Beamforming in Nonorthogonal Amplify-and-Forward Relay Networks

Beamforming in Nonorthogonal Amplify-and-Forward Relay Networks
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  1258 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 Beamforming in Nonorthogonal Amplify-and-ForwardRelay Networks Tung T. Pham, Ha H. Nguyen,  Senior Member, IEEE  , andHoang D. Tuan,  Member, IEEE   Abstract —This paper considers wireless amplify-and-forward (AF) re-lay networks in which the source communicates with the relays anddestination in the first phase and the relays  simultaneously  forward signalsto the destination in the second phase over uncorrelated Rayleigh fadingchannels. We examined one scenario in which each relay only knows theperfect information of its source–relay channel, whereas the destinationknows the exact information of the relay–destination channels and thestatistics of the source–relay channels. Based on a combiner that wasdeveloped at the destination, we propose an efficient beamforming schemeat the relays and develop its quantized version using Lloyd’s algorithmto work with a limited-rate feedback channel. Simulation results showthat the nonorthogonal relaying with the proposed beamforming schemeoutperforms the orthogonal relaying with power allocation in terms of the ergodic capacity. In terms of the signal-to-noise ratio (SNR), thenonorthogonal scheme also becomes superior to the orthogonal schemewhen the number of quantization regions increases.  Index Terms —Beamforming, nonorthogonal amplify and forward (AF),power allocation (PA), wireless relay networks. I. I NTRODUCTION In recent years, designing transmission methods for wireless relaynetworks that can adapt to partial knowledge of channel-state infor-mation (CSI) has gained a significant interest (see, e.g., [1]–[3]). Thisis because the tradeoff between a potential performance improvementoffered by a relay-assisted transmission and a large amount of feed-back overhead has been well recognized to be an important issue.To reduce the amount of feedback information from the destinationto the relays, a distributed suboptimal beamforming scheme is pro-posed in [4], whereas in [5] and [6], the relay selection scheme isrecommended to be a good choice. Although the benefits in terms of implementation complexity and overhead information exchange havebeen demonstrated, the common limitation of the methods in [4]–[6]is that the instantaneous CSI of all involved channels in the system isrequired at the destination.In our previous work [7], a wireless amplify-and-forward (AF)relay network model in which  orthogonal  transmissions are conductedbetween the relays and the destination in the second phase has beenconsidered.Itisassumedthateveryrelayknowsonlytheinstantaneouschannel from the source to itself, whereas the destination knows theinstantaneous channels from the source and all the relays to itself andthe first- and second-order statistics of the channels from the source toalltherelays.Thispracticalassumptionisalsoconsideredin[1].Wheneach relay is assigned an orthogonal channel, interrelay interference Manuscript received July 8, 2010; revised October 18, 2010 andJanuary 4, 2011; accepted January 5, 2011. Date of publication January 20,2011; date of current version March 21, 2011. This work was supported by aDiscovery Grant from Natural Sciences and Engineering Research Council of Canada. The review of this paper was coordinated by Dr. E. K. S. Au.T. T. Pham and H. H. Nguyen are with the Department of Electrical andComputer Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9,Canada (e-mail:, D. Tuan is with the School of Electrical Engineering and Telecommuni-cations, The University of New South Wales, Sydney, N.S.W. 2052, Australia(e-mail: versions of one or more of the figures in this paper are available onlineat Object Identifier 10.1109/TVT.2011.2107039 is completely eliminated, and the processing task of the destinationbecomes easier. One major