Performance Analysis of Massive MIMO Two-Way Relay Networks with Pilot Contamination, Imperfect CSI, and Antenna Correlation

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1 Performance Analysis of Massive MIMO Two-Way Relay etworks with Pilot Contamination, Imperfect CSI, and Antenna Correlation Shashindra Silva, Student Member, IEEE, Gayan Amarasuriya, Member, IEEE, Masoud Ardakani, Senior Member, IEEE, Chintha Teambura, Feow, IEEE Abstract We consider a multi-ce two-way relay network TWR consisting of single-antenna user nodes and amplify-andforward AF relay nodes having very large antenna arrays. We investigate the combined impact of co-channel interference CCI, imperfect channel state information CSI, pilot contamination, and the antenna correlation at the massive MIMO node. By using a large number of antennas at the relay, we show that the effect of CCI can completely be mitigated. However, the effects of imperfect CSI and pilot contamination degrades the performance even with a large antenna array. Yet, use of massive MIMO aows power scaling at the user nodes and relay and thus, even with channel imperfections, the benefits of employing a massive multiple-input multiple-output MIMO enabled relay on transmit power savings are significant. Furthermore, we derive closed-form approximations for the sum rate in our system model for the simplified setup when CCI and pilot contamination are absent and CSI is perfect. This result wi be useful to decide the required number of antennas at the relay node to obtain a certain percentage of the asymptotic sum rate. Also, our analysis of antenna correlation shows that the effect of antenna correlation can be mitigated by using a large antenna array. We, also find the optimal pilot sequence length to maximize the sum rate of the system. I. ITRODUCTIO Research on massive MIMO, an enabling technology for future fifth generation 5G wireless [], [3], has shown very high spectral efficiencies, low transmit powers per bit, and high energy efficiencies [4]. These advantages have greatly excited the research community. However, the main performance iting factor is pilot contamination, which is the residual interference caused by the reuse of non-orthogonal pilot sequences [4], [5]. In this paper, we consider massive MIMO with Two-way relay networks TWRs, which offer two fold increase in the achievable data rate compared to the one-way relay networks OWRs [6]. Multi-pair TWRs enable mutual data exchanges among multiple pairs of nodes with the aid of an intermediate relay [7]. We are motivated to consider massive MIMO TWRs due to their wide range of potential applications. For example, the system considered in this paper can be used as a heterogeneous wireless entity for the existing ceular network architecture for reducing the workload of the base-stations [8]. For example, bypassing the base-station and using the two-way relaying instead may be feasible. Service providers can thus use TWRs to Corresponding author. The authors Shashindra Silva, Masoud Ardakani, and Chintha Teambura are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada T6G V4, Phone: , Fax: {jayamuni,ardakani,chintha}@ece.ualberta.ca. Gayan Amarasuriya is with the Department of Electrical and Computer Engineering, Southern Iinois University, Carbondale, IL, USA 69, gayan.baduge@siu.edu This work in part was presented at the IEEE International Conference on Communications ICC, Kuala Lumpur, Malaysia, 6 []. improve the data throughput without making drastic changes to their existing infrastructure. Another potential application scenario is the internet of things IoT, which connects multiple wireless devices and sensors [9]. For example, the cooling system wi require data from temperature sensors while security system may require data from the motion sensors. This scenario fits the model of a multi-pair relay network with a central relay node. Moreover, when multiple IoT networks coexist, the effect of co-channel interference can be a significant impairment. Furthermore, these IoT devices often rely on battery power and thus, the energy and power efficiency is an important factor in IoT communications. Combining massive MIMO with TWRs provides the sum rate and energy efficiency performance gains that are required by the above mentioned applications. In the foowing, we classify existing works into general massive MIMO and massive MIMO TWRs. Related work: In [4], the asymptotic performance metrics of multi-user massive MIMO base stations BSs in non-cooperative ceular networks is investigated. Specificay, [4] concludes that whenever the number of BS antennas increases unbounded, simple linear precoders and decoders become asymptoticay optimal. Pilot contamination in multi-ce multi-user massive MIMO systems is investigated by deriving rigorous asymptotic signal-to-interference-plus-noise ratio SIR expressions [5]. In [], the asymptotic performance of multi-pair OWRs with very large relay antenna arrays is investigated when there is no co-channel interference CCI and the CSI is perfect. To this end, the asymptotic SIR and sum rate expressions are derived by considering three