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Effect of Feedback Delay on Amplify-and-Forward Relay Networks With Beamforming

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Effect of Feedback Delay on Amplify-and-Forward Relay Networks With Beamforming
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  See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/224218067 Effect of Feedback Delay on Amplify-and-Forward Relay Networks With Beamforming  Article   in  IEEE Transactions on Vehicular Technology · April 2011 DOI: 10.1109/TVT.2011.2112786 · Source: IEEE Xplore CITATIONS 41 READS 29 4 authors:Some of the authors of this publication are also working on these related projects: researching Millimeter Wave Communications for Future 5G Cellular Networks   View projectMassive MIMO wireless networks: Theory and methods   View projectHimal SuraweeraUniversity of Peradeniya 44   PUBLICATIONS   1,277   CITATIONS   SEE PROFILE Theodoros Tsiftsis 123   PUBLICATIONS   2,681   CITATIONS   SEE PROFILE George K. KaragiannidisAristotle University of Thessaloniki 399   PUBLICATIONS   7,345   CITATIONS   SEE PROFILE Arumugam NallanathanKing's College London 303   PUBLICATIONS   3,807   CITATIONS   SEE PROFILE   All content following this page was uploaded by George K. Karagiannidis on 04 November 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blueare linked to publications on ResearchGate, letting you access and read them immediately.  IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 1265 R EFERENCES [1] L. Piazzo, “Fast algorithm for power and bit allocation in OFDM sys-tems,”  Electron. Lett. , vol. 35, no. 25, pp. 2173–2174, Dec. 1999.[2] W. T. Vetterling, W. H. Press, S. A. Teukolsky, and B. P. Flannery,  Numer-ical Recipes in C  . Cambridge, U.K.: Cambridge Univ. Press, 1992.[3] M. Mohammadnia-Avval, C. Snow, and L. Lampe, “Error-rate analysisfor bit-loaded coded MIMO-OFDM,”  IEEE Trans. Veh. Technol. , vol. 59,no. 5, pp. 2340–2351, Jun. 2010.[4] L. Piazzo, “Fast optimal bit-loading algorithm for adaptive OFDMsystems,” Univ. Rome, Rome, Italy, INFOCOM Dept., Tech. Rep. 002-04-03, 2003. [Online]. Available: http://infocom.uniroma1.it/~lorenz/ rep02-04-03.ps[5] J. Campello, “Discrete bit loading for multicarrier modulation systems,”Ph.D. dissertation, Stanford Univ., Stanford, CA, 1999. Effect of Feedback Delay on Amplify-and-ForwardRelay Networks With Beamforming Himal A. Suraweera,  Member, IEEE  ,Theodoros A. Tsiftsis,  Senior Member, IEEE  ,George K. Karagiannidis,  Senior Member, IEEE  , andArumugam Nallanathan,  Senior Member, IEEE   Abstract —In this paper, the decremental effect of beamforming withfeedback delay on the performance of a two-hop amplify-and-forward(AF) relay network over Rayleigh-fading channels is investigated. An an-tenna configuration in which the source and the destination are equippedwith multiple antennas, whereas the relay is equipped with a single an-tenna, is assumed. We derive new expressions for the outage probabilityand the average bit error rate (BER), which are useful for a large numberof modulation schemes. To gain further insights, simple outage probabilityand average BER approximations at high signal-to-noise ratio (SNR) arealso presented. It is shown that, whenever a feedback delay exists, thenetworkisnotcapableofofferingdiversitygains.Furthermore,sourceandrelay power allocation results show significantly different behavior withfeedback delay. Numerical results supported by simulations are providedto show that feedback delay can severely degrade the performance of theconsidered AF relay system.  