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Evaluating the Performance of Full-Duplex Energy Harvesting Vehicle-to-Vehicle Communication System over Double Rayleigh Fading Channels


In this paper, we analyze the performance of vehicle-to-vehicle (V2V) communication system, which employs full-duplex (FD) and energy harvesting (EH) techniques at source and relay nodes from power beacon (PB) through radio frequency. Unlike previous systemswhere all nodes located at fixed locations, we investigate the case that three nodes (source, relay,and destination) are moving vehicles. Therefore, the channels between them follow double (cascade) Rayleigh fading distributions. Furthermore, the source and relay nodes can harvest the energy from PB for data transmission when they move on the road. We derive the exact expressions of the outage probability (OP) and symbol error probability (SEP) of the proposed system and then intensively study the impacts of various parameters such as the number transmission antennas of PB, the time duration for EH, the distances between nodes, and the residual self-interference (RSI) at the FD relay node on the system performance. Monte-Carlo simulations validate all theory analysis. Numerical results show that system performance is strongly impacted by the number of transmission antennas of the power beacon, the EH duration, the RSI, and the distances between nodes. Moreover, for a given transmission of power beacon and the SIC capability of the FD relay node, there exist optimal EH duration and optimal distance from the source to relay, which provide the best system performance.

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  1. Gazestani AH, Ghorashi SA, Mousavinasab B, Shikh-Bahaei M (2019) A survey on implementation and applications of full duplex wireless communications. Physical Communication 34:121–134

    Article  Google Scholar 

  2. Wang CX, Bian J, Sun J, Zhang W, Zhang M (2018) A survey of 5G channel measurements and models. IEEE Communications Surveys & Tutorials 20(4):3142–3168

    Article  Google Scholar 

  3. Zhang D, Liu Y, Dai L, Bashir AK, Nallanathan A, Shim B (2019) Performance analysis of FD-NOMA-based decentralized V2X systems. IEEE Transactions on Communications 67(7):5024–5036

    Article  Google Scholar 

  4. Kolodziej KE, Perry BT, Herd JS (2019) In-band full-duplex technology: techniques and systems survey. IEEE Transactions on Microwave Theory and Techniques 67(7):3025–3041

    Article  Google Scholar 

  5. Nguyen NP, Ngo HQ, Duong TQ, Tuan HD, da Costa DB (2017) Full-duplex cyber-weapon with massive arrays. IEEE Transactions on Communications 65(12):5544–5558

    Article  Google Scholar 

  6. Zhang T, Cai Y, Huang Y, Duong TQ, Yang W (2017) Secure full-duplex spectrum-sharing wiretap networks with different antenna reception schemes. IEEE Trans Commun 65(1):335–346

    Google Scholar 

  7. Deng Y, Kim KJ, Duong TQ, Elkashlan M, Karagiannidis GK, Nallanathan A (2016) Full-duplex spectrum sharing in cooperative single carrier systems. IEEE Trans Cognitive Commun Network 2 (1):68–82

    Article  Google Scholar 

  8. Nguyen NP, Kundu C, Ngo HQ, Duong TQ, Canberk B (2016) Secure full-duplex small-cell networks in a spectrum sharing environment. IEEE Access 4:3087–3099

    Article  Google Scholar 

  9. Doan XT, Nguyen NP, Yin C, Da Costa DB, Duong TQ (2017) Cognitive full-duplex relay networks under the peak interference power constraint of multiple primary users. EURASIP J Wirel Commun Netw 2017(1):1–10

    Article  Google Scholar 

  10. Nguyen BC, Hoang TM, Tran PT (2019) Performance analysis of full-duplex decode-and-forward relay system with energy harvesting over Nakagami-m fading channels. AEU-International Journal of Electronics and Communications 98:114–122

    Article  Google Scholar 

  11. Ba Cao Nguyen, Thang Nguyen Nhu, Tran Xuan Nam, Dung LT (2020) Impacts of imperfect channel state information, transceiver hardware, and self-interference cancellation on the performance of full-duplex MIMO relay system. Sensors 20(6):1671

