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Average symbol error rate and ergodic capacity of switch-and-examine combining diversity receivers over the α-μ fading channel

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Abstract

In this paper, we consider a switch-and-examine combining (SEC) diversity scheme operating over independent and identically distributed (i.i.d) branches assuming an α-μ fading channel. We derive expressions for the average symbol error rate (ASER) for a class of coherent modulation techniques considering this fading model as well as expressions for the ergodic capacity under the same assumptions. The results for the ASER and the ergodic capacity are shown to reduce to those previously reported in the literature for other channel models such as the Weibull model as a special case, which confirms the validity of the obtained expressions. Different aspects are studied including the effect of fading severity, the number of branches and the modulation scheme used. Also, insights on the optimal choice of the switching threshold are provided.

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References

  1. Yacoub MD (2007) The α−μ distribution: a physical fading model for the stacy distribution. IEEE Trans Veh Technol 56(1):27–34

    Article  Google Scholar 

  2. Magableh AM, Matalgah MM (2009) Moment generating function of the generalized α-μ distribution. IEEE Commun Lett 13(6):411–413

    Article  Google Scholar 

  3. Magableh AM, Matalgah MM (2011) Channel characteristics of the generalized alpha-mu multipath fading model. In: International wireless communications and mobile computing conference (IWCMC 2011), Istanbul, Turkey, July 4–8

  4. Mohamed R, Ismail MH., Newagy FA, Mourad H-A M (2012) Capacity of the α-μ fading channel with SC diversity under adaptive transmission techniques. In: Proc. of the 19th international conference on telecommunications (ICT), Jounieh, Lebanon

  5. Ismail MH, Mohamed R (2013) Outage probability analysis of selection and switch-and-stay combining diversity receivers over the α-μ fading channel with co-channel interference. Wirel Pers Commun 71(3):2181–2195. doi:10.1007/s11277-012-0930-x

  6. El Ayadi MMH, Ismail MH (2012) On the cumulative distribution function of the sum and harmonic mean of two α-μ random variables with applications. IET Commun 6(18):3122–3130

    Article  MATH  MathSciNet  Google Scholar 

  7. Xiao L, Dong X (2005) New results on the ASER of switched diversity combining over Nakagami-m fading channels. IEEE Commun Lett 9(2):136-138

    Article  Google Scholar 

  8. Alexandropoulos GC, Mathiopoulos PT, Sagias NC (2010) Switch-and-examine diversity over arbitrarliy correlated Nakagami-m fading channels. IEEE Trans Veh Technol 59(4):2080-2087

    Article  Google Scholar 

  9. Yang HC, Alouini M-S (2003) Performance analysis of multibranch switched diversity systems. IEEE Trans Commun 51(5):782-794

    Article  Google Scholar 

  10. Magableh AM, Matalgah MM (2011) Error probability performance and ergodic capacity of L-branch switched and examine combining in Weibull fading channels. IET Commun 5(9):11731181

    Article  Google Scholar 

  11. Chau YA, Chen Y-H (2012) Switch-and-examine combining with different new switch statistics for diversity over Rayleigh fading channels. Int J Comm Syst. doi:10.1002/dac.2442

  12. Simon MK, Alouini MS (2005) Digital communication over fading channels. Wiley, Hoboken

    Google Scholar 

  13. Goldsmith A (2005) Wireless communications. Cambridge University Press

  14. Gradshteyn IS, Ryzhik IM (2000) Table of integrals, series and products. Academic Press, San Diego

    MATH  Google Scholar 

  15. Prudnikov AP, Brychkov YA, Marichev OI (1990) Integral and series: more special functions, vol 3. Gordon and Breach Science Publishers

  16. Adamchik VS, Marichev OI (1990) The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system. In: Proceeding of international conference on symbolic and algebraic computation, Tokyo, Japan, pp 212–224

Download references

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Authors and Affiliations

  1. Department of Electronics and Communications Engineering, Faculty of Engineering, Cairo University, Giza, 12613, Egypt

    Refaat Mohamed, Mahmoud H. Ismail & Hebat-Allah M. Mourad

Authors
  1. Refaat Mohamed
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  2. Mahmoud H. Ismail
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  3. Hebat-Allah M. Mourad
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Correspondence to Mahmoud H. Ismail.

