On the Average Bit Error Rate and Average Channel Capacity over Generalized Fading Channels ‡
Osamah S. Badarneh∗ , Michel Kadoch‡ , and Ibrahem E. Atawi∗ ∗ Electrical Engineering Department, University of Tabuk Emails: {obadarneh, ieatawi}@ut.edu.sa ´ ´ D´epartement de G´enie Electrique, Ecole de Technologie Sup´erieure Email:
[email protected]
Abstract—In this paper, we present unified analytical expressions for the average bit error rate (BER) and the average channel capacity of single-branch receivers operating over generalized fading channels. More specifically, novel closed-form expressions for the average BER and the average channel capacity over generalized η−μ fading channels in terms of the bivariate Fox’s H-function/Meijer’s G-function are derived. The influence of the fading parameters η and μ on the system performance is analyzed and discussed through representative numerical examples. The correctness of our derivations is validated by means of Monte-Carlo simulations. In addition, our analytical results are in excellent agreement with the results that have been previously reported in literature. Index Terms—Average channel capacity, bit error rate, bivariate Fox’s H-function, generalized fading channels.
I. I NTRODUCTION
T
HE average bit error rate (BER) and the average channel capacity play an important role in the design of wireless communication systems as the demand for high quality of service (QoS) is growing rapidly. During the past years, there has been several research studies on evaluating the average BER and the average channel capacity over various types of fading channels [1–5]. The average BER of binary coherent signaling, PAM, and QAM coherent signaling over flat fading channels subject to additive white generalized Gaussian noise are derived and evaluated in [1, 2]. In [3], the authors provided an expression of the average channel capacity for arbitrary values of η and μ in terms of infinite series and Meijer’s Gfunction. However, for a given accuracy, the number of terms in the series depends on the values of η and μ. A closed-form expression for average channel capacity, which is only valid for integer values of μ, is derived in [4]. Based on MGF approach, Ermolova obtained closed-form expressions for the average symbol error rates (SER) of rectangular QAM modulation in terms of Appell’s and Lauricella’s hypergeometric functions over the generalized and κ−μ distributions [5]. However, the technique which is used to derived these expressions is complicated and the MGF of each channel must be derived first in order to obtain the desired results. In [6, 7], the average BER of some modulation techniques is numerically computed based on the MGF approach and there is no closed-form expression is provided. In this paper, we present novel closed-form expressions for the average BER and average channel capacity over generalized η−μ fading channels. The PDF-based approach
is adopted in our analysis. To this end, we rewrite the PDF of the η−μ fading channels in a new form which enable us to obtain closed-form expressions for average BER and average channel capacity. The derivation of both performance metrics is performed in a simple way. The expressions are obtained in terms of the well-known Fox’s function of two variables (bivariate Fox’s H-function), as well as in terms of bivariate Meijer’s G-function and they are valid for arbitrary values of the fading parameters η and μ. Using our derived expressions, the system performance over some well-known fading distributions such as Nakagami-m, Rayleigh, OneSided Gaussian, and Nakagami-q (Hoyt) can be studied. The rest of this paper is organized as follows. In Section II, we define the channel models under consideration. In Section III, we present our mathematical analysis. And, we then present some numerical and simulation results in Section IV. Finally, Section V concludes this paper. II. P RELIMINARIES The probability density function (PDF) of the instantaneous signal-to-noise ratio (SNR) γ of the η−μ distribution is given by [8] 2μHγ 2μhγ μ− 12 Iμ− 12 , exp − fγ (γ) = Ξγ γ¯ γ¯ (1) √ µ+ 1 µ ∞ 2h 2 πμ where Ξ = , Γ(x) = 0 tx−1 exp(−t)dt µ− 1 µ+ 1 Γ(μ)H
2
γ ¯
2
is the gamma function, γ¯ = Eγ denotes the average SNR, Iv (·) is the modified Bessel function of the first kind with order v, and μ > 0 represents the number of multipath cluster. The parameters h and H are functions of the fading parameter η, where η ∈ (0, ∞). The parameter η is represented in two different formats: In Format I, it represents the power ratio between the in-phase and quadrature scattered waves of the fading signal within each cluster. In this case, h = (2 + η −1 + η)/4 and H = (η −1 − η)/4. In Format II, the parameter η, where η ∈ (−1, 1), represents the correlation coefficient between the in-phase and quadrature phase components within each cluster with h = 1/(1 − η 2 ) and H = η/(1 − η 2 ). The η−μ distribution is better suited for non-line-of-sight applications. Besides, it includes Hoyt (Nakagami-q), Nakagami-m, Rayleigh and one sided Gaussian distributions as special cases.