Index Terms —Amplify-and-forward(AF),averagebiterrorrate(BER),beamforming, feedback delay, relays. I. I NTRODUCTION Wirelesscommunicationsystemscanbenefitfromrelaydeploymentsince the technology promises extended signal coverage, improvedthroughputs, and spatial diversity [1], [2]. One of the relaying pro-tocols described in the literature is amplify and forward (AF). The Manuscript received July 20, 2010; revised December 6, 2010 andJanuary 10, 2011; accepted January 26, 2011. Date of publication February 10,2011; date of current version March 21, 2011. This paper was presented in partat the IEEE Global Communications Conference, Honolulu, HI, Nov. 30–Dec.4, 2009. The review of this paper was coordinated by Prof. C. P. Oestges.H. A. Suraweera is with the Engineering Product Development, SingaporeUniversity of Technology and Design, Singapore (e-mail: himalsuraweera@sutd.edu.sg).T. A. Tsiftsis is with the Department of Electrical Engineering, Technolog-ical Educational Institute of Lamia, 35100 Lamia, Greece (e-mail: tsiftsis@teilam.gr).G. K. Karagiannidis is with the Department of Electrical and ComputerEngineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece(e-mail: geokarag@auth.gr).A. Nallanathan is with the Centre for Telecommunications Research, King’sCollege London, WC2R 2LS London, U.K. (e-mail: arumugam.nallanathan@kcl.ac.uk).Digital Object Identifier 10.1109/TVT.2011.2112786 performance of single-antenna AF relay networks has now been wellinvestigated [3]–[5].Work such as [6] has also demonstrated that significant benefitscan be gained if multiple antennas are deployed in relaying networks.A practical transmission scheme for AF relaying systems employingmultipleantennas isbeamforming[7]–[13].Theperformanceofbeam-formingagainstrelayselection,consideringbothunlimitedandlimitedfeedback, has been studied in [7]. In [8], the performance of a two-hoprelay network with transmit beamforming at the source and maximalratio combining (MRC) at the destination has been analyzed. Theperformance of the same system, by considering antenna correlationeffects at the source and the destination, is reported in [9]. In [10],the outage performance with beamforming, considering only limitedfeedback,hasbeenstudied.In[11],theperformanceofatwo-hopfixedgain network over Nakagami- m fading channels has been analyzed. In[12], assuming the absence/presence of the source–destination directlink, optimal beamforming codebook designs for an AF relay systemwith limited feedback was presented. In [13], a practical scenario inwhich a multiantenna-equipped source is communicating to a single-antenna-equipped destination via a relay has been considered. Despitethe significant practical interest, the authors in [13] have limited theiranalysis to a situation of   perfect   channel state information (CSI) at thesource.In beamforming systems, the received signal-to-noise ratio (SNR)maximization is achieved by providing CSI to the transmitter. Infrequency-division-duplex systems,such knowledge is provided by thefeedbackofCSIfromthereceivertothetransmitter.Feedbackinvolvesdelay, and as a result, in practice, the available CSI at the transmitterand the actual channel may be different. The use of outdated CSI forbeamforming degrades the system performance. Although this perfor-mance degradation for point-to-point systems is now well understood(see, e.g., [14]) and for partial relay selection [15], so far, in theexisting literature, the effect of feedback delay on the performance of AF relaying with beamforming has not been investigated.In this paper, the effect of feedback delay on the end-to-end perfor-mance of a two-hop AF relay network where multiple antennas at thesource are used for beamforming is investigated. We derive closed-form expressions for the system’s outage probability and the averagebit error rate (BER) applicable for a range of modulation schemes. Togain