Appendices

Appendix I: Evaluating the integral \(\int \text {erfc}(\sqrt {\psi \gamma }) ~\gamma ^{{\frac {\alpha \mu }{2}}-1}~e^{-\beta \gamma ^{\frac {\alpha }{2}}}d\gamma \)

To evaluate the ASER expression in (7), one can notice that we are in need to solve an integral in the form of \(\int \text {erfc}(\sqrt {\psi \gamma }) ~\gamma ^{{\frac {\alpha \mu }{2}}-1}~e^{-\beta \gamma ^{\frac {\alpha }{2}}}d\gamma \) with two different sets of limits of integration. We start evaluating this integral by performing integration by parts where \(\gamma ^{{\frac {\alpha \mu }{2}}-1}~e^{-\beta \gamma ^{\frac {\alpha }{2}}}\) is integrated yielding \(-\frac {2 \mathrm {\Gamma }(\mu ,\beta \gamma ^{\frac {\alpha }{2}})}{\alpha \beta ^{\mu }}\) [14, Eq. (2.33.11)] where Γ(s,x) is the upper incomplete Gamma function defined in [14, Eq. (8.350.2)] and \(\text {erfc}(\sqrt {\psi \gamma }) \) is differentiated yielding \(\sqrt {\frac {\psi }{\pi \gamma }}~e^{-\psi \gamma }\). Hence, one can obtain:

$$\begin{array}{@{}rcl@{}} \int \text{erfc}(\sqrt{\psi\gamma}) ~\gamma^{{\frac{\alpha\mu}{2}}-1}~e^{-\beta\gamma^{\frac{\alpha}{2}}}d\gamma &= -\frac{2}{\alpha\beta^{\mu}}\text{erfc}(\sqrt{\psi\gamma})\mathrm{\Gamma}(\mu,\beta\gamma^{\frac{\alpha}{2}}) \\ &+ \frac{2}{\alpha\beta^{\mu}} \sqrt{\frac{\psi}{\pi}}\int\frac{e^{-\psi\gamma}\mathrm{\Gamma}(\mu,\beta\gamma^{\frac{\alpha}{2}})}{\sqrt{\gamma}}d\gamma \end{array} $$
(16)

The problem now is reduced to that of solving an integral of the form above. Towards that end, we start by using the following series expansion of the upper incomplete Gamma function given in [14, Eq. (8.352.4)]:

$$\begin{array}{*{20}l} \mathrm{\Gamma}(n,x)=(n-1)! e^{-x} \sum\limits_{m=0}^{n-1} \frac{x^{m}}{m!}. \end{array} $$
(17)

By substituting from (17) in (16), one can easily get

$$\begin{array}{*{20}l} \int\frac{e^{-\psi\gamma}\mathrm{\Gamma}(\mu,\beta\gamma^{\frac{\alpha}{2}})}{\sqrt{\gamma}}d\gamma &= \frac{\psi}{\pi}(\mu-1)!\sum\limits_{m=0}^{\mu-1} \frac{\beta^{m}}{m!} \\ &\quad \times \underbrace{\int \gamma^{\frac{\alpha m - 1}{2}} e^{-\psi\gamma} e^{-\beta\gamma^{\frac{\alpha}{2}}} d\gamma}_{I_{5}}. \end{array} $$
(18)

It is worth noting here that we need to solve the integral I 5 in (18) using 2 sets of limits; one from 0 to ∞, while the other is from 0 to γ th.

