Simulated) δi= 15 dB (Simulated) δi= 20 dB (Simulated) Fig. 1. Average BEP of 64-QAM, for a Markov chain with two states, under the effects of noise GAWGN and η-µor κ-µ fading, for four different values of δi. δg(dB) 0 5 10 15 20 25 30 35 40 Pe 10-6 10-5 10-4 10-3 10-2 10-1 100 101 16-QAM (Theoretical) 64-QAM (Theoretical) 256-QAM (Theoretical) 1024-QAM (Theoretical) 16-QAM (AWGN) 64-QAM (AWGN) 256-QAM (AWGN) 1024-QAM (AWGN) 16-QAM (Simulated) 64-QAM (Simulated) 256-QAM (Simulated) 1024-QAM (Simulated) Fig. 2. Average BEP of M-QAM, for a Markov chain with two states, under the effect of noise GAWGN and η-µor κ-µ fading, for different values of modulation order M. and p1= 0.5. The greater the number of symbols Min the constellation, the closer the symbols affected by the noise and, consequently, the greater the average bit error probability for a fixed δg. An average bit error probability equal to 10−3is obtained with δg≈25.3 dB for M= 16 and with δg≈30.3 dB for M= 64. For δi= 20 dB, it is observed that the average bit error probability, Pe, is not less than 10−3for δg<40dB, for constellations with M≥64. In Fig. 6 average bit error probability curves of 64-QAM are presented for a Markov chain with two states, under the effects of noise G2AWGN and η-µor κ-µfading. The curves are plotted as a function of the signalto-permanent-noise ratio, δg, for different values of κ, with µ=2.0,η=1.5,Ω=1.0,αp=0.5,p1=0.5,p2=0.5and δi= 20dB. As κincreases, a greater bit error probability is obtained. It is observed that an average bit error probability of 10−3is obtained with δg≈25.0 dB for κ= 0.3, while it is obtained with δg≈33.0 dB for κ= 6.3. One observes in Fig. 6, by the dashed lines, that the Pecurves of 64-QAM for a Markov chain with two states under the effects of AWGN and η-µor κ-µfading are practically linear for δgin the range 20–40 dB. δg(dB) 0 5 10 15 20 25 30 35 40 Pe 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 κ= 6.3 (Theoretical) κ= 4.1 (Theoretical) κ= 2.7 (Theoretical) κ= 0.3 (Theoretical) κ= 6.3 (AWGN) κ= 4.1 (AWGN) κ= 2.7 (AWGN) κ= 0.3 (AWGN) κ= 6.3 (Simulated) κ= 4.1 (Simulated) κ= 2.7 (Simulated) κ= 0.3 (Simulated) Fig. 3. Average BEP of 64-QAM for a Markov chain with two states, under the effect of noise GAWGN and η-µor κ-µ fading, for different values of κ. δg(dB) 0 5 10 15 20 25 30 35 40 Pe 10-6 10-5 10-4 10-3 10-2 10-1 100 101 δi= 1 dB (Theoretical) δi= 10 dB (Theoretical) δi= 15 dB (Theoretical) δi= 20 dB (Theoretical) AWGN δi= 1 dB (Simulated) δi= 10 dB (Simulated) δi= 15 dB (Simulated) δi= 20 dB (Simulated) Fig. 4. Average BEP of 64-QAM, for a Markov chain with two states, under the effects of noise G2AWGN and η-µor κ-µfading, for different values of δi. RADIOENGINEERING, VOL. 27, NO. 4, DECEMBER 2018 1189 δg(dB) 0 5 10 15 20 25 30 35 40 Pe 10-4 10-3 10-2 10-1 100 101 16-QAM (Theoretical) 64-QAM (Theoretical) 256-QAM (Theoretical) 1024-QAM (Theoretical) 16-QAM (AWGN) 64-QAM (AWGN) 256-QAM (AWGN) 1024-QAM (AWGN) 16-QAM (Simulated) 64-QAM (Simulated) 256-QAM (Simulated) 1024-QAM (Simulated) Fig. 5. Average BEP of M-QAM, for a Markov chain with two states, under the effects of noise G2AWGN and η-µor κ-µfading, for different values of modulation order M. 6. Conclusion This article presents new exact expressions for the average bit error probability, Pe, of the M-QAM scheme for a wireless communication channel model with fading driven by a Markov chain of Nstates. For numerical evaluation of the mathematical expressions, a particular case of Markov chain with two states is considered, each one representing a different fading scenario. In the first state of the chain, the presence of the Gated Additive White Gaussian Noise (GAWGN) and fading η-µor κ-µis considered, whereas in the second state the presence of the Double Gated Additive White Gaussian Noise (G2AWGN) and fading η-µ or κ-µis considered. Average bit error probability curves under different values of the signal-to-impulsive-noise ratio (δi), fading parameters and modulation order Mare shown. In the simulation results, it has been observed that as the constellation order Mincreases, the performance of the receiver is worse, and consequently larger values of Peare obtained for fixed values of the