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DOI: 10.1109/ojcoms.2020.2996797
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DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
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Received 2 April 2020; revised 1 May 2020; accepted 8 May 2020. Date of publication 25 May 2020; date of current version 16 June 2020.
Digital Object Identifier 10.1109/OJCOMS.2020.2996797
Bit Error Probability for Large Intelligent Surfaces
Under Double-Nakagami Fading Channels
RICARDO COELHO FERREIRA1, MICHELLE S. P. FACINA1, FELIPE A. P. DE FIGUEIREDO 1,2,
GUSTAVO FRAIDENRAICH 1, AND EDUARDO RODRIGUES DE LIMA3
1DECOM/FEEC, State University of Campinas, Campinas 13083-970, Brazil 2Instituto Nacional de Telecomunicações, Santa Rita do Sapucaí 37540-000, Brazil 3Department of Hardware Design, Instituto de Pesquisas Eldorado, Campinas 3083-898, Brazil
CORRESPONDING AUTHOR: F. A. P. DE FIGUEIREDO (e-mail: felipe.figueiredo@inatel.br) This work was supported in part by the European Union’s
Horizon 2020 Research and Innovation Program (ORCA Project) under Grant 732174, in part by the Research and Development ANEEL-PROJECT COPEL and EMBRAPII-Empresa Brasileira de Pesquisa e Inovação Industrial under Grant 2866-0366/2013, in part by CNPq under Grant 313239/2017-7,
Grant 301777/2019-5, and Grant 304946/2016-8, and in part by the São Paulo Research Foundation (FAPESP) under Grant 2018/19538-4.
ABSTRACT In this work, we investigate the probability distribution function of the channel fading between a base station, an array of intelligent reflecting elements, known as large intelligent surfaces (LIS), and a single-antenna user. We assume that both fading channels, i.e., the channel between the base station and the LIS, and the channel between the LIS and the single user are Nakagami-m distributed. Additionally, we derive the exact bit error probability considering quadrature amplitude (M-QAM) and binary phase-shift keying (BPSK) modulations when the number of LIS elements, n, is equal to 2 and 3. We assume that the LIS can perform phase adjustment, but there is a residual phase error modeled by a Von Mises distribution. Based on the central limit theorem, and considering a large number of reflecting elements, we also present an accurate approximation and upper bounds for the bit error rate. Through several Monte Carlo simulations, we demonstrate that all derived expressions perfectly match the simulated results.
INDEX TERMS Bit error rate, large intelligent surfaces, massive MIMO, Nakagami-m fading, von Mises circular distribution.
I. INTRODUCTION
T
HE DEMAND for data rate has increased exponentially in the last years, mainly due to the emergence of new services such as the Internet of things (IoT) and the video on demand. Sensors, objects, and everyday items can communi-cate, generate, exchange, and consume data with minimal or no human intervention. One of the solutions to support this new increasing data rate relies on solutions such as antenna arrays, which can improve the signal-to-noise ratio (SNR) and therefore increase the spectral efficiency. However, they can be expensive and present high power consumption at the transmitter side [1].As an attempt to deal with this design compromise between increasing spectral efficiency and minimizing
transmission power consumption, several methods involv-ing large intelligent surfaces (LIS) have been proposed. A LIS, which is an array composed of a massive number of low-cost reflecting elements, is capable of reflecting the incident signals with adjustable phase-shifts [2]. As they leverage ultra-reliability, high spectral, and power efficiency of the digital communication systems, LIS-equipped systems meet the requirements posed by various emerging application scenarios such as the industrial IoT [3].
A concept, similar to LIS, was first mentioned in 2015 at the University of California Berkeley project [4]. The general idea consists of wallpapers that are electromagnetically active and have built-in process-ing power. There is a compact integration of large
numbers of tiny antennas with reconfigurable processing networks.
Since then, there are already some variants of this technol-ogy [5]. Intelligent reflecting surfaces (IRS) are composed of reflecting elements capable of changing the incident signal in frequency, amplitude, or polarization [6]. Large intelligent meta-surfaces (LIM) are reconfigurable surfaces composed of projected meta-materials that can change the incident sig-nals based on their physical properties [7]. On the other hand, software-defined surfaces (SDS) are capable of being configured by software solutions [8]. Passive intelligent sur-faces (PIS) reflect the incident signals without applying any power gain to the reflected signal [9].