Case 1: Evaluating I 5 from 0 to ∞:

We start by using the following expression given in [15, Eq. (2.3.2.13)]:

$$\begin{array}{*{20}l} \int\limits_{0}^{\infty} x^{\alpha-1} e^{-a x^{r}} e^{-sx} dx = U' \end{array} $$
(19)

where U ′ is defined as:

$$ U\prime =\left\{ \begin{array}{ll} \sum\limits_{j = 0}^{q - 1} \frac{(-a)^{j}}{j!} \mathrm{\Gamma}(\alpha + r j) s^{-\alpha - r j} \times \\ {_{p+1}}F_{q} \left(1, {\Delta}(p,\alpha+rj); ({-1})^{q} z;\Delta(q,1+j)\right), \,\,\,\,\,0< r \leq 1 \\ \sum\limits_{h = 0}^{p-1} \frac{(-1)^{h}}{h!r} \mathrm{\Gamma}(\frac{\alpha+h}{r}) a^{-\frac{\alpha+h}{r}} s^{h} \times \\ {_{q+1}}F_{p} \left(1, {\Delta}(q,\frac{\alpha+h}{r}); ({-1})^{p} {z}^{-1}, {\Delta}(p,1+h)\right) , \,\,\,\,\,r > 1 \end{array}\right. $$
(20)

and p F q (a 1,...,a ℓ ;b 1,...,b k ;ζ) is the generalised hypergeometric function defined in [14, Eq. (9.14)]. Also, Δ(x,y)=y/x,(y+1)/x,...,(y+x−1)/x, and the argument z is defined as \(z = \left (\frac {p}{s}\right )^{p} \left (\frac {a}{q}\right )^{q}\).

Case 2: Evaluating I 5 from 0 to γ th:

Using Taylor series expansion, one can replace \( e^{-\beta \gamma ^{\frac {\alpha }{2}}} \) with \( \sum \limits _{n = 0}^{\infty } \frac {(-1)^{n} \beta ^{n} \gamma ^{\frac {\alpha n}{2}}}{n!}\) and hence,

$$\begin{array}{*{20}l} \int\limits_{0}^{\gamma_{\text{th}}} \gamma^{\frac{\alpha m - 1}{2}} e^{-\psi\gamma} e^{-\beta\gamma^{\frac{\alpha}{2}}} d\gamma &= \sum\limits_{n = 0}^{\infty} \frac{(-1)^{n} \beta^{n}}{n!} \\ &\quad \times\underbrace{\int\limits_{0}^{\gamma_{\text{th}}} \gamma^{(\frac{\alpha}{2}m + \frac{\alpha}{2}n - \frac{1}{2})} e^{-\psi\gamma} d\gamma}_{I_{6}}. \end{array} $$
(21)

I 6 can be evaluated using the expression defined in [14, Eq. (3.351.1)] yielding

$$\begin{array}{*{20}l} I_{6} = \frac{\mathrm{\Upsilon} \left( \frac{1}{2}\left(\alpha m+\alpha n +1\right), \psi\gamma_{\text{th}} \right)}{ \psi^{\frac{1}{2}(\alpha m+\alpha n +1)}} \end{array} $$
(22)

where Υ(s,x) is the lower incomplete Gamma function defined in [14, Eq. (8.350.1)].

Substituting (22) into (21), one can get:

$$\begin{array}{*{20}l} \int\limits_{0}^{\gamma_{\text{th}}} \gamma^{\frac{\alpha m - 1}{2}} e^{-\psi\gamma} e^{-\beta\gamma^{\frac{\alpha}{2}}} d\gamma {}={} \sum\limits_{n = 0}^{\infty} \frac{(-1)^{n} \beta^{n}}{n!} \frac{\mathrm{\Upsilon} \left( \frac{1}{2}(\alpha m+\alpha n +1), \psi\gamma_{\text{th}} \right)}{ \psi^{\frac{1}{2}(\alpha m+\alpha n +1)}}. \end{array} $$
(23)

Substituting from (20) and (23) in the preceding equations, one can get the results presented in (8) and (10), respectively.

