SCIENCE CHINA Information Sciences, Volume 60 , Issue 10 : 102306(2017) https://doi.org/10.1007/s11432-016-0611-x

Throughput and BER of wireless powered DF relaying in Nakagami-${\boldsymbol m}$ fading

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  • ReceivedSep 14, 2016
  • AcceptedNov 21, 2016
  • PublishedJun 28, 2017


Energy harvesting provides a promising solution to the extra energy requirement at the relay due to relaying. In this paper, the throughput and bit error rate of a decode-and-forward relaying system are studied using power splitting wireless power. Three different transmission scenarios are considered: instantaneous transmission, delay- or error-constrained transmission and delay- or error-tolerant transmission. For each scenario, exact expressions for the throughput and bit error rate are derived. Numerical results show that, for instantaneous transmission, the optimum splitting factor is not sensitive to the channel gain of the source-to-relay link. For delay- or error-constrained transmissions, the optimum splitting factor increases with the quality of the source-to-relay link and decreases with the quality of the relay-to-destination link. For delay- or error-tolerant transmissions, the optimum splitting factor is insensitive to the quality of the source-to-relay link.


The work of Yan Gao was financially supported by Open Foundation of Engineering Research and Development Center for Nanjing College of Information Technology (Grant No. KF20150104), Research Project of Nanjing College of Information Technology (Grant No. YK20150102), Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (Grant No. PPZY2015C242). The work of Aiqun Hu was supported in part by National Natural Science Foundation of China (Grant No. 61571110).



Derivation of (29)

In this appendix, the ergodic capacity of the DF relaying system will be derived. From (28), one has \begin{align}\bar{C} =&\int_0^{\infty}\int^{\infty}_{\frac{\sigma_{da}^2+\sigma_{dc}^2}{(1-\rho)\sigma_{ra}^2+\sigma_{rc}^2}\frac{1-\rho}{\eta\rho}d_r^v}\ln\left(1+\frac{(1-\rho)P_Sx/d_s^v}{(1-\rho)\sigma_{ra}^2+\sigma_{rc}^2}\right)f_{|h|^2}(x)f_{|g|^2}(y){\rm d}x{\rm d}y \\ &+\int_0^{\infty}\int_0^{\frac{\sigma_{da}^2+\sigma_{dc}^2}{(1-\rho)\sigma_{ra}^2+\sigma_{rc}^2}\frac{1-\rho}{\eta\rho}d_r^v}\ln\left(1+\frac{\eta\rho P_Sxy/(d_s^vd_r^v)}{\sigma_{da}^2+\sigma_{dc}^2}\right)f_{|h|^2}(x)f_{|g|^2}(y){\rm d}x{\rm d}y. \tag{37} \end{align} Since $h$ and $g$ are independent, the first term can be calculated as \begin{align}&\int_0^{\infty}\int^{\infty}_{\frac{\sigma_{da}^2+\sigma_{dc}^2}{(1-\rho)\sigma_{ra}^2+\sigma_{rc}^2}\frac{1-\rho}{\eta\rho}d_r^v}\ln\left(1+\frac{(1-\rho)P_Sx/d_s^v}{(1-\rho)\sigma_{ra}^2+\sigma_{rc}^2}\right)f_{|h|^2}(x)f_{|g|^2}(y){\rm d}x{\rm d}y \\ & =\int_0^{\infty}\ln\left(1+\frac{(1-\rho)P_Sx/d_s^v}{(1-\rho)\sigma_{ra}^2+\sigma_{rc}^2}\right)f_{|h|^2}(x){\rm d}x\cdot\int^{\infty}_{\frac{\sigma_{da}^2+\sigma_{dc}^2}{(1-\rho)\sigma_{ra}^2+\sigma_{rc}^2}\frac{1-\rho}{\eta\rho}d_r^v} f_{|g|^2}(y){\rm d}y. \tag{38} \end{align} Using the CDF of $|g|^2$ and $W(\cdot,\cdot,\cdot)$ solved in (30) using a variable transformation of $t=1+ax$ and [-1], one has (29).

Derivation of (33)

The key to derive (33) is to solve two integrals as \begin{align}U(a,b,c) =& \int_0^{\infty}{\rm erfc}\left(\sqrt{ax}\right)x^{b-1}{\rm e}^{-cx}{\rm d}x, \tag{39} \\ V(a,b_1,c_1,b_2,c_2) =& \int_0^{\infty}{\rm erfc}\left(\sqrt{axy}\right)x^{b_1-1}{\rm e}^{-c_1x}y^{b_2-1}{\rm e}^{-c_2y}{\rm d}x{\rm d}y. \tag{40} \end{align} For $U(a,b,c)$, by letting $\sqrt{x}=t$, one has \begin{eqnarray}U(a,b,c) &=& 2\int_0^{\infty}{\rm erfc}\left(\sqrt{a}t\right)t^{2b-1}{\rm e}^{-ct^2}{\rm d}t=2\int_0^{\infty}\left[1-{\rm erf}(\sqrt{a}t)\right]t^{2b-1}{\rm e}^{-ct^2}{\rm d}t \\ &=&\frac{\Gamma(b)}{c^b}-2\int_0^{\infty}{\rm erf}\left(\sqrt{a}t\right)t^{2b-1}{\rm e}^{-ct^2}{\rm d}t, \tag{41} \end{eqnarray} where ${\rm erf}(\cdot)$ is the error function given by ${\rm erf}(x)=1-{\rm erfc}(x)$. Then, using Eq. (4.3.8) in Geller's work 1), (34) can be derived. For $V(a,b_1,c_1,b_2,c_2)$, using (34) for the inner integral over $x$, one has \begin{eqnarray}V(a,b_1,c_1,b_2,c_2) &=&\frac{\Gamma(b_1)\Gamma(b_2)}{c_1^{b_1}c_2^{b_2}}-\frac{2\sqrt{\frac{a}{\pi}}(c_1/a)^{b_2+0.5}}{c_1^{b_1+0.5}} \\ & &\int_0^{\infty}t^{b_2-0.5}{\rm e}^{-\frac{c_1c_2}{a}t}{_2F_1(0.5,b_1+0.5;1.5;-t)}{\rm d}t. \tag{42} \end{eqnarray} Using [-1] in the above, one can derive (35). Using $U(a,b,c)$ and $V(a,b_1,c_1,b_2,c_2)$, one has (33).

Geller A, Ng E W. A table of integrals of the error functions. J Res Nation Bur Stand, 1969, 73B: 1–20


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