signal-to-permanent-noise ratio (δg). Concerning the increase in the signal-to-impulsivenoise ratio, it was observed that some average bit error probability curves remained practically constant at certain intervals with the increase of δg, for δiequal to 1dB and 10 dB. As future works, we aim to determine closed expressions for the average bit error probability of the modulation scheme M-QAM for a Nstates Markov chain under the effects of GAWGN or G2AWGN and α-µfading [22]. Acknowledgments This study was financed in part by the Coordenação de Aperfeiçoamento de Níve Superior – Brasil (CAPES) – Finance Code 001 and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). δg(dB) 0 5 10 15 20 25 30 35 40 Pe 10−4 10−3 10−2 10−1 100 101 κ= 6.3 (Theoretical) κ= 4.1 (Theoretical) κ= 2.7 (Theoretical) κ= 0.3 (Theoretical) κ= 6.3 (AWGN) κ= 4.1 (AWGN) κ= 2.7 (AWGN) κ= 0.3 (AWGN) κ= 6.3 (Simulated) κ= 4.1 (Simulated) κ= 2.7 (Simulated) κ= 0.3 (Simulated) Fig. 6. Average BEP of 64-QAM for a Markov chain with two states, under the effects of noise G2AWGN and η-µor κ-µfading, for different values of κ. References [1] TANGHE, E., JOSEPH, W., VERLOOCK, L., MARTENS, L., et al. The industrial indoor channel: Large-scale and temporal fading at 900, 2400, and 5200 MHz. IEEE Transactions on Wireless Communications, 2008, vol. 7, no. 7, p. 2740–2751. ISSN: 1536-1276. DOI: 10.1109/TWC.2008.070143. [2] CHEFFENA, M. 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SILVA received his B.Sc. and M.Sc degrees in Electrical Engineering from Federal University of Campina Grande (UFCG), Brazil, in 2014 and 2016, respectively. His main research interests include optical communication systems, digital communication systems and coding theory. Marcelo S. de ALENCAR received his Bachelor Degree in Electrical Engineering, from Federal University of Pernambuco (UFPE), Brazil, his Master Degree in Electrical Engineering, from Federal University of Paraíba (UFPB), Brazil, and his Ph.D. from the University of Waterloo, Canada. He worked for the Federal University of Paraiba (UFPB), for the Federal University of Campina Grande (UFCG), and for the State University of Santa Catarina (UDESC). Since 2017 he is with the Federal University of Bahia, Brazil. He is founder and President of the Institute for Advanced Studies in Communications (Iecom). He received achievement awards from the Brazilian Telecommunications Society (SBrT), from the Medicine College of the Federal University of Campina Grande (UFCG), and from the College of Engineering of the Federal University of Pernambuco. He published over 450 engineering and scientific papers and 22 books. He also wrote chapters for 11 books. Wamberto J. L. de QUEIROZ received the B.Sc. degree in 1997 and the M.Sc. degree in 2002 from Federal University of Paraíba and the D.Sc. degree from Federal University of Campina Grande, Campina Grande, PB, in 2004, all in Electrical Engineering. From July 2004 to January 2005 he was with University Tiradentes, Aracaju, SE. From February 2005 to January 2007, he was with University of Fortaleza, CE, and from February 2007 to May 2010 he was with Federal University of Ceará. Since June 2010 he has been with the Department of Electrical Engineering of Federal University of Campina Grande. His research interests include signal spatial processing, diversity systems, channel modeling and estimation of parameters for wireless communication. Rodrigo de A. COELHO received his B.Sc. degrees in Electrical Engineering from Federal University of Campina Grande (UFCG), Brazil, in 2015. His main research interests include power quality and power theory. Francisco MADEIRO was born in Fortaleza, Ceará, Brazil, in 1972. He received his B.Sc., M.Sc and D.Sc. degrees in Electrical Engineering from Federal University of Paraíba (UFPB), Brazil, in 1995, 1998 and 2001, respectively. Since 2006 he is with University of Pernambuco (UPE), Brazil, where he is Associate Professor. His main research interests include signal processing, communications systems and computational intelligence. He was recipient of the Distinguished Award in Teaching at Polytechnic School of Pernambuco (POLI), UPE, in 2008 and 2013. He was recipient of the Distinguished Award in Research at POLI, in 2013.