arXiv:1403.3461v1 [cs.IT] 13 Mar 2014 h wds eerhCucl(R,teSeihFudto f Foundation Swedish ELLIIT. and the (SSF), (VR), Research Council Research Swedish the ik ihlna emomn ehius h aestation base the techniques, beamforming down- the linear On out. with fil- canceled matched link, be the can as interference such and detector noise ter, linear simple th a on explicitly, with More uplink, performance. optimal achieve can called ing the is and This station propagation orthogonal. able base nearly the be between should vectors terminals channel the that is [4]. processing linear with a efficiency reliability, power communication systems high throughput, MIMO massive high at precisely, provide but can More MIMO, traditional scale. larger of de- can much benefits systems attractive Such the all [1–3]. liver terminals of tens simultaneousl several antennas massiveserves hundred few in a interest with equipped of station bas a deal where great systems (MIMO) a multiple-output multiple-input been has there Recently, value well. singular model UR-LoS the the explains urns-and-balls of and characteristic an simple spread is propose model we This model, model. UR-LoS Furthermore, the propagation. analyze environ- favorable (nearly) Both offer fadin (UR-LoS). ments Rayleigh line-of-sight i.i.d. random scenarios: uniform extreme and two for chan- be the can favorable nels how sum- examine we the favorable favorable Secondly, under between propagation. from obtained capacity gap “distance maximum the the a and is capacity propose which we measure, offere is, propagation” propagation channel the the favorable For how by evaluate equal. be to case not the may case, vectors for this channel good the con- of not norms channel is the proxy the where this is However, not number. or dition favorable for proxy is useful propagation that i One whether show scenario sum-capacity. first the desirable maximizing we most of paper, terms the this offers In that propagation exploited detail. work favorable is in little that topic been this channel has studies there radio one However, the is MIMO. of Massive terminals, in properties orthogonality the key to mutual the channels of as vector-valued the defined among propagation, Favorable h oko .Q g n .G aso a upre npr by part in supported was Larsson G. E. and Ngo Q. H. of work The n ftekyasmtosepotdb asv MIMO massive by exploited assumptions key the of One ∗ eateto lcrclEgneig(S) Link¨oping Un (ISY), Engineering Electrical of Department ihfvrbepoaain ierprocess- linear propagation, favorable With . .INTRODUCTION 1. SET FFVRBEPOAAINI ASV MIMO MASSIVE FAVORABLE PROPAGATION IN OF ASPECTS ABSTRACT † elLbrtre,ActlLcn uryHl,N 77,U 07974, NJ Hill, Murray Alcatel-Lucent Laboratories, Bell inQo Ngo Quoc Hien ∗ rkG Larsson G. Erik , rStrategic or favor- nd to n g y d a e e fthe If all and etra h aesainis station base the at vector aat h aesain h aesainhas station base The station. base the to data trans simultaneously and independently terminals antenna of and minal, where stecanlvco ewe h aesainadthe and station base the between vector channel the is osdrteuln fasnl-elsse where system single-cell a of uplink the Consider system. single-cell argu- a the of of uplink sake the the consider will For we case. ment, UR-LoS the analyze to urns-and-bal simple model a propose also We f propagation. substantially vorable line-of-sig offer scenarios random both scenar- uniform that show extreme We and (UR-LoS). two fading consider Rayleigh favorabl i.i.d. we to end, ios: lead this will To also scenarios We practical propagation. so. not not or is whether this fa- ask norms, the how different when evaluate have However, vectors to the channel is. proxy environment i.i.d., useful propagation are the a vectors vorable is channel spread the value singular When a under constraint. fa- sum-capacity under power the that maximize show we first propagation, We vorable channels. different of condition not or whether for case, proxy this propagation. good in favorable a have here, we not see is will th number we of condition As the norms identical. the not locations, are different channels have f the terminals are to the owing channels practice, that the in However, that