Ergodic Sum Rate Evaluation of Cellular Multiuser Two-Way Relaying with Beamforming and Antenna Selection Over Nakagami-m Fading

Abstract

In this paper, we evaluate the ergodic sum rate (ESR) performance of cellular multiuser two-way relaying networks (CMTWRNs), where a multiantenna base station (BS) exchanges information bidirectionally with one of the several single-antenna mobile stations (MSs) with the help of a single-antenna relay terminal. Specifically, we adopt two transmission schemes (i.e., beamforming (BF) and antenna selection (AS)) at the BS and user selection at MSs to maximize the end-to-end signal-to-noise ratios. Under such transmission schemes, we derive new tight closed-form ESR expressions for the CMTWRNs in the presence of Nakagami-m fading environment. Further, based on the numerical results, we conduct a comparative study between the ESR performances of the two schemes, which indicates that the AS scheme provides approximately equal ESR performance as BF. Therefore, AS scheme can be a good alternative to the BF for the CMTWRNs as it reduces the transceiver complexity and signaling cost. Finally, simulation results are presented to corroborate our theoretical findings.

Appendices

Appendix 1: Proof of Theorem 1

The nth-order moment of the end-to-end SNR $\varLambda _{ab_{k^{*}}}$ is given by

$$\begin{aligned} {\mathbb {E}}\left\{ \varLambda _{ab_{k^{*}}}^{n} \right\} =\int _{0}^{\infty }z^{n}f_{\varLambda _{ab_{k^{*}}}}(z)dz. \end{aligned}$$

(29)

To obtain ${\mathbb {E}}\left\{ \varLambda _{ab_{k^{*}}}^{n} \right\}$, we first need to derive the PDF of $\varLambda _{ab_{k^{*}}}$, which can be evaluated by deriving the CDF of $\varLambda _{ab_{k^{*}}}$ as

$$\begin{aligned} F_{\varLambda _{ab_{k^{*}}}}(z)&=\text {Pr}\left[ \varLambda _{ab_{k^{*}}}<\lambda _{1}z \right] \nonumber \\&=\text {Pr}\left[ \varLambda _{1a}<\frac{\lambda _{1}\varLambda _{1b}z}{\varLambda _{1b}-\lambda _{1}z}\big | \varLambda _{1b} \right] \nonumber \\&=\int _{0}^{\lambda _{1}z}f_{\varLambda _{1b}}(y)\int _{0}^{\infty }f_{\varLambda _{1a}}(x)dx\,dy+\int _{\lambda _{1}z}^{\infty }f_{\varLambda _{1b}}(y)\int _{0}^{\frac{\lambda _{1}zy}{y-\lambda _{1}z} }f_{\varLambda _{1a}}(x)dx\,dy , \end{aligned}$$

(30)

where $\lambda _{1}=\frac{P_{b}+P_{r}}{P_{r}}$. Now, invoking the PDFs of $\varLambda _{1a}$ and $\varLambda _{1b}$ from (9) and (10), respectively into (30), and performing the required integrations with the aid of [21, eqs. (8.350.1), (8.352.6)], the CDF of $\varLambda _{ab_{k^{*}}}$ can be obtained as

$$\begin{aligned} F_{\varLambda _{ab_{k^{*}}}}(z)&=\frac{K\left( \frac{m_{b}}{{\overline{\varLambda }}_{1b}}\right) ^{m_{b}}}{\varGamma (m_{b})}\sum _{p=0}^{K-1}\sum _{r=0}^{p(m_{b}-1)}{K-1 \atopwithdelims ()p}(-1)^{p}b_{r}^{p}\Biggl [ \left( \frac{{\overline{\varLambda }}_{1b}}{m_{b}(p+1)}\right) ^{m_{b}+r}\!\!\!\!\varGamma (m_{b}+r) \nonumber \\&\quad -\,\sum _{m=0}^{m_{a}N_{a}-1}\sum _{j=0}^{m_{b}+r-1}\sum _{i=0}^{m}\frac{{m_{b}+r-1 \atopwithdelims ()j}}{m!}{m \atopwithdelims ()i}\left( \frac{m_{a}}{{\overline{\varLambda }}_{1a}} \!\right) ^{m}{\text {e}}^{-\alpha _{1}z}\left( \lambda _{1}z\right) ^{\delta } \nonumber \\&\quad \times \,2\left( \frac{{\overline{\varLambda }}_{1b}m_{a}}{{\overline{\varLambda }}_{1a}m_{b}(p+1)} \right) ^{\frac{j-i+1}{2}}{\mathcal {K}}_{j-i+1}(2\beta _{1}z)\Biggr ], \end{aligned}$$