valuable insights, in the high-SNR regime, we also present as-ymptotic outage probability and average BER expressions. The impactof different antenna configurations, feedback delay, and SNR imbal-ance on the performance is illustrated through some analytical results.II. S YSTEM  M ODEL Consider a wireless network where a source  S   equipped with  N  t antennas communicates with a destination D  equipped with N  r  anten-nas through a single antenna relay  R  [8], [9], [11]. In this network, we assume that  S   does not have a direct link to  D . The communicationfrom  S   to  D  via relay  R  takes place in two time slots. In the first timeslot,  S   beamforms its signal to  R . The received signal at  R  can bewritten as y R ( t ) = √  P  1 w † t h sr ( t ) x ( t ) + n 1 ( t )  (1)where  x ( t )  is the data symbol,  P  1  is the transmit power,  h sr ( t ) =[ h 1 sr ( t ) ,...,h N  t sr  ( t )] T  is the channel vector from S   to R with Rayleighfading entries, and n 1 ( t )  is the additive white Gaussian noise (AWGN)at  R  with one-sided power spectrum density  σ 21 . The transpose andthe conjugate transpose are denoted by  ( · ) T  and  ( · ) † , respectively.According to the principles of maximal ratio transmission, we choose 0018-9545/$26.00 © 2011 IEEE  1266 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 w t  = ( h sr ( t − T  d ) /  h sr ( t − T  d )  F  ) , where  · F   is the Frobeniusnorm. Note that  w t  is calculated from the  outdated channel  h sr ( t − T  d ) , instead of the channel  h sr ( t ) , with  T  d  being the feedback delay. The received scalar signal at  R  is then multiplied by a gain G  / and transmitted to the destination. The received signal at  D  isgiven by y D ( t ) =   P  2 h rd ( t ) G   P  1 w † t h sr ( t ) x + n 1 ( t )  + n 2 ( t )  (2)where  P  2  is the relay transmit power,  h rd ( t ) = [ h 1 rd ( t ) ,...,h N  r rd  ( t )] is the channel vector from  R  to  D  with Rayleigh fading entries, and n 2 ( t )  is the AWGN vector at  D  with one-sided power spectral density E  [ n 2 ( t ) n † 2 ( t )] =  σ 22 I N  r . The expectation operator is  E  [ · ] , and  I N  r denotes the identity matrix of size  N  r . According to the principles of MRC, we then multiply the received signal  y D ( t )  by a vector  w r  =( h rd ( t ) /  h rd ( t )  F  )  and write r D ( t ) =   P  1 P  2 w † r h rd ( t ) G w † t h sr ( t ) x +   P  2 w † r h rd ( t ) Gn 1 ( t ) + w † r n 2 ( t ) .  (3)Using the CSI-assisted AF relay gain [5], [8] 1 G 2 = 1 P  1  w † t h sr  2 F   + σ 21 (4)and after some manipulations, the end-to-end SNR can be written as γ  eq1  =  γ  1 γ  2 γ  1  + γ  2  + 1  (5)where  γ  1  =   w † t h sr ( t )  2 F  ¯ γ  1 ,  γ  2  =   h rd ( t )  2 F  ¯ γ  2 , and  ¯ γ  i  = ( P  i /σ 2 i ) for  i  = 1 , 2 .III. P ERFORMANCE  A NALYSIS In this section, we derive important performance metrics, e.g., theoutage probability and the average BER for the two-hop network underinvestigation.  A. Outage Probability The outage probability  P  o  is an important quality-of-service mea-sure defined as the probability that  γ  eq1  drops below an acceptableSNR threshold  γ  th . To study the outage probability of (5), it isnecessary to obtain the cumulative distribution function (cdf) F  Z ( γ  th ) of the random variable, which is defined as Z   =  γ  1 γ  2 γ  1  + γ  2  + c  (6)where  c  is a positive constant. After applying some algebraic manipu-lations along similar lines as in [8],  F  Z ( γ  th )  can be expressed as F  Z ( γ  th )=1 − ∞   0 f  γ  1 ( x + γ  th )  1 − F  γ  2  γ  th + γ  2th + cγ  th x  dx (7)where  f  γ  1 ( · )  and  F  γ  2 ( · )  denote the probability density function (pdf)of   γ  1  and the cdf of   γ  2 , respectively. 