fading. case Rayleigh is. the i.i.d. channel considered the favorable only how papers evaluate to These matri proxy channel a the as of used number pa- was condition the the in There, discussed [4,5]. was Favo pers MIMO massive interference. of propagation mutual able causing mul- without to terminals streams data tiple multiple beamform simultaneously can i SR.Tecanlvector channel The (SNR). tio signal-to-no “transmit” normalized of interpretation the x 1 x , . . . , w ∗ nti ae,w netgt h aoal propagation favorable the investigate we paper, this In hmsL Marzetta L. Thomas , r i.i.d. are K x K [ = K .SNL-ELSSE MODEL SYSTEM SINGLE-CELL 2. emnl iutnosytasi the transmit simultaneously terminals emnl hr h aetm-rqec resource. time-frequency same the share terminals y w hr E where , = x vriy 8 3Link¨oping, Sweden 83 581 iversity, sanievco.W sueta h elements the that assume We vector. noise a is 1 CN x , . . . , √ ρ k X (0 K =1 , K | 1) g x ] k k T V.Wt hsassumption, this With RVs. x | , 2 k G + 1 = † [ = g w k SA hnthe then , = g noprtsteefcsof effects the incorporates 1 √ , . . . , ρ Gx g K M + ] , w × g M K , k 1 K ∈ antennas received symbols k single- C s ra- ise hter- th ρ M mit has (1) act × a- ht r- ls x 1 e e large-scale fading and small-scale fading. More precisely, the Secondly, we consider a more relaxed constraint on the 2 2 mth element of gk is modeled as: channel G: constraint G F instead of gk . From (6), by using Jensen’s inequality,k k we get {k k } gm = β hm, k =1,...,K, m =1,...,M, (2) k k k K K 2 1 2 mp C log2 1+ ρ gk =K log2 1+ ρ gk where hk is the small-scale fading and βk represents the ≤ k k · K k k large-scale fading which depends on k but not on m. kX=1 kX=1 ρ K ρ 3. PRELIMINARIES OF FAVORABLE Klog 1+ g 2 =Klog 1+ G 2 , (7) ≤ 2 K k kk 2 K k kF PROPAGATION k=1 ! X In favorable propagation, we can obtain optimal performance where the equality in the first step holds when (3) satisfied, 2 with simple linear processing techniques. To have favorable and the equality in the second step holds when all gk are propagation, the channel vectors g , k =1,...,K, should equal. So, for this case, C is maximized if (3) holdsk andk g { k} k be pairwisely orthogonal. More precisely, we say that the have the same norm. The constraint on G that results in{ (7) is} channel offers favorable propagation if more relaxed than the constraint on G that results in (6), but the bound in (7) is only tight if all g have the same norm. 0, i,j =1,...,K, i = j { k} gH g = 6 (3) i j g 2 =0, k =1,...,K. k kk 6 3.2. Measures of Favorable Propagation In practice, the condition (3) will never be exactly satisfied, Clearly, to check whether the channel can offer favorable but (3) can be approximately achieved. For this case, we say propagation or not, we can check directly the condition (3) or that the channel offers approximately favorable propagation. (4). However, to do this, we have to check all (K 1)K/2 Also, under some assumptions on the propagation environ- possible pairs. This has computational complexity.− Other ment, when M grows large and k = j, it holds that 6 simple methods to measure whether the channel offers favor- 1 able propagation is to consider the condition number, or the gH g 0,M . (4) distance from favorable propagation (to be defined shortly). M k j → → ∞ These measures will be discussed in more detail in the fol- For this case, we say that the channel offers asymptotically lowing subsections. favorable propagation. The favorable propagation condition (3) does not offer 3.2.1. Condition Number only the optimal performance with linear processing but also represents the most desirable scenario from the perspectiveof Under the favorable propagation condition (3), we have maximizing the information rate. See the following section. GH G = Diag g 2, , g 2 . (8) {k 1k ··· k K k } 3.1. Favorable Propagation and Capacity We can see that if gk have the same norm, the condition { } H Consider the system model (1). We assume that the base sta- number of the Gramian matrix G G is equal to 1: tion knows the channel G. The sum-capacity is given by σmax/σmin =1, (9)
H where σmax and σmin are the maximal and minimal singular val- C = log2 I + ρG G . (5) ues of GH G.