drawback of the orthogonal relayingscheme is that it requires more channel uses for each transmission(and, therefore, may not offer high bandwidth efficiency) comparedwith the  nonorthogonal  scheme. However, developing an appropriaterelaying scheme for the nonorthogonal case under such a partial CSIassumption has not been adequately investigated.This paper extends the model examined in [7] to the nonorthogonalAF relaying scheme. Using the proposed signal processing approach,we cophase the signal received at each relay to prevent the destructiveeffects that are propagated from the source–relay channels over therelay–destination channels. Under the assumption of uncorrelatedRayleigh fading channels, an  approximate  maximal-ratio combiner isemployed at the destination, which allows us to derive a beamformingscheme for the relays based on an approximate averaged signal-to-noise ratio (SNR). Because the destination has to compute the optimalbeamforming scheme and then send this information back to all relays,vector quantization using Lloyd’s algorithm [8] will be implemented.With this scheme, the destination broadcasts only some bits that repre-sent the index of the best beamforming vector in the codebook througha finite-rate feedback channel. A comparison in terms of the reliability(i.e.,theSNR)andbandwidthefficiency(withtheergodiccapacity[9])between the orthogonal and nonorthogonal relaying schemes is carriedout. The special case of relay selection is also examined and compared.The rest of this paper is organized as follows. Section II summarizesthe signal processing model proposed in [7], with some modificationsfor the  nonorthogonal  AF relaying scheme. Section III provides abeamforming scheme and its quantized version and discusses a relayselection scheme. A comparison of different relaying schemes in termsof the average SNR and ergodic capacity is presented in Section IV.Finally, some conclusions are given in Section V.  Notations:  Italic, bold lowercase, and bold uppercase letters de-note scalars, vectors, and matrices, respectively. The superscripts ( · ) T  ,  ( · ) H  , and  E {·}  stand for the transpose, Hermitian transpose,and statistical expectation operations, respectively. The notation  x  ∼CN  ( µ , Σ )  means that x is a vector of complex Gaussian random vari-ables with mean vector  µ  and covariance matrix  Σ . The notation  I L stands for an identity matrix of size  L × L , whereas  diag( x 1 ,...,x L ) is a diagonal matrix with the diagonal elements  x 1 ,...,x L .II. S YSTEM  M ODEL This sectionsummarizes the signalmodel considered in[7,Sec. IV]and explains key differences when the nonorthogonal AF relay-ing scheme is employed. Fig. 1 shows a wireless relay systemin which the source terminal  S   communicates with the destina-tion terminal  D  through  L  relay terminals  R 1 ,...,R L . All termi-nals are equipped with one antenna and operate in a half-duplexmode. The transmission for every information symbol  s  happensin two phases. In the first phase, the source transmits the sig-nal to the destination through the direct channel  h s  ∼ CN  (0 , Σ ( s ) ) and to the relays through the “uplink” (source–relay) channels h u  = [ h u 1 ,...,h uL ] T  ∼ CN  ( 0 , Σ ( u ) ) , where  h ul  is the instanta-neous channel coefficient between the source and the  l th relay. In thesecond phase, the relays  simultaneously  forward their received signalsto the destination through the “downlink” (relay–destination) channels h d  = [ h d 1 ,...,h dL ] T  ∼ CN  ( 0 , Σ ( d ) ) . All channels are assumed tobe uncorrelated Rayleigh fading (i.e., Σ ( u ) and Σ ( d ) are diagonal).The signal received at the destination in this phase is d 1  =  h s s  +  n s  (1) 0018-9545/$26.00 © 2011 IEEE  IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 1259 Fig. 1. System model of the wireless relay network under consideration. where  n s  ∼ CN  (0 ,σ 2 s )  represents additive white Gaussian noise(AWGN) with variance  σ 2 s . Similarly, the signals received at the relayscan collectively be written as r  = [ r 1 ,r 2 ,...,r L ] T  =  h u s  + n u  (2)where  n u  ∼ CN  ( 0 ,σ 2 u I L )  accounts for AWGN with the same vari-ance  σ 2 u  at the relays.Assuming that the  l th relay only knows  h ul , whereas the destinationknows  h s ,  h d , and  Σ ( u ) , appropriate signal processing operationsneed to be carried out at both the relays and the destination. First,because every relay cannot compute the beamforming scheme, thedestination shall determine it and then send the result back to the relaysthrough some feedback channel. 1 Second, the random phase that wasintroduced by the uplink channel is corrected by setting the  l th relaygain as g l  =  w l e − jθ l  P  S | h ul | 2 +  σ 2 u  − 1 / 2 (3)where  P  S  = E {| s | 2 }  is the average transmitted power of the source,and  θ l  is the phase of the uplink channel coefficient  h ul . With theaforementioned scaling factor, the transmitted power of the  l th relayis limited to | w l | 2 . The received signal at the destination in the secondphase can be expressed as 2 d 2  = L  l =1 g l r l  +  n d = wH d F u ¯ h u  · s  + wH d F u ¯ n u  +  n d  (4)where  w  = [ w 1 ,...,w L ]  is the beamforming vector that was usedby the  L  relays, H d  = diag( h d 1 ,...,h dL ) , F u  = diag(( P  S | h u 1 | 2 + σ 2 u ) − 1 / 2 ,..., ( P  S | h uL | 2 + σ 2 u ) − 1 / 2 ) ,  ¯ h u =[ | h u 1 | ,..., | h uL | ] T  ,  ¯ n u =[ n u 1 e − jθ 1 ,...,n uL e − jθ L ] T  ∼ CN  ( 0 ,σ 2 u I L ) , and  n d  ∼ CN  (0 ,σ 2 d ) is the AWGN that the destination experienced in the secondphase.The received signals at the destination in two phases can be repre-sented as the following standard Gaussian vector form: d  = [ d 1 ,d 2 ] T  = ¯ a · s  + ¯ n  (5)where  ¯ a  = [ h s , wH d F u ¯ h u ] T  , and  ¯ n  = [ n s , wH d F u ¯ n u  +  n d ] T  . 1 The feedback information may be in the form of analog feedback, i.e., thetrue beamforming vector, or in the form of finite-rate feedback, i.e., a quantizedversion of the true beamforming vector. 2 In [7], the destination receives  L  orthogonal signals from  L  relays in thesecond phase. Therefore, a power allocation scheme at the relays is sufficientto help the destination constructively combine those signals. However, in thenonorthogonal case, power allocation alone is not the optimal scheme. Last, because the instantaneous amplitudes of the uplinks  h u  arenotknownatthedestination,an approximate maximal-ratiocombining(MRC) filter that combines the signals received after two phases isgiven as [7] f   =   R − 1 E h u { ¯ a }  (6)where   R is the covariance matrix of the noise vector  ¯ n . It is computedas follows:  R  = E { ¯ n ¯ n H  }  = diag  σ 2 s ,σ 2 u wH d T u H H d  w H  +  σ 2 d   (7)where T u  = 1 σ 2 u E h u  F u ¯ n u ¯ n H u  F H u  = E h u  F u F H u  = diag( T  u 1 ,...,T  uL )  (8) T  ul  = E h ul  F  2 ul  = 2 ξ l σ 2 u exp(2 ξ l ) E  1 (2 ξ l )  (9)where  ξ l  =  σ 2 u / (2 P  S Σ ( u ) ll  ) , and  E  1 ( x ) =   + ∞ x  ( e − t /t ) dt ,  x >  0  isthe exponential integral.Under uncorrelated Rayleigh fading uplink channels, the term  ˜ a  = E h u { ¯ a } in (6) is found as ˜ a  =  h s , wH d E h u { F u ¯ h u }  T  = [ h s , wH d q u ] T  (10)where q u  =[ q  u 1 ,...,q  uL ] T  q  ul  = E h ul  { F  ul | h ul |} = 1 √  P  S ξ l  exp( ξ l )[ K  1 ( ξ l ) − K  0 ( ξ l )]  (11)where  ξ l  is defined in (9), and  K  α ( x )  is the  α th-order modified Besselfunction of the second kind.  Lemma 1: With the approximate MRC filter given in  (6),  the esti-mate of the data symbol  s  at the filter  ’ s output is  ˆ s  =  f  H  d ,  and theresulting average SNR can be shown to be SNR non  ≈ P  S ˜ a H   R − 1 ˜ a = P  S  | h s | 2 σ 2 s +  wH d q u q H u  H H d  w H  σ 2 u wH d T u H H d  w H  +  σ 2 d  .  (12)The proof can be similarly carried out as shown in [7] and is omittedhere for brevity. Note that, unlike the orthogonal relaying scheme withpower allocation (PA) in [7], the average SNR that was obtained inthe nonorthogonal relaying scheme has a special form. This form isexploited in the next section to compute an efficient beamformingscheme.III. R ELAY  B EAMFORMING  S CHEMES  A. Proposed Relay Beamforming Based on the average SNR expression in (12), a general beam-forming scheme can numerically be obtained under the total andindividual power constraints at the relays through second-order coneprogramming [10]. For the case that only the total transmit powerat all the relays is limited, a closed-form beamforming scheme canbe derived using the following lemma (the proof directly follows byapplying the Rayleigh–Ritz theorem [11]).  1260 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011  Lemma 2: Based on  [1],  let   A  be an  n × n  Hermitian matrix   and B  be an  n × n  positive definite Hermitian .  Furthermore ,  let   B  bedecomposed as B  =  LL H  .  Then x H  Axx H  Bx  ≤  λ max  for all  x  ∈ C n (13) where  λ max  is the largest eigenvalue of   L − 1 A ( L H  ) − 1 .  The equalityholds if   x  =  c ( L H  ) − 1 u max ,  where  c  is any nonzero constant  ,  and  u max  is the eigenvector of  L − 1 A ( L H  ) − 1 that corresponds to  λ max .With the total relay power constraint ww H  =  P  R , the average SNRcan be rewritten as SNR non ≈ P  S  | h s | 2 σ 2 s +  wH d q u q H u  H H d  w H  w  σ 2 u H d T u H H d  +  σ 2 d P  R I L  w H   .  (14)Because the second term in (14) is exactly the special case (withequality) that was considered in Lemma 2, we can readily find thefollowing beamforming vector 3 to maximize the average SNR, i.e., SNR non : w opt  =  c ˜ w  =  c q H u  H H d  V  (15)where  V  = ( σ 2 u H d T u H H d  + ( σ 2 d /P  R ) I L ) − 1 , and  c  =   P  R / (˜ w ˜ w H  ) . It follows that SNR non  ≈  P  S σ 2 s | h s | 2 + L  l =1 P  S σ 2 u q  2 ulP  R σ 2 d | h dl | 2 T  ulP  R σ 2 d | h dl | 2 +  1 σ 2 u .  (16)With the average SNR given in (16), one interesting SNR compar-ison between the nonorthogonal and orthogonal relaying schemes isstated in the following corollary. Corollary 1: With the analog feedback  ,  the average SNR that wasachieved by the nonorthogonal relaying transmission with the optimalbeamforming scheme is always higher than the average SNR that wasachieved by the orthogonal relaying transmission with optimal PA. The proof of Corollary 1 is given in Appendix A. Note that theaforementioned SNR comparison is also true for the case with perfectand full CSI assumption, in which the instantaneous SNRs of bothschemes have the same form as given in [1, eq. (10)]. For the systemmodel with a finite-rate feedback, further analysis and comparison arecarried out in Section IV.Given the beamforming scheme (15), the ergodic capacity of thetransmission model under consideration can be calculated as  I   = 12 E h s , h u , h d  log 2  1 +  S  N    (17)where the received signal power is S   =  P  S  | h s | 2 σ 2 s + q H u  H H d  VH d F u ¯ h u  2 (18)and the averaged noise power is  N   = | h s | 2 σ 2 s + q H u  H H d  V 2 ×  σ 2 u H d F u F H u  H H d  +  σ 2 d P  R I L  H d q u . (19) 3 Note that a beamforming scheme usually includes the following two com-ponents: 1) the phase compensation and 2) the power allocation (see, e.g., [1],[10], and [12]). The beamforming scheme (15) is also in this form. Obtaining a closed-form expression of the ergodic capacity asdefined in (17) appears to be very difficult. Nevertheless, a tight upperbound of the ergodic capacity can be found as follows. Because thefunction  log 2 (1 +  x )  is concave in  x >  0 , applying Jensen’s inequal-ity to (17) yields  I ≤  12 log 2  1 + E h s , h u , h d  S  N   ≈  12 log 2 (1 +    SNR)  (20)where    SNR = E h s , h d { SNR non }≈  P  S σ 2 s Σ ( s ) + L  l =1 P  S q  2 ul σ 2 u T  ul [1 − χ l  exp( χ l ) E  1 ( χ l )]  (21)and  χ l  =  σ 2 d / ( σ 2 u P  R T  ul Σ ( d ) ll  ) . The simulation results in Section IVwill validate the tightness of the aforementioned upper bound.  