Optimal Beamforming and Performance Analysis of Wireless Relay Networks with Unmanned Aerial Vehicle

Jian Ouyang; Min Lin

doi:10.1515/freq-2014-0065

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Optimal Beamforming and Performance Analysis of Wireless Relay Networks with Unmanned Aerial Vehicle

Jian Ouyang und Min Lin

Veröffentlicht/Copyright: 28. Januar 2015

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Abstract

In this paper, we investigate a wireless communication system employing a multi-antenna unmanned aerial vehicle (UAV) as the relay to improve the connectivity between the base station (BS) and the receive node (RN), where the BS–UAV link undergoes the correlated Rician fading while the UAV–RN link follows the correlated Rayleigh fading with large scale path loss. By assuming that the amplify-and-forward (AF) protocol is adopted at UAV, we first propose an optimal beamforming (BF) scheme to maximize the mutual information of the UAV-assisted dual-hop relay network, by calculating the BF weight vectors and the power allocation coefficient. Then, we derive the analytical expressions for the outage probability (OP) and the ergodic capacity (EC) of the relay network to evaluate the system performance conveniently. Finally, computer simulation results are provided to demonstrate the validity and efficiency of the proposed scheme as well as the performance analysis.

Keywords: unmanned aerial vehicle; wireless relay network; optimal beamforming; performance analysis

Appendix A

According to the definition of CDF, Fγtu can be obtained as

(A.1)Fγt(u)=Pr(γ1,tγ2,tγ1,t+γ2,t+1<u) =∫0uPr(γ2,t>u(γ1,t+1)γ1,t−u|γ1,t)fγ1,t(x)dx+∫u∞Pr(γ2,t<u(γ1,t+1)γ1,t−u|γ1,t)fγ1,t(x)dx =1−∫u∞[1−Fγ2,t((x+1)ux−u)]fγ1,t(x)dx

where fγ1,tx is the probability density function (PDF) of γ1,t and Fγ2,tx is the CDF of γ2,t. Here, g1,t experiences correlated Rician fading, the PDF expression of γ1,t=τtg1,tF2γˉ1 is expressed as [22]

(A.2)fγ1,t(x)=∑m=0∞∑n=0m(mn)(−1)nξmxN+n−1Γ(N+n)(2τtβγ¯1)N+nexp(−x2τtβγ¯1)(x≥0)

where β is the convergence parameter (1≤β≤4) and ξ0=1, ξm=∑n=0m−1δm−nξn/m for n≥1, with δm given in eq. (33). Meanwhile, since g2,t is subject to correlated Rayleigh fading, the PDF of γ2,t=1−τtg2,tF2γˉ2 has the formula [22]

(A.3)fγ2,t(x)= ∑i=1μ∑j=1viϑi,jxn−1Γ(j)((1−τt)λ2,t,iγ¯2)jexp(−x(1−τt)λ2,t,iγ¯2) (x≥0)

where ϑi,j is given in eq. (32). With the help of the identity [20, eq. 3.351.1]

∫0uxne−αxdx=n!αn+1−e−uα∑k=0nn!k!ukαn−k+1

, the CDF of γ2,t can be further calculated as

(A.4)Fγ2,t(x)=∫0xfγ2,t(y)dy=∑i=1μ∑j=1viϑi,jΓ(j)((1−τt)λ2,t,iγ¯2)j×∫0xyn−1exp(−y(1−τt)λ2,t,iγ¯2)dy =1−∑i=1μ∑j=1vi∑k=0j−1ϑi,jΓ(k+1)(x(1−τt)λ2,t,iγ¯2)k×exp(−x(1−τt)λ2,t,iγ¯2)

By using eqs (A.2) and (A.3) into (A.1), the CDF of γt can be written as

(A.5)Fγt(u)=1−∑i=1μ∑j=1vi∑k=0j−1∑m=0∞∑n=0m(mn)(−1)nϑi,jξmΓ(N+n)Γ(k+1)(12τtβγ¯1)N+n(1(1−τt)λ2,t,iγ¯2)k×∫u∞(u(x+1)x−u)kxN+n−1exp(−u(x+1)(x−u)(1−τ)λ2,t,iγ¯2−x2τtβγ¯1)dx︸I1

Following the change of variables as y=x−u, after some calculations, one can yield

(A.6)I1=exp(−(12τtβγ¯1+1(1−τ)λ2,t,iγ¯2)u)∑p=0k∑q=0N+n−1(kp)×(N+n−1q)uN+j+k−q−1(1+u)p×∫0∞yq−pexp(−u2+u(1−τ)λ2,t,iγ¯21y−y2τtβγ¯1)dy

Following the identity [20, eq. 3.471.9]

∫0∞xv−1exp(−βx−γx)dx=2(βγ)v2Kv(2βγ)

where Kv⋅ represents the vth order modified Bessel function of the second kind [20], after some mathematical computations, I1 can be computed as

(A.7)I1=2exp(−(12τtβγ¯1+1(1−τ)λ2,t,iγ¯2)u)∑p=0k∑q=0N+n−1(kp) ×(N+n−1q)(1(1−τ)λ2,t,iγ¯2)q−p+12 ×(12τtβγ¯1)p−q−12u2N+2N+2k−p−q−12 ×(1+u)p+q+12Kq−p+1(2u2+u2τt(1−τ)βγ¯1λ2,t,iγ¯2)

Finally, by substituting eq. (A.7) into eq. (A.5), we can obtain the CDF of γt as in eq. (30).

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 61271255), the Natural Science Foundation of Jiangsu Province (No. BK20131068 and BK20140883), the Jiangsu Planned Projects for Postdoctoral Research Funds (No. 1402068B) and the Open Research Fund of National Mobile Communications Research Laboratory in Southeast University (No. 2012D15).

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Published Online: 2015-1-28

Published in Print: 2015-3-31

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https://doi.org/10.1515/freq-2014-0065

Schlagwörter für diesen Artikel

unmanned aerial vehicle; wireless relay network; optimal beamforming; performance analysis