    Article  Google Scholar 

  12. Bharadia D, McMilin E, Katti S (2013) Full duplex radios. In: ACM SIGCOMM computer communication review, vol 43. ACM, pp 375–386

  13. Li X, Tepedelenlioglu C, Senol H (2017) Channel estimation for residual self-interference in full duplex amplify-and-forward two-way relays. IEEE Transactions on Wireless Communications 16(8):4970–4983

    Article  Google Scholar 

  14. Antonio-Rodríguez E, López-Valcarce R, Riihonen T, Werner S, Wichman R (2013) Adaptive self-interference cancellation in wideband full-duplex decode-and-forward MIMO relays. In: 2013 IEEE 14th workshop on signal processing advances in wireless communications (SPAWC). IEEE, pp 370–374

  15. Hoang TM, Tan NT, Cao NB, Dung LT (2017) Outage probability of MIMO relaying full-duplex system with wireless information and power transfer. In: 2017 conference on information and communication technology (CICT), pp 1–6

  16. Dung LT, Nguyen BC, Hoang TM, Choi SG (2018) Full-duplex relay system with energy harvesting: outage and symbol error probabilities. In: 2018 international conference on advanced technologies for communications (ATC). IEEE, pp 360–365

  17. Li S, Kun Y, Zhou M, Wu J, Song L, Li Y, Li H (2017) Full-duplex amplify-and-forward relaying: power and location optimization. IEEE Transactions on Vehicular Technology 66(9):8458–8468

    Article  Google Scholar 

  18. Li C, Chen Z, Wang Y, Yao Y, Xia B (2017) Outage analysis of the full-duplex decode-and-forward two-way relay system. IEEE Transactions on Vehicular Technology 66(5):4073–4086

    Article  Google Scholar 

  19. Van Nguyen L, Nguyen BC, Tran XN, Dung LT (2020) Transmit antenna selection for full-duplex spatial modulation multiple-input multiple-output system. IEEE Systems Journal 14(4):4777–4785

  20. Campolo C, Molinaro A, Berthet AO, Vinel A (2017) Full-duplex radios for vehicular communications. IEEE Communications Magazine 55(6):182–189

    Article  Google Scholar 

  21. Chen Y, Wang L, Ai Y, Jiao B, Hanzo L (2017) Performance analysis of NOMA-SM in vehicle-to-vehicle massive MIMO channels. IEEE Journal on Selected Areas in Communications 35(12):2653–2666

    Article  Google Scholar 

  22. Mao CX, Gao S, Wang Y (2018) Dual-band full-duplex tx/rx antennas for vehicular communications. IEEE Transactions on Vehicular Technology 67(5):4059–4070

    Article  Google Scholar 

  23. Yang M, Jeon SW, Kim DK (2018) Interference management for in-band full-duplex vehicular access networks. IEEE Transactions on Vehicular Technology 67(2):1820–1824

    Article  Google Scholar 

  24. Nguyen KK, Duong TQ, Vien NA, Le-Khac NA, Nguyen LD (2019) Distributed deep deterministic policy gradient for power allocation control in D2D-based V2V communications. IEEE Access 7:164533–164543

    Article  Google Scholar 

  25. Ai Y, Cheffena M, Mathur A, Lei H (2018) On physical layer security of double rayleigh fading channels for vehicular communications. IEEE Wireless Communications Letters 7(6):1038–1041

    Article  Google Scholar 

  26. Kovacs IZ, Eggers PCF, Olesen K, Petersen LG (2002) Investigations of outdoor-to-indoor mobile-to-mobile radio communication channels. In: Proceedings IEEE 56th vehicular technology conference. IEEE, pp 430–434

  27. Zhong C, Suraweera HA, Zheng G, Krikidis I, Zhang Z (2014) Wireless information and power transfer with full duplex relaying. IEEE Transactions on Communications 62(10):3447–3461

