CSI is not needed for Optimal Scaling in Multiuser Massive SIMO ...

CSI is not needed for Optimal Scaling in Multiuser Massive SIMO Systems Alexandros Manolakos

Mainak Chowdhury

Andrea J. Goldsmith

Dep. of Electrical Engineering Stanford University, Stanford, CA Email: [email protected]

Dep. of Electrical Engineering Stanford University, Stanford, CA Email: [email protected]

Dep. of Electrical Engineering Stanford University, Stanford, CA Email: [email protected]

Abstract—An uplink system with a fixed number of single antenna transmitters and a single receiver with a large number of antennas is considered. For this system we propose an energy-based noncoherent communication scheme that does not use instantaneous channel state information at either the transmitter or the receiver: only the channel and noise statistics are used. We show that, in terms of the scaling law of achievable symmetric rates for two users, our scheme’s performance is no different from that achievable with perfect CSI (channel state information) at the transmitters and the receiver. We also provide a simple constellation design using the design criterion of minimum distance and present numerical results on how these designs perform in non-asymptotic regimes with typical channel and noise statistics.

I. I NTRODUCTION Massive MIMO systems with large numbers of transmit or receive antennas have recently emerged as a novel communication paradigm. In these systems, the large number of transmit or receive antennas makes significant performance gains feasible [1], in terms of array gain or diversity gain. However, as described in [2], [3], an important challenge in the implementation of such systems is the Channel State Information acquisition (CSI), since even for moderately large multiantenna systems (such as LTEA systems), this acquisition can take up a significant part of the communication resources (≈ 15% [4]). While in time division duplexed (TDD) systems, one can use channel estimates for the reverse link to alleviate this problem, in frequency division duplexed (FDD) systems, channel state acquisition becomes even more difficult and the complexity in general scales with the number of antennas. Moreover, channel estimation requires the use of orthogonal training sequences, in the absence of which the estimates will be erroneous. One manifestation of this is the pilot contamination problem [1] which has been shown to be an important limiting factor on the performance gains of massive MIMO systems. Moreover, in fast time-varying environments, channel

estimation and subsequent coherent communication may be impossible due to the extremely short coherence time of the channel. All of these motivate moving away from channel estimation and coherent schemes. A line of work in this direction which examines the performance of general MIMO noncoherent communication schemes exploits simplifications afforded either by spatial diversity or high SNR. Developed in [5] and [6], these schemes look at space-time coding over the Grassman manifold associated with the channel matrix of a block fading channel model. Subsequent work showed that Grassman manifold signaling can be effective in multiuser systems also [7] [8]. A fundamental feature of this line of work is the use of high SNR approximations to derive analytical insights. While insights from the asymptotic regimes of high SNR are critical to characterizing the fundamental limits of such systems, the transmission and decoding schemes to achieve optimality (e.g. Grassman manifold signaling) have a high computational complexity. In this line of work, we look into ways of transmitting information without the use of CSI, assuming known channel statistics and independent and identically distributed channel. This work builds on our previous results in [9] where we considered a simple energybased single shot transmission scheme in the regime of a large number of receiver antennas. For this system we showed that, for a single user, we can achieve rates that are no different from coherent communication schemes, in a scaling law sense, in the limit of an infinite number of receive antennas. This work generalizes the suggested scheme in [9] to multiuser SIMO systems. For simplicity, we consider two users, but the same ideas may be generalized to more users. Our results hold for finite SNR in the limit of a large number n of receiver antennas. We show that with our energy-based noncoherent detection scheme, large spatial diversity is sufficient to achieve gains that are close, in a scaling law sense, to what would be achieved

using a coherent system. The rest of the paper is organized as follows. Section II describes the system model and Section III describes upper bounds on the symbol error probability. Section IV compares the asymptotic behavior of our scheme with the optimal coherent scheme. Section V describes explicitly the constellation design problem, which is then relaxed to more tractable problems in some special cases. Finally, we conclude with numerical evaluations in Section VI.