The work of Hum and Perruisseau-Carrier [10], for exam-ple, proposes a solution very similar to LIS technology. It discusses, from a theoretical point of view, the future challenges and the possibility of using array lenses, micro-electro-mechanical systems (MEMS), and reconfigurable reflect-arrays to make electronic beamforming by changing the polarization, amplifying signals and increasing the sig-nal to noise ratio. The use of reflecting surfaces to perform beamforming has emerged in the last three years as an alter-native to the sixth generation of mobile communications. Hu et al. [11] use adjacent surfaces, covered with a mag-netic and active material capable of being electronically and intelligently manipulated, to solve wireless communication tasks [12].
LIS technology can be divided into two types, passive and active. Although both of them can constructively combine the signals, minimizing the noise effect, there are funda-mental differences. The LIS with passive elements can only reflect the incident signals, modifying their phases without any reflection amplitude control. On the other hand, active LIS achieves better signal to noise ratio due to reflection amplitude control. It is necessary to perform power allo-cation at the reflectors, which requires high computational complexity [13].
Despite recent proposals in the literature to tackle all these challenges [14], achieving the optimal power allocation for active LIS beamforming is a complex optimization problem for itself. The search for the precoder vector and the design of the matrix with phase-shifts involves the optimization of non-convex functions, which is an intricate task [15].
Regarding the phase adjustments common to the two types of LIS, they are designed for two reasons. To cancel the channel phase, improving the SNR or even destructively at the non-intended receiver to avoid interference and enhance security/privacy. According to [16], the LIS can control and optimize the refraction and reflection towards anoma-lous directions, thus altering the spatial distribution of the intended and interfering signals. But, it is worth mentioning that this process is susceptible to phase errors.
According to Zhang et al. [17] the matrix of reflectors operates by controlling diodes on each LIS element alternat-ing between the states on and off to control the bias voltage. Since the diodes are digitally controlled, there is a limited
amount of bits to represent the reflection angles. Therefore the phase is quantized, and there are random phase errors.
Considering that such errors do not exist, the frame-work investigated in [13], [18] takes into account a LIS composed of passive antenna elements with reconfigurable characteristics. They do not require any dedicated energy source for either decoding, channel estimation, or transmis-sion. The authors came across some non-convex problems that are solved through the use of optimization techniques. Based on majorization-maximization alternated by fractional programming, it is possible to optimize energy efficiency.
Taha et al. [19] propose a training process based on deep learning. Using a massive number of passive reflectors and a small number of active ones, the LIS can learn the channel parameters and autonomously optimize the data transmission. Following this idea, Basar et al. [6] present closed-form solutions for optimization problems. Parameters related to LIS design, power allocation, precoder, and phase shift matrix are derived and influence the performance of the system directly.
Besides, LIS can also be useful in smart radio environ-ments employing millimeter-wave band (frequencies around 30-100 GHz). In the absence of line of sight (LoS) paths, systems operating at such bands cannot work correctly due to the high attenuation caused by the absorption of atmospheric gases. Then, LIS-equipped systems can create LoS paths connecting the base station to devices [20]. These and other advantages are evident in [21], in which the authors present a comparative performance study between classic relaying and reflecting surfaces at high-frequency bands when there is no LoS path.
Yang et al. [22] proposed a transmission protocol for a system that combines LIS and orthogonal frequency division multiplexing (OFDM) with channel estimation, the reflectors were divided into clusters, and the adopted methodology considers that adjacent reflectors share the same reflection coefficient. Only the channel resulting from the combination of the clusters needs to be estimated.
Some authors have employed a similar approach to the one used here, but most of them adopt restrictive models. In [23], the authors have used the central limit theorem (CLT) to obtain an approximate probability distribution and calculate the outage probability in scenarios involving many reflectors and only one antenna at the base station. The authors only consider a phase error with uniform distribution (worst case of phase estimation) and a channel subjected to the Rayleigh fading. They do not provide any expression for a small number of reflectors.