    Article  Google Scholar 

  28. Tam HHM, Tuan HD, Nasir AA, Duong TQ, Poor HV (2017) MIMO energy harvesting in full-duplex multi-user networks. IEEE Trans Wirel Commun 16(5):3282–3297

    Article  Google Scholar 

  29. Nguyen VD, Duong TQ, Tuan HD, Shin OS, Poor HV (2017) Spectral and energy efficiencies in full-duplex wireless information and power transfer. IEEE Trans Commun 65(5):2220–2233

    Article  Google Scholar 

  30. Hoang TM, El Shafie A, Da Costa DB, Duong TQ, Tuan HD, Marshall A (2020) Security and energy harvesting for MIMO-OFDM networks. IEEE Trans Commun 68(4):2593–2606

    Article  Google Scholar 

  31. Nasir AA, Hoang TD, Duong TQ, Hanzo L (2020) Transmitter-side wireless information- and power-transfer in massive MIMO systems. IEEE Transactions on Vehicular Technology 9(2):2322–2326

    Article  Google Scholar 

  32. Zhou X, Zhang R, Ho CK (2013) Wireless information and power transfer: architecture design and rate-energy tradeoff. IEEE Transactions on Communications 61(11):4754–4767

    Article  Google Scholar 

  33. Yang K, Cui H, Song L, Li Y (2015) Efficient full-duplex relaying with joint antenna-relay selection and self-interference suppression. IEEE Transactions on Wireless Communications 14(7):3991–4005

    Article  Google Scholar 

  34. Nguyen BC, Tran XN (2019) Performance analysis of full-duplex amplify-and-forward relay system with hardware impairments and imperfect self-interference cancellation. Wireless Communications and Mobile Computing 29(Article ID 4946298):1–10

  35. Nguyen LV, Nguyen BC, Tran XN (2019) Closed-form expression for the symbol error probability in full-duplex spatial modulation relay system and its application in optimal power allocation. Sensors 19 (24):5390

    Article  Google Scholar 

  36. Nguyen BC, Tran XN, Tran DT, Dung LT (2019) Full-duplex amplify-and-forward relay system with direct link: performance analysis and optimization. Physical Communication 37:100888

    Article  Google Scholar 

  37. Hong S, Brand J, Choi JI, Jain M, Mehlman J, Katti S, Levis P (2014) Applications of self-interference cancellation in 5G and beyond. IEEE Communications Magazine 52(2):114–121

    Article  Google Scholar 

  38. Lu X, Wang P, Niyato D, Kim DI, Han Z (2015) Wireless networks with RF energy harvesting: a contemporary survey. IEEE Communications Surveys & Tutorials 17(2):757–789

    Article  Google Scholar 

  39. Sabharwal A, Schniter P, Guo D, Bliss DW, Rangarajan S, Wichman R (2014) In-band full-duplex wireless: challenges and opportunities. IEEE Journal on Selected Areas in Communications 32(9):1637–1652

    Article  Google Scholar 

  40. Goldsmith A (2005) Wireless communications. Cambridge University Press, Cambridge

    Book  Google Scholar 

  41. Leon-Garcia A, Leon-Garcia A (2008) Probability, statistics, and random processes for electrical engineering, 3rd edn. Pearson/Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  42. Jeffrey A, Zwillinger D (2007) Tableof integrals, series, and products. Academic Press, Cambridge, Massachusetts

    Google Scholar 

  43. Shankar PM (2017) Fading and shadowing in wireless systems. Springer, Berlin

    Book  Google Scholar 

  44. Duy TT, Alexandropoulos GC, Tung VT, Son VN, Duong TQ (2016) Outage performance of cognitive cooperative networks with relay selection over double-rayleigh fading channels. IET Communications 10(1):57–64

    Article  Google Scholar 

  45. Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol 9. Dover, New York

    MATH  Google Scholar 

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Appendix A

This appendix presents step-by-step derivations of the mathematical expression of the outage probability in Theorem 1. Firstly, the probability Pr{γR < x} is calculated as Eq. A1.

