Skip to main content
Log in

Energy Efficient Relay Positioning and Power Allocation for Multi-Relay Symmetric Channel with Analog Network Coding

  • Published:
Wireless Personal Communications Aims and scope Submit manuscript

Abstract

In this paper, we focus on studying the optimal joint relay positioning and power allocation for two serial relays’ transmission. We consider that information exchanges in Rayleigh flat-fading channels between two end-users which have symmetric traffic requirements in terms of the data rate. Multiple serial half-duplex relay nodes are employed to extend the communication coverage and assist the bidirectional communication between two end-users using the analog network coding protocol. With the objective of minimizing the total transmitting energy at the required data rate c, we investigate the optimal relay positioning and power allocation and then propose sub-optimal solutions for a two-serial-relay one-way channel and a two-serial-relay two-way channel due to no close-form optimal solution. Simulation results demonstrate a consistency with our proposed schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Cover, T. M., & El Gamal, A. A. (1979). Capacity theorems for the relay channel. IEEE Transactions on Information Theory, 25(5), 572–584.

    Article  MATH  Google Scholar 

  2. Parkvall, S., Dahlman, E., Furuskar, A., Jading, Y., Olsson, M., Wanstedt, S., & Zangi, K. (2008). LTE-advanced- evolving LTE towards IMT-advanced. In Proceedings of th IEEE VTC, pp. 1–5.

  3. Soldani, D., & Dixit, S. (2008). Wireless relays for broadband access [radio communications series]. Communications Magazine, IEEE, 46(3), 58–66.

    Article  Google Scholar 

  4. Zhou, M., Cui, Q., Valkama, M., & Tao, X. (2012). Energy-efficient resource allocation for OFDMA-based two-way relay channel with physical-layer network coding. EURASIP Journal on Wireless Communications and Networking, 2012(66), 1–11.

    Google Scholar 

  5. Li, Y., Vucetic, B., Zhou, Z., & Dohler, M. (2007). Distributed adaptive power allocation for wireless relay networks. IEEE Transactions on Wireless Communications, 6(3), 948–958.

    Article  Google Scholar 

  6. Chen, M., Serbetli, S., & Yener, A. (2008). Distributed power allocation strategies for parallel relay networks. IEEE Transactions on Wireless Communications, 7(2), 552–561.

    Article  Google Scholar 

  7. Hasna, M.O., & Alouini, M.S. (2004). Optimal power allocation for relayed transmissions over Rayleigh-fading channels. IEEE Transactions on Wireless Communications, 3(6), 1999–2004.

  8. Host-Madsen, A., & Zhang, J. (2005). Capacity bounds and power allocation for wireless relay channels. IEEE Transactions on Information Theory, 51(6), 2020–2040.

    Article  MathSciNet  Google Scholar 

  9. Zhang, X., & Gong, Y. (2009). Joint power allocation and relay positioning in multi-relay cooperative systems. Communications IET, 3(10), 1683–1692.

    Article  Google Scholar 

  10. Lu, Y., Wang, D., & Fattouch, M. (2014). Novel spectrum sensing scheme in cognitive radio by simultaneously sensing/transmitting at full-duplex Tx and BER measurements at Rx. In Proceedings of the IEEE PIMRC, pp. 1–5.

  11. Shannon, C.E. (1961). Two-way communication channels. In Proceedings of the 4th Berkeley symposium mathematical statistics probability, pp. 611–644.

  12. Zhou, M., Cui, Q., Jntti, R., & Tao, X. (2012). Energy-efficient relay selection and power allocation for two-way relay channel with anolog network coding. Communications Letters, IEEE, 16(6), 816–819.

    Article  Google Scholar 

  13. Li, Y., Zhang, X., Peng, M., & Wang, W. (2011). Power provisioning and relay positioning for two-way relay channel with analog network coding. Signal Processing Letters, IEEE, 18(9), 517–520.

    Article  Google Scholar 

  14. Dohler, M., Gkelias, A., & Aghvami, H. (2004). Resource allocation for FDMA-based regenerative multihop links. EEE Transactions on Wireless Communications, 3(6), 1989–1993.

    Article  Google Scholar 

  15. Luenberger, D. G., & Ye, Y. (2009). Linear and nonlinear programming . New York: Springer.

    Google Scholar 

Download references

Acknowledgments

This work is financially supported by NSFC No. 61071214.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Canyan Zhu.

Appendices

Optimal Relay Positioning and Power Allocation in One-Way One-Relay Transmission

In one-way transmission between \(R_1\) and \(S_2\) via \(R_2\), the optimization problem on relay positioning and power allocation can be expressed as

$$\begin{aligned}&\min _{\left\{ d_2, d_3, P_{R_1}, P_{R_2}, P_{S_2}\right\} } \ \ \ \xi 1 = P_{R,1} + 2P_{R,2} + P_{S,2}, \\&{\text {subject \;to}} \ \ d_2+d_3 = d-d_1, \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{P_{R,1}P_{R,2}|\alpha _2|^2|\alpha _3|^2}{P_{R,2}|\alpha _3|^2+1} \ge 2^{2c}-1, \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{P_{S,2}P_{R,2}|\alpha _2|^2|\alpha _3|^2}{P_{R,2}|\alpha _2|^2+1} \ge 2^{2c}-1, \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_{R, 1}, P_{R, 2}, P_{S, 2} \ge 0. \end{aligned}$$
(40)