In most previous research, it is assumed that the channel fading is Rayleigh distributed. Of course, the Rayleigh-fading model is known to be a reasonable assumption for the fading encountered in many wireless communications systems. However, many measurement campaigns [24], [25] show that the Nakagami-m distribution provides a much better fitting for the fading channel distribution. Since the Nakagami-m distribution has one more free parameter, it
FERREIRA et al.: BIT ERROR PROBABILITY FOR LARGE INTELLIGENT SURFACES UNDER DOUBLE-NAKAGAMI FADING CHANNELS
FIGURE 1. System model.
allows for more flexibility. It moreover contains the Rayleigh distribution (m = 1), the one-sided Gaussian distribution (m→ 0.5), and the uniform distribution on the unit circle1 (m→ 1) as special (extreme) cases. The Nakagami-m dis-tribution is a general, but an approximate solution to the random phase problem [26]. The exact solution involves the knowledge of the distribution and the correlations of all of the partial waves composing the total signal. It becomes infeasible due to its complexity. This has been circumvented by Nakagami [26], who, through empirical methods based on field measurements followed by a curve-fitting process, obtained the approximate distribution.
In this work, we consider passive elements and focus on the bit error rate. Our main contribution is the derivation of exact expressions for the BPSK and M-QAM bit error proba-bility for an arbitrary number of reflectors n. The expressions derived here allow us to predict how the channel parameters and the phase error distribution impact the performance of the communication between the base station and the end-user. One of the main problems solved here is indeed of a very general statistical application that can be used in countless applications. In particular, we find an exact solu-tion to the problem of obtaining the distribusolu-tion of the sum of random vectors whose amplitudes follow the product of two Nagakami-m distributions and whose phases follow the Von Mises distribution. Note that the literature solves this problem but for a very limited scenario, namely vectors with fixed amplitudes and uniform phases. Certainly, our solution is much more general than this. Of course, in both cases, as the number of vectors increases the CLT can be used. But, for arbitrary finite number of vectors, our solution is much more general.
The remainder of this paper is organized as follows. Section II describes the channel characteristics, LIS param-eters, and the communication system’s physical structure. Section III presents the probability density function and error probability for low and large values of n. Section IV presents the proposed formulas for the error probability. Finally, Section V compares the simulated and analytical results. We close the discussion in Section VI, summarizing our conclusions.
II. SYSTEM MODEL
As shown in Fig. 1, we consider a single-input single-output (SISO) system formed by a base station (BS) with a single antenna sending signals to LIS. The array is composed of n reflector elements, in which the incident signals are reflected with a calculated phase-shift. Only a single-antenna user is considered.
The system is composed of two channels. The single-input multiple-output (SIMO) channel between BS and LIS is modeled by the fading coefficient H1k∈ C1×n. The
multiple-input single-output (MISO) channel between LIS and the user is modeled by the coefficient H2k ∈ Cn×1. Both of
them are independent random variables modeled by the Nakagami-m distribution.
Let X be the transmitted symbol with zero mean and unit variance. According to Badiu and Coon [27], the incident signal on the single-antenna user can be expressed as
Y =√γ0 n k=1 ejφkH 1kH2kX+ W, (1)
in whichφk is the phase adjustment performed by the LIS, W is the additive noise and follows a Gaussian distribution, W ∼ N (0, σW2), γ0 is the average signal to noise ratio for
a single reflector, |H1k| and |H2k| for ∀k are Nakagami-m
distributed with the following probability density function f|Hik|(x) = 2m
m (m)mx
2m−1e−mx2
, (2)
in which m is the shape parameter and is the spread parameter, and|.| denotes the absolute value.
We can assume that the phase introduced by the product of the two complex fading coefficients, H1kand H2kare almost
entirely canceled by LIS. But there are remaining errors due to the quantization bit limitation, estimation of the channel phases, and synchronism error. The total residual phase error can be modeled as a Von Mises random variableθk. The zero-mean Von Mises distribution has real support in the range from−π to π, and the κ parameter describes how scattered the probability density function is from the mean.
The Von Mises distribution is a continuous probability distribution on a circle analogous to the Gaussian distribution but with the entire non-zero part of the PDF distributed in a range of size 2π centered on the mean. This distribution was previously used to simulate the azimuth angles [8] and model the phase error [27] in LIS-based systems. It is versatile to represent random angles since the concentration parameter is related to the distribution variance and models how closely the angle is from the mean.