$$ \begin{array}{@{}rcl@{}} \text{Pr}\{\gamma_{\mathrm{R}} < x\} &=& \text{Pr}\left\{\frac{\eta \alpha P |h_{\text{SR}}|^{2} }{(1-\alpha)d_{\text{SR}}^{\beta}(\gamma_{\text{RSI}} +\sigma^{2})} \sum\limits_{i = 1}^{\mathrm{K}} |h_{i\mathrm{S}}|^{2} < x\right\}\\ &=& \text{Pr}\left\{|h_{\text{SR}}|^{2} < \frac{(1-\alpha)d_{\text{SR}}^{\beta}(\gamma_{\text{RSI}} +\sigma^{2})x }{\eta \alpha P \sum\limits_{i = 1}^{\mathrm{K}} |h_{i\mathrm{S}}|^{2} } \right\} \end{array} $$

Due to the fact that \(|h_{\text {SR}}|^{2}=|h_{\text {SR}_{1}}|^{2}|h_{\text {SR}_{2}}|^{2}\), the CDF of |hSR|2 is computed as

$$ \begin{array}{@{}rcl@{}} F_{|h_{\text{SR}}|^{2}}(x)&=& \Pr(|h_{\text{SR}_{1}}|^{2}|h_{\text{SR}_{2}}|^{2} < x)\\ &=&{\int}_{0}^{\infty}\Pr\Big(|h_{\text{SR}_{2}}|^{2} < \frac{x}{y}\Big)f_{|h_{\text{SR}_{1}}|^{2}}(y)dy. \end{array} $$

From Eqs. 17 and 18, we can rewrite (A2) as

$$ \begin{array}{@{}rcl@{}} F_{|h_{\text{SR}}|^{2}}(x)&=&1-\frac{1}{{{\varOmega}}_{3}}{\int}_{0}^{\infty}\exp\left( -\frac{y}{{{\varOmega}}_{3}}-\frac{x}{y{{\varOmega}}_{4}}\right)dy \\ &=&1-\sqrt{\frac{4x}{{{\varOmega}}_{3}{{\varOmega}}_{4}}} K_{1}\Bigg(\sqrt{\frac{4x}{{{\varOmega}}_{3}{{\varOmega}}_{4}}}\Bigg). \end{array} $$

We can see from Eq. A3 that, the CDF of |hSR|2 implicitly takes into account the effects of scatterers around the transmitter and the receiver which include the fluctuating amplitude and phase of signal, the Doppler shifts caused by the movement of vehicles [26].

Using Eqs. A320, and A1 becomes (A4). It is also noted that, we set \({{\varOmega }}_{1} =\mathbb {E}\{|h_{i\mathrm {S}}|^{2}\}\), \({{\varOmega }}_{3} =\mathbb {E}\{|h_{\text {SR}_{1}}|^{2}\}\) and \({{\varOmega }}_{4} =\mathbb {E}\{|h_{\text {SR}_{2}}|^{2}\}\) in Eq. A4.