1 In the presence of a feedback delay, the ability of the relay to perfectlytrack the current CSI is subjective to the pilot placement arrangement employedfor channel estimation at the relay. Analyzing the impact of any specific pilotplacement on the relay gain and system performance is beyond the scope of thecurrent work. To obtain  F  Z ( γ  th ) , expressions for the pdf of   γ  1  and the cdf of   γ  2 are needed. To model the relationship between  h sr ( t )  and  h sr ( t − T  d ) , we employ the following widely adopted time-varying channelmodel: h sr ( t ) =  ρ d h sr ( t − T  d ) +   1 −| ρ d | 2 e ( t )  (8)where  ρ d  is the normalized correlation coefficient between  h jsr ( t ) and  h jsr ( t − T  d ) , with  j  = 1 ,...,N  t . The error vector  e ( t )  ∼ CN ( 0 , I N  t ) , where CN ( ϕ , Ξ )  denotes the complex Gaussian distrib-ution with mean ϕ and covariance Ξ , is uncorrelated with h sr ( t ) . ForJake’s fading spectrum,  ρ d  =  J  0 (2 πf  d T  d ) , where  f  d  is the Dopplerfrequency, and  J  0 ( · )  is the zeroth-order Bessel function of the firstkind [16, Sec. (9.1)].The pdf of   γ  1  using [14, (15)] can be written as 2 f  γ  1 ( x ) = 1¯ γ  N  t 1 N  t − 1  i =0  N  t − 1 i  ( | ρ d | 2 ) N  t − i − 1 (¯ γ  1  (1 −| ρ d | 2 )) i ( N  t  − i − 1)! × x N  t − i − 1 e −  x ¯ γ 1  (9)and the cdf of   γ  2  is given by F  γ  2 ( x ) = 1 − e −  x ¯ γ 2 N  r − 1  j =0  x ¯ γ  2  j  j !  .  (10)Substituting (9) and (10) into (7), we obtain F  Z ( γ  th ) =1 −  e −   1¯ γ 1 +  1¯ γ 2  γ  th ¯ γ  N  t 1 × N  t − 1  i =0  N  t − 1 i  ( | ρ d | 2 ) N  t − i − 1 (¯ γ  1  (1 −| ρ d | 2 )) i ( N  t  − i − 1)! × N  r − 1  j =0  γ  th ¯ γ  2  j  j !  I  1  (11)where  I  1  =    ∞ 0 ( x + γ  th ) N  t − i − 1 ( x + γ  th  + c ) j x − j e − ( x/ ¯ γ  1 ) − ( γ  2th +  cγ  th /x ¯ γ  2 ) dx. Using the binomial theorem and with the help of the identity [17, eq.(3.471.9)], we obtain  I  1 = N  t − i − 1  p =0  N  t  − i − 1  p  γ  N  t − i − p − 1th j  q =0   jq   ( γ  th  + c ) j − q × 2  ¯ γ  1  ( γ  2th  + cγ  th )¯ γ  2  p + q − j +12 K  p + q − j +1  2   γ  2th + cγ  th ¯ γ  1 ¯ γ  2  (12)where  K  ν ( · )  is the  ν  th-order modified Bessel function of the secondkind [16, Sec. (9.6)]. Now, the outage probability follows by substitut-ing  c  = 1  into (12) and then using (11).Note that the derived outage probability (valid for arbitrary  ρ d ) canbe further simplified in many special cases. For example, when  D  hasa single antenna and the feedback delay goes to infinity, the outage 2 It is noted that, [14, eq. (15)] includes a small typo, which we have correctedin (9). The power of the term  1 / ¯ γ  1  outside the  must be  N  t  and not  N  t − 1 .  IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 1267 probability can be obtained from (11) and (12) by substituting N  r  = 1 , i  =  N  t  − 1 , and | ρ d | 2 = 0  and is given by F  γ  eq1 ( γ  th ) = 1 − 2 e −   1¯ γ 1 +  1¯ γ 2  γ  th   γ  2th  + γ  th ¯ γ  1 ¯ γ  2 × K  1  2   γ  2th  + γ  th ¯ γ  1 ¯ γ  2  .  (13)  B. Outage Probability at High SNR Although (11) is exact and valid for any given SNR, it is difficult togain insights such as the effect of parameters  N  t  and  ρ d  on the outageprobability. Therefore, as  ¯ γ  1  and  ¯ γ  2  tend to infinity with fixed ratio µ , we can analyze the system’s asymptotic outage probability with thehelp of   Theorem 1 . Theorem 1:  The outage probability at high SNR for  ρ d  = 1  isgiven by 3 P  ∞ o  =  1 N  t !  γ  th ¯ γ  1  N  t +  o  γ  th ¯ γ  1  N  t +1  , N  t  < N  r 1 N  !  1 +  1 µ N   γ  th ¯ γ  1  N  +  o  γ  th ¯ γ  1  N  +1  , N  t  =  N  r  =  N  1 µ N r N  r !  γ  th ¯ γ  1  N  r +  o  γ  th ¯ γ  1  N  r +1  , N  t  > N  r .