Similarly, if the channel offers asymptotically favorable Next, we will show that, subject to a constraint on G, under favorable propagation conditions (3), C achieves its largest propagation, then we have 2 H possible value. Firstly, we assume gk are given. For G G D,M , (10) this case, by using the Hadamard inequality,{k k } we have → → ∞ where D is a diagonal matrix whose kth diagonal element is K βk. So, if all βk are equal, then the condition number is H H { } C = log2 I + ρG G log2 [I + ρG G]k,k asymptotically equal to 1. ≤ ! k=1 Therefore, when the channel vectors have the same norm Y K K (the large scale fading coefficients are equal), we can use = log [I +ρGH G] = log 1+ρ g 2 . (6) 2 k,k 2 k kk the condition number to determine how favorable the channel kX=1 kX=1 propagation is. Since the condition number is simple to evalu- ate, it has been used as a measure of how favorable the propa- We can see that the equality of (6) holds if and only if GH G gation offered by the channel G is, in the literature. However, is diagonal, so that (3) is satisfied. This means that, given 2 it has two drawbacks: i) it only has a sound operational mean- a constraint on gk , the channel propagation with the {k k } ing when all g have the same norm or all βk are equal; condition (3) provides the maximum sum-capacity. { k} { } and ii) it disregards all other singular values than σmin and σmax . H 2 3.2.2. Distance from Favorable Propagation Furthermore, in Massive MIMO, the quantity gk gj is As discussed above, when gk have different norms or βk of particular interest. For example, with matched filtering, the 4 are different, we cannot use{ the} condition number to measure{ } power of the desired signal is proportionalto g , while the k kk how favorable the propagationis. For this case, we propose to H 2 power of the interference is proportional to gk gj , where use the distance from favorable propagation which is defined k = j. For k = j, we have that 6 6 as the relative gap between the capacity C obtained by this 1 propagation and the upper bound in (6): gH g 2 0, (15) M 2 | k j | → K 2 H V 1 H 2 M +2 1 log 1+ρ g log I +ρG G ar gk gj = . (16) , k=1 2 k kk − 2 M 2 | | M 3 ≈ M 2 ∆C . (11) H P log2 I + ρG G Equation (15) shows the convergence of the random quanti- 2 ties gH g when M which represents the asymp- { k j } → ∞ The distance from favorable propagation represents how far totical favorable propagation of the channel, and (16) shows from favorable propagation the channel is. Of course, when the speed of the convergence. ∆C =0, we have favorable propagation. 4. FAVORABLE PROPAGATION: RAYLEIGH 4.2. Uniform Random Line-of-Sight FADING AND LINE-OF-SIGHT CHANNELS We consider a scenario with only free space non-fadingline of One of the key properties of Massive MIMO systems is that sight propagation between the base station and the terminals. the channel under some conditions can offer asymptotically We assume that the antenna array is uniform and linear with favorable propagation. The basic question is, under what antenna spacing d. Then in the far-field regime, the channel conditions is the channel favorable? A more general question vector gk can be modelled as: is what practical scenarios result in favorable propagation. In T −i2π d sin(θ ) −i2π(M−1) d sin(θ ) g = 1 e λ k e λ k , (17) practice, the channel properties depends a lot on the prop- k ··· agation environment as well as the antenna configurations. h i where θk is the arrival angle from the kth terminal measured Therefore, there are varieties of channel models such as relative to the array boresight, and λ is the carrier wavelength. Rayleigh fading, Rician, finite dimensional channels, keyhole For any fixed and distinct angles θk , it is straightfor- channels, LoS, etc. In this section, we will consider two ward to show that { } particular channel models: independent Rayleigh fading and 1 1 uniform random line-of-sight (UR-LoS). These channels rep- g 2 =1, and gH g 0,M , k = j, (18) M k M k j resent very different physical scenarios. We will study how k k → → ∞ 6 favorable these channels are and compare the singular value so we have asymptotically favorable