Incremental Fixed-Gain Opportunistic AF in Two-Way Relaying Networks

Abstract

In this paper, we investigate an incremental semi-blind opportunistic amplify-and-forward (AF) protocol in two-way relaying communication. This protocol is analyzed in terms of the average sum-rate and average symbol error rate considering independent Rayleigh fading channels. Bounds of these performance criteria are provided in closed-form expressions for the semi-blind and channel state information (CSI)-assisted relaying. The performance of the incremental semi-blind opportunistic AF relaying is compared to the performance of incremental CSI-assisted opportunistic AF relaying in order to prove the validity of the proposed analysis. We illustrate that the incremental semi-blind opportunistic AF relaying reduces significantly the system complexity for the cost of a slight decrease in the system performance.

Appendices

Appendix 1

The average sum-rate of the direct transmission in two-way system over Rayleigh fading channels can be written as

$$\begin{aligned} C_{D} =\frac{1}{2ln\left( 2 \right) } \int _{0}^{\infty } ln\left( x^{2}\right) f_{\varGamma }\left( x \right) dx \end{aligned}$$

(71)

where $f_{\varGamma }\left( x\right)$ is the PDF of $\varGamma$ given as

$$\begin{aligned} f_{\varGamma }\left( x\right) =\frac{1}{\overline{\varGamma }} \exp \left( \frac{-x}{\overline{\varGamma }} \right) \end{aligned}$$

(72)

(71) can be rewritten after simple manipulations as

$$\begin{aligned} C_{D} =\frac{1}{4ln\left( 2 \right) }\int _{0}^{\infty } ln\left( t \right) \frac{f_{\varGamma }\left( t^{\frac{1}{2}}\right) }{t^{\frac{1}{2}}} dt=\frac{1}{4ln\left( 2 \right) \overline{\varGamma }_{N} }\int _{0}^{\infty } \frac{ln\left( t \right) }{\sqrt{t}} \exp \left( \frac{-\sqrt{t}}{\overline{\varGamma }_{N}}\right) dt \end{aligned}$$

(73)

The above integral can be written as

$$\begin{aligned} \int _{0}^{\infty } \frac{ln\left( t \right) }{\sqrt{t}} \exp \left( \frac{-\sqrt{t}}{\overline{\varGamma }_{N}}\right) dt=4 \overline{\varGamma }_{N}\left( -\zeta +ln\left( \overline{\varGamma }_{N}\right) \right) \end{aligned}$$

(74)

By substituting (74) into (73), we can obtain (43).

Appendix 2

The end-to-end SNRs given in (30) and (31) [16] can be approximated by their upper bounds as

$$\begin{aligned} \varGamma ^{SB }_{N_2\rightarrow N_1}\le \frac{P\sqrt{P_{R_{i}}}}{2\sqrt{2}}\sqrt{\frac{C_{1}}{C_{2}\left( C_{1}+C_{2} \right) }} \varGamma _{R_{i}N_{2}} \sqrt{\varGamma _{N_{1}R_{i}}} \end{aligned}$$

(75)

and

$$\begin{aligned} \varGamma ^{SB}_{N_1\rightarrow N_2} \le \frac{P\sqrt{P_{R_{i}}}}{2\sqrt{2}}\sqrt{\frac{C_{2}}{C_{1}\left( C_{1}+C_{2} \right) }} \varGamma _{N_{1}R_{i}} \sqrt{\varGamma _{R_{i}N_{2}}} \end{aligned}$$

(76)

respectively.

The equivalent SNR can be approximated by its upper bound as

$$\begin{aligned} \frac{\frac{P^{2}P^{2}_{R_{i}}}{4} \varGamma ^{2}_{N_{1}R_{i}} \varGamma ^{2}_{R_{i}N_{2}}}{\kappa _{i} } \le \frac{P^{2}P_{R_{i}}}{8\left( C_{1}+C_{2} \right) } \left( \varGamma _{N_{1}R_{i}} \varGamma _{R_{i}N_{2}}\right) ^{\frac{3}{2}} \end{aligned}$$

(77)

where $\kappa _{i}$ is defined in (49).