(31)

where $\alpha _{1}=\lambda _{1}\left( \frac{m_{a}}{{\overline{\varLambda }}_{1a}}+\frac{m_{b}(p+1)}{{\overline{\varLambda }}_{1b}}\right)$, $\beta _{1}=2\sqrt{\!\frac{m_{a}m_{b}\lambda _{1}^{2}(p+1)}{{\overline{\varLambda }}_{1a}{\overline{\varLambda }}_{1b}}}$, and $\delta =m_{b}+r+m$. Then, differentiating (31) with respect to z and applying the derivative property of ${\mathcal {K}}_{\nu }(\cdot )$ [21, eq. (8.486.11)], the corresponding PDF can be expressed as

$$\begin{aligned} f_{\varLambda _{ab_{k^{*}}}}(z)&=\frac{K\left( \frac{m_{b}}{{\overline{\varLambda }}_{1b}}\right) ^{m_{b}}}{\varGamma (m_{b})}\sum _{p=0}^{K-1}\sum _{r=0}^{p(m_{b}-1)}{K-1 \atopwithdelims ()p}(-1)^{p}b_{r}^{p}\sum _{m=0}^{m_{a}N_{a}-1} \nonumber \\&\quad \,\times \sum _{j=0}^{m_{b}+r-1}\sum _{i=0}^{m}\!\frac{{m_{b}+r-1 \atopwithdelims ()j}}{m!}{m \atopwithdelims ()i}\left( \frac{m_{a}}{{\overline{\varLambda }}_{1a}} \right) ^{m}\left( \lambda _{1}z\right) ^{\delta } 2\!\left( \frac{{\overline{\varLambda }}_{1b}m_{a}}{{\overline{\varLambda }}_{1a}m_{b}(p+1)} \right) ^{\frac{j-i+1}{2}} \nonumber \\&\quad \,\times \text {e}^{-\alpha _{1}z}\Biggl [ \alpha _{1}{\mathcal {K}}_{j-i+1}(\beta _{1}z)-\frac{\delta }{z}{\mathcal {K}}_{j-i+1}(\beta _{1}z)+\frac{\beta _{1}}{2}( {\mathcal {K}}_{j-i}(\beta _{1}z)+{\mathcal {K}}_{j-i+2}(\beta _{1}z))\Biggr ]. \end{aligned}$$

(32)

Invoking $f_{\varLambda _{ab_{k^{*}}}}(z)$ from (32) into (24) and simplifying the integral with the help of [21, eq. (6.621.3)], we can get ${\mathbb {E}}\left\{ \varLambda _{ab_{k^{*}}}^{n} \right\}$ as given in (24).

Appendix 2: Proof of Theorem 2

The $n^{\text {th}}$-order moment of the end-to-end SNR $\varPsi _{a_{l^{*}}b_{k^{*}}}$ is given by

$$\begin{aligned} {\mathbb {E}}\left\{ \varPsi _{a_{l^{*}}b_{k^{*}}}^{n} \right\} =\int _{0}^{\infty }z^{n}f_{\varPsi _{a_{l^{*}}b_{k^{*}}}}(z)dz. \end{aligned}$$

(33)

To obtain ${\mathbb {E}}\left\{ \varPsi _{a_{l^{*}}b_{k^{*}}}^{n}\right\}$, we first need to derive the PDF of $\varPsi _{a_{l^{*}}b_{k^{*}}}$, which can be evaluated by deriving the CDF of $\varPsi _{a_{l^{*}}b_{k^{*}}}$ as

$$\begin{aligned} F_{\varPsi _{a_{l^{*}}b_{k^{*}}}}(z)&=\text {Pr}\left[ \varPsi _{a_{l^{*}}b_{k^{*}}}<\lambda _{1}z \right] \nonumber \\&=\text {Pr}\left[ \varPsi _{1a}<\frac{\lambda _{1}\varPsi _{1b}z}{\varPsi _{1b}-\lambda _{1}z}\big | \varPsi _{1b} \right] \nonumber \\&=\int _{0}^{\lambda _{1}z}f_{\varPsi _{1b}}(y)\int _{0}^{\infty }f_{\varPsi _{1a}}(x)dx\,dy+\int _{\lambda _{1}z}^{\infty }f_{\varPsi _{1b}}(y)\int _{0}^{\frac{\lambda _{1}zy}{y-\lambda _{1}z} }f_{\varPsi _{1a}}(x)dx\,dy. \end{aligned}$$