propagation. Now assume that the K angles θk are randomly and in- spread. For simplicity, we set βk =1 for all k in this section. { } dependently chosen such that sin(θk) is uniformly distributed 4.1. Independent Rayleigh Fading in [ 1, 1]. We refer to this case as uniformly random line-of- − m sight. In this case, and if additionally d = λ/2, then Consider the channel model (2) where hk are i.i.d. CN(0, 1) RVs. By using the law of large numbers,{ we} have 1 1 1 Var gH g = . (19) M k j M − M 2 1 g 2 1,M , and (12) Comparing (14) and (19), we see that the inner products M k kk → → ∞ g g 1 between different channel vectors k and j converge to zero gH g 0,M , k = j, (13) with the same rate for both i.i.d. Rayleigh fading and in UR- M k j → → ∞ 6 LoS. Interestingly, for finite M, the convergence is slightly so we have asymptotically favorable propagation. faster in the UR-LoS case. H 2 In practice, M is large but finite. Equations (12)–(13) Now consider the quantity gk gj . For the UR-LoS sce- show the asymptotic results when M goes to infinity. nario, with k = j, we have → ∞ 6 But, they do not give an account for how close to favorable 1 gH g 2 0, (20) propagation the channel is when M is finite. To study this M 2 | k j | → fact, we consider V 1 gH g . For finite , we have ar M k j M 1 (M 1)M(2M 1) 2 Var gH g 2 = − − . (21) M 2 | k j | 3M 4 ≈ 3M 1 1 Var gH g = . (14) M k j M We next compare (16) and (21). While the convergence of the inner products between gk and gj has the same rate in 1 H both i.i.d. Rayleigh fading and UR-LoS, the convergence of We can see that, M gk gj is concentrated around 0 (for k = j 6 H 2 or 1 (for k = j) with the variance is proportional to 1/M. gk gj is considerably slower in the UR-LoS case.
1,0 1,0 M = 100 M = 100 0,8 0,8 K = 10 K = 10 0,6 0,6
0,4 0,4
0,2 0,2
10 100 1000 10 100 1000 1,0 1,0 M = 200 M = 200 0,8 0,8 K = 20 K = 20
Cumulative Distribution 0,6 Cumulative Distribution 0,6
0,4 0,4
0,2 0,2
10 100 1000 10 100 1000 Singular Values (Sorted) Singular Values (Sorted)
Fig. 1. Singular values of GH G for i.i.d. Rayleigh fading. Fig. 2. Same as Figure 1, but for UR-LoS. Here, (M = 100,K = 10) and (M = 200,K = 20).
gation with high probability in the UR-LoS case, we propose 4.3. Urns-and-Balls Model for UR-LoS to use the following simplified model. The base station array can create M orthogonal beams with the angles θm : In Section 4.2, we assumed that angles θk are fixed and dis- { } { } tinct regardless of M. With this assumption, we have asymp- 2m 1 totically favorable propagation. However, if there exist θ sin (θm)= 1+ − , m =1, 2, ..., M. (23) { k} − M and θj such that sin(θk) sin(θj ) is in the order of 1/M, then{ we} cannot have favorable− propagation. To see this, as- Suppose that each one of the K terminals is randomly and in- sume that sin(θk) sin(θj )=1/M. Then dependentlyassigned to one of the M beams given in (23). To − guarantee the channel is favorable, each beam must contain at iπ(sin(θk)−sin(θj ))M iπ 1 H 1 1 e 1 1 e most one terminal. Therefore, if there are two or more termi- gk gj = − − = − M M 1 eiπ(sin(θk) sin(θj )) M 1 eiπ/M nals in the same beam, all but one of those terminals must be 2i − − dropped from service. Let N , M K N < M, be the =0,M . (22) 0 − ≤ 0 → π 6 → ∞ number of beams which have no terminal. Then, the number of terminals that have to be dropped from service is In practice, M is finite. If the number of terminals K is in order of tens, then the probability that there exist and θk N = N (M K) . (24) θ such that sin(θ ) sin(θ ) 1/M cannot be neglected.{ } drop 0 − − { j} k − j ≤ This makes the channel unfavorable. This insight can be con- By using a standard combinatorialresult given in [6, Eq. (2.4)], firmed by the following examples. Let consider the singular we obtain the probability that terminals, , are H n 0 n 1.0 computed by using (25). We can see that the probability that i.i.d. Rayleigh, exact 0.8 i.i.d. Rayleigh, bound three terminals (for the case of M = 100, K = 10) and UR-LoS, exact four terminals (for the case of M = 200, K = 20) must 0.6 UR-LoS, bound be dropped is less than 1%. This is in line with the result in Fig. 2 where three (for the case of M = 100, K = 10) 0.4 ρ = -10 dB ρ = 0 dB or four (for the case of M = 200, K = 20) of the singular values are substantially smaller than the rest, with probability 0.2 CumulativeDistribution less than 1%. Note that, to guarantee favorable propagation, the number of terminals must be dropped is small ( 20%). 