Impact of ine-of-sight and Unequa Spatia Correation on Upin MU- MIMO Systems Tataria,., Smith, P. J., Greenstein,. J., Dmochowsi, P. A., & Matthaiou, M. (17). Impact of ine-of-sight and Unequa Spatia Correation on Upin MU-MIMO Systems. IEEE Wireess Communications etters, 6(99). https://doi.org/1.119/wc.17.766 Pubished in: IEEE Wireess Communications etters Document Version: Peer reviewed version Queen's University Befast - Research Porta: in to pubication record in Queen's University Befast Research Porta Pubisher rights 17 IEEE. This wor is made avaiabe onine in accordance with the pubisher s poicies. Pease refer to any appicabe terms of use of the pubisher. Genera rights Copyright for the pubications made accessibe via the Queen's University Befast Research Porta is retained by the author(s) and / or other copyright owners and it is a condition of accessing these pubications that users recognise and abide by the ega requirements associated with these rights. Tae down poicy The Research Porta is Queen's institutiona repository that provides access to Queen's research output. Every effort has been made to ensure that content in the Research Porta does not infringe any person's rights, or appicabe UK aws. If you discover content in the Research Porta that you beieve breaches copyright or vioates any aw, pease contact openaccess@qub.ac.u. Downoad date:5. Apr. 19
Impact of ine-of-sight and Unequa Spatia Correation on Upin MU-MIMO Systems arsh Tataria, Member, IEEE, Peter J. Smith, Feow, IEEE, arry J. Greenstein, ife Feow, IEEE, Pawe A. Dmochowsi, Senior Member, IEEE, and Michai Matthaiou, Senior Member, IEEE Abstract Cosed-form approximations of the expected pertermina signa-to-interference-pus-noise-ratio (SINR) and ergodic sum spectra efficiency of a mutiuser mutipe-input mutipe-output system are presented. Our anaysis assumes spatiay correated Ricean fading channes with maximum-ratio combining on the upin. Unie previous studies, our mode accounts for the presence of unequa correation matrices, unequa Rice factors, as we as unequa in gains to each termina. The derived approximations end themseves to usefu insights, specia cases and demonstrate the aggregate impact of ine-of-sight (os) and unequa correation matrices. Numerica resuts show that whie unequa correation matrices enhance the expected SINR and ergodic sum spectra efficiency, the presence of strong os has an opposite effect. Our approximations are genera and remain insensitive to changes in the system dimensions, signato-noise-ratios, os eves and unequa correation eves. Index Terms Ergodic sum spectra efficiency, expected SINR, ine-of-sight, MU-MIMO, unequa correation. I. INTRODUCTION The ac of rich scattering and insufficient antenna spacing at a ceuar base station (BS) eads to increased eves of spatia correation 1. For mutiuser mutipe-input mutipeoutput (MU-MIMO) systems, this is nown to negativey impact the signa-to-interference-pus-noise-ratio (SINR) of a given termina, as we as the sum spectra efficiency of the system. Numerous wors have investigated the SINR and spectra efficiency performance of MU-MIMO systems with spatia correation (see e.g., 4 and references therein). owever, very few of the above mentioned studies consider the effects of ine-of-sight (os) components, iey to be a dominant feature in future wireess access with the rise of smaer ce sizes 5. Thus, understanding the performance of such systems with Ricean fading is of particuar importance. The upin Ricean anaysis presented in 6 does not consider the effects of spatia correation at the BS. On the other hand, the reated iterature (see e.g., 3, 7) routiney assumes that on the upin, a