The second term in (77) can be written as

$$\begin{aligned} \left( \varGamma _{N_{1}R_{i}} \varGamma _{R_{i}N_{2}}\right) ^{\frac{3}{2}}\, =\, \left( \frac{\varGamma _{N_{1}R_{i}} \varGamma _{R_{i}N_{2}}}{\varGamma _{N_{1}R_{i}} +\varGamma _{R_{i}N_{2}}}\right) ^{\frac{3}{2}}\left( \varGamma _{N_{1}R_{i}} +\varGamma _{R_{i}N_{2}}\right) ^{\frac{3}{2}} \end{aligned}$$

(78)

The first term in (78) can be approximated by its upper bound as in [18]

$$\begin{aligned} \left( \frac{\varGamma _{N_1R_i}\varGamma _{R_iN_2}}{\varGamma _{N_1R_i}+\varGamma _{R_iN_2}}\right) ^{\frac{3}{2}} \le \left( min\left( \varGamma _{N_1R_i}, \varGamma _{R_iN_2}\right) \right) ^{\frac{3}{2}} \end{aligned}$$

(79)

The last term in (78) can be rewritten using (24) and (25) as

$$\begin{aligned} \left( \varGamma _{N_1R_i}+\varGamma _{R_iN_2} \right) ^{\frac{3}{2}}\,= \, \left( \frac{C_{1}+C_{2}-2}{P}\right) ^{\frac{3}{2}} \end{aligned}$$

(80)

By substituting (78), (79) and (80) into (77), we can obtain (51).

Appendix 3

The upper bound of the average sum-rate for incremental semi-blind opportunistic AF relaying over Rayleigh fading channels can be written as

$$\begin{aligned} C_{R}=\frac{1}{3ln\left( 2 \right) } \int _{0}^{\infty } ln\left( A_{i} x^{\frac{3}{2}}\right) f_{\varGamma }\left( x \right) dx \end{aligned}$$

(81)

where $A_{i}=\frac{\sqrt{P}\;P_{R_i}\left( C_{1}+C_{2}-2\right) \sqrt{C_{1}+C_{2}-2}}{8\left( C_{1}+C_{2}\right) }$ and $f_{\varGamma }\left( x\right)$ is the PDF of $\varGamma$ given as $\varGamma =max\left( \varGamma _{1},\ldots ,\varGamma _{L}\right)$.

Equation (81) can be rewritten after simple manipulations as

$$\begin{aligned} C_{R}&=\frac{1}{3ln\left( 2 \right) }\int _{0}^{\infty } ln\left( A_{i} \; t \right) \frac{2 f_{\varGamma }\left( t^{\frac{2}{3}}\right) }{3t^{\frac{1}{3}}} dt \nonumber \\&=\frac{L}{3ln\left( 2 \right) }\int _{0}^{\infty }\left( ln\left( A_{i}\right) +ln\left( t \right) \right) \left( 1- \exp \left( \frac{-t^{\frac{2}{3}}}{\overline{\varGamma }_{b}}\right) \right) ^{L-1} \frac{2\exp \left( \frac{-t^{\frac{2}{3}}}{\overline{\varGamma }_{b}}\right) }{3\overline{\varGamma }_{b}t^{\frac{1}{3}}} dt \nonumber \\&=\frac{ln\left( A_{i}\right) }{3ln\left( 2 \right) }+\frac{L}{3ln\left( 2 \right) }\int _{0}^{\infty } ln\left( t \right) \sum _{l=0}^{L-1}\left( _{l}^{L-1}\right) \frac{2 \left( -1\right) ^{l} \exp \left( \frac{-\left( l+1\right) t^{\frac{2}{3}}}{\overline{\varGamma }_{b}}\right) }{3\overline{\varGamma }_{b}t^{\frac{1}{3}}} dt \nonumber \\&=\frac{ln\left( A_{i}\right) }{3ln\left( 2 \right) }+\frac{L}{2ln\left( 2 \right) } \sum _{l=0}^{L-1}\left( _{l}^{L-1}\right) \left( -1\right) ^{l+1} \frac{\xi +ln\left( \frac{1+l}{\overline{\varGamma }_{b}} \right) }{1+l} \end{aligned}$$

(82)

Appendix 4

The upper bound of the average sum-rate for incremental CSI-assisted opportunistic AF relaying over Rayleigh fading channels can be written as

$$\begin{aligned} C_{R}=\frac{1}{3ln\left( 2 \right) } \int _{0}^{\infty } ln\left( \frac{P^{2}}{\mu _{i}} x^{2}\right) f_{\varGamma }\left( x \right) dx \end{aligned}$$