transmit does not matter, since the receiver uses energy measurements. Keeping this in mind, we define the following equivalent constellation containing the power levels Pj = {p1,j , p2,j , . . . , pLj ,j : pi,j < pi+1,j ∀i ∈ {1, . . . , Lj − 1}}. The set of all possible transmitted power levels is denoted by P = {p ∈ Rm , pj ∈ Pj , ∀ j ∈ {1, . . . , m}}. By definition we Q can verify that m the cardinality of P is given by |P| = j=1 Lj , L. In order to detect the transmitted p∗ ∈ P, the decoder computes

II. S YSTEM M ODEL

||y||2 /n ∈ R+ ,

We consider m single antenna transmitters and one receiver with n antennas. The system is represented as

i.e., it estimates the average received power across all the antennas. Based on its knowledge of the statistics of the channels and of the constellations {Pj }m j=1 , it then divides the positive real line into non-intersecting intervals or decoding regions {Ip }p∈P , corresponding 2 ˆ ∈ {p : ||y|| ∈ Ip }. Then to each p ∈ P, and returns p n the probability of error given that p∗ is transmitted is ˆ }. given by Pe (p∗ ) , P r{p∗ 6= p

y = Hx + ν n×1

(1)

n×1

with y ∈ C , ν ∈ C , H = [h1 , h2 , · · · , hm ], 2 hj ∈ C n×1 and each νi ∼ CN q(0, σ ), Hi,j ∼ f (h), such Kj that E[Re(Hi,j )] = µj , Kj +1 , E[Im(Hi,j )] = 0, Var[Hi,j ] = σj2 = Kj1+1 , for some known Kj > 0 and f (h) is the probability density function of any element of H. Assume that the moment generating function of |yi |2 , 2 i.e., M (θ) , E[eθ|yi | ], exists and is twice differentiable in an interval around θ = 0. Many fading distributions fall within this model, e.g., Rayleigh and Rician fading [10], in which case Hi,j ∼ CN (µj , σj2 ). Note that the assumption E[Im(Hi,j )] = 0 is for notational simplicity; the same analysis holds for complex E[Hi,j ]. We assume that the instantaneous channel realization is unknown to both the transmitters and the receiver and investigate energy-based detection schemes for recovering transmitter data, based only on the knowledge of the statistics 2 of the system, i.e., the parameters {Kj }m j=1 , and σ . We also assume that every transmission will be associated with an independent channel realization. To be more specific, if Tc is the coherence time of the channel, we assume that Ts = Tc , where Ts is the symbol time. If the coherence time Tc > Ts , we could employ time coding schemes in non-asymptotic regimes similar to the ideas described in energy-time coding for the single-user systems in [9]. For simplicity of presentation we consider Ts = Tc in the analysis that follows. We focus on the following encoding and decoding procedure. The j th transmitter transmits symbols from √ √ √ the constellation Cj = { p1,j , p2,j , · · · , pLj ,j }, subject constraint PLj to an individual average power 1 + p ≤ 1, ∀j. Here p (∈ R ) is the energy i,j i,j i=1 Lj level of the ith constellation point of the j th user and Lj is the constellation size of the j th transmitter. The above inequalities imply that pi,j ≤ Lj . Note that the phase of the particular symbols that the transmitter chooses to

(2)

