# General Order Antenna Selection in MIMO Cooperative Relay Network

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2 II. SYSTEM DESCRIPTION A dual-hop relay system, as shown in Fig., consists of the source S sending signals towards the destination D via the assistance of the relay R. In Amplify-Forward AF) relay systems, R amplifies the received signal before forwarding it to the D node. The al transmission involves two timeslots. The source has M s antennas where the relay and the destination are uipped with M r, and M d antennas. Note that unlike in [6], we assumed same set of antennas at Relay for receiving and transmission. H denotes M r M s channel matrix for the first hop whereas H 2 is M d M r matrix for the second hop, both matrix elements are i.i.d complex Gaussian random variables CGRVs) with mean zero and variance.5 per dimension. The channel elements are ordered in the decreasing order of their absolute magnitudes. The n th and n th 2 best links Tx/Rx antenna pairs) are then selected for transmission in S R and R D hops respectively, therefore corresponds to the channel gains h n and h n2. The received signal at R is, y r = h n x + n r ) where x is the transmit signal with normalized power, n r is the additive white Gaussian noise vector with power N at the relay antenna. The R amplifies the signal by G and transmits it through the n th 2 best link to the D. Therefore we can write the signal at D as, y d = h n2 Gy r + n d = Gh n2 h n x + Gh n2 n r + n d 2) n d is the AWGN noise at the destination with same power N. For a fixed-gain relaying, the uivalent SNR can be shown as [2, n. 6], γ γ 2 γ = 3) C + γ 2 where the fixed Gain G 2 =/CN ), C being a constant; the first hop SNR γ = N h n 2 = γ h n 2, and the second hop SNR γ 2 = N h n2 2 = γ 2 h n2 2. III. SNR STATISTICS ANALYSIS In this section, we study the statistical behaviour of the general order antenna selection GOAS) multi-antenna dualhop amplify-forward fixed-gain cooperative relay network. In particular, we derive closed form expressions for cumulative distribution function, probability distribution function, moment generation function, and general moments for the relayed link. A. Statistical characterization of the received SNR In this subsection, we derive closed form expressions for CDF, and PDF of γ defined in 3). From the CDF expression, we easily derive the outage probability. The outage probability is an important system performance metric, and defined as the probability that the instantaneous SNR falls below a predefined threshold, γ th. The CDF of γ th is given by the following theorem. CDF {n,n 2 } = {,},{,2},{2,},{2,2}, M s = M r = M d =3 γ = γ 2 =5dB γ, db Fig. 2: Cumulative Distribution Function for General Order Antenna Selection in MIMO Relay, indicates Monte-Carlo simulation points Theorem : The c.d.f. of γ is given by Msr Mrd F γ γ) = 2M sr M rd M sr n = ) k+k2 e n+k) γ γ k 2= k 2 K λ) n + )n 2 + k 2 ) γ γ 2 where K.) denotes the modified Bessel of second kind, λ = 2 n+)n 2+k 2) γ γ 2, M sr = M s M r, and M rd = M r M d. Proof: See Appendix I. Corollary : The PDF of γ can be obtained by taking derivative of 22) with respect to γ as follws Msr n Msr Mrd fγ GOS γ) =2M sr M rd = k 2= k 2 n + )n 2 + k 2 ) γ γ 2 ) k+k2 exp γn )[ + ) n + K λ)+ λ ] γ γ 2γ K λ) 5) Proof: The proof is straightforward, by applying the identity z d dz K vz) +vk v z) +zk v z) = [5, ]. Corollary 2: The outage probability of γ for general order antenna selection multi-antenna dual-hop amplify-forward fixed-gain cooperative relay network can be given by substituting γ = γ th. For special cases, when M s = M r = M d = indicating single-antenna nodes, 4) specializes to [2, n. 9]. Furthermore when n = n 2 =highest order statistics i.e. best 4)