terminas are seen by the BS via the same set of incident directions, resuting in equa correation structures. In reaity, a different set of incident directions are iey to be observed by mutipe terminas, due to their different geographica ocations, eading to variations in the oca scattering. This gives rise to wide variations in the. Tataria and M. Matthaiou are with the Schoo of Eectronics, Eectrica Engineering and Computer Science, Queen s University Befast, Befast, BT3 9DT, UK (e-mai: {h.tataria, m.matthaiou}@qub.ac.u). P. J. Smith is with the Schoo of Mathematics and Statistics, Victoria University of Weington, Weington 614, New Zeaand (e-mai: peter.smith@vuw.ac.nz). P. A. Dmochowsi is with the Schoo of Engineering and Computer Science, Victoria University of Weington, Weington 614, New Zeaand (e-mai: pawe.dmochowsi@ecs.vuw.ac.nz).. J. Greenstein was with the Wireess Information Networ aboratory, Rutgers University, North Brunswic, NJ 89, USA (e-mai: jg@winab.rutgers.edu). correation patterns across mutipe terminas 4. ence, we consider unequa correation matrices from each termina. Motivated by this, with a uniform inear array (UA) and maximum-ratio combining (MRC) at the BS, we present insightfu cosed-form approximations of the expected pertermina SINR and ergodic sum spectra efficiency of an upin MU- MIMO system. Unie previous resuts, for both microwave and miimeter-wave (mmwave) propagation parameters, the cosed-form expressions consider unequa correation matrices, Rice (K) factors and in gains for each termina. The approximations are shown to be extremey tight for sma and arge system dimensions, as we as, arbitrary signa-to-noise- ratios (SNRs). To the best of our nowedge, this eve of accuracy over such a genera channe mode capturing a wide range of scenarios has not been achieved previousy. Numerica resuts show the aggregate impact of os and unequa spatia correation. Specia cases are presented for Rayeigh fading channes with equa and unequa correation matrices, as we as, for Ricean fading channes with equa correation matrices. II. SYSTEM MODE The upin of a MU-MIMO system operating in an urban microceuar environment (UMi) is considered. The BS is ocated at the center of a circuar ce with radius R c, and is equipped with a M eement UA simutaneousy communicating with singe-antenna terminas (M ). Channe nowedge is assumed at the BS, as the prime focus of the manuscript is on performance anaysis with genera fading channes and not on system eve imperfections. The composite M 1 received signa at the BS is given by y = ρ 1 GD 1 s + n, where ρ is the average upin transmit power, G is the M fast-fading channe matrix between the M BS antennas and terminas, D is an diagona matrix of in gains, where the in gain for termina is given by D, = β. The arge-scae fading effects for termina in geometric attenuation and shadow-fading are captured in β = ϱζ (r /r ) α. In particuar, ϱ is the unit-ess constant for geometric attenuation at a reference distance of r, r is the distance between the -th termina and the BS, α is the attenuation exponent and ζ captures the effects of shadowfading, modeed via a og-norma density, i.e., 1 og 1 (ζ ) N (, σsh). Moreover, s is the 1 vector of upin data symbos from terminas to the BS, such that the -th entry of s, s has an expected vaue of one, i.e., E s = 1. The M 1 vector of additive white Gaussian noise at the BS is denoted by n, such that the -th entry of n, n CN (, σ ). We assume that σ = 1. ence, the average upin SNR is defined as ρ/σ = ρ. The M 1 channe vector from termina to the BS is denoted by g, which forms the -th coumn of G = g 1,..., g. 1