transmit power scaling laws. In [], the sum rate of a MIMO TWR has been analyzed under ZF beamforming at the relay or user nodes and closed-form results and approximations for sum rate are obtained. Further, in [], the channel aging effects of multi-ce multi-way massive MIMO relaying are investigated. In [3], the multi-pair TWRs with massive MIMO is investigated by employing linear precoders and detectors, where again, perfect CSI and CCI/pilot contamination free scenario is assumed. There are some recent publications which analyze multi-pair massive MIMO TWRs. In [4] and [5], multi-pair massive MIMO TWRs have been analyzed for maximum ratio combining/transmission MRC/MRT beamforming. While [4] assumes perfect CSI, [5] analyzes the system under imperfect CSI scenario. Furthermore, [6] analyzes multi-pair massive MIMO TWR with imperfect CSI under ZF beamforming. In [7], a fu-duplex multi-pair massive MIMO system is analyzed under imperfect CSI with ZF beamforming. However none of these work analyze the system under CCI and pilot contamination in the context of two-way multi-pair massive

2 MIMO relaying. Problem statement and our contribution: With dense deployment of wireless systems, CCI and pilot contamination are dominant performance iting factors [5]. Further, because precoders/detectors need CSI, channel estimation errors imperfect CSI severely degrade the overa performance. However, perfect CSI is assumed in the analysis of [3] [7]. In contrast, we analyze the effects of CCI, imperfect CSI, pilot contamination, and antenna correlation for multi-pair massive MIMO TWRs. Although our previous work [] analyzed these effects separately, this paper provides a complete analysis with a more generalized system model which contains the effects of CCI, imperfect CSI, and pilot contamination simultaneously. Furthermore, this paper provides extended proofs of the SIR results []. In this work, we study these imperfections in massive MIMO TWRs. More specificay, the contributions of this paper can be listed as foows. The asymptotic SIR and sum rate expressions are derived for three transmit power scaling laws: namely i power scaling at user nodes, ii power scaling at the relay, and iii power scaling at both the relay and user nodes. We show that for the CCI case, the asymptotic SIR expressions asymptoticay become independent of the number of co-channel interferers L, and consequently, the CCI degradation can be canceed asymptoticay, whenever the relay antenna count grows unbounded. evertheless, the asymptotic performance is ited by the residual interference incurred due to pilot contamination, and its impact cannot be completely mitigated even in the it of infinitely many relay antennas. otably, the asymptotic performance metrics are independent of the fast fading component of the wireless channel, and hence, the cross-layer resource scheduling becomes simple. Our analysis and Monte-Carlo simulations reveal that substantial sum-rate gains can be achieved via very large relay antenna arrays. The asymptotic results are valid only when the number of antennas at the relay is infinite. However, for practical purposes, the sum-rate results for a finite number of antennas is important. Thus, we obtain closed-form sum rate results for the finite relay antenna array. To make analysis tractable we assume perfect channel conditions i.e. no CCI, no pilot contamination and perfect CSI. The benefits of this result are that it can be used to decide the optimal number of relay antennas to obtain a certain percentage of the asymptotic performance and to determine how fast the performance of the system approaches the asymptotic performance. 3 We obtain asymptotic SIR and sum-rates for multipair massive MIMO TWRs with relay antenna correlation. Its effect on the performance of massive MIMO systems can be significant. Fortunately, our analysis shows that the correlation impact can be mitigated by using a large antenna arrays in TWRs. otation: Z H, [Z] k, and [Z] i,j denote the Hermitian-transpose, the kth diagonal element of the matrix, Z, and the i, j th diagonal element of the matrix, Z, respectively. The diagonal matrix D with kth diagonal element d k is denoted as diag d k. I M and O M are the M M Identity matrix and M matrix of a zeros, respectively. A complex Gaussian random variable RV X with mean µ and standard deviation σ is denoted as X C µ, σ. Further, E z, Rz >, is the exponential integral function [8, Eqn. 8.]