(14)The outage probability for  ρ d  <  1  is given by P  ∞ o  =  (1 −| ρ d | 2 ) N  t − 1 +  1 µ  γ  th ¯ γ  1  +  o  γ  th ¯ γ  1  2  , N  r  = 1  (1 −| ρ d | 2 ) N  t − 1  γ  th ¯ γ  1  +  o  γ  th ¯ γ  1  2  , N  r  >  1 (15)where  µ  = (¯ γ  2 / ¯ γ  1 ) . Proof:  See the Appendix.   We observe that a longer delay (low  ρ d ) degrades the systemperformancebyincreasingtheoutageprobability.Moreimportantly,inthe presence of feedback delay, no diversity gains can be achieved. Inthis two-hop system, the worst link dominates the performance at highSNR. According to (8), coefficients of the noise vector  e ( t ) , althoughundesirable, contributes in determining the beamforming weights of the S   − R channel h sr ( t ) . As such, although signals from N  t  differentpaths are received at R , we do not expect an improvement in the signalquality since  e ( t )  is uncorrelated with  h sr ( t ) . C. Average BER For many modulation formats used in wireless applications, theaverage BER expressed as  E  [ a Q (   bγ  eq1 )]  is given by P  b  =  a √  2 π ∞   0 F  γ  eq1  t 2 b  e − t 22  dt  (16)where  a ,  b >  0 , and  Q ( x ) = (1 / √  2 π )   ∞ x  e − ( y 2 / 2) dy  is theGaussian  Q -function. Moreover, our BER derivations can also be 3 Wehavedefined f  ( x ) =  o ( g ( x )) , x → x 0 ,if  lim x → x 0 ( f  ( x ) /g ( x )) = 0 . extended to square/rectangular  M  -QAM since  P  b  can be written asa finite-weighted sum of   a Q (   bγ  eq1 )  terms.To the best of our knowledge, (16) does not have a closed-formsolution. To overcome this, we substitute  c  = 0  into (11) and tightlylower bound the average BER as P  b  ≥  a 2  −   2 π N  t − 1  i =0  N  t − 1 i  ( | ρ d | 2 ) N  t − i − 1 ( b ¯ γ  1  (1 −| ρ d | 2 )) i ( b ¯ γ  1 ) N  t ( N  t  − i − 1)! × N  r − 1  j =0 ( b ¯ γ  2 ) − j  j ! N  t − i − 1  p =0  N  t  − i − 1  p  × j  q =0   jq   ¯ γ  1 ¯ γ  2  p + q − j +12  I  2 .  (17)In (17),  I  2  =   ∞ 0  t 2( N  t − i + j ) e − ((1 /b ¯ γ  1 )+(1 /b ¯ γ  2 )+(1 / 2)) t 2 K  p + q − j +1 ((2 t 2 /b √  ¯ γ  1 ¯ γ  2 )) dt . Applying a simple variabletransformation and using [17, eq. (6.621.3)],  I  2  has a closed-formsolution given by  I  2  = √  π (2 α 2 ) η 2 2( α 1  + α 2 ) η 1 + η 2 Γ( η 1  + η 2 )Γ( η 1  − η 2 )Γ  η 1  +  12  × 2 F  1  η 1  + η 2 ,η 2  + 12; η 1  + 12;  α 1  − α 2 α 1  + α 2   (18)where  η 1  =  N  t  − i +  j  + (1 / 2) ,  η 2  =  p + q  −  j  + 1 ,  α 1  =(1 /b ¯ γ  1 ) + (1 /b ¯ γ  2 ) + (1 / 2) ,  α 2  = (2 /b √  ¯ γ  1 ¯ γ  2 ) ,  Γ( z  )  is the gammafunction, and  2 F  1 ( · , · ; · ; · )  is the Gauss hypergeometric function[16, eq. (15.1.1)].  D. Average BER at High SNR Asymptotic analysis on relaying networks are important since suchstudies provide valuable insights that are useful to design engineers.It is shown in [18] that, at high SNRs, the error performance canbe calculated based on the behavior of the pdf of the instantaneouschannel power gain around zero. First, consider the case of   ρ d  = 1 .By following [18, Prop. (1)] and using (14), the average BER in thehigh-SNR regime can be expressed as 4 P  ∞ b  =2 q a Ψ Γ  q   +  32  √  π  ( b ¯ γ  1 ) − ( q +1) + o  ¯ γ  − ( q +2)1   (19)where  q   = min( N  t ,N  r ) − 1 , and Ψ =  1 N  t ! , N  t  < N  r 1 N  !  1 +  1 µ N   , N  t  =  N  r  =  N  1 µ N r N  r ! , N  t  > N  r .(20)Equation (19) implies that the array gain  G a  and the diversity gain  G d can be written as G a  = b  2 q a Ψ Γ  q   +  32  √  π  −  1 q +1 (21) G d  = min( N  t ,N  r ) .  (22) 4 A high-SNR average-symbol-error-rate approximation has also been pre-sented in [8, eq. (18)]. However, our expression is much simpler and clearlyshows how network parameters affect the system’s error performance athigh SNR.  1268 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 Fig. 1. Outage probability for various antenna configurations.  