Beamforming with Limited Feedback in Amplify-and-Forward ...

3Note that, interestingly, this received SNR expression has thesame form as in TDMA scheduled AF systems [4, Eq.(1)].Therefore, the results in [3], [4] directly lead to the followingapproximations on the outage probability and probability oferror of the optimal beamforming with unlimited feedback AFnetwork at high SNRs:whereP optout (γ t )≃P opte (γ t ) ≃(ζ 2 (m+1)R ) m+1− 1,m!(m + 1) γ t(12)ζ(2m + 1)!,m!(m + 1)!(2cγ t )m+1(13)(1m∏ 1ζ =E s σs,d2 E s σs,i2i=1)+ 1E r σi,d2(14)is a constant determined by transmission power and channelvariances σs,d 2 , σ2 s,i and σ2 i,d, R is the target transmission rateand c is a constant determined by the modulation scheme.The idea of an AF network with relay selection wasintroduced in [3] as an improvement to the conventionalTDMA-based AF networks. The high-SNR approximation ofthe outage probability and probability of error of the selectionAF network are [3], [4](Pout(γ s ζ 2 (m+1)R ) m+1− 1t ) ≃, (15)(m + 1) γ tPe s ζ(2m + 1)!(γ t ) ≃. (16)(m + 1)!(2cγ t )m+1From (12), (13) and (15), (16) with the same transmit SNR,the performances of the two schemes satisfyP s outP optout= P esPeopt = m! (17)Since the high SNR approximation of the two schemesare parallel lines with slope −(m + 1)/10 in Log-dB scale,the SNR difference for the two schemes to achieve the sameperformance is thereforeδ γ =log m!(m + 1)/10 = 10m + 1 log 10 m! (dB). (18)Therefore, the asymptotic SNR gap 1 between the selection AF10scheme and the B-UF scheme ism+1 log 10 m! dB.Applying √ Stirling’s formulalim m→∞ 2πme −m m m /m! = 1 to (18) and retainingonly the most significant term, we can see that as the numberof relay nodes m increases, the asymptotic SNR gap betweenselection AF and B-UF is 10 log 10 m/e dB. This gap is aquickly increasing function of m and may seem to be severein large networks. For instance, the loss is 1dB when m = 2and 1.95dB when m = 3, but it grows to 5.96dB whenm = 10. However, the optimal beamforming scheme is highlyimpractical for any realistic application, especially when mis large, since it involves feedback of m complex numbersin real time and strict synchronization. Therefore we treatit as a performance bound only and compare more practical1 Defined as the difference in SNR required to achieve the same asymptoticperformance with selection AF and with B-UF.schemes using their performance loss to this optimal one.IV. BEAMFORMING WITH LIMITED FEEDBACKIn Section III we treated the selection AF scheme asone special case of beamforming with limited feedback andstudied its performance loss in relation to the B-UF scheme.Clearly, this loss can be reduced by increasing the amountof feedback. In this section, we study the performance of AFbeamforming schemes as a function of feedback available. Insuch a network, a codebook whose size is determined by theamount of feedback is first established at both the destinationand the relay nodes. For each transmission, the destinationselects the optimal codeword based on the CSI, and feedsback its index to the relays. The relays perform beamformingusing the corresponding codeword as the weight vector.A. Optimal Codebook DesignOptimal transmit beamforming codebook design has beenstudied in the context of single-user MISO (multi-input singleoutput)systems [7], [8], [12], where GLP provided the optimalcodebook design for both average received SNR [7] and outageprobability [8]. In the AF networks, however, GLP is no longeroptimal due to the noise amplification by relay nodes. TheGeneralized Lloyd Algorithm (GLA) [13] can still be usedin the optimal codebook design. In particular, assuming Bbits of feedback, the algorithm starts with a set of randomlyselected 2 B vectors, and repeats the following two steps untilconvergence:1) For each codeword in the current codebook, find theregion in C m for which it is the γ r -maximizing beamformingvector.2) For each region, find a new codeword to replace thecurrent one by maximizing the average γ r over thatregion.Due to the complex form of γ r , only numerical resultsare available for AF relays (shown later in simulations), evenfor the special case of i.i.d channels among the network. Weare particularly interested in log 2 m bits of feedback, for theselection scheme is a possible candidate in this case. Unlike inMISO systems, where any orthogonal basis of m-dimensionalspace is an optimal codebook, selection (identity codebook)is the unique optimal for AF beamforming. The reason forthe uniqueness is straightforward: although all orthogonalcodebooks achieve the same maximal received signal power,selection is the one that minimizes noise amplification.Figure 1 shows the outage probability of these schemes ina network with 3 potential relays. All the channel gains inthe network are i.i.d. CN (0, 1) random variables. From thefigures we can see the 1.95 dB performance loss betweenselection and B-UF as predicted in Table I. It also showsthat optimal beamforming with limited feedback achieveslittle improvement over selection, while entailing significantsystem-wide disadvantages such as extremely high complexityand slow convergence rate for GLA and synchronization ofrelays. In other words, although increasing the amount offeedback and designing a near-optimal codebook throughthe GLA can improve performance, this approach involvespractical difficulties and yields only small performance gains.

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