. Γz is the Gamma function [8, Eqn. 8.3.], and Γa, z is the upper incomplete Gamma function [8, Eqn ]. II. SYSTEM, CHAEL, AD SIGAL MODEL A. System and channel model The system model consists of L adjacent TWRs having K number of users each. The users in the lth TWR are denoted as U l,,, U l,k, where U l,k exchange data signals with its paired-user U l,k via the half-duplex AF relay R l for k, k {,, K} and l {,, L}. Users are singleantenna terminals, and the relays are equipped with antennas. The number of relay antennas are unbounded with respect to the total number of users K. The channel matrix from K users in the jth TWR to the lth relay is defined as G jl = F jl D jl, where F jl C K K, I I K accounts for sma-scale fading, and D jl = diag η j,l,,, η j,l,k represents large-scale fading. As customary, the channel gains are independent and identicay distributed i.i.d. and are assumed to remain fixed over two consecutive time-slots and reciprocal, and hence, relay-to-user channel matrix becomes G T jl. The CCI on the lth TWR occurs due to the data transmissions of the other L TWRs with relay R j, where j {,, L} and j l. B. Channel Estimation The lth relay estimates G by using the pilot sequences transmitted by the users. To this end, K users in each TWR transmit mutuay orthogonal pilot sequences of length τ. Yet, due to the unavailability of orthogonal pilot sequences for a users in L TWRs, same sequence is reused by the L adjacent cochannel TWRs. Thus, the channel estimation at the relay not only contains the desired channel, but also the channels belonging to K users in each of the adjacent TWRs. The corresponding minimum mean square error MMSE estimation of the users-torelay channel is written as [9] L Ĝ = G jl + V l / P p D, for l {, L}, j= where P p is the transmit power of the pilot sequence, the elements of V l are distributed as independent C, random variables, L D = P p j= jl D + IK, and G jl is defined in the previous section. The estimation error is E l = Ĝ L j= G jl. The receive-zf detector and transmit-zf precoder is constructed at the relay by using the CSI with estimation errors. The elements of the kth column of E l is Gaussian distributed with mean zero and variance L j= η L j,l,k/p p j= η j,l,k +. Due to the MMSE properties, the matrices E l and Ĝ are statisticay independent. C. Signal model During two time-slots, K users in each TWR exchange their information pair-wisely via their assigned relay. Specificay, the paired users U l,i, U l,i exchange their data signals

3 3 γ l,k = ˆβ l P S K m=,m k glk T ˆβ l P S glk T Ŵ l g lk L Ŵ l g lm + ˆβ l P S j=,j l glk T Ŵ l G jl + ˆβ l σr L l j=,j l glk T. 7 Ŵ l + σ nl,k ˆβ l = L P S j= G Tr jl G H jlĝ[ĝh Ĝ P R ] P ] P [ĜT Ĝ [ĜH Ĝ ] ĜH ] P ] P. 9 +σr l Tr [ĜH [ĜT Ĝ Ĝ x l,i, x l,i, where i {,, K} and l {,, L}. In the first time-slot, users transmit K signal vector x l, which is the concatenated data signals of K users in the lth TWR, towards the relay R l. The signal vector x l satisfies E [ ] x l x H l = IK. The received signal at R l is written as y Rl = L P S G jl x j + n Rl, j= where n Rl is the additive white Gaussian noise AWG vector at the relay satisfying E [ ] n Rl n H R l = I σr l and P S is the transmit power of the users. During the second-time slot, relay first amplifies and then forwards its received signal towards the users. The transmitted signal from the relay is y R l = ˆβ l Ŵ l y Rl, where Ŵl is the concatenated beamforming-and-amplification matrix at the relay and β l is the amplification factor to satisfy the relay power constraint which is presented in the sequel. Here, Ŵ l is designed to cancel sub-channel interference within a given TWR, and hence, it is constructed by using receive-zf and transmit-zf precoding and detection concepts as foows []: Ŵ l = Ĝ ĜT Ĝ PĜH Ĝ Ĝ H, 3 where P is the block diagonal permutation matrix for user pairing, constructed as P = diagp,, P K and P i = for i {,, K}. Further, the relay power constraint is P R = Tr y R l y H R l. 4 ext, the received signal at U l,k is given as y l,k = g T lk y R l + n l,k = ˆβ l PSg T lk ŴlG x l + ˆβ L l PSg T lk Ŵl G jl x j + ˆβ l g T lk Ŵln Rl + n l,k, 5 j=,j l where U l,k and U l,k are the paired-users exchanging their data signals with k, k = i, i for i {,, K}. Further, g lk is the k th column vector of the matrix G for The results obtained in this paper can also be obtained for other detection and precoding methods such as matched filter and MMSE. However due to the page restrictions we have only included results for ZF detection and ZF precoding. k {,, K} and n l,k is the AWG at the k th user of the lth TWR with variance σn l,k. 5 is further simplified as y l,k = ˆβ l PS g T lk Ŵlg lk x l,k + ˆβ l PS g T lk Ŵ l K +ˆβ l PS g T lk Ŵ l L j=,j l m=,m k g lm x l,m G jl x j + ˆβ l g T lk Ŵ l n Rl +n l,k, 6 where the first term is the desired signal at U l,k and other terms are the interferences and noise. Thus the end-to-end SIR at U l,k, γ l,k is given in 7 at the top of this page. We derive the asymptotic SIR for three power scaling scenarios in next sections. D. Calculation of value ˆβ l By using and y R l = ˆβ l Ŵ l y Rl, we simplify the power constraint 4 as P R = ˆβ l Tr = ˆβ l P STr Ŵ l Ŵ l P S L j= L j= G jl G H jl + σ R l I Ŵ H l G jl G H jlŵh l + ˆβ l σ R l TrŴl Ŵ H l. 8 By substituting Ŵl into 8, ˆβl is obtained in 9 at the top of this page. E. Overa sum rate of the system The average sum rate for the lth K-user TWR with estimated CSI at the relay is defined as [4] R l = T C τ T C K k= E [log + γ l,k ], where T C and τ are the coherence time of the wireless channel and length of the pilot sequence used for channel estimation. In particular, the pre-log factor T C τ/t C accounts for the pilot overhead [4]. The two time-slots required for the data transmission between the paired users results in the pre-log factor of /.