Letθkbe the phase error modeled by a Von Mises random variable and concentration parameter κ, In other words, ∼ Von Mises(κ). The Von Mises distribution has the following PDF [28]
f(θ) = 1
2πI0(κ)
eκ cos(θ), (3) in which I0(κ) is the modified Bessel function of first kind
and order 0. The parameter κ indicates the spread of the distribution. Largeκ means that the phase error is concen-trated in a small interval. Whenκ = 0, the phase errors are equally probable and follow the uniform distribution.
Consider thatθH1k = arg(H1k) is the phase of the channel
between BS and the LIS elements andθH2k = arg(H2k) is the
phase of the link between each LIS element and the user. The
LIS attempts to perform phase cancellation with respect to the composite channel H1kH2k, by rotating the incident signal
by a phase shiftφk, in such a way thatθH1k+ θH2k+ φk= 0. In practice, the phase correction is not perfect and residual errors are left, that is
θk= φk+ θH1k+ θH2k. (4)
Isolating the variable φk in (4) and substituting in (1), we have that Y =√γ0 n k=1 ejφkH 1kH2kX+ W =√γ0 n k=1 ejθke−j θH1k+θH2kH 1kH2kX+ W =√γ0 n k=1 ejθke−jθH1k+θH2k|H 1k||H2k| · · · × ejθH1k+θH2kX+ W =√γ0 n k=1 ejθk|H 1k||H2k|X + W. (5)
We can define the composite fading coefficient H as H= 1 n n k=1 ejθk|H 1k||H2k|, (6)
and therefore, the output at the receiver (5) can be written as
Y = n√γ0HX+ W, (7)
The distribution of H, given in (6), is of fundamental importance to our problem. Since we have sum of terms |H1k||H2k|, we have to derive the distribution of the product
of two Nakagami-m distribution. Fortunately, from [26], we know that the distribution of Z= |H1k||H2k|, is given by
fZ(z) = 4zm1+m2−1K m1−m2 2z2i=1 mi i 2 i=1(mi) i mi m2+m1 2 , (8)
in which(.) represents the Gamma function, Z is a random variable with Double-Nakagami distribution, while mi and i are the parameters of each Nakagami-m coefficient.
It is important to highlight that the phase distribution of the complex fading coefficients is not relevant in our analysis, but only the magnitude.
III. FADING DISTRIBUTION
In this section, we derive the distributions of |H| using the channel parameters and the characteristic function associated with the random phase error for n= 2 and n = 3. Moreover, it is also proposed an approximate distribution for large values of n.
It is possible to rewrite (6) as H= 1n nk=1|H12k|ejθk, in
which|H12k| = |H1k||H2k| is the magnitude of the combined
fading coefficients. The squared fading coefficient magnitude can be written as:
|H|2= n k=1 Rkcosθk 2 + n k=1 Rksinθk 2 , (9)
in which Rk= 1n|H12k|, θ1. . . θnare independent Von Mises
random variables. Rk is the k-th sample direction, and the resultant direction is|H|. Note that (9) can be written as
|H|2= C2+ S2,
(10) in which C and S are the in-phase and quadrature components given by C= n k=1 Rkcosθk, (11) and S= n k=1 Rksinθk. (12)
A. EXACT DISTRIBUTION OF|H|FOR N=2 AND N=3 The general solution to the computation problem of R = | ni=1Riejθi|, i.e., the magnitude of a linear com-bination of complex variables with constant magnitudes and random phases uniformly distributed, was obtained by Maghsoodi [29]. However, our problem requires that R1, R2, . . . , Rn be random variables. Therefore, in order to
apply this result to our problem, we multiply the solution given in [29] by the distribution of the product of two Nakagami-m distributions, i.e., the double-Nakagami distri-bution given in (8). Then, we perform the integration with respect to the variables R1, R2, . . . , Rn.
The exact PDF of|H| for n = 2 and n = 3 is given by (13) and (14), respectively, on the bottom of the next page. The symbol U(·) denotes the Heaviside unit step function, and
¯r2(φ) =
r21+ r22+ 2r1r2cosφ.
Fig. 2 shows a comparison between the exact and sim-ulated PDF for m = 1, 2, 3. As can be seen, as the value of m increases, the mean of the distribution also increases, that is, the larger the m parameter, the better the condition of the channel.