3 .5 3 PDF {n,n 2 } = {2,2} γ = γ 2 =5dB line M s = M d =,M r =4 circle M s = M r = M d =2 {n,n 2 } = {,} Average End end SNR n =n 2 = γ 2 =2 γ solid line dash line dot line M s =M r =M d =2 M s =4,M d =M r = M s =M d =,M r =4 5 5 γ, db Fig. 3: Probability Density Function for General Order Antenna Selection in MIMO Relay for same al number of antennas 5 5 γ,db Fig. 4: Average End-end SNR for General Order Antenna Selection in MIMO Relay for same al number of antennas Tx/Rx antennas pair in both hops), we obtain the CDF for antenna selection in MIMO dual-hop amplify-forward system with a fixed-gain relay. Note that [6] [7] obtained results for an ideal-gain relay. Fig. 2 shows the CDF plot for varying order statistics. The first and the second hop average SNRs are assumed ual, i.e. γ = γ 2 = 5dB, and the number of antennas at relay M =4. One can see that how the lower order statistics result a loss in the system performance. The Monte-Carlo simulation results validate the analysis. Fig. 3 shows the PDFs for two different settings with a al number of antenna in the system fixed to 6 - first setting involves evenly distributed antenna i.e. M s = M r = M d = 2, and; in the second one there are 4 antennas at the relay and antenna each at the source and destination, i.e. M s = M d =,M r =4. The system performance is similar in both cases as illustrated in Fig. 3. B. Moment generating function MGF) MGF is useful to compute error rates as shown in [6]. Since MGF Mγ GOS s) =E[e sγ ],weget Msr n Msr Mrd Mγ GOS s) =M sr M rd k = n + )n 2 + k 2 )C exp k 2 2 γ 2 n + + s γ ) [ W,/2 σ) n 2 + k 2 )n + + s γ ) + Cn + n 2 + s γ ) n + )n 2 + k 2 ) γ 2 ] W /2, σ) k 2= ) ) ϕ where σ = n+k)n2+k2)c) γ 2n ++s γ ), and ϕ = + k 2.Wehave used [5, ] to arrive at the final expression, and W.,..) is Whittaker function defined in [5, 9.22]. When M s = M r = M d =, 6) is uivalent to [2, n. 2]. 6) C. General Moments In this subsection, we characterize the general moments of the end-to-end SNR γ. The general moments are important measure matric, which can be used to obtain the end-to-end SNR γ, variance, and amount of fading AoF). By definition, the generalized moments of γ can be given by, μ n = n γ n [ F GOS γ γ)]dγ 7) To this end, substituting the CDF expression given by 22) and with the help of [5, ], the closed-from expression for the n-th moments of γ th can be expressed as follows: Msr n Msr Mrd μ n = M sr M rd = γ ) k+k2 k 2 n + nn + 2)n!) 2 ) n + )n 2 + k 2 ) exp Cn2 + k 2 ) ) 2 γ 2 k 2= ) n+ n2 + k 2 )C W n+),/2 8) γ 2 As a direct application, the average end-to-end SNR can be obtained as n =, and the AoF, which quantifies of fading severity, can be obtained by, AoF = E[γ2 ] {E[γ ]} 2 {E[γ ]} 2 = μ 2 μ 2 9) Fig. 4 shows the average end-end SNR of the system for the highest order antenna selection at both hops again for a fixed number of antennas in the system. The evenly distributed system offers the highest average SNR, whereas the system with multi-antenna relay performs poorly.

