Evaluating the effect of self-Interference on the performance of full-duplex two-way relaying communication with energy harvesting

ewritten as 0
 ∞
 τ(bx+c) Z 4λ τc
 x0 ax − 2 −λ x
 Z Z = e a4x 1 dx
 P1 = fx,y (x, y)dydx
 0 (72)
 0 y0 ∞
 x x Z β
 0 − 1 −λ x
 = e 4x 1 dx
 τ(bx+c)
 x
 Z 0 Zax 0
 −λ1x −λ2y
 = λ1e λ2e dydx s
 β1 p 
 y =
 0 0 x K1 β1λ1 ,
 x0 λ1
  τ(bx+c) 
 x 4λ2τc
 Z 0 Zax where β1 = a .
  −  −
 = λ λ  e λ2ydye λ1xdx
 1 2  
 y We rewrite T12 in (71) as
 0 0 x
 x0 ∞
 Z λ2τc
 x0 − −λ1x
 Z  λ τ(bx+c) λ y  T = e ax e dx
 − 2 − 2 0 x −λ x 12
 = −λ e ax − e x0 e 1 dx
 1 x0
 0 ∞ (73)
 Z φ1
 x0 x0 − x −λ1x
 Z ( + ) Z λ y = e e dx.
 − λ2τ bx c − − 2 0 x −
 ax λ1x x0 λ1x
 = −λ1 e e dx + λ1 e e dx x0
 0 0
 x0 x0 λ2τc
 λ bτ Z λ τc Z λ2y0 Let φ = . Then, applying the Taylor series
 − 2 − 2 −λ x − x −λ x 1 a
 a ax 1 x0 1 ∞ t
 = −λ1e e e dx + λ1 e e dx − φ1 (−1) φ t
 x = 1
 expansion for e ∑ t!xt , one obtain
 0 0 t=0
 x0 x0
 λ y ∞
 λ2bτ Z λ2τc Z − 2 0 x ∞ t t
 − − −λ1x x −λ1x Z (−1) φ
 = −λ1e a e ax e dx + λ1 e 0 e dx T = 1 e−λ1xdx
 12 ∑ t!xt
 0 0 x t=0
 | {z } | {z } 0
 T T (74)
 1 2 t ∞
 ∞ (−1) φ t Z e−λ1x
 (69) = 1 dx.
 ∑ t! xt
 t=0 x
 The T2 term is given by 0
 x0
 Z − λ2y0
 x x −λ1x Using the exponent integral
 T2 = λ1 e 0 e dx
 ∞
 0 Z e−zt
 x0 Ek (z) = dt (75)
  λ y  k
 Z − 2 0 + x t
 x λ1 (70) 1
 = λ1 e 0 dx
 0 to write (74) in the closed form as
  
  − λ2y0 +  ∞ t t
 λ1 x λ1 x0 (−1) φ −
 = − e 0 − 1 T = 1 (x )1 kE (λ x ) . (76)
 λ2y0 + λ 12 ∑ t! 0 k 1 0
 x0 1 t=0
P. Nguyen-Huu et al.: Evaluating the Effect of Self-Interference. . . 75
 4λ1τc
 From (69), we have where β2 = d and
 ∞
 P = T + T2 λ τc
 1 1 Z − 1
 H = e dy e−λ2ydy
 − λ2bτ 12
 = −λ e a (T + T )
 1 11 12 y0
 (82)
 λ1  −(λ y +λ x )  ∞
 − e 2 0 1 0 − 1 Z φ2
 − y −λ2y
 λ2y0/x0 + λ1 = e e dy.
 s y0
 − λ2bτ β1 p 
 = −λ e a K β λ (77) φ t
 1 1 1 1 − 2 ∞ (−1) φ t
 λ1 = λ1τc y = 2
 Let φ2 d . Then, we have e ∑ t!yt and
 ! t=0
 ∞ (−1)tφ t
 1 1−t ∞ t
 − ∑ (x0) Et (λ1x0) Z ∞ (−1) φ t
 t! = 2 −λ2y
 t=0 H12 ∑ t e dy
 t=0 t!y
 λ   y0
 − 1 e−(λ2y0+λ1x0) − 1 .
 + t ∞
 λ2y0/x0 λ1 ∞ (−1) φ t Z e−λ2y
 = 2 (83)
 ∑ t dy
 t=0 t! y
 y0
 Following the same derivation as P1, we have
 ∞ t t
skipped some manipulation of P2 in (78) (−1) φ2 1−t
 = ∑ (y0) Et (λ2y0)
 τ(by+c) t=0 t!
 y0 dy
 Z Z Inserting (79) and (80) into (78), we have
P2 = fx,y (x, y)dxdy
 0 x0 P2 = H1 + H2
 y y
 0 s
 τ(by+c) λ1bτ β2 p 
 − d
 y0 dy = −λ2e K1 β2λ2 −
 Z Z λ2
 −λ1x −λ2y
 = λ1e λ2e dxdy
 ∞ t t ! (84)
 x (− )
 0 0 y 1 φ2 1−t
 y0 ∑ (y0) Et (λ2y0)
 t=0 t!
 y0 y0
 λ τc λ x
 λ1bτ Z − 1 Z − 1 0 y
 − dy −λ2y y −λ2y λ2  −(λ x +λ y ) 
 = −λ2e d e e dy + λ2 e 0 e dy − e 1 0 2 0 − 1 .