B. APPROXIMATED DISTRIBUTION OF|H|FOR LARGE N Unfortunately, the exact distribution for large n is very intricate and its evaluation becomes computationally pro-hibitive. However, as n increases, the distributions of C and S, given in (11) and (12) respectively, become approx-imately Gaussian due to central limit theorem [30], that is C ∼ N (μC, σC2), and S ∼ N (μS, σS2). Since C and S are uncorrelated (see Appendix C), for the Gaussian case, this condition implies that they are also independent random variables [31], and therefore their joint distribution can be given as fC,S(x, y) = 1 2πσCσSe − (x−μC)2 2σ2C e − (y−μS)2 2σ2S . (15) VOLUME 1, 2020 753
FERREIRA et al.: BIT ERROR PROBABILITY FOR LARGE INTELLIGENT SURFACES UNDER DOUBLE-NAKAGAMI FADING CHANNELS
FIGURE 2. Exact PDF of|H|for n=2 double-Nakagami.
The analytical expression for the mean μC and variance σ2
C of C, are expressed by (34) and (43), respectively in Appendix A. In the same way, the meanμSand varianceσS2 of S are given by (44) and (49), respectively in Appendix B. Since we are interested in the distribution of the fading envelope, that is ρ = |H|, we can perform a change of variables as x = ρ cos φ and y = ρ sin φ, and then fP, (ρ, φ) = ρfC,S(x, y) can be written as fP, (ρ, φ) = ρ 2πσCσSe − (ρ cos(φ)−μC)2 2σ2C e − (ρ sin(φ)−μS)2 2σ2S . (16)
1) UNIFORMLY DISTRIBUTED PHASE ERROR
Considering that the phase error is uniformly distributed, the mean and variance of C and S can be further simplified. Assume κ = 0 and θk is a zero mean random variable, therefore according to (34) and (44), the mean of C is given by
μC= μ1μ2E[ cosθk]= μ1μ2α1= 0, (17)
in which E[.] is the expected value of a random variable andαp is the real part for the characteristic functionϕpthat can be defined as
ϕp= E ejpθ
= αp+ jβp. (18) Note that βp= 0, because the imaginary part of the char-acteristic function is associated with a sine function, which
is an odd function and the PDF is symmetrical. Then,ϕp is a real number and can be given as ϕp= αp=Ip(κ)
I0(κ).
The mean of S is given by
μS= μ1μ2E[ sinθk]= μ1μ2β1= 0. (19)
Using (43) and (49), the variance of C and S can be calculated as σ2 C= σS2= σ2 12+ μ1μ2 2n . (20)
For these values of mean and variance, (16) simplifies to fPU, (ρ, φ) = ρ
2πσC2e
−2ρ2σ2
C. (21)
IV. BIT ERROR PROBABILITY
The bit error probability for the M-QAM and BPSK modu-lations can be found according to [32], [33] respectively as
PQAMe ( ¯γ) = 1 − 1− 2 1−√1 M Q 3¯γlog2M (M − 1) 2 , (22) and PBPSKe ( ¯γ) = Q¯γ , (23)
in which Q(·) is the Gaussian error function, ¯γ = Eb
2σW2, Eb
is the energy per bit, and M is the number of symbols. Under multipath propagation, the mean probability of error can be written as ¯PQAM e ( ¯γ) = ∞ 0 PQAMe n2¯γr2 f|H|(r)dr, (24) ¯PBPSK e ( ¯γ) = ∞ 0 PBPSKe n2¯γr2 f|H|(r)dr, (25) in which f|H|(r) is given by (13) and (14) for n = 2 and n= 3, respectively.