B. Quantized Relay Beamforming The beamforming scheme in (15) is computed at the destinationand is sent back to the relays through some feedback channel. Forpractical implementation, we sketch in this section the vector quan-tization procedure based on Lloyd’s algorithm to design the quantizedrelay beamforming codebook. This procedure can also be appliedfor the PA codebook design in the orthogonal relaying scheme (seeAppendix B). Let ¯ γ  ( w | h d ) =  wH d q u q H u  H H d  w H  w  σ 2 u H d T u H H d  +  σ 2 d P  R I L  w H  .  (22)Given a codebook of beamforming vectors W   =  { w 1 , w 2 ,..., w Z , } ∈ C L (23)where  Z   is the number of quantization regions, and ⌈ log 2  Z  ⌉ , in which ⌈ x ⌉  is the smallest integer that is greater than or equal to  x , is thenumber of feedback bits, the average distortion with respect to theoptimal beamforming vector can be defined as δ  ( W  ) = E h d  ¯ γ  ( w opt | h d ) −  max 1 ≤ z ≤ Z { ¯ γ  ( w z | h d ) }  .  (24)The codebook design using Lloyd’s algorithm vector quantizationcan be summarized as follows [8].1) Randomly generate a codebook   W  (0) =  { w (0)1  , w (0)2  ,..., w (0) Z  , } . Set  t  = 1 .2) Generate a set of   N   test channel realization vectors { h (1) d  , h (2) d  ,..., h ( N  ) d  } . Divide this set into  Z   quantization re-gions, with the  z  th region defined as C  z  =  h ( n ) d  | ¯ γ   w ( t − 1) i  | h ( n ) d  ≤  ¯ γ   w ( t − 1) z  | h ( n ) d   (25)for all  i   =  z,  1  ≤  n  ≤  N  .3) Construct a new codebook   W  ( t ) , with the  z  th codewordgiven as w ( t ) z  = argmax w E h ( n ) d  ∈C z  ¯ γ   w | h ( n ) d   (26)  IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 1261 where E h ( n ) d  ∈C z  ¯ γ   w | h ( n ) d  = E h ( n ) d  ∈C z  wH ( n ) d  q u q H u  H ( n ) d  H  w H  w  σ 2 u H ( n ) d  T u  H ( n ) d  H  +  σ 2 d P  R I L  w H   ≈ w E h ( n ) d  ∈C z  H ( n ) d  q u q H u  H ( n ) d  H   w H  w  σ 2 u E h ( n ) d  ∈C z  H ( n ) d  T u  H ( n ) d  H   +  σ 2 d P  R I L  w H  and E h ( n ) d  ∈C z { ¯ γ  ( w | h ( n ) d  ) }  is approximated by the first term of its Taylor series [1]. 4 4) If   δ  ( W  ( t − 1) ) − δ  ( W  ( t ) ) >ǫ , set  t  ←  t +1 , and go back to step 2.Otherwise, set W   =  W  ( t ) and stop.Depending on the initial codebook in step 1 and the test set of channel vectors that were generated in step 2, Lloyd’s iterative pro-cedure can result in local maxima. Therefore, to obtain a more reliablesolution, a large test set is preferred. Although the offline designof the codebook appears to be computationally intensive, codewordselection is really simple. Because the destination and all relays knowthe codebook, the destination only needs to broadcast the index of theoptimal codeword to all the relays. This approach significantly reducesthe number of feedback bits compared with the optimal beamformingscheme with  analog  feedback. C. Relay Selection Relay selection can be considered a special scheme of both beam-forming and PA. When full CSI is available at the destination, relayselection has been proven an efficient scheme with a simple feedback strategy [13]. In our scenario, the selection algorithm can only beimplementedbasedontheinstantaneousdownlinkchannelcoefficientsand the statistics of the uplink channels. Based on (14), it can beinferred that the  l ⋆ th relay is selected to forward the signal in thesecond phase if  l ⋆ = argmax l ∈ [1 ,L ] P  R | h dl | 2 q  2 ul σ 2 u P  R | h dl | 2 T  ul  +  σ 2 d (27)and the feedback channel needs  ⌈ log 2  L ⌉  bits to inform all the relaysthrough a broadcast channel.IV. N UMERICAL  R ESULTS AND  D ISCUSSION All the uplink and downlink channels are assumed to be underuncorrelated Rayleigh fading but with different variances, i.e.,  Σ ( u )11  = − 10  dB,  Σ ( u )22  = 0  dB,  Σ ( u )33  = 10  dB,  Σ ( u )44  = 20  dB,  Σ ( d )11  = 20  dB, Σ ( d )22  = 10  dB,  Σ ( d )33  = 0  dB, and  Σ ( d )44  =  − 10  dB. Similarly, the directchannel is