4 4 ˆβ l = L K E S j= i= η j,l,i +η j,l,i τe S ˆη l,i ˆη l,i E R + σr K l i= τe S ˆη l,i ˆη l,i. 4 g T lk Ŵ l n Rl = 4 glk T Ĝ ĜT Ĝ PĜH Ĝ Ĝ H n Rl = gt lk Ĝ ĜT Ĝ ĜH P Ĝ ĜH n Rl 4. 8 F. Energy efficiency of the system As mentioned in the introduction, for applications with lowpower sensors, the power/energy efficiency of the system wi be very important. Thus, we analyze the energy efficiency of the system defined as [3] K k= ρ = E [log + γ l,k], KP S + P R where the denominator consists of the total power consumption of the system and the numerator consists of the overa sum rate. III. ASYMPTOTIC PERFORMACE AALYSIS In this section, we derive asymptotic SIR and sum rate for the three power scaling laws [3] at the user nodes and relay whenever the relay antenna count grows unbounded. Moreover, we investigate the detrimental impacts of CCI, imperfect CSI, and pilot contamination on the SIR of the system. A. Transmit power scaling at the user nodes Whenever the transmit power at the user nodes is scaled inversely proportional to the relay antenna count, the power of the pilot sequence is also scaled accordingly. Thus, the overa transmit power can only be scaled inversely proportional to the square-root of the relay antenna count for estimated CSI. We obtain the normalized relay gain for united number of relay antennas by letting P S = E S /, P P = τe S / and P R = E R while keeping E S and E R fixed, as given at the top of this page see Appendix B for the proof. Here, we define ˆη l,k as L ˆη l,k = η j,l,k. 3 j= The parameter ˆη l,k, is a measure of the pilot contamination experienced by U l,k and wi extensively appear in SIR equations we obtain in the sequel. Further, if pilot contamination is absent, then ˆη l,k = η l,l,k. We rewrite 6 by dividing both sides by 4 and substituting E S values as y l,k 4 = ˆβ l ESg T lk Ŵlg lk x l,k + ˆβ l ESg T lk Ŵl + ˆβ l ESg T lk Ŵl L j=,j l G jl x j K m=,m k g lm x l,m + ˆβ l 4 g T lk Ŵln Rl + 4 n l,k. 4 ext, we obtain the asymptotic it of the intended signal term in 4 by using the its given in Appendix A, as ESg T lk Ŵlg lk x l,k = g T lk E S Ĝ P ĜT Ĝ ĜH Ĝ Ĝ H g lk x l,k = η l,l,k η l,l,k E S x ˆη l,k l,k. 5 Similarly we obtain the asymptotic it of the second term, which represents the inter-user interference in 4, as ES glk T Ŵ K l g lm x l,m =. 6 m=,m k Above, inter-user interferences are removed due to ZF receiving and transmission at the relay. Even with a different beamforming method, it can be shown that this value asymptoticay goes to zero. Further, we derive the asymptotic it of the third term in 4, which is the co-channel interference from the nearby jth TWR as ESg T lk ŴlG jl x j = g T lk E S Ĝ ĜT Ĝ ĜH P Ĝ Ĝ H G jl x j = η l,l,k η j,l,k E S x ˆη l,k j,k. 7 Also, we rewrite the fourth term in 4, which represents the added noise at the relay as 8 given at the top of this page. Based on 8, we can obtain the asymptotic distribution of the fourth term as 4 g T lk Ŵ l n Rl d η l,l,k τe S ˆη l,k 3 ñ l, 9, τe S ˆη l,k σ. Furthermore, we obtain the Rl where ñ l C asymptotic it of the last term in 4 as 4 n l,k =. By using the above asymptotic values and distributions of the terms in 4, we derive the asymptotic SIR for transmit power scaling at the user nodes as

5 5 ˆβ l = ˆβ l 4 = L K E S j= i= L K E S j= i= E R η j,l,i +τe S ˆη l,i +η j,l,i +τe S ˆη l,i τe S ˆη l,i ˆη l,i η j,l,i +η j,l,i τe S ˆη l,i ˆη l,i E R + σr K l i= τe S ˆη l,i ˆη l,i.. 4 γ l,k = = E S η l,l,k η l,l,k ˆη 4 l,k E S η L l,l,k j=,j l η j,l,k ˆη l,k 4 τes η l,l,k τe S B. Transmit power scaling at the relay + η l,l,k τe S ˆη 4 l,k σ R l L j=,j l η j,l,k + σ R l = γ. In this case, we derive the asymptotic SIR by scaling transmit power at the relay inversely proportional to the number of relay antennas. Thus, we substitute P R = E R /, P S = E S and P p = τe S to obtain the asymptotic SIR. We omit the proofs in this section due to their similarity to the results in Section III-A. We obtain the asymptotic value of the normalized relay gain as on the top of this page. Proof of is based on the results in Appendix C. Similar to the previous case, received signal at U l,k is given by 6. We use the its on each term of 6 to obtain the SIR value at U l,k as γ l,k = Ψ l E Sηl,l,k η l,l,k Ψ l E Sηl,l,k L j=,j l η j,l,k +, 3 ˆη4 l,k σ n l,k where Ψ l = ˆβl and given in. C. Transmit power scaling at the user nodes and relay In this section, we obtain the asymptotic SIR for the transmit power scaling at the relay and user nodes where P S = E S /, P P = τe S / and P R = E R /. As in the previous section, we omit the proofs due to their similarity to the results in Section III-A. We obtain the asymptotic value of the normalized