III. E RROR P ROBABILITY U PPER B OUND We first give an intuitive description of our system through an equivalent expression for the statistic (2) computed by the receiver. We have kyk2 kHx + νk2 kHxk2 kνk2 Re((Hx)∗ ν) = = + +2 . n n n n n For a transmitted p = [p1 , · · · , pm ]T ∈ P, we observe that, in the limit of large n (due to the law of large numbers and the central limit theorem), the following hold: Pm ||y||2 n→∞ = r(p) + σ 2 a.s., where r(p) , j=1 pj + n 2 2 P √ ) n→∞ √ ∼ 2 j1 0. With the above observations, we have the following model: y˜ = r(p) + 2 ˜ √ν for the received statistic y˜ = kyk for a large n n enough n, where the distribution of ν˜ approaches a zero mean Gaussian distribution (with variance being a function of p) in the limit. We note that the “noise”, i.e., the factor causing deviations from the value r(p) + σ 2 , is due to the deviations of the empirical means like kHxk2 /n, kνk2 /n, from their respective ensemble means. To characterize this “noise”, we start with a lemma [11]: Lemma 1. Let zero mean, i.i.d. random variables (r.v.) u1 , . . . , un ∼ f (u), and denote as U P a r.v. such that n i=1 ui U ∼ f (u). Then, for any d > 0, P ≥d ≤ n
−nI(d) θU e , where I(d) = supθ>0 θd − log(E[e ]) . We now show how to get bounds on the error probability based on the above lemma. Denote yi = 2

√ hi,j pj + vi and define ui , |yi |2 − σ 2 − r(p). Thus, from Lemma 1, we get that

Pm

follows along similar lines and is omitted due to space constraints. The main idea in the proof is to demonstrate that as the number of antennas n increases, the optimal minimum distance as a function of the constellation size is polynomial in the size of the constellation. We use the following constellations P1 = {0, 2/M, 4/M, . . . , 2(M − 1)/M } , P2 =  2/M 2 , 4/M 2 , . . . , 2M/M 2 . We note that Pboth constellations satisfy the power constraints |P1i | p∈Pi p ≤ 1, ∀i ∈ {1, 2}. Moreover, the minimum distance is

j=1

 P

||y||2 − r(p) − σ 2 ≥ d n



≤ e−nI1,p (d) ,

(3)


where I1,p (d) , supθ>0 θd − log(E[eθU ]) . Similarly, by considering −U , it follows that  P

||y||2 − r(p) − σ 2 ≤ −d n



≤ e−nI2,p (d) ,

(4)


where I2,p (d) , supθ>0 θd − log(E[e−θU ]) . Defining the rate function Ip (d) , min(I 1,p  (d), I2,p (d)) we get ||y||2 2 that P n − r(p) − σ ≥ d ≤ 2e−nIp (d) . Then, an upper bound on the symbol error probability, assuming equiprobable signaling and a decoder that uses the following decoding intervals Ip = (r(p) + σ 2 − d2,p , r(p) + σ 2 + d1,p ], is:  1 X  −nI1,p (d1,p ) UL , e + e−nI2,p (d2,p ) . (5) L

dmin =

We can show that the rate function Ip (d) is monotonically increasing in d for d > 0 and in |d|, for d < 0. We further state a result about the small d asymptotics of Ip (d) (proof in [12]), used in Section IV: Lemma 2. It holds that

=

|q1 + q2 − q3 − q4 | =

2 . M2

By choosing M = nK , and noting that maxp (E[U 2 ]) < c in Lemma 2, for some c independent of n, we get that the symbol error rate in the system is upper bounded −K 2 1−2K )/c by c1 e−n(n ) /c = c1 e−(n (Lemma 2) for a large enough n and some c1 > 0. Choosing K such that 1 − 2K > 0, we get that the symbol error probability vanishes as n → ∞, and the rate per user is log2 M = K log2 n.

p∈P

I (d) limd→0 pd2

min

q1 ,q2 ,q3 ,q4 :q1 ,q3 ∈P1 , q2 ,q4 ∈P2 ,(q1 ,q2 )6=(q3 ,q4 )

V. C ONSTELLATION DESIGN PROBLEM A. Probability of Symbol Error Minimization Let’s assume that the transmitters use all possible combinations of power levels in P with equal probability. The constellation design problem for minimum average symbol error is 1 X Pe , minimize Pe (p). L {p, Ip }p∈P

1 2E[U 2 ] .

In the rest of the paper, for clarity of presentation, we assume we have 2 users. Most of the ideas generalize to more users [12] and are omitted due to space constraints. Details of this more general multiuser case can be found in [12].