 λ1x0/y0 + λ2
 0 0
 | {z } | {z } Therefore, combining (77) with (84) results in
 H1 H2
 (78)
 A3 = Pr ({γ1 < τ} ∩ {γ2 < τ})
where s
 − λ2bτ β1 p 
 y0 = −λ1e a K1 β1λ1
 Z − λ1x0 λ
 y y −λ2y 1
 H2 = λ2 e 0 e dy
 ∞ t t !
 0 (79) (−1) φ1 1−t
 − ∑ (x0) Et (λ1x0)
 λ   t=0 t!
 = − 2 e−(λ1x0+λ2y0) − 1
 λ1x0/y0 + λ2 λ  
 − 1 e−(λ2y0/x0+λ1) − 1
and λ2y0/x0 + λ1 (85)
 s
 y0 λ bτ
 λ bτ Z λ τc − 1 β2 p 
 − 1 − 1 −λ y − d
 H = −λ e d e dy e 2 dy λ2e K1 β2λ2
 1 2 λ2
 0 !
 ∞ (−1)tφ t
   − 2 (y )1−tE (λ y )
 ∑ t! 0 t 2 0
  ∞ ∞  t=0
 Z λ τc Z λ τc
 − λ1bτ  − 1 −λ y − 1 − 
 d  dy 2 dy λ2y  λ2  −( x y + ) 
 = −λ2e  e dy + e e dy − e λ1 0/ 0 λ2 − 1 .
   λ x /y + λ
 0 xy  1 0 0 2
 | {z } | {z } OWFD TWFD
 H11 H12 This finishes the proof of Pout,12 in (29) and Pout,12
 (80) in (41).
with
 ∞
 Z λ1τc References
 − dy −λ2y
 H11 = e dy
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P. Nguyen-Huu et al.: Evaluating the Effect of Self-Interference. . . 77
 two-way full duplex relay networks with amplify-and- Khuong Ho-Van received the B.E. (with the
 forward protocol,” IEEE Transactions on Wireless Commu- first-rank honor) and the M.S. degrees in Elec-
 nications, vol. 13, no. 7, pp. 3768–3777, 2014. tronics and Telecommunications Engineering
[37] H. Chen, G. Li, and J. Cai, “Spectral–energy efficiency from Ho Chi Minh City University of Technol-
 tradeoff in full-duplex two-way relay networks,” IEEE ogy, Vietnam, in 2001 and 2003, respectively,
 and the Ph.D. degree in Electrical Engineering
 Systems Journal, vol. 12, no. 1, pp. 583–592, 2015. from University of Ulsan, Korea in 2007. Dur-
[38] Z. Chen, B. Xia, and H. Liu, “Wireless information and ing 2007-2011, he joined McGill University,
 power transfer in two-way amplify-and-forward relaying Canada as a postdoctoral fellow. Currently, he
 channels,” in Proceedings of the Global Conference on Signal is an Associate professor at Ho Chi Minh City
 and Information Processing (GlobalSIP). IEEE, 2014, pp. University of Technology. His major research
 168–172. interests are modulation and coding techniques, diversity techniques,
[39] Y. Liu, L. Wang, M. Elkashlan, T. Q. Duong, and A. Nal- digital signal processing, energy harvesting, physical layer security,
 lanathan, “Two-way relay networks with wireless power and cognitive radio.
 transfer: design and performance analysis,” IET Commu-
 nications, vol. 10, no. 14, pp. 1810–1819, 2016.
[40] A. Jeffrey and D. Zwillinger, Table of integrals, series, and
 products. Elsevier, 2007. Vo Nguyen Quoc Bao was born in Nha Trang,
 Khanh Hoa Province, Vietnam. He received
 the B.E. and M.Eng. degree in electrical engi-
 neering from Ho Chi Minh City University of
 Technology (HCMUT), Vietnam, in 2002 and
 Phong Nguyen-Huu received the B.E. degree 2005, respectively, and Ph.D. degree in elec-
 in Telecommunications Engineering from Uni- trical engineering from University of Ulsan,
 versity of Transport and Communi cations- South Korea, in 2009. In 2002, he joined the
 Campus II (UTC2), Vietnam in 2006, and the Department of Electrical Engineering, Posts
 Master’s degree in Telecommunications Engi- and Telecommunications Institute of Technol-
 neering from Posts and Tele communications ogy (PTIT), as a lecturer. Since February 2010,
 Institute of Technology (PTIT), Vietnam in he has been with the Department of Telecommunications, PTIT,
 2014. Currently, he is working towards a Ph.D. where he is currently an Assistant Professor. His major research
 degree in Ho Chi Minh City University of interests are modulation and coding techniques, MIMO systems,
 Technology (HCMUT), Vietnam. His research combining techniques, cooperative communications, and cognitive
 interests are two-way communications, full- radio. Dr. Bao is a member of Korea Information and Communi-
duplex transmission, and energy harvesting. cations Society (KICS), The Institute of Electronics, Information and
 Communication Engineers (IEICE) and The Institute of Electrical and
 Electronics Engineers (IEEE). He is currently serving as the Editor of
 Transactions on Emerging Telecommunications Technologies (Wiley
 ETT). He is also a Guest Editor of EURASIP Journal on Wireless
 Communications and Networking, special issue on “Cooperative
 Cognitive Networks” and IET Communications, special issue on
 “Secure Physical Layer Communications”.