2 4 6 8 ≈ bits/channel use/terminal 6. CONCLUSION Fig. 3. Capacity per terminal for i.i.d. Rayleigh fading and Both i.i.d. Rayleigh fading and LoS with uniformly random UR-LoS channels. Here M = 100 and K = 10. angles-of-arrival offer asymptotically favorable propagation. In i.i.d. Rayleigh, the channel singular values are well spread 0 10 out between the smallest and largest value. In UR-LoS, al- M =100, K =10 most all singular values are concentrated around the maxi- 10 -1 ) M =200, K =20 mum singular value, and a small number of singular values n = = are very small. Hence, in UR-LoS, by dropping a few termi- -2 drop 10 nals, the propagation is approximately favorable. N ( P The i.i.d. Rayleigh and the UR-LoS scenarios represent -3 10 two extreme cases: rich scattering, and no scattering. In prac- tice, we are likely to have a scenario which lies in between 10 -4 0 2 4 6 8 10 of these two cases. Thus, it is reasonable to expect that in Number of Terminals Dropped from the Service (n) most practical environments, we have approximately favor- able propagation. Fig. 4. The probability that n terminals must be dropped from The observations made regarding the UR-LoS model also service, using urns-and-balls model for UR-LoS propagation. underscore the importance of performing user selection in massive MIMO. Remark 1 The result obtained in this subsection yields an important insight: for Rayleigh fading, terminal selection REFERENCES schemes will not substantially improve the performance since [1] T. L. Marzetta, “Noncooperative cellular wireless with the singular values are uniformly spread out. By contrast, in unlimited numbers of base station antennas,” IEEE UR-LoS, by dropping some selected terminals from service, Trans. Wireless Commun., vol. 9, no. 11, pp. 3590–3600, we can improve the worst-user performance significantly. Nov. 2010. 5. EXAMPLES AND DISCUSSIONS [2] J. Hoydis,S. ten Brink, andM. Debbah,“MassiveMIMO in the UL/DL of cellular networks: How many anten- Figure 3 shows the cumulative probability of the capacity per nas do we need?” IEEE J. Sel. Areas Commun., vol. 31, terminal for i.i.d. Rayleigh fading and UR-LoS channels, no. 2, pp. 160–171, Feb. 2013. when M = 100 and K = 10. The “exact” curves are obtained [3] E. G. Larsson, F. Tufvesson, O. Edfors, and T. L. by using (5), and the “bound” curves are obtained by us- Marzetta, “Massive MIMO for next generation wireless ing the upper bound (6) which is the maximum sum-capacity systems,” IEEE Commun. Mag., vol. 52, no. 2, pp. 186– achieved under favorable propagation. For both Rayleigh fad- 195, Feb. 2014. ing and UR-LoS, the sum-capacity is very close to its upper [4] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy bound with high probability. This validates our analysis: both and spectral efficiency of very large multiuser MIMO independent Rayleigh fading and UR-LoS channels offer fa- systems,” IEEE Trans. Commun., vol. 61, no. 4, pp. vorable propagation. Note that, despite the fact that the con- 1436–1449, Apr. 2013. dition number for UR-LoS is large with high probability (see Fig. 1), we only need to drop a small number of terminals (2 [5] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. terminals in this case) from service to have favorable propa- Marzetta, O. Edfors, and F. Tufvesson, “Scaling up gation. As a result, the gap between capacity and its upper MIMO: Opportunities and challenges with very large ar- bound is very small with high probability. rays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40– Figure 4 shows the probability that n terminals must be 60, Jan. 2013. dropped from service, P (Ndrop = n), for two cases: M = [6] W. Feller, An Introduction to Probability Theory and Its 100,K = 10 and M = 200,K = 20. This probability is Applications,2nd ed. New York: Wiley, 1957,vol. 1.