When n is large and from some assumptions, we can derive elegant expressions for the bit error probability as follows. f|H|(r) = 2 π ∞ 0 ∞ 0 U4r12r22− (r2− r12− r22)2 4r21r22− (r2− r2 1− r 2 2)2 2 k=1 ⎛ ⎜ ⎜ ⎝ 4rm1+m2−1 k Km1−m2 2rk m1m2 12 m1 1 m1+m2 2 m2 2 m1+m2 2 ⎞ ⎟ ⎟ ⎠drk, n = 2 (13) f|H|(r) = 2 π2 π 0 ∞ 0 ∞ 0 ∞ 0 U4r32¯r2(φ)2− (r2− ¯r2(φ)2− r32)2 4r23¯r2(φ)2− (r2− ¯r2(φ)2− r23)2 3 k=1 ⎛ ⎜ ⎜ ⎝ 4rm1+m2−1 k Km1−m2 2rk m1m2 12 drk m1 1 m1+m2 2 m2 2 m1+m2 2 ⎞ ⎟ ⎟ ⎠dφ, n = 3 (14) 754 VOLUME 1, 2020
A. BIT ERROR PROBABILITY FOR LARGE N
Considering that C and S are Gaussian distributed, then the error probability can be calculated as
¯PQAM e ( ¯γ) = π −π ∞ 0 PQAMe n2¯γρ2 fP, (ρ, φ)dρdφ. (26) The same way, the bit error probability for BPSK can be written as ¯PBPSK e ( ¯γ) = π −π ∞ 0 PBPSKe n2¯γρ2 fP, (ρ, φ)dρdφ. (27) Considering the Chernoff bound Q(x) ≤ 1
2e− 1
2x2, then the
upper bound of the error probability can be calculated by (28) on the bottom of the page. Here, k1= M+3γ log2(M)n2σS2−
1, k2= M + 6γ log2(M)n2σS2− 1, and σmin= min(σC, σS). Following the same reasoning in (27), and defining σmin= min(σC, σS), we obtain an upper bound for the BPSK modulation as ¯PBPSK e ( ¯γ) ≤ e− γ n2(μC2+μS2) 2γ n2σ 2min+1 22γ n2σ2 min+ 1 . (29)
B. UNIFORM PHASE DISTRIBUTION
The worst case for the phase error is the uniform distribution, in which all angles are equally likely. Considering BPSK modulation, the bit error probability can be written as
¯PBPSK e ( ¯γ) = 2π ∞ 0 PBPSKe n2¯γρ2 fPU, (ρ, φ)dρ, (30) where fPU, (ρ, φ) is defined in (21). Its closed-form resem-bles the channel under Rayleigh fading presented in [32]. For the sake of clarity, it is repeated here as
¯PBPSK e ( ¯γ) = 1 2 ⎛ ⎜ ⎜ ⎝1 − n √ ¯γ n2¯γ + 1 σ2 C ⎞ ⎟ ⎟ ⎠. (31)
For the general M-QAM modulation, the error probability considering uniform phase error is given as
¯PQAM e ( ¯γ) = 2π ∞ 0 PQAMe n2¯γr2 fPU, (ρ, φ)dρ. (32) Unfortunately, there is no closed-form solution to (32). However, we can neglect the terms Q(.)2and then obtain an
FIGURE 3. PDF of|H|for n=128 and different values ofκ.
TABLE 1.Mean squared error of the approximated PDF.
approximation for the M-QAM bit error probability as
¯PQAM e ( ¯γ) ≈ 2√M− 1 σ2 C (M−1) log(8) n2¯γσ2 Clog(M)+ 9 − 3 √ M (M−1) log(8) n2¯γσ2 Clog(M) + 9 . (33) V. NUMERICAL RESULTS
In this section, we present some numerical results to validate the Monte Carlo simulations obtained from 106realizations. For simplicity, the variablesσ12andσ22assume unitary value. Fig. 3 shows the probability density function of |H| for κ ∈ {0, 1, 2} and n = 128. We can observe that the simulated curves match very well the approximated results.
Table 1 presents the mean squared error (MSE) between the approximate the simulated distributions for different val-ues of n. It is evident that the accuracy of the proposed approach becomes higher as n increases. For example, the error is less than 5% for n ≥ 32 and practically zero for n> 128.
Considering n= 2, Fig. 4 shows the exact bit error prob-ability given by (24) and (25), for 4-QAM and BPSK, respectively. As expected, the theoretical and simulated curves are indistinguishable.
¯PQAM e ( ¯γ) ≤ − √ M− 1 2√ M+ 1 e− μ2C +μ2 S 2σ2min ⎛ ⎝k1 √ M− 1 e (M−1)(μ2C +μ2 S) 2k2σmin2 − 2k2√Me (M−1)(μ2C +μ2 S) 2k1σmin2 ⎞ ⎠ k1k2M (28) VOLUME 1, 2020 755
FERREIRA et al.: BIT ERROR PROBABILITY FOR LARGE INTELLIGENT SURFACES UNDER DOUBLE-NAKAGAMI FADING CHANNELS
FIGURE 4. Exact bit error probability for n=2.