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The PDF in (1) can be rewritten as 1 2μ(h − H)γ 2μHγ fγ (γ) = Ξγ μ− 2 exp − exp − γ¯ γ¯ 2μHγ . (2) ×Iμ− 12 γ¯ By using the representation of e−x Iv (x) in terms of Meijer’s G-function [9, Eq. (8.4.22.3)] and with the help of [9, Eq. (8.3.2.21)], the PDF in (2) can be represented in terms of Fox’s H-function as 1 Ξ 2μ(h − H)γ fγ (γ) = √ γ μ− 2 exp − γ¯ π 1 , 1 1, 1 4μH 2 .(3) γ ×H1, 2 μ− 12 , 1 , 12 −μ, 1 γ¯ Note that, rewriting the PDF of the η−μ fading channels as described in (3) simplifies the derivation and enables us to derive closed-form expressions for the average BER and the average channel capacity using the PDF-based approach.
Finally, using [11, Eq. (2.6.2)] then a closed-form expression for the average BER in terms of the bivariate Fox’s HL, N, N1 , M, M1 function is obtained as in (11), where HE, [A: C], F, [B: D] [·] is the generalized (bivariate) Fox’s H-function which is defined in terms of a double Mellin-Barnes type integral as in (12). To the best of the author’s knowledge (11) is new. The bivariate Fox’s H-function in (12) converges if the following conditions are satisfied [11]: ρ1
=
M j=1
+
B
βj −
L
E
θj −
=
M1
D
δj −
L
θj −
In this section, and based on the PDF-based approach, analytical derivations for the average BER of coherent and non-coherent modulation schemes as well as the average channel capacity are presented. A. Average Bit Error Rate for Some Modulation Schemes 1) Coherent Detection: For a large variety of coherent modulation schemes, a unified conditional bit error rate can be expressed in terms of Gaussian Q-function [10], that is √ (4) Pr(e | γ) = Am Q(Bm γ), where the values of the parameters Am and Bm dependon the ∞ modulation schemes and Q(x) = √12π x exp −t2 /2 dt. For binary phase shift keying (BPSK), binary frequency shift keying (BFSK) and for high values of average SNR for Gaussian minimum shift keying (GMSK), M -arydifferentially encoded PSK, quadrature PSK, M -PSK, M FSK, square M -ary-quadrature amplitude modulation (M QAM), and M -ary-differential PSK, the average BER is given by
∞ Pr(e | γ)fγ (γ)dγ Pr(e) = 0
∞ √ = Am Q(Bm γ)fγ (γ)dγ, (5) 0
where fγ (γ) is the PDF of the instantaneous SNR γ. Now plugging (3) into (5) results in (9), at the top of the next page (Equs. (9)-(15) are listed in Table I). However, in order to obtain a closed-form expression for the average BER, we represent the Gaussian Q-function in terms of Fox’s H-function as follows: Using the relation between the complementary error function √ erfc(x) and the Gaussian Qfunction, 2Q(x) = erfc(x/ 2) and with the help of [9, 8.4.14.2], then (9) can be rewritten as in (10).
αj −
E j=L+1
A
αj
j=N +1
F
φj > 0
(6)
j=1
δj +
j=M1 +1
j=1
III. P ERFORMANCE A NALYSIS
θj −
j=L+1
j=1
+
N j=1
j=M +1
j=1
ρ2
βj +
N1
γj −
j=1
θj −
F
C
γj
j=N1 +1
φj > 0
(7)
j=1
πρ2 πρ1 and |arg y| < . (8) 2 2 However, it is straightforward to show that the parameters of the Fox’s H-function in (11) and (24) satisfy these sufficient conditions, and therefore the Fox’s H-function converges. |arg x|