(34)

Substituting the PDFs of $\varPsi _{1a}$ and $\varPsi _{1b}$ from (18) and (19), respectively into (34), and then performing the required integration with the help of [21, eqs. (8.350.1), (8.352.6)], the CDF of $\varPsi _{a_{l^{*}}b_{k^{*}}}$ after some mathematical simplifications can be obtained as

$$\begin{aligned} F_{\varPsi _{a_{l^{*}}b_{k^{*}}}}(z)&=\frac{K\left( \frac{m_{b}}{{\overline{\varPsi }}_{1b}}\right) ^{m_{b}}}{\varGamma (m_{b})}\sum _{p=0}^{K-1}{K-1 \atopwithdelims ()p}(-1)^{p}\sum _{r=0}^{p(m_{b}-1)}b_{r}^{p}\Biggl [ \left( \!\frac{{\overline{\varPsi }}_{1b}}{m_{b}(p+1)}\right) ^{m_{b}+r} \nonumber \\&\quad \times \,\Upsilon \left( m_{b}+r,\frac{m_{b}(p+1)\lambda _{1}z}{{\overline{\varPsi }}_{1b}}\right) +\frac{N_{a}\left( \frac{m_{a}}{{\overline{\varPsi }}_{1a}}\right) ^{m_{a}}}{\varGamma (m_{a})}\sum _{s=0}^{N_{a}-1}{N_{a}-1 \atopwithdelims ()s}(-1)^{s}\sum _{t=0}^{s(m_{a}-1)} \nonumber \\&\quad \times \,c_{t}^{s} \left( \!\frac{{\overline{\varPsi }}_{1a}}{m_{a}(s+1)}\!\right) ^{m_{a}+t}\!\!\!\varGamma (m_{a}+t)\Biggl (\!\left( \frac{{\overline{\varPsi }}_{1b}}{m_{b}(p+1)}\!\right) ^{m_{b}+r}\!\!\!\varGamma \left( \!m_{b}+r,\frac{m_{b}(p+1)\lambda _{1}z}{{\overline{\varPsi }}_{1b}}\!\right) \nonumber \\&\quad -\,2\sum _{m=0}^{m_{a}+t-1}\sum _{j=0}^{m_{b}+r-1}\sum _{i=0}^{m}\!\frac{{m_{b}+r-1 \atopwithdelims ()j}{m \atopwithdelims ()i}}{m!}\left( \frac{m_{a}(s+1)}{{\overline{\varPsi }}_{1a}} \right) ^{m}{\text {e}}^{-\sigma _{1}z}\left( \lambda _{1}z \right) ^{m_{b}+r+m} \nonumber \\&\quad \times \,\left( \frac{{\overline{\varPsi }}_{1b}m_{a}(s+1)}{{\overline{\varPsi }}_{1a}m_{b}(p+1)}\right) ^{\frac{j-i+1}{2}}{\mathcal {K}}_{j-i+1}(\rho _{1}z)\Biggr )\Biggr ], \end{aligned}$$

(35)

where $\sigma _{1}=\lambda _{1}\left( \frac{m_{a}(s+1)}{{\overline{\varPsi }}_{1a}}+\frac{m_{b}(p+1)}{{\overline{\varPsi }}_{1b}}\right)$ and $\rho _{1}=2\sqrt{\frac{m_{a}m_{b}(p+1)(s+1)\lambda _{1}^{2}}{{\overline{\varPsi }}_{1a}{\overline{\varPsi }}_{1b}}}$. Further, differentiating (35) with respect to z and applying the derivative property of ${\mathcal {K}}_{\nu }(\cdot )$ [21, eq. (8.486.11)], the corresponding PDF can be expressed as