FIGURE 5. Error probability for 4-QAM and different values of reflecting elements, n.
FIGURE 6. Error probability for different values of n.
In Fig. 5, we present the bit error probability for 4-QAM modulation for n = 32 and n = 128. As can be seen, the proposed approximated distribution curve is very close to the simulated curve. Its accuracy increases for larger values of n, as predicted by the law of large numbers.
FIGURE 7. Approximate bit error probability for 4-QAM and n=32.
FIGURE 8. Error probability for 4-QAM and different values ofκ.
FIGURE 9. Comparison between the BPSK upper bound given in (29) and the exact result for n=16.
Fig. 6 shows the behavior of BER of 4-QAM modulation as a function of the number of reflectors, n, for two values of SNR. As expected, the error decreases as n increases.
Fig. 7 shows the theoretical and simulated BER for
4-QAM, n = 32 and different values of κ. It is possible
to see that BER decreases asκ increases.
Fig. 8 shows the bit error probability for 4-QAM modu-lation as a function of the concentration parameterκ for a
FIGURE 10. Approximate bit error probability and upper bound for n=16 using 16-QAM modulation.
FIGURE 11. Bit error probability for channels with different parameters m1 and m2, assuming 4-QAM modulation, n=64, andκ =4.
fixed signal to noise ratio of −25 dB and n = 256. Note that as κ increases, the BER decreases fast, since the phase errors are more concentrated around the mean.
In Fig. 9, we can verify that the upper bound, proposed in (29), for the bit error rate for BPSK modulation and n = 16, works correctly and can be useful to approximate the integral formula of (27).
Fig. 10 shows (26) and its approximation proposed in (33), considering the 16-QAM modulation with n = 16. As can be seen, the approximation is very accurate, especially for large SNR values. Also, we present the result for the upper bound proposed in (28). It is possible to see that the upper bound is very tight to the numerical solution.
Finally, Fig. 11 shows the bit error probability for differ-ent combinations of m1 and m2 as a function of the signal
to noise ratio. As well known in the literature, for a conven-tional single input single output channel, the greater the m parameter, the better the performance in terms of BER [34]. It is interesting here to observe that the equivalent parameter m is the product m1m2, that is, the greater the product, the
better the performance.
VI. CONCLUSION
In this work, we have proposed an approximated and upper bound expressions for calculating the bit error probability of LIS-assisted systems. We have considered BPSK and M-QAM modulations under the effect of Nakagami-m fading channels.
We have obtained the exact distribution of the channel coefficient for n< 4. However, as the exact formula gets too intricate for large values of n, we have employed a Gaussian approximation for the in-phase and quadrature components. The approximation for the PDF of|H| converges very fast, and even for low values of n, the mean square error is small. We have analyzed different scenarios regarding the Nakagami m parameter, the concentration parameter κ, and the number of reflecting elements n. In short, the BER decreases when at least one of these parameters increases. All the results were validated by numerical simulations and have shown an excellent agreement.
In conclusion, we have proved that the proposed approx-imation is a simple and efficient tool to evaluate the performance of LIS-assisted systems.