More specificay, g = η h + γ R 1 h. (1) The M 1 os and the non os (NoS) components of the channe are denoted by h and h. Note that γ = (1/ (1 + K )) 1/ and η = (K / (K + 1)) 1/, with K being the Ricean K-factor for the -th termina. R is the receive correation matrix specific to termina, h CN (, I M ) and h = 1, e jπd cos(φ ),..., e jπd(m 1) cos(φ ). ere, d is the equidistant inter-eement antenna spacing normaized by the carrier waveength and φ U, π is the azimuth ange-ofarriva of the os component for the -th termina. We empoy a inear receiver at the BS array in the form of a MRC fiter, where G is the M fiter matrix used to separate y into data streams by r = G y = ρ 1/ G GD 1/ s+ G n. ence, the combined signa from termina is given by r = ρ 1/ β 1/ g g s + ρ 1/ =1 β 1/ g g s + g n. Thus, the corresponding SINR for termina is given by ρβ g 4 SINR = g + ρ =1 β g g. () As such, the instantaneous upin spectra efficiency for the -th termina (measurabe in bits/sec/z) is given by R se = og (1 + SINR ). From here, the ergodic sum spectra efficiency over a terminas is given by E R sum = E, (3) =1 Rse where the expectation is performed over the fast-fading. III. EXPECTED PER-TERMINA SINR AND ERGODIC SUM SPECTRA EFFICIENCY ANAYSIS The expected SINR of termina can be obtained by evauating the expected vaue of the ratio in (). Exact evauation of this is extremey cumbersome, as shown in 6. ence, we resort to the first-order Deta method expansion, as shown in the anaysis methodoogy of 6. This gives ρβ E g 4 E SINR E g + ρ =1 β E g g. (4) Remar 1. The approximation in (4) is of the form of EX EY. The accuracy of such an approximation reies on Y having a sma standard deviation reative to its mean. This can be seen by appying a mutivariate Tayor series expansion of X Y around EX EY, as shown in the methodoogy of 6. Both X and Y are we suited to this approximation as M and start to increase. This is evident from the presented numerica resuts in Section V. In emmas 1, and 3 which foow, we derive the expected vaues in the numerator and denominator of (4). emma 1. For a UA with M receive antennas at the BS, considering a correated Ricean fading channe, g, from the -th termina to the BS δ = E g 4 { = (η ) 4 M + tr (R ) } +M (η ) (γ ) + (γ ) (η ) h R h + (γ ) 4 M, (5) where each parameter is defined after (1). Proof: See Appendix A. emma. Under the same conditions as emma 1, ϕ, = E g g =(η ) (η ) tr R R + (η ) (γ ) tr h R h + (γ ) (η ) tr h h R + (γ ) (γ ) h h. (6) Proof: See Appendix B. emma 3. Under the same conditions as emma 1, χ = E g = M (γ ) + (η ) = M. (7) Proof: We begin by recognizing that χ = E g = E g g. Substituting the definition of g into (7) and performing the expectations in with respect to h yieds the desired resut. Ony a setch of the proof is given here, as it reies on straightforward agebraic manipuations. Theorem 1. With MRC and a UA at the BS, the expected upin SINR of termina undergoing spatiay correated Ricean fading can be approximated as ρβ δ E SINR χ + ρ, (8) =1 β ϕ, where δ, ϕ, and χ are given by (5), (6) and (7), respectivey. Proof: Substituting the resuts from emmas 1, and 3 for δ, χ and ϕ, yieds the desired expression. Remar. Further agebraic manipuations aows us to express (8) as (1), shown on top of the next page for reasons of space. Note that (1) can be used to approximate the ergodic sum spectra efficiency of the system by stating E R sum =1 og ( 1 + E SINR ). (9) Whie the accuracy of (1) and (9) is demonstrated in Section V, in the seque, we present the impications and specia cases of (1) to demonstrate its generaity. IV. IMPICATIONS AND SPECIA CASES A. Impications of (1) Both the numerator and the denominator of (1) contain quadratic forms of the