6 Outage Probability and, dash line n = n 2 = γ th =5dB M s = M r = M d =2 solid line n = n 2 = γ = γ 2 = γ 3,dB Fig. 8: Impact of General Order Antenna Selection on Outage Probability in MIMO Selection Relaying; N =, 2, 3, 4 ) M n M ) M n ) k F γn γ) = M n k n + k k= [ ] n+k) e γ γ 2) To this end, from the definition of CDF of γ,wehave ) ) Fγ GOS γ) = Prγ <xγ )=Pr γ < + 2 γ. 2) Conditioned on γ, and γ 2, the CDF of γ can be expressed ) ) Fγ GOS γ) = Pr γ < + 2 γ f γ γn2 ) 2)dγ 2 ) Msr Mrd = M sr M rd ) +k 2 ) Msr n = k 2= [4] S. Chen, W. Wang, X. Zhang, and D. Zhao, Performance of Amplifyand-Foward MIMO Relay Channels with Transmit Antenna Selection and Maximal-Ratio Combining, in Proc. of WCNC 29, April 29. [5] A. K. Gurung, F. S. Al-Qahtani, and Z. M. Hussain, Outage Behaviour of Dual-Hop Amplify and Forward Cooperative Transmission with Multi Antenna Relay, submitted to GLOBECOM 2. [6] J.-B. Kim and D. Kim, BER analysis of dual-hop amplify-and-forward MIMO relaying with best antenna selection in Rayleigh fading channels, IEICE Trans. Commun., vol. E9-B, pp , Aug. 28 [7] Himal A. Suraweera, George K. Karagiannidis, Yonghui Li, Hari K. Garg, A. Nallanathan, and Branka Vucetic, Amplify-and-Forward Relay Transmission with End-to-End Antenna Selection, in Proc. of WCNC 2, April 2. [8] S. Choi and Y. C. Ko, Performance of Selection MIMO Systems with Generalized Selection Criterion over Nakagami-m Fading Channels, IECE Trans. Commun., vol. E89-B, no. 2, pp , Dec. 26. [9] I. Lee and D. Kim, Outage Probability of Multi-Hop MIMO Relaying with Transmit Antenna Selection and Ideal Relay Gain over Rayleigh Fading Channels, IEEE Trans. Commun., vol. 57, no. 2, Feb. 29. [] M. Elkashlan, T. Khattab, C. Leung, and R. Schober, Stastistics of General Order Selection in Correlated Nakagami Fading Channels, IEEE Trans. Commun., vol. 56, no. 3, pp , March 28. [] S. S. Ikki, and M. H. Ahmed, On the Performance of Amplifyand-Forward Cooperative Diversity with the Nth Best-Relay Selection Scheme, in Proc. of ICC 29, May 29. [2] M. O. Hasna and M. S. Alouini, A Performance Study of Dual-Hop Transmissions With Fixed Gain Relays, IEEE Trans. Wireless Commun., vol. 3, no. 6, pp , Nov. 24. [3] T. A. Tsiftsis, G. K. Karagiannidis, P. T. Mathiopoulos, and S. A. Kotsopoulos, Nonregenerative dual-hop cooperative links with selection diversity, EURASIP J. Wireless Commun. Networking, vol. 26, Article ID [4] H. A. David and H. N. Nagaraja, Order Statistics, 3rd Ed., John Wiley & Sons, New York, NY, 23. [5] I. S. Gradhsteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th Edition, Academic Press, 27. [6] M. K. Simon and M. S. Alouini, Digital Communication over Fading Channels: A Unified Approach to Performance Analysis. New York: Wiley, 2. e n+k) γ γ k 2 γ 2 n + ) γc n+k) γ e γ n 2 2+k 2) γ 2 γ2 dγ 2 } {{ } I 22) To this end, the desired result can be obtained after some simple algebraic manipulations with the help of formula [5, ]. REFERENCES [] Y. Fan, A. Adinoyi, J. S. Thompson, and H. Yanikomeroglu, Antenna combining for multi-antenna multi-relay channels, Eur. Trans. Telecomms., Aug. 27, 8:67626, Wiley InterScience. [2] R. H. Y. Louie, Y. Li, and B. Vucetic, Performance analysis of beamforming in two hop amplify and forward relay networks, in Proc. of ICC 28, pp , May 28. [3] D. B. Costa and S. Aïssa, Beamforming in Dual-Hop Fixed Gain Relaying Systems, in Proc. of ICC 29, May 29.

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