APPENDIX A
MEAN AND VARIANCE OF C The mean value of C is given by
μC= E[C] = E n k=1 1 n|H12k| cos θk . (34)
Assuming that all coefficients are independent and have the same statistics, the following can be written
μC= n ×1 n × E[|H12k| cos θk] = μ1μ2 I1(κ) I0(κ) , (35)
in whichμ1= E[H1k] andμ2= E[H2k]. The variance of C
is given by σ2 C= var n k=1 1 n|H12k| cos θk . (36)
Note that the variance computation in (36) involves the product of two independent variables. Let the variables u and v be independent. Therefore, the variance of the prod-uct is given by var(uv) = var(u)var(v) + var(u)E[v]2+ var(v)E[u]2 [31]. Additionally, since all the variables are independent, the following can be written
σ2 C= 1 nvar(|H12k| cos θk) = 1 n σ2 12σC2k+ σ 2 12μ2Ck+ σ 2 Ckμ 2 12 , (37)
whose parameters can be calculated as follows σ2 12= σ 2 1σ 2 2 + μ 2 1σ 2 2 + μ 2 2σ 2 1, (38) VOLUME 1, 2020 757
FERREIRA et al.: BIT ERROR PROBABILITY FOR LARGE INTELLIGENT SURFACES UNDER DOUBLE-NAKAGAMI FADING CHANNELS
in which σ12= var(H1k) and σ22= var(H2k),
σ2 Ck = E cos2θk − E2 [cosθk], (39) μCk = E[cos θk]= α1= I1(κ) I0(κ), (40) E cos2θk =1 2 + 1 2E[cos 2θk]= 1 2+ 1 2α2. (41) and σ2 Ck = 1 2+ 1 2 I2(κ) I0(κ)+ I1(κ) I0(κ) 2 . (42)
Therefore, the variance of C is defined as σ2 C= 1 2nσ 2 12 1+I2(κ) I0(κ)+ 4 I1(κ) I0(κ) 2 + · · · + μ1μ2 1 2n 1+I2(κ) I0(κ) . (43) APPENDIX B
MEAN AND VARIANCE OF S
The mean value of S can be written as μS= E[S] = E n k=1 1 n|H12k| sin θk = 1 n× n × μ1μ2E[sinθk], (44) Using the same assumptions as in the previous case, the mean value of S can be calculated as
μSk= E[sin θk]= β1= 0. (45) Therefore,
μS= 0 (46)
On the other hand, the variance of S can be computed as σ2 S = var n k=1 1 n|H12k| sin θk = 1 n σ2 12σS2k+ σ 2 12μ2Sk+ σ 2 Skμ 2 12 , (47)
whose parameters can be calculated as follows σ2 Sk = E sin2θk − E2[sinθk] = 1 2 − 1 2 I2(κ) I0(κ) E sin2θk = 1 2 − 1 2E[cos 2θk]= 1 2 − 1 2α2. (48)
Therefore, the variance of S is defined as σ2 S = 1 2nσ 2 12 1−I2(κ) I0(κ) + 1 2n(μ1μ2) 2 1−I2(κ) I0(κ) . (49) APPENDIX C
MEAN OF THE PRODUCT OF C AND S
This Appendix shows that C and S are uncorrelated. Considering a possible correlation between C and S, the bivariate Gaussian joint distribution can be written as fC,S(x, y) = 1 2πσCσS1− ρ2· · · × e− 1 2√1−ρ2 (x−μC)2 σ2C + ( y−μS)2 σ2S − 2ρ(x−μC)(y−μS) σCσS (50) where ρ is defined as [31] ρ =E[CS]− E[C]E[S] σCσS (51)
As it has been calculated from (46), the term E[S] = 0, that is, the second term of the numerator of (51) is zero. Therefore in order to prove thatρ = 0, we need to compute the mean of the product between C and S and show that is also zero.
Departing from (11) and (12), the mean of the product between C and S can be written as
E[CS]= E ⎡ ⎣n i=1 n j=1
RiRjcos(θi) sin
θj ⎤
⎦ (52) Since Rk is independent ofθk for all k, then
E[CS]= 2 n i=1 n j>i
E#RiRj$E#cos(θi) sinθj$
+ n k=1 E R2i
E[sin(θi) cos(θi)] (53) In the sequel, the following equalities will be proven E[ cos(θi) sin(θj)] = 0 and E[ sin(θi) cos(θi)] = 0, and
therefore (53) will be null.
Using the very definition of the mean, the term E[ cos(θi) sin(θj)] can be computed as
E#cos(θi) sinθj$
= 2π
0
2π 0
cos(θi) sinθj e
κ cos(θi) 2πI0(κ) eκ cos(θj) 2πI0(κ) dθidθj= 0, (54) in the same way, the mean with respect to θi in the second term of (53), can be calculated as
E[cos(θi) sin(θi)] =
2π 0
cos(θi) sin(θi) 2πI0(κ)
eκ cos(θi)dθi= 0, (55) therefore E[CS]= 0 and consequently ρ, given in (51), will beρ = 0. From this, (50) can be written as in (15). ACKNOWLEDGMENT
The authors would like to thank the editor, Prof. Daniel Benevides da Costa, and the anonymous reviewers for their constructive comments, which helped them to improve the manuscript.
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