assumed to be  h s  ∼ CN  (0 , 1) . Binary phase-shift keyingmodulation is used at the source. Assuming that  σ 2 s  =  σ 2 u  =  σ 2 d  = 1 , 4 Note that, if   |C z |  = 0 , then  w  is randomly regenerated. If   |C z |  = 1 , usethe special case in Lemma 2 to find the closed-form solution; otherwise, usethe general solution that was given in Lemma 2. In addition, using a higherorder Taylor-series approximation for  E h ( n ) d  ∈C z { ¯ γ  ( w | h ( n ) d  ) }  may be moredesirable but appears to be complicated.Fig. 2. SER of the nonorthogonal relaying model with the optimal beam-forming scheme and the orthogonal relaying scheme with the power allocationscheme in [7]. The number of relays is given as  L  = 2 , 3 , 4 . the average channel signal-to-noise ratio (CSNR) is simply definedas  P  S /σ 2 s  =  P  S /σ 2 u  =  P  S /σ 2 d  =  P  S . The total transmit power at therelays is chosen to be  P  R  =  P  S .Fig. 2 compares the symbol error rate (SER) performances achievedwith the nonorthogonal relaying with optimal beamforming schemeand the orthogonal relaying with PA scheme in [7]. The numberof relays is given as  L  = 2 , 3 , 4 . It is shown in the figure thatthe performance improvement (from the “ equivalent  ” coding gain)with nonorthogonal relaying is significant: about 3 dB/relay at highSNR. However, because the destination does not know full CSI, thediversity order does not increase with the number of relays. Thiscondition also agrees with the results of the orthogonal case in [7]and of the nonorthogonal case in [1], where the diversity order isproven to be only 2. As a reference, the SER that was obtained bydirect transmission with a total transmit power of   P   =  P  S  +  P  R  isalso shown in Fig. 2. Even with such a large transmit power, theperformance of the direct transmission is significantly worse than therelay-assisted transmission. Note that, in general, the performanceimprovement of the relay-assisted transmission strongly depends onthe channel conditions among the source, relays, and the destination.The performance curves shown in Fig. 2 should be interpreted for thespecific channel conditions under consideration.Next, we compare the nonorthogonal relaying scheme under con-sideration with the orthogonal relaying scheme considered in [7]under the same CSI assumption. A broadcasting feedback channelthat accommodates 2 or 6 feedback bits is assumed. To adapt withthese limited feedback rates, the quantized version of the beamformingvector (given in Section III-B) and of the PA scheme used in [7] (givenin Appendix B) are employed. In the procedure, the number of testchannel vectors is  N   =  20000, and the termination criterion is chosenas  ǫ  = 10 − 4 δ  ( W  ( t − 1) ) .As predicted from Corollary 1, the simulation results in Fig. 3confirm that, when infinite-rate (i.e., analog) feedback is available, thebeamforming scheme in the nonorthogonal case is superior in termsof the average SNR compared with the PA scheme in the orthogonalcase. This is because the phases of both uplink and downlink channelscan perfectly be matched at each relay, resulting in  a constructivesuperposition  ( basically the addition of signal amplitudes )  of all thecophased signals at the destination . Consequently, the energy of thecombined signal is higher than the orthogonal case, where  energies of   1262 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011 Fig. 3. Average SNRs of the nonorthogonal relaying scheme with beamform-ing and the orthogonal relaying scheme with the power allocation scheme in[7]. The number of relays is given as  L  = 4 .Fig. 4. Ergodic capacity of the nonorthogonal relaying scheme with beam-forming and the orthogonal relaying scheme with the power allocation schemein [7]. The number of relays is given as  L  = 4 . different signals, and not their amplitudes, are accumulated  . When nofeedback is available, it is expected that the nonorthogonal relayingscheme is inferior to the orthogonal