$$ \begin{array}{@{}rcl@{}} \text{Pr}\{\gamma_{\mathrm{R}} < x\} &=& \int\limits_{0}^{\infty} \Bigg[1-\sqrt{\frac{4(1-\alpha)d_{\text{SR}}^{\beta}(\gamma_{\text{RSI}} +\sigma^{2})x}{{{\varOmega}}_{3}{{\varOmega}}_{4}\eta \alpha P y}} K_{1}\Bigg(\sqrt{\frac{4(1-\alpha)d_{\text{SR}}^{\beta}(\gamma_{\text{RSI}} +\sigma^{2})x}{{{\varOmega}}_{3}{{\varOmega}}_{4}\eta \alpha P y}}\Bigg) \Bigg] \frac{y^{\mathrm{K}-1}}{{{\varOmega}}_{1}^{\mathrm{K}} {\Gamma}(\mathrm{K})} \exp\Big(-\frac{y}{{{\varOmega}}_{1}}\Big) \\ &=& \int\limits_{0}^{\infty} \Bigg[1-\sqrt{\frac{Ax}{y}} K_{1}\Bigg(\sqrt{\frac{Ax}{y}}\Bigg) \Bigg] \frac{y^{\mathrm{K}-1}}{{{\varOmega}}_{1}^{\mathrm{K}} {\Gamma}(\mathrm{K})} \exp\Big(-\frac{y}{{{\varOmega}}_{1}}\Big)\\ &=& \frac{1}{{{\varOmega}}_{1}^{\mathrm{K}} {\Gamma}(\mathrm{K})} \Bigg[\int\limits_{0}^{\infty} y^{\mathrm{K}-1} \exp\Big(-\frac{y}{{{\varOmega}}_{1}}\Big)dy - \sqrt{Ax} \int\limits_{0}^{\infty} K_{1}\Bigg(\sqrt{\frac{Ax}{y}}\Bigg) y^{\mathrm{K}-\frac{3}{2}} \exp\Big(-\frac{y}{{{\varOmega}}_{1}}\Big) dy\Bigg] . \end{array} $$

For the first integral in Eq. A4, we apply [42, 3.351.3] to have

$$ \begin{array}{@{}rcl@{}} \int\limits_{0}^{\infty} y^{\mathrm{K}-1} \exp\Big(-\frac{y}{{{\varOmega}}_{1}}\Big)dy ={{\varOmega}}_{1}^{\mathrm{K}} {\Gamma}(\mathrm{K}). \end{array} $$

For the second integral in Eq. A4, we set \(t = \exp \Big (-\frac {y}{{{\varOmega }}_{1}}\Big )\). After some algebra calculations, we obtain the following integral

$$ \begin{array}{@{}rcl@{}} &&{}\int\limits_{0}^{\infty} K_{1}\Bigg(\sqrt{\frac{Ax}{y}}\Bigg) y^{\mathrm{K}-\frac{3}{2}} \exp\Big(-\frac{y}{{{\varOmega}}_{1}}\Big) dy \\ &=& {{\varOmega}}_{1} {\int\limits_{0}^{1}} K_{1}\Bigg(\sqrt{\frac{Ax}{-{{\varOmega}}_{1} \ln t}}\Bigg) (-{{\varOmega}}_{1} \ln t)^{\mathrm{K}-\frac{3}{2}} dt. \end{array} $$

Applying the Gaussian-Chebyshev quadrature method in [45], Eq. A6 becomes

$$ \begin{array}{@{}rcl@{}} &&{}{{\varOmega}}_{1} {\int\limits_{0}^{1}} K_{1}\Bigg(\sqrt{\frac{Ax}{-{{\varOmega}}_{1} \ln t}}\Bigg) (-{{\varOmega}}_{1} \ln t)^{\mathrm{K}-\frac{3}{2}} dt \\ &=& \frac{{{\varOmega}}_{1} \pi}{2M} \sum\limits_{m=1}^{M} \sqrt{1-{\phi_{m}^{2}}} K_{1}\Bigg(\sqrt{\frac{Ax}{-{{\varOmega}}_{1}\ln u}}\Bigg) (-{{\varOmega}}_{1}\ln u)^{{\mathrm{K}-\frac{3}{2}}}. \end{array} $$

Substituting Eqs. A5 and A7 into Eq. A4, we get (A8).