1 Lj

IV. A SYMPTOTIC T HROUGHPUT C HARACTERIZATION

PLj

i=1

pi,j ≤1, 0≤pi,j , ∀j

p∈P

This optimization problem is in general NP hard, but we can relax the problem by focusing on maximizing the minimum distance between the constellation points. Note that in a real system this optimization can be done offline. Whenever a user becomes inactive or a new user wants to transmit, the receiver would need to send everyone their new (optimized) constellations.

first define the symmetric ergodic capacity C , PWe 2 E[log(1 + σi (H)2 /σ 2 )] of the coherent channel i=1 where σi (H) is the ith largest singular value of H and σi+1 ≥ σi ≥ 0. In the limit of large n, it can σ2 be shown that ni → c > 0 almost surely for some . c > 0. This gives us that C = log n for coherent systems. The following theorem shows that this same scaling behaviour can be achieved by our noncoherent scheme.

B. Constellation Design for 2 users We now indicate how the insights about the rate function can be used to design transmission and decoding schemes for 2 users. To simplify the constellation design problem we look at the following quand2 tity minp∈P Ip (d) ≈ 2α for small d, where α , 2 maxp (E[U ]); i.e., the worst error exponent over our constellation points. It may be shown that Ip (d) is always positive if p1 , p2 < ∞. Thus, in order to guarantee vanishing Pe with increasing n, we treat the constellation design as a code design problem and we maximize the

Theorem 1. There is a 2-user constellation with bounded average power per user which achieves a rate of K log2 n bits per transmission per transmitting user for some K > 0 with vanishing probability of error with increasing n. Proof. We prove here the statement for the case when µ1 µ2 = 0 (i.e., for Rayleigh fading in one of the channels). The general proof for µ1 µ2 6= 0 3

minimum distance dmin between received constellation points {r(p), ∀p ∈ P}. As described in [9], this criterion is optimal with increasing σ 2 . The distance between r(q1 ) and r(q2 ) where q1 , q2 ∈ P is

Therefore, in these special cases a way to approach the problem is to enumerate all possible total orderings that agree with the initial partial ordering (referred to as linear extensions [13]) and keep the solution that gives the largest objective function. However, since the problem of generating the set of linear extensions is a hard problem, we were able to identify the optimal total ordering only for small constellation sizes. As an example, consider the case of µ1 = µ2 = 0, i.e., no LOS in any of the channels (e.g. Rayleigh fading). Then, for the case of L1 = 3, L2 = 4, two total orderings that lead to the best objective function are depicted in Fig. 1. Optimal orderings have been identified through exhaustive search for all the cases for L1 ≤ 4, L2 ≤ 4. Fig. 2 shows the minimum distance achieved for the total orderings depicted in Fig. 1. We also plot an upper bound on the minimum distance, d¯min = L1 L42 −1 , obtained by considering a single superuser with L1 L2 constellation points and the same total power constraint derived in our companion paper [9].

d(q1 , q2 ) , |q1,2 + q2,2 − q1,1 − q2,1 √ √ + 2( q1,2 q2,2 − q1,1 q2,1 )µ1 µ2 |. The minimum distance is defined as dmin , minp1 ,p2 ∈P d(p1 , p2 ). Maximizing the minimum distance involves putting the power levels in P as far apart as possible. Using this heuristic, together with the total power constraint, we get the following non convex problem max

t

{pi,j }1≤i≤Lj ,1≤j≤2

Lj 1 X pi,j ≤ 1, 0 ≤ pi,j , ∀j, Lj i=1

s.t.