(83)

(83) can be rewritten after simple manipulations as

$$\begin{aligned} C_{R}&=\frac{1}{3ln\left( 2 \right) }\int _{0}^{\infty } ln\left( \frac{P^{2}}{\mu _{i}} \; t \right) \frac{f_{\varGamma }\left( \sqrt{t}\right) }{2\sqrt{t}} dt \nonumber \\&=\frac{L}{3ln\left( 2 \right) }\int _{0}^{\infty }\left( ln\left( \frac{P^{2}}{\mu _{i}} \right) +ln\left( t \right) \right) \left( 1- \exp \left( \frac{-\sqrt{t}}{\overline{\varGamma }_{b}}\right) \right) ^{L-1} \frac{\exp \left( \frac{-\sqrt{t}}{\overline{\varGamma }_{b}}\right) }{2\overline{\varGamma }_{b}\sqrt{t}} dt\nonumber \\&=\frac{ln\left( \frac{P^{2}}{\mu _{i}}\right) }{3ln\left( 2 \right) }+\frac{L}{3ln\left( 2 \right) }\int _{0}^{\infty } ln\left( t \right) \sum _{l=0}^{L-1}\left( _{l}^{L-1}\right) \frac{\left( -1\right) ^{l} \exp \left( \frac{-\left( l+1\right) \sqrt{t}}{\overline{\varGamma }_{b}}\right) }{2\overline{\varGamma }_{b}\sqrt{t}} dt \nonumber \\&=\frac{ln\left( \frac{P^{2}}{\mu _{i}}\right) }{3ln\left( 2 \right) }+\frac{2L}{3ln\left( 2 \right) } \sum _{l=0}^{L-1}\left( _{l}^{L-1}\right) \left( -1\right) ^{l+1} \frac{\xi +ln\left( \frac{1+l}{\overline{\varGamma }_{b}} \right) }{1+l} \end{aligned}$$

(84)

Appendix 5

1.1 Case of $x< \varGamma _{th}$

The $f_{\varGamma _{N_{2}}} \left( x\right)$ can be written as

$$\begin{aligned} f_{\varGamma _{N_{2}}} \left( x\right) =\int _{0}^{x} f_{\varGamma _{N}}\left( x-\varphi \right) f_{\varGamma ^{SB}_{N_1\rightarrow N_2}}\left( \varphi \right) d\varphi \end{aligned}$$

(85)

By substituting (60) and (61) into (85), we get

$$\begin{aligned} f_{\varGamma _{N_{2}}} \left( x\right)&= \frac{L }{\overline{\varGamma }_{s1} \overline{\varGamma }_{N}\left( 1-\exp \left( \frac{-\varGamma _{th}}{\overline{\varGamma }_{N}}\right) \right) } \int _{0}^{x} \exp \left( \frac{-\left( x-\varphi \right) }{\overline{\varGamma }_{N}}\right) \exp \left( \frac{-\varphi }{\overline{\varGamma }_{s1}}\right) \nonumber \\&\quad \times \,\left( 1- \exp \left( \frac{-\varphi }{\overline{\gamma }_{s1}}\right) \right) ^{L-1} d\varphi \end{aligned}$$

(86)

With the help of the binomial expansion $\left( 1-x \right) ^{k}=\sum _{i=0}^{k} \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( -1 \right) ^{i} x^{i}$ and after simple mathematical manipulations, we can obtain (62) for $x<\varGamma _{th}$.

1.2 Case of $x\ge \varGamma _{th}$

The $f_{\varGamma _{N_{2}}} \left( x\right)$ can be written as

$$\begin{aligned} f_{\varGamma _{N_{2}}} \left( x\right)&=\int _{0}^{x} f_{\varGamma _{N}}\left( \varphi \right) f_{\varGamma ^{SB }_{N_1\rightarrow N_2}}\left( x-\varphi \right) d\varphi \nonumber \\&=\int _{0}^{\varGamma _{th}} f_{\varGamma _{N}}\left( \varphi \right) f_{\varGamma ^{SB }_{N_1\rightarrow N_2}}\left( x-\varphi \right) d\varphi +\int _{\varGamma _{th}}^{x} f_{\varGamma _{N}}\left( \varphi \right) f_{\varGamma ^{SB }_{N_1\rightarrow N_2}}\left( x-\varphi \right) d\varphi \end{aligned}$$