relay gain in 4 at the top of this page. Then by substituting E S values, we rewrite 6 for the received signal as y l,k = ˆβ l 4 ESg T lk Ŵlg lk x l,k + ˆβ l 4 ESg T lk Ŵl K m=,m k g lm x l,m + ˆβ l 4 ESg T lk Ŵl L j=,j l G jl x j + ˆβ l 4 4 g T lk Ŵln Rl + n l,k. 5 By using the same procedure as in the Section III-A, we obtain the SIR as γ l,k = τλ l ESη l,l,k η l,l,k τλ L,6 l E S η l,l,k j=,j l η j,l,k +Λ l η l,l,k σ R l +τe S ˆη l,k 4 σ n l,k ˆβl where Λ l = 4 and given in 4. Remark V.: We can clearly see that the asymptotic SIRs, 3, and 6 are affected by the pilot contamination. Interestingly, whenever η j,l,k = for j l or when L =, the asymptotic SIRs results in, 3, and 6 approaches the same SIRs provided in [3]. Furthermore, by substituting asymptotic SIRs, 3, and 6 into and, overa sum rate and the energy efficiency of the system is obtained. IV. FIDIG THE OPTIMAL PILOT SEQUECE LEGTH In this section we analyze the optimal pilot sequence length to maximize the sum rate of the system for the case with L = under power scaling at the user nodes scenario. For this case, can be written as γ l,k = τe S η l,l,k σr = τξ l,k, 7 l where ξ l,k = E S η l,l,k. By substituting this value to, we obtain σr l the overa sum rate of the system as R l = T C τ T C K k= log + τξ l,k. 8 By taking the partial derivation of 8 with respect to τ and by equating it to zero, we obtain the foowing equation. K k= T C τξ l,k K = ln + τξ l,k. 9 + τξ l,k k= Since the total sum rate can be maximized by maximizing the sum rates of individual user nodes, above equation can be rewritten and rearranged as TC τξ l,k exp + = + τξ l,k exp, 3 + τξ l,k + T C ξ l,k + TC ξ l,k exp = + T C ξ l,k exp. 3 + τξ l,k + τξ l,k By using the Lambert-W function [], 3 is solved as W [ + T C ξ l,k ] exp = + T Cξ l,k + τξ l,k. 3 The optimal value for τ to maximize the sum rate can be derived as τk + T C ξ l,k = W [ + T C ξ l,k ] exp, 33 ξ l,k

6 6 β l = K K P R K K P S i= η l,l,i + K. 4 K σ R l i= η l,l,i η l,l,i where. is the floor function. Analysis on the second partial derivative of R l with respect to τ shows that the solution in 33, maximizes the sum rate of the system. V. AALYSIS FOR A FIITE UMBER OF ATEAS In this section, we derive the approximate closed-form sum rate for the three power scaling laws at the user nodes and relay with a finite number of antennas. Further, for mathematical tractability we it our analysis for CCI free, perfect CSI and no pilot contamination case. Furthermore, we obtain the cumulative distribution function CDF and the probability distribution function PDF of the end-to-end SIR at U l,k. For this case, the beamforming matrix Ŵl can be rewritten as Ŵ l = W l = G G T G PG H G G H, 34 By using W l, the amplification factor β l is given as β l = P STr [G T G ] + σr l Tr P R [G H G ] P[G T G ] P.35 Using W l and the absence of co-channel interfering TWRs, 5 is further simplified as y l,k = β l PS x l,k + β l k PG H G G H n Rl + n l,k, 36 where k represents K vector with value at the k location and zeros in a other places. Thus the end-to-end SIR at U l,k, γ l,k is derived as βl γ l,k = P S [ G βl σ H 37 R l G + σ ]k n l,k [ G ] H where G is the k th diagonal entry of the matrix k G H. G To begin the analysis of SIR value we use the long term power amplification factor β l. Here, we use the foowing matrix identities from []. [ G H ] Tr E G = Tr D K K = i= η l,l,i K. 38 [ ] Tr E G H G P G T G P = Tr PD PD K + Tr PTr PD D K K = K i= η l,l,i η l,l,i K. 39 By using the above equations the value of β l is rewritten as 4 at the top of this page. By substituting the value of β l to 37, we obtain α l,k X γ l,k =, 4 η l,k X + ζ l,k where X = given as foows: α l,k = ζ l,k = [G H G ] and other symbols are k,k K K P R P S. 4 K K P R σr l. 43 η l,k = K P S σn l,k K i= η l,l,i K + K σn l,k σ R l η l,l,i η l,l,i. 44 By using distribution of the kth diagonal element of the inverse Wishart matrix [3], the CDF of γ l,k was obtained as []: F γl,k x = Γ ζ K+, l,k x α l,k η l,k x Γ K+, < x < α l,k η l,k 45, x α l,k η. l,k By differentiating 45 by using the Leibniz integral rule the PDF is obtained as f γl,k x= d [ ζ l,k x dx α l,k η l,k x ] i= ζ l,k x K e ζ l,k x α l,k η l,k x α l,k η l,k x Γ K + ζ l,k x = α l,k ζ l,k K+ x K e α l,k η l,k x, 46 K+ Γ K +α l,k η l,k x where x < α l,k η. The average sum rate can be approximated l,k by solving R l = ln ln + xf γl,k x dx as R L ln where I is defined as foows: I = α l,k ζ K+ l,k exp α l,k η l,k K! I, 47 x K α l,k η l,k x K+ ζ l,k x ln + x dx, 48 α l,k η l,k x By substituting the dummy variable t = ζ l,k x/α l,k η l,k x into 48 the integral I can be simplified as I = t K e t ζl,k +α l,k + η l,k t ln dt, 49 ζ l,k +η l,k t ext, I in 49 can be solved in closed-form as foows: I = J K, ζ l,k, α l,k + η l,k J K, ζ l,k, η l,k 5,