f  d T  d  = 0 . 1 , ¯ γ  1  = 3  dB, and  ¯ γ  2  = 7  dB. From (22), we can infer that beamforming with no feedback de-lay can achieve the maximum possible diversity order of thissystem [8].Nowconsiderthecaseof  ρ d  <  1 .TheaverageBERinthehigh-SNRregime is given by P  ∞ b  =  aψ 2 b ¯ γ  1 + o  ¯ γ  − 21   (23)where ψ  =  (1 −| ρ d | 2 ) N  t − 1 +  1 µ  , N  r  = 1  (1 −| ρ d | 2 ) N  t − 1  , N  r  >  1 .(24)Equation (23) implies that the  G a  and  G d  of the system withfeedback delay can be written as  G a  = (2 b/aψ )  and  G d  = 1 ,respectively.IV. N UMERICAL AND  S IMULATION  R ESULTS In this section, we confirm the derived analytical results throughcomparison with Monte Carlo simulations and discuss the impact of feedback delay on system’s performance. We assume Jake’s fadingspectrum, and in the examples,  ρ d  = 0 . 903 , 0 . 642 , and  0 . 472  corre-spond to  f  d T  d  = 0 . 1 , 0 . 2 , and  0 . 3 , respectively.Fig. 1 shows the outage probability for various antenna configura-tions. We see that the analytical results obtained using (11) exactlymatch the simulations. As expected, the outage probability is signifi-cantly improved as the number of antenna increases. Thus, deploymentof multiple antennas improve the performance of this network.Fig. 2 shows the outage probability for various feedback delays.We see that a large feedback delay can significantly degrade systemperformance. When the feedback delay is large, the system’s outageperformance is mostly determined by low SNR thresholds. The outagecurves shift to high SNR thresholds when the feedback delay tendsto zero, indicating that the system performance improves for smallfeedback delays.Fig. 3 shows the outage probability using source and relay powerallocations  β   using our analytical results in (11). In particular, weset  ¯ γ  1  + ¯ γ  2  = ¯ γ   and have  ¯ γ  1  =  β  ¯ γ   and  ¯ γ  2  = (1 − β  )¯ γ  . As expected,with no feedback delay  ( ρ d  = 1) , equal power allocation  ( β   = 0 . 5)  is Fig. 2. Outage probability for different  ρ d ’s.  N  t  =  N  r  = 3 ,  ¯ γ  1  = 3  dB, and ¯ γ  2  = 7  dB.Fig. 3. Outage probability for different  β  ’s with  N  t  =  N  r  = 2 ,  γ  th  = − 5  dB, and  ¯ γ   = 20  dB. optimal, i.e., yields the lowest outage probability. This observation isexpected since, for the considered antenna configuration (equal num-ber of antennas at the source and the destination), the network is sym-metrical with respect to S   − R and R − D  link SNRs. However, inter-estingly, when there is a feedback delay,  β   = 0 . 5  yields a suboptimalresult. The optimal  β   is close to 0.9. The optimal  β   slightly changesfor different  ρ d ’s. In the presence of feedback delays, the  S   − R  and R − D linksarenotsymmetrical.The S   − R linkisweaker,comparedwith the  R − D  link, and becomes the bottleneck for the network’sperformance. Therefore, to improve the performance, more powermust be allocated to the  S   − R  link. For large feedback delays  ( ρ d  =0 . 2 − 0 . 6) , the outage probability is barely affected by the variationsof   β  . For example, when  ρ d  = 0 . 2 , the outage probabilities at  β   =0 . 1  and  0 . 9  are approximately  3 × 10 − 2 and  4 × 10 − 3 , respectively.However, when the feedback delay is decreased  ( ρ d  = 0 . 8  →  1) , thesituationissignificantlyimproved.Whenalargefeedbackdelayexists,the  S   − R  link is severely weakened, and although allocating morepower to the  S   − R  link improves the system’s outage probability, itdoessomarginally.Fig.3alsoshowsthat,forthegiven β  ’sintherangeof 0.1–0.9, when a large feedback delay exists  ( ρ d  = 0 . 2 − 0 . 6) , only
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