$$ \begin{array}{@{}rcl@{}} \text{Pr}\{\gamma_{\mathrm{R}} < x\} &=& \frac{1}{{{\varOmega}}_{1}^{\mathrm{K}} {\Gamma}(\mathrm{K})} \left[{{\varOmega}}_{1}^{\mathrm{K}} {\Gamma}(\mathrm{K}) - \sqrt{Ax} \frac{{{\varOmega}}_{1} \pi}{2M} \sum\limits_{m=1}^{M} \sqrt{1-{\phi_{m}^{2}}}\right.\\ &&{\kern3.4pc}\left.\times K_{1}\Bigg(\sqrt{\frac{Ax}{-{{\varOmega}}_{1}\ln u}}\Bigg) (-{{\varOmega}}_{1}\ln u)^{{\mathrm{K}-\frac{3}{2}}} \right]\\ &=& 1 - \frac{\pi\sqrt{Ax}}{2M({{\varOmega}}_{1})^{\mathrm{K}-1} {\Gamma}(\mathrm{K})} \sum\limits_{m=1}^{M} \sqrt{1-{\phi_{m}^{2}}}\\ &&\times K_{1}\Bigg(\sqrt{\frac{Ax}{-{{\varOmega}}_{1}\ln u}}\Bigg) (-{{\varOmega}}_{1}\ln u)^{{\mathrm{K}-\frac{3}{2}}} \end{array} $$

By doing the same calculations as for Pr{γR < x} for Pr{γD < x}, we have (A9).

$$ \begin{array}{@{}rcl@{}} \text{Pr}\{\gamma_{\mathrm{D}} < x\} &=& \text{Pr}\left\{\frac{\eta \alpha P |h_{\text{RD}}|^{2} }{(1-\alpha)d_{\text{RD}}^{\beta} \sigma^{2}} \sum\limits_{i = 1}^{\mathrm{K}} |h_{i\mathrm{R}}|^{2} < x\right\}\\ &=& \text{Pr}\left\{|h_{\text{RD}}|^{2} < \frac{(1-\alpha)d_{\text{RD}}^{\beta}\sigma^{2}x }{\eta \alpha P \sum\limits_{i = 1}^{\mathrm{K}} |h_{i\mathrm{R}}|^{2} } \right\}\\ &=& 1 - \frac{\pi\sqrt{Bx}}{2N({{\varOmega}}_{2})^{\mathrm{K}-1} {\Gamma}(\mathrm{K})} \sum\limits_{n=1}^{N} \sqrt{1-{\phi_{n}^{2}}}\\ &&\times K_{1}\Bigg(\sqrt{\frac{Bx}{-{{\varOmega}}_{2}\ln v}}\Bigg) (-{{\varOmega}}_{2}\ln v)^{{\mathrm{K}-\frac{3}{2}}}. \end{array} $$

Then, plugging Eqs. A8 and A9 into Eq. 15, we obtain (16) in Theorem 1 which is the closed-form expression of the outage probability of the proposed system. The proof is complete.

Appendix B

In this section, we provide the detailed derivations of Eq. 22 in Theorem 2.

Replacing F(x) in Eq. 23 with Pout in Eq. 16, we obtain (B1).

$$ \begin{array}{@{}rcl@{}} \text{SEP} &=& \frac{a \sqrt b}{2\sqrt {2\pi }}\int\limits_{0}^{\infty} \frac{e^{-b x/2}}{\sqrt x} \Bigg[ 1-\frac{\pi^{2} \sqrt{AB}x}{4MN({{\varOmega}}_{1}{{\varOmega}}_{2})^{\mathrm{K}-1} {\Gamma}^{2}(\mathrm{K})}\\ &&\times\sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} \sqrt{(1-{\phi_{m}^{2}})(1-{\phi_{n}^{2}})} K_{1}\Bigg(\sqrt{\frac{Ax}{-{{\varOmega}}_{1}\ln u}}\Bigg) \\ &&\times K_{1}\Bigg(\sqrt{\frac{Bx}{-{{\varOmega}}_{2}\ln v}}\Bigg) ({{\varOmega}}_{1}{{\varOmega}}_{2}\ln u\ln v)^{\mathrm{K}-\frac{3}{2}}\Bigg]dx \\ &=& \frac{a \sqrt b}{2\sqrt {2\pi }} \Bigg[\int\limits_{0}^{\infty} \frac{e^{-b x/2}}{\sqrt x} dx -\frac{\pi^{2} \sqrt{AB} ({{\varOmega}}_{1}{{\varOmega}}_{2}\ln u\ln v)^{\mathrm{K}-\frac{3}{2}}}{4MN({{\varOmega}}_{1}{{\varOmega}}_{2})^{\mathrm{K}-1} {\Gamma}^{2}(\mathrm{K})}\\ &&\times\sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} \sqrt{(1-{\phi_{m}^{2}})(1-{\phi_{n}^{2}})}\\ &\times& \int\limits_{0}^{\infty} x^{\frac{1}{2}} e^{-b x/2} K_{1}\Bigg(\sqrt{\frac{Ax}{-{{\varOmega}}_{1}\ln u}}\Bigg) K_{1}\Bigg(\sqrt{\frac{Bx}{-{{\varOmega}}_{2}\ln v}}\Bigg)dx \Bigg]. \end{array} $$