type h R h. Via the Rayeigh quotient resut, such quadratic forms are maximized when h is parae (aigned) to the maximum eigenvector of R. From this, an interesting observation can be made: Aignment of h and R ampifies the expected signa power, whie aignment of h with R, h with R and h with h increases the expected interference power, eading to a ower SINR. iewise, if R and R become simiar, then tr R R increases, degrading the SINR. The goba observation is that the SINR reduces by virtue of channe simiarities of various types (os and correation) and increases if the channes are more diverse. B. Specia Cases of (1) Coroary 1. In pure NoS conditions (i.e., Rayeigh fading) with unequa correation matrices, (1) reduces to E { SINR c1 ρβ M + tr R M + ρ }. (11) =1 β {tr R R Proof: Substituting K = K =,, = {1,..., } in (1) yieds the desired resut. Coroary (Proposition 1 in 3). In pure Rayeigh fading with equa correation matrices, (1) coapses to E { SINR c ρβ M + tr R M + ρ { }. (1) =1 β tr R } }
E SINR M + ρ =1, ρβ ) + tr R + K h R h } { ( (K +1) M 1 + K + K ) ) }. (1) {tr R R + K ( h R h + K ( h R h + K K h h β (K +1)(K +1) 3 Proof: Setting R = R,, = {1,..., } in (11) gives the desired resut. The resut is consistent with 3. Coroary 3. With os presence and equa correation matrices, (1) can be approximated with E SINR c3 ρβ δ χ + ρ, (13) =1 β ϕ, where ϕ, = (η ) (η ) tr R + (η ) (γ ) tr h R h + (γ ) (η ) tr h h R +(γ ) (γ ) h h. Proof: Repacing R with R and substituting the definition of δ and χ from (5) and (7) yieds the desired resut. V. NUMERICA RESUTS We empoy a statistica approach to determine whether a given termina experiences os or NoS propagation. The NoS and os probabiities are governed by the in distance, from which other in parameters such as the attenuation exponent and shadow-fading standard deviation are seected. We consider the UMi propagation parameters for microwave 8 and mmwave 9, 1 frequencies at and 8 Gz, respectivey. For both cases, the ce radius (R c ) and excusion area (r ) are fixed to 1 m and 1 m. The terminas are randomy ocated outside r and inside R c with a uniform distribution with respect to the ce area. The os and NoS attenuation exponents (α) are given by., 3.67 and,.9 at microwave and mmwave frequencies, whie the parameter ϱ is chosen such that the fifth percentie of the instantaneous SINR of termina is db at ρ = db, for the system dimensions of M = 64, = 4. Moreover, the os and NoS shadow-fading standard deviations (σ sh ) are 3 db, 4 db and 5.8 db, 8.7 db for the microwave and mmwave cases. The Ricean K-factor has a og-norma density with a mean of 9 and standard deviation of 5 db for microwave (K n (9, 5)) 8 and a mean of 1 with standard deviation of 3 db for the mmwave (K n (1, 3)) cases 1. With microwave parameters, the probabiity of termina experiencing os is given by P os (r ) = (min(18/r, 1)(1 e r/36 )) + e r /36 8. Equivaenty, at mmwave, P os = (1 P out(r ))e ιosr, where 1/ι os = 67.1 m and P out, the outage probabiity, is set to for simpicity 9. For both cases, P NoS = 1 P os. Due to its generaity in modeing spatiay correated fading, the one-ring mode is chosen to generate unequa spatia correation at the BS, as in, 4, 11. The (i, j) entry in the correation matrix of termina is given by 11 +φ R i,j = 1 e jπd(i j) sin(θ) dθ, (14) +φ where denotes the azimuth anguar spread, φ is the centra azimuth ange from termina to the BS array, θ is the actua ange-of-arriva (AoA) and d (i j) captures the inter-eement spacing normaized by the carrier waveength between i-th and j-th antenna eements. Uness expicity stated, we set d (1) =.5 and assume that φ U, π. The instantaneous vaue of θ is aso drawn