7 7 Jx, y, z= x λ x exp λ lny+zλ dλ=γx+ lny+ Γx p+ p= y z x p y y exp E z z x p q y Γq. 5 z x p + q= where the function Jx, y, z is defined in 5 at the top of this page. By substituting 5 into 47, an approximation of the sum rate can be derived in closed-form as in 5. R L = J K, ζ l,k, α l,k + η l,k J K, ζ l,k, η l,k. 5 ln K! The sum rates under different power scaling scenarios can be obtained by substituting the P S and P R values to α l,k, ζ l,k and η l,k. The results obtained in 5 wi be useful to identify the number of antennas required to obtain a certain percentage of the asymptotic sum rate. VI. ASYMPTOTIC AALYSIS FOR ATEA CORRELATIO AT THE RELAY In this section, we derive asymptotic results for the SIR and sum-rate under the antenna correlation at the relay nodes under the three power scaling scenarios identified in section III. Further, for mathematical tractability we it our analysis for CCI free, perfect CSI and no pilot contamination case. When there is antenna correlation at the relay the channel vector to the relay from the user k is written as ḡ T,k = Ψ l,k f,k T d lk, where Ψ l,k is the correlation matrix at the relay. Furthermore, f,k T is the kth column vector in the matrix F and d lk is the kth diagonal entry of the matrix D that is given in II-A. Accordingly the channel [ matrix between a relay ] and a the users is given as Ḡ = ḡ, T ḡ, T ḡ,k T. This corresponds to the max-semi-correlated Rayleigh fading scenario presented in [4]. For this case, the beamforming matrix Ŵl can be rewritten as W l = Ḡ ḠT Ḡ PḠH Ḡ Ḡ H, 53 By using W l, the amplification factor β l is given as P R β l = [ḠT ] [ḠH P STr Ḡ + σr l Tr Ḡ] P [ Ḡ ] T.54 Ḡ P By using the similar steps as in section V, we obtain the endto-end SIR at U l,k, γ l,k as γ l,k = [ ḠH ] where Ḡ ḠH. Ḡ β l P S β l σ R l [ ḠH Ḡ ] k k + σ n l,k 55 is the k th diagonal entry of the matrix To analyse the asymptotic performance of antenna correlation at the relay, we first look at the it results relevant to the channel matrix Ḡ. Here, we obtain [ḠH ] Ḡ i,j = d / li f H,i / Ψ H l,i l,j Ψ f,j d / lj. 56 Spectral efficiency bps/hz L =,P S = ES Asymptotic it for L =,P S = ES L =4,P S = ES Asymptotic it for L =4,P S = ES L =8,P S = ES Asymptotic it for L =8,P S = ES umber of antennas Fig.. Spectral efficiency versus the number of relay antennas of an -user TWR with different L values. The channels in G jl are i.i.d. Rayleigh RVs with D = I K and D jl = I K, where j, l {,, L} and j l. By using the it results, it can be shown that if i j, then the value of 65 goes to zero. If i = j then the above value equals TrΨ l,j which is equal to for correlation matrices. Based on this, the it result can be given as [ḠH ] Ḡ D, 57 and coincidently the asymptotic results for the case with antenna correlation is equal to the results obtained for the case without any antenna correlation. This shows that by using massive MIMO, the degenerative effect of antenna correlation can be removed in our system model. VII. SIMULATIO RESULTS This section presents our simulation results and comparisons with the derived asymptotic results. The power at the user nodes and the relay nodes is taken as E S = E R =, and the noise powers at user and relay nodes is taken as σn l,k = σn R =. The pathloss exponent η is assumed to be two. The normalized pilot sequence power factor τ/t C is.8. Spectral and energy efficiencies under different power scaling scenarios, different K values, and different L values are presented in the sequel. What is the effect of having multiple TWRs on the spectral efficiency and the energy efficiency? In Fig. and Fig., the spectral efficiency and the energy efficiency are presented for power scaling at user nodes case, for L =, L = 4 and L = 8 values, when K = 6, respectively. The obtained asymptotic values are also plotted for comparison. We can see in Fig. that for a L values, the spectral efficiency asymptoticay reaches our analytical results, validating our derived results. Furthermore, as the number of TWRs L in the system is increased, the