Applying KKT conditions into (40), its solution can be obtained as

$$\begin{aligned} \left( \xi 1\right) _{min} = \left( 2^{2c}-1\right) \left( d^n_2+d^n_3\right) +4\sqrt{2^{2c}-1} (d_2d_3)^{\frac{n}{2}}. \end{aligned}$$
(41)

Therefore,

$$\begin{aligned} \frac{\partial \xi 1_{min}}{\partial d_2}= &\, {} n \left( 2^{2c}-1\right) \left( d^{n-1}_2 - d^{n-1}_3\right) \\&+ 2n\sqrt{2^{2c}-1} (d_2d_3)^{\frac{n}{2}-1} (d-d_1-2d_2). \\ \end{aligned}$$
(42)

Let \(\frac{\partial \xi 1_{min}}{\partial d_2}=0\), it is concluded that

$$\begin{aligned} d_2=d_3=\frac{d-d_1}{2}. \end{aligned}$$
(43)

By substituting (43) into (41), the transmit power becomes

$$\begin{aligned} \xi 1 \ge \left( \xi 1\right) _{min} = \left( 2\left( 2^{2c}-1\right) +4\sqrt{2^{2c}-1} \right) \left( \frac{d-d_1}{2}\right) ^n. \end{aligned}$$
(44)

And the corresponding transmit power at \(R_1, R_2\) and \(S_2\) is respectively obtained as

$$\begin{aligned} P_{R, 1}(d_1)\ge & {} \left( \left( 2^{2c}-1\right) + \sqrt{2^{2c}-1} \right) \left( \frac{d-d_1}{2}\right) ^n, \\ P_{R, 2}(d_1)\ge & {} \sqrt{2^{2c}-1} \left( \frac{d-d_1}{2}\right) ^n, \end{aligned}$$

and

$$\begin{aligned} P_{S, 2}(d_1)\ge & {} \left( \left( 2^{2c}-1\right) + \sqrt{2^{2c}-1} \right) \left( \frac{d-d_1}{2}\right) ^n. \end{aligned}$$
(45)

Optimal Relay Positioning and Power Allocation in Two-Way One-Relay Transmission

In two-way data transmission between \(R_1\) and \(S_2\) via \(R_2\), the optimization problem on relay positioning and power allocation can be expressed as

$$\begin{aligned}&\min _{\left\{ d_2, d_3, P_{R_1}, P_{R_2}, P_{S_2}\right\} } \ \ \ \xi 3 = P_{R,1} + P_{R,2} + P_{S,2}, \\&{\text {subject\; to}} \ \ d_2+d_3 = d-d_1, \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{P_{R,1}P_{R,2}|\alpha _2|^2|\alpha _3|^2}{P_{R,2}|\alpha _3|^2+1} \ge 2^{2c}-1, \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{P_{S,2}P_{R,2}|\alpha _2|^2|\alpha _3|^2}{P_{R,2}|\alpha _2|^2+1} \ge 2^{2c}-1, \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ P_{R, 1}, P_{R, 2}, P_{S, 2} \ge 0. \end{aligned}$$
(46)

Applying KKT conditions into (46), its solution can be obtained as

$$\begin{aligned} \left( \xi 3\right) _{min} = \left( 2^{2c}-1\right) \left( d^n_2+d^n_3\right) +2\sqrt{2}\sqrt{2^{2c}-1} (d_2d_3)^{\frac{n}{2}}. \end{aligned}$$
(47)

Therefore,

$$\begin{aligned} \frac{\partial \xi 3_{min}}{\partial d_2}= &\; {} n \left( 2^{2c}-1\right) \left( d^{n-1}_2 - d^{n-1}_3\right) \\&+ \sqrt{2}n\sqrt{2^{2c}-1} (d_2d_3)^{\frac{n}{2}-1} (d-d_1-2d_2). \\ \end{aligned}$$
(48)

Let \(\frac{\partial \xi 3_{min}}{\partial d_2}=0\), it is concluded that

$$\begin{aligned} d_2=d_3=\frac{d-d_1}{2}. \end{aligned}$$
(49)

By substituting (49) into (47), the transmit power becomes

$$\begin{aligned} \xi 3 \ge \left( \xi 3\right) _{min} = \left( 2\left( 2^{2c}-1\right) +2\sqrt{2}\sqrt{2^{2c}-1} \right) \left( \frac{d-d_1}{2}\right) ^n. \end{aligned}$$
(50)

And the corresponding transmit power at \(R_1, R_2\) and \(S_2\) is respectively obtained as

$$\begin{aligned} P_{R, 1}(d_1)\ge & {} \left( \left( 2^{2c}-1\right) + \sqrt{2^{2c}-1} \right) \left( \frac{d-d_1}{2}\right) ^n, \\ P_{R, 2}(d_1)\ge & {} \sqrt{2}\sqrt{2^{2c}-1} \left( \frac{d-d_1}{2}\right) ^n, \end{aligned}$$

and

$$\begin{aligned} P_{S, 2}(d_1)\ge & {} \left( \left( 2^{2c}-1\right) + \sqrt{2^{2c}-1} \right) \left( \frac{d-d_1}{2}\right) ^n. \end{aligned}$$
(51)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, J., Zhu, C., Wang, D. et al. Energy Efficient Relay Positioning and Power Allocation for Multi-Relay Symmetric Channel with Analog Network Coding. Wireless Pers Commun 84, 2735–2755 (2015). https://doi.org/10.1007/s11277-015-2764-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11277-015-2764-9

Keywords

Navigation