from a uniform distribution on,, i.e., θ U,. As such, represents the tota anguar spread, naturay bounded from to π radians ( to 36 ). Note that the one-ring mode captures a genera physica scenario and is not intended to be specific for a particuar carrier frequency. Naturay, one can fix d (1) and the distribution of φ, and seect vaues for from channe measurements at both microwave and mmwave frequencies. owever, is varied deibratey to understand its impact with os on the expected SINR and ergodic sum spectra efficiency. With M = 3, = 3, Fig. 1 iustrates the expected pertermina SINR of a given termina as a function of ρ. In addition to the microwave and mmwave cases, we consider the two extremes in uncorreated Rayeigh fading and pure os channes. Furthermore, unequay correated Rayeigh and Ricean fading cases are considered, where the Ricean case has a fixed K-factor of 5 db for each termina. Three trends can be observed: (1) Transitioning from arger to smaer anguar spread ( = 9 to = ) significanty reduces the expected SINR for a cases. This is despite the fact that the UA is equipped with a moderate number of receive antennas, and is due to the reduction in the spatia seectivity of the channe, enforcing the UA to see a narrower spread of the incoming power. () Increasing the mean of K has a negative impact on the expected SINR, as stronger os presence tends to reduce the mutipath diversity and the ran of the composite channe. (3) The proposed expected SINR approximations in (1) are seen to remain extremey tight for the entire range of ρ for a cases. The approximations can aso be seen to remain tight for the specia case of Rayeigh fading with unequa correation matrices in (11). Furthermore, the expected SINR in each case is seen to saturate with ρ, as the MRC fiter is unabe to mitigate mutiuser interference. Considering the specia cases in (1) and (13), we now examine the aggregate impact of os, as we as equa and unequa correation on the ergodic sum spectra efficiency, as shown in Fig.. With M = 56 and = 3, using the same propagation parameters as in Fig. 1, at ρ = 1 db, we compare the cumuative distribution functions (CDFs) of the derived ergodic sum spectra efficiency approximation in (9) with its simuated counterparts. Each CDF is obtained by averaging over the fast-fading, with each vaue representing the variations in the in gains and the K-factors. The derived approximations remain tight with changes in the system size. Moreover, irrespective of the underying propagation characteristics, unequa correation matrices resut in higher ergodic sum spectra efficiency, aowing the UA to everage more spatia diversity. This is noticed when comparing the K = 5 db curves with a fixed φ = π/16 (equa correation) and variabe φ (unequa correation) for each termina. In contrast to the correated Rayeigh case, a dominant os component is again seen to be detrimenta to system performance. VI. CONCUSION We have presented a genera, yet insightfu approximation to the expected per-termina SINR and ergodic sum spectra efficiency of an upin MU-MIMO system. With a UA and MRC at the BS, the approximation is robust to equa and unequa correation matrices, unequa eves of os, unequa in gains, unequa operating SNRs and system dimensions.