8 Energy efficiency bps/hz/j L =,K =6,P S = ES L =4,K =6,P S = ES L =8,K =6,P S = ES Spectral efficiency bps/hz L =4,K =,P S = E S, P R = ER K = Asymptotic result L =4,K =6,P S = E S, P R = ER K=6 Asymptotic result 5 5 umber of antennas. 5 5 umber of antennas Fig.. Energy efficiency versus the number of relay antennas of -user, L TWRs. The channels in G jl are i.i.d. Rayleigh RVs with D = I K and D jl = I K, where j, l {,, L} and j l. Fig. 3. Spectral efficiency versus the number of relay antennas of a L = 4 TWR systems with different number of user pairs. The channel gains of G jl are i.i.d. Rayleigh RVs. with D = I K and D jl = I K, where j, l {,, L} and j l. achievable spectral efficiency and the energy efficiency of a single TWR decreases due to the interference and pilot contamination introduced by other TWRs. As an example, a L = system can obtain 4.9bps/Hz efficiency while an L = 8 system can only achieve a spectral efficiency of 3bps/Hz. However, if we consider the spectral efficiency of the whole system by multiplying the spectral efficiency of a single TWR by L, we can conclude that the bandwidth can be utilized further by increasing the number of TWRs. According to the values obtained in Fig., the total spectral efficiency is 9.8 bps/hz when L = and approximately 5 bps/hz when L = 8. Thus we can conclude that by using multipair massive MIMO TWRs for pairwise communications between nodes, that the ited bandwidth can be utilized effectively. We analyze the effect of number of users in a single system K in Fig. 3. Specificay, the sum rate is plotted for a system with eight relay networks under power scaling at the relay nodes for K = and K = 6 case. A four-user TWR achieves.6bps/hz while a -user TWR obtains.8 bps/hz. Once again, we have plotted our analytical results 3, which match the simulated values. ote that the spectral efficiency increases as the number of users increases, as the same bandwidth is used by the additional users. Thus, a massive MIMO pairwise TWR improves bandwidth utilization by serving more users. However, there wi be countering factors that wi it the number of user pairs in a network, such as the number of available orthogonal pilot sequences which is ited by the coherence time of the system. We compare the spectral efficiency gains and energy efficiency gains of different power scaling scenarios in Fig. 4 and Fig. 5 for -user TWRs L = 8. The analytical results, 3, and 6 are also plotted for comparison purposes. Power scaling at the user nodes has the highest asymptotic of 7. bps/hz out of a the three cases while case- and case-3 achieved asymptoticay 5. bps/hz and.8 bps/hz, respectively. Moreover, Spectral efficiency bps/hz PS = ES, PR = ER Simulation Asymptotic it for PS = ES, PR = ER PS = ES, PR = ER Asympttotic it for PS = ES, PR = ER PS = ES, PR = ER Asymptotic it for PS = ES, PR = ER 5 5 umber of antennas Fig. 4. Spectral efficiency versus the number of relay antennas of -user, L = 8 TWRs. The channels in G jl are i.i.d. Rayleigh RVs with D = I K and D jl = I K, where j, l {,, L} and j l. in Fig. 5 power scaling at both user and relay nodes obtains the highest energy efficiency as expected. Furthermore, the energy efficiency of case- power scaling at the relay only is very low compared to other two cases. This result is expected as the power of user nodes are kept unchanged in this scenario. Thus the numerator in is relatively constant for different values, and thus the power efficiency wi be low. How accurate are our analytical results when the number of antennas is finite? In Fig. 6, we answer this question by plotting the sum rate results. The simulated sum rate values match with our closed-form result in 5, justifying the accuracy of our approximation. For example, as few as 4 relay antennas yields about 85% of the asymptotic performance =. Moreover,

9 9.4 Energy efficiency bps/hz/j L =8,K =6,P S = ES L =8,K =6,P S = E S, P R = ER L =8,K =6,P S = ES, P R = ER 5 5 umber of antennas Fig. 5. Energy efficiency versus the number of relay antennas of -user, 8 TWRs. The channels in G jl are i.i.d. Rayleigh RVs with D = I K and D jl = I K, where j, l {,, L} and j l. Spectral efficiency bps/hz K = P S = ES P R = ER K = P S = E S P R = ER Low correlation l =.8, θ = π 3, σ = π 8 o correlation High correlation l =.4, θ = π 3, σ = π Asymptotic value umber of Antennas at the relay Fig. 7. Sum-rate versus the number of relay antennas of a 4-user TWR L = under relay antenna correlation. Spectral efficiency bps/hz K = P S = ES P R = ER K = P S = E S P R = ER Simulation Results Asymptotic Value Analytical Approximation umber of Antennas Fig. 6. Sum rate versus the number of relay antennas of a 4-user TWR L = under different power scaling scenarios. The channel gains of G are i.i.d. Rayleigh RVs. increasing to = relay antennas yields a 9%. This is good news because more or less the performance of massive MIMO is possible with a finite number of antennas. How much degradation of the sum rate occurs due to antenna correlation? In Fig. 7, we have used the antenna correlation model given in [5, Eqn. 4]. Here, the p, qth element of correlation matrix is given as e jπp ql cosθ e πp ql sinθσ [4], where l is