$$\begin{aligned} f_{\varPsi _{a_{l^{*}}b_{k^{*}}}}(z)&={\mathcal {D}}_{1}\left( \frac{z^{m}m_{b}(p+1)\lambda _{1}}{{\overline{\varPsi }}_{1b}}{\text {e}}^{-\frac{m_{b}(p+1)\lambda _{1}z}{{\overline{\varPsi }}_{1b}}}-mz^{m-1}{\text {e}}^{-\frac{m_{b}(p+1)\lambda _{1}z}{{\overline{\varLambda }}_{1b}}}\right) \nonumber \\&\quad +\,{\mathcal {D}}_{1}{\mathcal {D}}_{2}\left( mz^{m-1}{\text {e}}^{-\frac{m_{b}(p+1)\lambda _{1}z}{{\overline{\varPsi }}_{1b}}}-\frac{z^{m}m_{b}(p+1)\lambda _{1}}{{\overline{\varPsi }}_{1b}}{\text {e}}^{-\frac{m_{b}(p+1)\lambda _{1}z}{{\overline{\varPsi }}_{1b}}}\right) \nonumber \\&\quad +\,{\mathcal {D}}_{3}z^{m_{b}+r+m}{{\text {e}}}^{-\left( \frac{m_{a}(s+1)}{{\overline{\varPsi }}_{1a}}+\frac{m_{b}(p+1)}{{\overline{\varPsi }}_{1b}}\right) \lambda _{1}z}\bigg [ \sigma _{1}{\mathcal {K}}_{j-i+1}(\rho _{1}z ) \nonumber \\&\quad -\,(m_{b}+r+m) z^{-1}{\mathcal {K}}_{j-i+1}(\rho _{1}z )+\frac{\rho _{1}}{2}\bigg ({\mathcal {K}}_{j-i}(\rho _{1}z)+{\mathcal {K}}_{j-i+2}(\rho _{1}z)\bigg )\bigg ], \end{aligned}$$

(36)

where

$$\begin{aligned} {\mathcal {D}}_{1}&=\frac{K\left( \frac{m_{b}}{{\overline{\varPsi }}_{1b}}\right) ^{m_{b}}}{\varGamma (m_{b})}\sum _{p=0}^{K-1}{K-1 \atopwithdelims ()p}(-1)^{p}\sum _{r=0}^{p(m_{b}-1)}b_{r}^{p}\left( \frac{{\overline{\varPsi }}_{1b}}{m_{b}(p+1)}\right) ^{m_{b}+r} \nonumber \\&\quad \times \,\varGamma (m_{b}+r)\sum _{m=0}^{m_{b}+r-1}\frac{1}{m!}\left( \frac{m_{b}(p+1)\lambda _{1}}{{\overline{\varPsi }}_{1b}}\right) ^{m} \end{aligned}$$

(37)

$$\begin{aligned} {\mathcal {D}}_{2}=\frac{N_{a}\left( \frac{m_{a}}{{\overline{\varPsi }}_{1a}}\right) ^{m_{a}}}{\varGamma (m_{a})}\sum _{s=0}^{N_{a}-1}{N_{a}-1 \atopwithdelims ()s}(-1)^{s}\sum _{t=0}^{s(m_{a}-1)}c_{t}^{s}\left( \frac{{\overline{\varPsi }}_{1a}}{m_{a}(s+1)}\right) ^{m_{a}+t}\!\!\!\varGamma (m_{a}+t) \end{aligned}$$

(38)

$$\begin{aligned} {\mathcal {D}}_{3}&={\mathcal {D}}_{2}\frac{K\left( \frac{m_{b}}{{\overline{\varPsi }}_{1b}}\right) ^{m_{b}}}{\varGamma (m_{b})}\sum _{p=0}^{K-1}{K-1 \atopwithdelims ()p}(-1)^{p}\sum _{r=0}^{p(m_{b}-1)}b_{r}^{p}\sum _{m=0}^{m_{a}+t-1}\sum _{j=0}^{m_{b}+r-1} \nonumber \\&\quad \times \,\sum _{i=0}^{m}\frac{{m \atopwithdelims ()i}{m_{b}+r-1 \atopwithdelims ()j}}{m!}\left( \frac{m_{a}(s+1)}{{\overline{\varPsi }}_{1a}}\right) ^{m}2\left( \frac{{\overline{\varPsi }}_{1b}m_{a}(s+1)}{{\overline{\varPsi }}_{1a}m_{b}(p+1)}\right) ^{\frac{j-i+1}{2}}\!\!\!\lambda _{1}^{m_{b}+r+m} \end{aligned}$$

(39)

Now, substituting the PDF of $f_{\varPsi _{a_{l^{*}}b_{k^{*}}}}(z)$ from (36) into (33) and evaluating the integral with the help of [21, eq. (6.621.3)], we can get ${\mathbb {E}}\left\{ \varPsi _{a_{l^{*}}b_{k^{*}}}^{n} \right\}$ as shown in (28).