d(q1 ,q2 ) ≥ t, ∀q1 , q2 ∈ P. One source of hardness of this problem is the non-convex and (in general) non differentiable constraints. A simplified version of this problem is obtained by reparametrizing and solving over a restricted set. Specifically, to √ simplify the problem, let’s first define ri,j = pi,j and 2 2 sk,l = rk,1 +rl,2 +2rk,1 rl,2 µ1 µ2 . The minimum distance between two codewords in P, pk = {rk21 ,1 , rk22 ,2 }, and pl = {rl21 ,1 , rl22 ,2 } is then given by d(pk , pl ) = |sk1 ,k2 − sl1 ,l2 |. Observe that if l1 > l2 , k1 > k2 and ∀l, k, sk,l1 < sk,l2 , and sk1 ,l < sk2 ,l since pi1 ,j < pi2 ,j for i1 < i2 , ∀j. This partial ordering is depicted with red arrows in Fig. 1 for L1 = 3, L2 = 4. By expanding the partial order to a total ordering we formulate the following (possibly non convex) quadratic problem, where the ordering information is used in the first constraint. max

{ri,j }1≤i≤Lj ,1≤j≤2 {si,j }1≤i≤L1 ,1≤j≤L2

s.t.

s1,1$

s1,2$

s1,3$

s1,4$

s1,1$

s1,2$

s1,3$ s1,4$

s2,1$

s2,2$

s2,3$ s2,4$

s2,1$

s2,2$

s2,3$

s2,4$

s3,1$

s3,2$

s3,3$ s3,4$

s3,1$

s3,2$

s3,3$

s3,4$

Fig. 1. An arrow from si,j to sk,l implies si,j ≤ sk,l . The two total orderings in green lead to the same optimal constellation design for L1 = 3, L2 = 4, µ1 = µ2 = 0. Initial partial ordering is in red.

log10(d min)

0

L1 =4

−1

Achievable d¯min

−2

L1 =16 0

t

10

20 L2

30

Fig. 2. The minimum distance achieved by the two orderings shown in Fig. 1 for different constellation sizes.

sk1 ,l1 + t < sk2 ,l2 if sk1 ,l1 < sk2 ,l2 , 2 2 sk,l = rk,1 + rl,2 + 2rk,1 rl,2 µ1 µ2 ,

D. General Case

Lj 1 X 2 r ≤ 1, and ri,j ≥ 0 ∀i, j. Lj i=1 i,j

In the general case, we solve the non convex quadratic program in Subsection V-B to find a local minimum which we then use in the numerical results in Section VI. In our plots we used the following total ordering: sk1 ,l1 < sk2 ,l2 if k1 < k2 or l1 < l2 .

C. Special Cases: No LOS, Only LOS Imposing a total ordering significantly simplifies the problem in the case of µ1 = 0 or µ2 = 0 (i.e., zero LOS in any one channel), or µ1 = µ2 = 1 (i.e., only LOS in both channels), where they are equivalent to a linear program and a convex quadratic program, respectively.

VI. N UMERICAL RESULTS Fig. 3 presents the simulation estimates of the probability of symbol error (Pe ) for 2 users with Rayleigh 4

significant gains possible, even without the large beamforming gain associated with knowing the small scale fading perfectly. Future work will focus on investigating these gains further especially in channels with small coherence times. Our preliminary results suggest that using the phase of the channel output instead of just the energy can also yield significant gains over coherent schemes (with associated training overhead) in block fading models with small coherence times.

fading for both channels and L1 = L2 . We use the constellation described in Subsection V-C and minimum distance decoding. Fig. 4 presents the performance results 100 10-1

Pe

10-2 10-3 10-4 10-5 1 10

ACKNOWLEDGEMENTS

L =2,K1 = K2 =0 L =4,K1 = K2 =0 L =8,K1 = K2 =0 L =16,K1 = K2 =0 UL for L =2,K1 = K2 =0 UL for L =4,K1 = K2 =0 UL for L =8,K1 = K2 =0 UL for L =16,K1 = K2 =0

102 103 Number of receiver antennas n

This work was supported by a 3Com Corporation and an Alcatel-Lucent Stanford Graduate Fellowship, an A.G. Leventis Foundation Scholarship, the NSF Center for Science of Information (CSoI): NSF-CCF-0939370, and by CableLabs. The second author would like to thank Yair Yona and Nima Soltani for helpful discussions.