scheme. For the case with only2 feedback bits, the performance of the orthogonal scheme is stillbetter. This condition also reflects the sensitivity of the nonorthogonalrelaying scheme to the quality of the phase estimates available atthe relays. However, the gap is very small, and it can practically beremoved and even reversed with more feedback bits (e.g., with 6feedback bits or more in our simulation setup). In addition, note that,under the assumed channel model, the performance of the PA schemedoes not improve much as the number of feedback bits increases.Because the average SNRs of the two relaying schemes are notmuch different, their bandwidth efficiencies mostly depend on thenumber of channel uses (which is also the number of relays) requiredin the second phase. As shown in Fig. 4, the ergodic capacity of thenonorthogonal scheme is about 2.5 times larger than the ergodic capac-ity of the orthogonal scheme at any SNR region (i.e., approximately ( L  + 1) / 2  times for the cases with 2 or 6 feedback bits). Withoutfeedback information, the nonorthogonal scheme performs a little bitworse at low- and medium-SNR regions. For benchmark comparison,the capacity of the model in [1] under the assumption of full CSIavailable at the destination is also plotted. As expected, a decrease incapacity can clearly be observed when less CSI is available at both thetransmitters and receiver.Figs. 3 and 4 also plot curves (marked with filled circles) to validatethe approximate    SNR  given in (21) and the upper bound of the ergodiccapacity given in (20), respectively. It is observed that both    SNR  andthe upper bound of the ergodic capacity are very close to the actualvalues. Other performance curves in Figs. 3 and 4 also show thatthe relay selection scheme in (27) does not perform as well as thebeamforming scheme with the 2-bit feedback channel. However, it issuperior to the PA scheme (even with analog feedback case). Thus,relay selection is a suitable solution when we need to balance betweenthe implementation complexity and performance gain.V. C ONCLUSION Extending our previously proposed orthogonal relaying model, theproblem of joint design of signal processing at the relays and destina-tion has been considered for wireless  nonorthogonal  relay networks.With a highly accurate approximation of the SNR, a closed-formbeamforming scheme for the relays has been proposed. Although nodiversity gain can be obtained as the number of relays increases, theerrorperformanceofthesystemcanstillbesignificantlyimproved.Fora practical implementation with limited feedback from the destination,the quantized beamforming scheme is also developed. Simulationresults have shown that the proposed nonorthogonal relaying withbeamforming scheme can outperform the orthogonal relaying with PAin terms of the SNR and, consequently, the error performance, as wellas the bandwidth efficiency.A PPENDIX  AP ROOF OF  C OROLLARY  1Using the same notations in [7], let  W  = diag( w 1 ,w 2 ,...,w L ) , ˜ a  = [ h s , ( WH d q u ) T  ] T  , and   R  = diag( σ 2 s ,σ 2 u WH d T u H H d  W H  + σ 2 d I L ) . The average SNR for the orthogonal relaying transmissiongiven in [7, eq. (41)] can be rewritten as SNR ortho ≈  P  S ˜ a H   R − 1 ˜ a =  P  S  | h s | 2 σ 2 s + q H u  H H d  W H  ×  σ 2 u WH d T u H H d  W H  +  σ 2 d I L  − 1 WH d q u  =  P  S σ 2 s | h s | 2 + L  l =1 P  S σ 2 u q  2 ul | w l | 2 σ 2 d | h dl | 2 T  ul | w l | 2 σ 2 d | h dl | 2 +  1 σ 2 u .  (28)For the nonorthogonal relaying transmission with our proposed beam-forming scheme, the average SNR is given in (16). Both  SNR non given in (16) and  SNR ortho  given in (28) have the same form,which are increasing functions of   P  R  and | w l | 2 , respectively. Because  Ll =1 | w l | 2 =  P  R , i.e.,  | w l | 2 ≤  P  R  ∀ l , it is obvious that substituting | w l | 2 in (28) by  P  R  leads to  SNR ortho  ≤  SNR non . The equality holdswhen  L  = 1 .
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