Appendix II: Evaluating the integral \(\int \ln \left (1+\gamma \right ) \gamma ^{\frac {\alpha \mu }{2} -1} e^{-\beta \gamma ^{\frac {\alpha }{2}}} d\gamma \)

We first consider the integral with the limits 0 and ∞. We will represent each function within this integral in terms of the Meijer’s G function. We then use [16, Eq. (21)] to obtain:

$$\begin{array}{*{20}l} &\int\limits_{0}^{\infty} \ln\left(1+\gamma\right) \gamma^{\frac{\alpha \mu}{2} -1} e^{-\beta\gamma^{\frac{\alpha}{2}}} d\gamma = \\ &\frac{\ell^{-1}\left(k\right)^{\frac{1}{2}}}{\left(2\pi\right)^{(\ell-1)+\frac{1}{2}(k-1)}} \mathbf{G}_{2\ell,k+2\ell}^{k+2\ell,\ell}\left({} \left(\frac{\beta}{k}\right)^{k}{}\Bigg\arrowvert{}\begin{array}{c} \Delta(\ell,\frac{-\alpha\mu}{2}),\Delta(\ell,1+\frac{\alpha\mu}{2}) \\ \Delta(k,0),\Delta(\ell,\frac{-\alpha\mu}{2}),\Delta(\ell,\frac{-\alpha\mu}{2})\\ \end{array}{}\right). \end{array} $$
(24)

We next consider the same integral but with the limits 0 and γ th. First, we start executing the integral by parts to get :

$$\begin{array}{*{20}l} \int\limits_{0}^{\gamma_{\text{th}}} \mathrm{\ln}\left(1+\gamma\right) \gamma^{\frac{\alpha \mu}{2} -1} e^{-\beta\gamma^{\frac{\alpha}{2}}} d\gamma = &- \frac{\mathrm{\Gamma}\left(\mu, \beta \gamma_{\text{th}}^{\frac{\alpha}{2}}\right)}{\frac{\alpha}{2} \beta^{\mu}} \text{ln}(1 + \gamma_{\text{th}})\\& + \frac{1}{\frac{\alpha}{2} \beta^{\mu}} \underbrace{\int\limits_{0}^{\gamma_{\text{th}}} \frac{\mathrm{\Gamma}(\mu,\beta \gamma^{\frac{\alpha}{2}})}{1+\gamma} d\gamma}_{I_{7}} \end{array} $$
(25)

To solve I 7, we start by using the series expansion of the upper incomplete Gamma function given in (17) along with the Taylor series expansion for \( e^{-\beta \gamma ^{\frac {\alpha }{2}}}\) to get

$$\begin{array}{*{20}l} I_{7} = \mathrm{\Gamma}(\mu) \sum\limits_{m = 0}^{\mu - 1}\sum\limits_{n = 0}^{\infty} \frac{(-1)^{n} \beta^{m+n}}{m! n!} \underbrace{\int\limits_{0}^{\gamma_{\text{th}}} \frac{\gamma^{\frac{\alpha}{2}(m+n)}}{1+\gamma} d\gamma}_{I_{8}}. \end{array} $$
(26)

Using [14, Eq. (3.194.4)], which states that

$$\begin{array}{*{20}l} {\int\limits_{0}^{U}} \frac{x^{\zeta-1}}{\left(1+\rho x\right)^{\nu}}dx = \frac{U^{\zeta}}{\zeta} {_{2}}F_{1}\left(\nu,\zeta;1+\zeta;-\rho U\right), \end{array} $$
(27)

one can solve I 8 easily. By substituting from the preceding equations, (14) and (15) are straightforwardly obtained.

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Mohamed, R., Ismail, M.H. & Mourad, HA.M. Average symbol error rate and ergodic capacity of switch-and-examine combining diversity receivers over the α-μ fading channel. Ann. Telecommun. 70, 37–48 (2015). https://doi.org/10.1007/s12243-014-0432-9

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