4 Expected Per-Termina SINR db 1 1 8 6 4 - -4-5 5 1 15 Fig. 1: Expected per-termina SNR vs. ρ (SNR) with M = 3, = 3 and = and 9. CDF 1.9.8.7.6.5.4.3..1..15.1.5 1 3 4 5 6 1 3 4 5 6 7 8 9 1 Ergodic Sum Spectra Efficiency bps/z Fig. : Ergodic sum spectra efficiency CDF with M = 56, = 3 at ρ = 1 db and =. With both microwave and mmwave parameters, our resuts show that unequa correation matrices yied higher expected SINRs and ergodic sum spectra efficiency in comparison to equa correation. Moreover, increasing the os component of the channe reduces the expected SINR and ergodic sum spectra efficiency due to the oss of spatia diversity. APPENDIX A PROOF OF EMMA 1 We begin by recognizing that δ = E g 4 = E ( g ). Substituting the definition of g and denoting v = γ R 1 h and q = η h aows us to state ( g δ = E ) (v =E v +v q +q v +q ) q. (15) Expanding (15) aows us to write (v ) + ( ) ( ) ( δ = E v v v q q + v q q ) v + ( q v v ) ( ) q + q q. (16) Performing the expectations over v in the ast four terms of (16) and simpifying yieds (v ) δ = E v + M (η ) ( q ) q + (η ) q R q + ( q q ). (17) After noting that E ( v v ) = E v v v v, substituting the definition of v and extracting the reevant constants yieds E ( v v ( ) 4E ) ) = η ( h R h, where R = ΦΛΦ via an eigenvaue decomposition. ence, (v ) ( M ) E v = (η ) 4 E ( h ). (18) i=1 Λ i,i Performing the expectation with respect to h and simpifying yieds E(v v ) = (η ) 4 {(tr R ) + tr R }. As tr R = M, E(v v ) = (η ) 4 {M + tr(r ) }. Substituting the right-hand side aong with the definition of q into (17), recognizing h h = M and simpifying yieds emma 1. APPENDIX B PROOF OF EMMA Appying the definition of g and g into ϕ, = E g g and denoting v = η R 1 h and q = γ h ( ) ( yieds ϕ, = E v + q v + h ). Expanding and simpifying further gives ϕ, = E v v v v + E v q q v + E q v v q + E q q q q. (19) Recognizing that E v v = E η R 1 h η h R 1 = (η ) R, substituting bac the definitions of v, v, q and q in (19) and extracting the reevant constants yieds ϕ, = (η ) (η ) tr R R + (η ) (γ ) E tr R 1 h h R 1 h h + (γ ) (η ) E tr h h R 1 h h R 1 + (γ ) (γ ) h h. () Taing the trace and simpifying yieds (6). REFERENCES 1 F. Ruse, D. Persson, B. au, E. G. arsson, T.. Marzetta, O. Edfors, and F. Tufvesson, Scaing up MIMO: Opportunities and chaenges with very arge arrays," IEEE Signa Process. Mag., vo. 3, no. 1, pp. 4-6, Nov. 13. J. oydis, S. ten Brin, and M. Debbah Massive MIMO in the U/D of ceuar networs: ow many antennas do we need?," IEEE J. Se. Areas Commun., vo. 31, no., pp. 16-171, Feb. 13. 3 J. Zhang,. Dai, M. Matthaiou, C. Masouros, and S. Jin, On the spectra efficiency of space-constrained massive MIMO with inear receivers," in Proc. IEEE ICC, May 16, pp. 1-6. 4 J. Nam, G. Caire, and J. a, On the roe of transmit correation diversity in mutiuser MIMO systems," IEEE Trans. Inf. Theory, vo. 63, no. 1, pp. 336-354, Jan. 17. 5. Tataria, P. J. Smith,. J. Greenstein, and P. A. Dmochowsi, Zero-forcing precoding performance in mutiuser MIMO systems with heterogeneous Ricean fading," IEEE Wireess Commun. ett., vo. 6, no. 1, pp. 74-77, Feb. 17. 6 Q. Zhang, S. Jin, K-K. Wong,. Zhu, and M. Matthaiou, Power scaing of upin massive MIMO systems with arbitrary-ran channe means," IEEE J. Se. Topics Signa Process., vo. 8, no. 5, pp. 966-981, Nov. 14. 7. Faconet and. Sanguinetti, A. Kammoun, and M. Debbah, Asymptotic anaysis of downin MISO systems over Rician fading channes," in Proc. IEEE ICASSP, Mar. 16, pp. 396-393. 8 3GPP TR 36.873 v1.., Study on 3D channe modes for TE, 3GPP, Jun. 15. 9 M. R. Adeniz, Y. iu, M. K. Samimi, S. Sun, S. Rangan, T. S. Rappaport, and E. Erip, Miimeter wave channe modeing and ceuar capacity evauation," IEEE J. Se. Areas Commun., vo. 3, no. 6, pp. 1164-1179, Jun. 14. 1 T. Thomas,. C. Nguyen, G. R. MacCartney, and T. S. Rappaport, 3D mmwave channe mode proposa," in Proc. IEEE VTC-Fa, Sep. 14, pp. 1-6. 11 Z. Jiang, A. F. Moisch, G. Caire, and Z. Niu, Achievabe rates of FDD massive MIMO systems with spatia channe correation," IEEE Trans. Wireess Commun., vo. 14, no. 5, pp. 86-88, May 15. i