the relative antenna spacing, θ is the average angle of arrival/departure, and σ is the standard deviation of the angle of arrival/departure. We have plotted the sum rate of the system under low correlation and high correlation at the relay as we as with no correlation. As seen from Fig. 7, although antenna correlation degrades the sum rate, as the number of relay antennas increases, the sum rate approaches to that of the uncorrelated antenna case. This observation corroborates our analysis that the effect of antenna correlation can be mitigated by a massive relay antenna array. VIII. COCLUSIO For multi-pair massive MIMO TWRs, we have investigated the impact of several key impairments, namely co-channel interference, imperfect CSI, antenna correlation and pilot contamination. We derived the asymptotic SIR, sum rate, and energy efficiency for the three transmit power scaling laws. Importantly, we show that the user transmit power can be scaled down inversely proportional to the square-root of the number of relay antennas. otably, if power scaling is ited to the relay node, power can be scaled down inversely proportional to the number of relay antennas without any performance penalty. Our analytical and simulation results reveal that substantial sum-rate gains and energy-efficiency gains can be achieved via a massive antenna array at the relay. Also, the closed-form results obtained for a finite number of antennas at the relay help wireless engineers to compute the number of antennas required to obtain a certain percentage of the asymptotic sum rate. Further, our results show that massive MIMO mitigates the degenerative effects of antenna correlation. In terms of future research, further work is required to rigorously analyze by considering a imperfections the sum-rate results for multiple massive MIMO TWRs with not so large number of antennas, especiay in a practical range between to. APPEDIX A PROOF OF LIMITS In this section, several important it results are provided. We begin with expressing the foowing three results [6].

10 G H jl Ĝ L ηj,l, E j= P diag η j,l, L + E P j= η,, η L j,l,k j= η j,l,k L j,l, + E P j= η, for P P = E P. 64 j,l,k G H jl Ĝ E P diag L η j,l, j= η j,l,,, η j,l,k L η j,l,k j=, for P P = E P. 65 ] P ] P ] ] ] P ] ĜH Tr G jl G H jl [ĜH Ĝ Ĝ [ĜT Ĝ [ĜH Ĝ = Tr GH jlĝ [ĜH Ĝ P[ĜT Ĝ [ĜH Ĝ ĜH G jl.66 Tr G jl G H jl Ĝ[ĜH Ĝ ] P [ĜT Ĝ ] P [ĜH Ĝ ] ĜH E P K η j,l,i i= ˆη l,i ˆη l,i + η j,l,i ˆη l,i ˆη l,i. 67 Tr G jl G H jl Ĝ[ĜH Ĝ ] P [ĜT Ĝ K i= + P P ˆη l,i + P P ˆη l,i P P ˆη l,i ˆη l,i ] P [ĜH Ĝ ] ĜH [ ηj,l,i + P P ˆη l,i P P ˆη l,i + η j,l,i + P P ˆη l,i P P ˆη l,i ]. 69 For two independent vectors, p C, σ p and q C, σ q, the foowing identities are valid. p H p / σ p and p H q /, 58 p H q / d C, σpσ q, 59 where subscripts and d stands for almost sure convergence and the convergence of distributions, respectively. By using the aforementioned identities, it can be shown that G H jlg jl G H jlg ml = D jl F H jl F jl = D jl F H jl F ml D jl D ml D jl, and 6 K, for j m. 6 Furthermore, by using the above two results, foowing it can be obtained. [ Ĝ H Ĝ = L ] H G jl + V L l D G jl + V l D Pp Pp j= j= L = D L H G H L jlg ml + VH l j= G jl j= m= Pp L j= + GH jlv l + VH l V l D PP L = D H D jl + I K D H, 6 j= where H is a diagonal matrix with kth diagonal entry given by P P L j= η j,l,k. Furthermore when transmit power scaling is +P P L j= η j,l,k done at the users ie. P P = E P /, the foowing it can be obtained. Ĝ H Ĝ Ĥ, 63 where Ĥ is a diagonal matrix with kth diagonal entry given by E P L j= η j,l,k. Furthermore, the it results 64 and 65 that are shown at the top of this page are obtained using the same procedure. APPEDIX B PROOF OF LIMITS FOR POWER SCALIG AT THE USER ODES This section provides a sketch of the proof of SIR for the transmit power scaling scenario. First we prove the its for the power normalizing factor ˆβ l. The first term in the denominator in 9 can be written as shown in 66 at the top of this page. By using the it results 6 and 63 given in Appendix A on each term in the above equation, foowing result is formulated as 67 shown at the top of this page. Here in 67, ˆη l,k = L j= η j,l,k. Similarly, the it of the second term in the denominator in 9 is derived as [ĜH P Tr Ĝ] [ĜT Ĝ ] P K i= By using the above two results, is obtained. E P ˆη l,i ˆη l,i. 68 APPEDIX C PROOF OF LIMITS FOR POWER SCALIG AT THE RELAY ODES This section provides a sketch of the proof of SIR for the transmit power scaling at the relay. The first term in the denominator in 9 can be written in a similar way by replacing by in 66 in Appendix B. By using the it results 6

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