For the first integral in Eq. B1, we use [42, 3.361.2] to solve it, i.e.,

$$ \begin{array}{@{}rcl@{}} \int\limits_{0}^{\infty} \frac{e^{-b x/2}}{\sqrt x}dx = \sqrt{\frac{2\pi}{b}}. \end{array} $$

For the second integral in Eq. B1, we set z = ebx/2. After some mathematical calculations, we get (B3).

$$ \begin{array}{@{}rcl@{}} &&{}\int\limits_{0}^{\infty} x^{\frac{1}{2}} e^{-b x/2} K_{1}\Bigg(\sqrt{\frac{Ax}{-{{\varOmega}}_{1}\ln u}}\Bigg) K_{1}\Bigg(\sqrt{\frac{Bx}{-{{\varOmega}}_{2}\ln v}}\Bigg)\\ {\kern12pt}dx &=& \frac{2}{b} {\int\limits_{0}^{1}} \sqrt{\frac{-2 \ln z}{b}} K_{1}\Bigg(\sqrt{\frac{2A\ln z}{b{{\varOmega}}_{1}\ln u}}\Bigg) K_{1}\Bigg(\sqrt{\frac{2B \ln z}{b{{\varOmega}}_{2}\ln v}}\Bigg)dz. \end{array} $$

Due to the complexity of the integral in Eq. B3, it is hard to find the closed-form expression for this integral. Therefore, we use the Gaussian-Chebyshev quadrature method in [45] to solve this integral. After that, we have (B4).

$$ \begin{array}{@{}rcl@{}} &&{}\frac{2}{b} {\int\limits_{0}^{1}} \sqrt{\frac{-2 \ln z}{b}} K_{1}\Bigg(\sqrt{\frac{2A\ln z}{b{{\varOmega}}_{1}\ln u}}\Bigg) K_{1}\Bigg(\sqrt{\frac{2B \ln z}{b{{\varOmega}}_{2}\ln v}}\Bigg) dz = \frac{\pi}{Jb} \\ &&{\kern3pt}\times\sum\limits_{j=1}^{J} \sqrt{1-{\phi_{j}^{2}}} \sqrt{\frac{-2 \ln w}{b}} K_{1}\Bigg(\sqrt{\frac{2A\ln w}{b{{\varOmega}}_{1}\ln u}}\Bigg) K_{1}\Bigg(\sqrt{\frac{2B \ln w}{b{{\varOmega}}_{2}\ln v}}\Bigg). \end{array} $$

Substituting Eqs. B2 and B4 into Eq. B1, we obtain (22) in Theorem 2 which is the closed-form expression of the SER of the proposed system. The proof is complete.

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Nguyen, B.C., Hoang, T.M., Tran, X.N. et al. Evaluating the Performance of Full-Duplex Energy Harvesting Vehicle-to-Vehicle Communication System over Double Rayleigh Fading Channels. Mobile Netw Appl 26, 1777–1787 (2021).

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  • Vehicle-to-vehicle communication
  • Full-duplex
  • Energy harvesting
  • Self-interference cancellation
  • Outage probability
  • Symbol error probability