(87)

From (61), $f_{\varGamma _{N}}\left( x \right) =0$ for $x\ge \varGamma _{th}$. Hence, (87) can be rewritten as

$$\begin{aligned} f_{\varGamma _{N_{2}}} \left( x\right) =\int _{0}^{\varGamma _{th}} f_{\varGamma _{N}}\left( \varphi \right) f_{\varGamma ^{SB }_{N_1\rightarrow N_2}}\left( x-\varphi \right) d\varphi \end{aligned}$$

(88)

By substituting (60) and (61) into (88), we get

$$\begin{aligned} f_{\varGamma _{N_{2}}} \left( x\right)&= \frac{L }{\overline{\varGamma }_{s1} \overline{\varGamma }_{N}\left( 1-\exp \left( \frac{-\varGamma _{th}}{\overline{\varGamma }_{N}}\right) \right) } \int _{0}^{\varGamma _{th}} \exp \left( \frac{-\varphi }{\overline{\varGamma }_{N}}\right) \exp \left( \frac{-\left( x-\varphi \right) }{\overline{\varGamma }_{s1}}\right) \nonumber \\&\quad \times \,\left( 1- \exp \left( \frac{-\left( x-\varphi \right) }{\overline{\varGamma }_{s1}}\right) \right) ^{L-1} d\varphi \end{aligned}$$

(89)

Using the binomial expansion given after (86) and after some mathematical manipulations, we can obtain (62) for $x\ge \varGamma _{th}$.

Appendix 6

In this paper, we consider incremental relaying when the decision is operated at the end-receiver. In this case, (63) can be rewritten for BPSK modulation as

$$\begin{aligned} P_{coop} = \underbrace{\int _{0}^{\varGamma _{th}} \frac{1}{2} \;erfc\left( \sqrt{x} \right) f_{\varGamma _{N_{2}}} \left( x\right) dx }_{I_{1}} +\underbrace{\int _{\varGamma _{th}}^{\infty } \frac{1}{2} \;erfc\left( \sqrt{x} \right) f_{\varGamma _{N_{2}}} \left( x\right) dx }_{I_{2}} \end{aligned}$$

(90)

$I_{1}$ and $I_{2}$ can be resolved using (62) as

$$\begin{aligned} I_{1}&= \frac{L}{2 \left( 1-\exp \left( \frac{-\varGamma _{th}}{\overline{\varGamma }_{N}}\right) \right) }\sum _{i=0}^{L-1} \left( {\begin{array}{c}L-1\\ i\end{array}}\right) \left( -1\right) ^{i} \frac{1}{\left( 1+i\right) \overline{\varGamma }_{N}-\overline{\varGamma }_{s1}} \nonumber \\&\quad \times \,\left[ l\left( \frac{1}{\overline{\varGamma }_{N}},\varGamma _{th}\right) -l\left( \frac{1+i}{\overline{\varGamma }_{s1}},\varGamma _{th}\right) \right] \end{aligned}$$

(91)

and

$$\begin{aligned} I_{2}&= \frac{L}{2 \left( 1-\exp \left( \frac{-\varGamma _{th}}{\overline{\varGamma }_{N}}\right) \right) }\sum _{i=0}^{L-1} \left( {\begin{array}{c}L-1\\ i\end{array}}\right) \left( -1\right) ^{i} \frac{1}{\overline{\varGamma }_{s1}-\left( 1+i\right) \overline{\varGamma }_{N}} \nonumber \\&\quad \times \,\left[ 1- \exp \left( -\varGamma _{th} \left( \frac{1}{\overline{\varGamma }_{N}}- \frac{\left( 1+i\right) }{\overline{\varGamma }_{s1}} \right) \right) \right] \lambda \left( \frac{1+i}{\overline{\varGamma }_{s1}},\varGamma _{th}\right) \end{aligned}$$

(92)

respectively.

Where $l\left( \cdot ,\cdot \right)$ and $\lambda \left( \cdot ,\cdot \right)$ are defined in (65) and (66), respectively.

By substituting (91) and (92) into (90), we can obtain (64).