104

Fig. 3. Simulated Probability of Symbol Error for 2 users with Rayleigh fading. The dotted lines represent UL computed from (5).

for K1 = K2 = 50 (Rician fading) and constellations as described in Subsection V-D. We observe that increasing Kj has significant benefits over the case when there is no LOS component.

R EFERENCES [1] T. L. Marzetta, “Noncooperative Cellular Wireless with Unlimited Numbers of Base Station Antennas,” IEEE Transactions on Wireless Communications, vol. 9, no. 11, pp. 3590–3600, 2010. [2] ——, “How much Training is Required for Multiuser MIMO?” in IEEE Fortieth Asilomar Conference on Signals, Systems and Computers, 2006. ACSSC’06, 2006, pp. 359–363. [3] J. Hoydis, S. Ten Brink, and M. Debbah, “Massive MIMO: How many Antennas do we need?” in IEEE 49th Annual Allerton Conference on Communication, Control, and Computing, 2011, pp. 545–550. [4] 3GPP, “6.10.1.2 Mapping to Resource Elements,” ETSI, Tech. Rep., Dec. 2010. [Online]. Available: http://www.3gpp.org/ftp/ Specs/html-info/36211.htm [5] L. Zheng and D. N. C. Tse, “Communication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel,” IEEE Transactions on Information Theory, vol. 48, no. 2, pp. 359–383, 2002. [6] B. M. Hochwald and T. L. Marzetta, “Unitary Space-Time Modulation for Multiple-Antenna Communications in Rayleigh Flat Fading,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 543–564, 2000. [7] S. Shamai and T. L. Marzetta, “Multiuser Capacity in Block Fading with no Channel State Information,” IEEE Transactions on Information Theory, vol. 48, no. 4, pp. 938–942, 2002. [8] S. Murugesan, E. Uysal-Biyikoglu, and P. Schniter, “Optimization of Training and Scheduling in the Non-Coherent SIMO Multiple Access Channel,” IEEE Journal on Selected Areas in Communications, vol. 25, no. 7, pp. 1446–1456, 2007. [9] M. Chowdhury, A. Manolakos, and A. J. Goldsmith, “Design and Performance of Noncoherent Massive SIMO Systems,” in IEEE 48th Annual Conference on Information Sciences and Systems (CISS), 2014. [10] A. Goldsmith, Wireless Communications. Cambridge University Press, 2005. [11] A. Dembo et al., Large Deviations Techniques and Applications. Springer, 2010, vol. 38. [12] M. Chowdhury, A. Manolakos, and A. J. Goldsmith, “Design and Performance of Multiuser Noncoherent Massive SIMO Systems,” Under preparation, 2014. [13] G. Pruesse and F. Ruskey, “Generating Linear Extensions Fast,” SIAM Journal on Computing, vol. 23, no. 2, pp. 373–386, 1994.

100 10-1

Pe

10-2 10-3 10-4 10-5 1 10

L =2,K1 = K2 =50 L =4,K1 = K2 =50 L =8,K1 = K2 =50 L =16,K1 = K2 =50 UL for L =2,K1 = K2 =50 UL for L =4,K1 = K2 =50 UL for L =8,K1 = K2 =50 UL for L =16,K1 = K2 =50

102 103 Number of receiver antennas n

104

Fig. 4. Simulated Probability of Symbol Error for 2 users with Rician fading with K1 = K2 = 50. The dotted lines represent UL computed from (5).

VII. C ONCLUSIONS AND F UTURE W ORK In this work we have presented a novel noncoherent multiuser SIMO system with a large number of receive antennas and characterized its performance analytically and via numerical simulations. The proposed system uses a very simple encoding and decoding procedure. Specifically, encoding entails transmitting from a specific constellation and decoding is done at a receiver which knows just the statistics of the channel and performs only energy measurements. Through explicit constellation designs, we show that the achievable throughput of the proposed system, in the limit of a large number of receiver antennas, scales as if the receiver had perfect CSI. The results discussed here suggest that the diversity already present in systems with many antennas can make 5