Massive MIMO in Sparse Channels - IEEE Xplore

2014 IEEE 15th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

Massive MIMO in Sparse Channels Nikolaos Kolomvakis∗ , Michail Matthaiou†∗ , and Mikael Coldrey‡ ∗ Department † School

of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, U.K. ‡ Ericsson Research, Ericsson AB, Gothenburg, Sweden E-mail: [email protected], [email protected], [email protected]

This can be attributed to the fact that accurate physical models, which capture the scattering environment, exhibit nonlinear dependence on the propagation parameters. To the best of our knowledge, the only relevant study is [14], which considered non-idealistic scenarios, such as propagation environments without rich enough scattering. However, such a scenerio was modeled through rich scattering P -dimensional channel vectors for different MSs, with P much smaller than the number of BS antennas. Given the high complexity of accurate physical models, [12] proposed a linear virtual MIMO channel representation for uniform linear arrays (ULAs) in a pointto-point MIMO system, that keeps the essence of physical modeling and provides a simple geometric interpretation of the scattering environment. Motivated by the above discussion, we study the performance of a single-cell uplink massive multiuser single-input multiple-output (MU-SIMO) system under a parametrized channel model that captures the sparse multipath structure of the environment. This setup can occur, for instance, in macrocell urban environments where the propagation links between the users and BS are often blocked by large buildings. More specifically, we study analytically the impact of the sparse channel on the achievable rate of a massive MU-SIMO system for both linear MRC and ZF receivers with full CSI.1 Compared to Rayleigh fading conditions, sparse channels incur a substantial achievable rate loss. Moreover, we investigate asymptotically the performance of ZF and MRC receivers, when both the number of antennas at the BS and the number of the MSs go to infinity, under the assumption that N/K → c, c ≥ 1. An interesting observation is that for MRC receivers the achievable rate per MS reaches a saturation point, while for ZF receivers it grows without bound on a logarithmic scale. Notation: We use uppercase and lowercase to denote matrices and vectors, respectively. The superscripts (·)∗ and (·)T stand for the conjugate and conjugate-transpose, respectively. The operators tr(·), det(·), E{·} and ⊙ denote the trace of a matrix, the determinant of a matrix, the expectation, and Hadamard product, correspondingly.

Abstract—Massive multi-user multiple-input multiple-output (MU-MIMO) systems are cellular networks where the base stations (BSs) are equipped with hundreds of antennas, N , and communicate with tens of mobile stations (MSs), K, such that, N ≫ K ≫ 1. Contrary to most prior works, in this paper, we consider the uplink of a single-cell massive MIMO system operating in sparse channels with limited scattering. This case is of particular importance in most propagation scenarios, where the prevalent Rayleigh fading assumption becomes idealistic. We derive analytical approximations for the achievable rates of maximum-ratio combining (MRC) and zero-forcing (ZF) receivers. Furthermore, we study the asymptotic behavior of the achievable rates for both MRC and ZF receivers, when N and K go to infinity under the condition that N/K → c ≥ 1. Our results indicate that the achievable rate of MRC receivers reaches an asymptotic saturation limit, whereas the achievable rate of ZF receivers grows logarithmically with the number of MSs.

I. I NTRODUCTION Multiuser multiple-input multiple-output with very large antenna arrays at BSs, (a.k.a. massive MIMO), has emerged as one of the most promising technologies with several attractive features [1], [2]. For instance, large-scale antenna systems bring huge improvements on the spectral-efficiency compared to 4G wireless technologies and simultaneously improve the radiated energy-efficiency [3], [4]. In particular, when the number of antennas, N , of the BS is very large, and assuming perfect channel state information (CSI), the transmit power per MS can be scaled down as 1/N with minor performance loss. Furthermore, in a massive MIMO system, the simplest linear receivers e.g, MRC and ZF, are nearly optimal [5], while thermal noise, interference and channel estimation errors vanish and the only performance limitation is pilot contamination [6], [7]. Yet, the features described above can be reaped under favorable propagation conditions, i.e., the channel vectors between different MSs should become pairwisely orthogonal as N grows [8]. In essence, this corresponds to independent and identically distributed (i.i.d.) Rayleigh fading conditions, which correspond to rich scattering environments with large angular spreads. However, measurement results [9]–[11] have shown that physical MIMO channels very often have a sparse multipath structure, due to insufficient richness in the scattering environment, even for relatively small antenna dimensions [12]. Although most physical channels have an underlying sparse structure, little is still known about the performance of massive MIMO systems in such propagation scenarios [13].

978-1-4799-4903-8/14/$31.00 ©2014 IEEE

II. S YSTEM M ODEL We consider the uplink of a MU-SIMO system, which includes a BS equipped with an uniform linear array of N 1 The perfect CSI assumption, though simplifying, is adopted herein for the clarity of exposition and ease of the analysis

21


The {θn = n/N } are fixed virtual receive angles that uniformly sample θ, resulting in a N ×N unitary discrete Fourier transform (DFT) matrix AR . The N × K matrix HV is the virtual channel representation and its coefficients [HV ]nk represent the coupling between the kth MS and the nth virtual receive angle θn , n = 0, ..., N − 1. Next, we will model sparse channels in the virtual domain following the methodology of [15].

antennas communicating with K single-antenna MSs. The N × 1 received vector at the BS is √ (1) r = ρu Yx + w where Y is the N × K channel matrix that characterizes the propagation environment, x is the K × 1 vector of symbols transmitted simultaneously by the K MSs, with the average transmit power of each MS being ρu . Finally, w ∼ CN (0, I) is the additive white Gaussian noise (AWGN). More specifically, Y models the composite propagation channel affected by small-scale fading, geometric attenuation and lognormal shadow fading. Its elements [Y]nk are given by p [Y]nk = [H]nk βk (2)

B. Sparse Channel Modeling In [15], the notion of degrees of freedom (DoF) in pointto-point MIMO systems was proposed in order to develop a model for sparse multipath channels. The number of independent DoF of the kth MS, fk , is defined as the number of entries in the kth column of HV with non-vanishing power. P The matrix HV is sparse if it contains k fk ≪ N K nonvanishing coefficients. In general, a sparse virtual channel HV can be modeled as [15]:

where [H]nk is the small-scale channel coefficient from the kth user to the nth antenna element. The term βk models geometric attenuation and shadow fading between the k-th MS and BS. The large-scale fading is modeled via βk = ζk /rkα , where ζk is a lognormal random variable with mean µ > 0 and variance σ 2 . That is, ζk ∼ ln N (µ, σ 2 ). Finally, the term rk is the distance between BS and the k-th MS, and α is the path loss exponent. We can alternatively express Y as follows Y = HD1/2

HV = M ⊙ Hiid

(7)

where Hiid contains i.i.d. entries which are distributed as P CN (0, 1). Moreover, M is a mask matrix with f unit k k entries and zeros elsewhere. The matrix M defines the structure and level of sparsity of the virtual channel representation. We next introduce a parametrized stochastic model for the mask matrix M.

(3)

where D is a K ×K diagonal matrix, whose diagonal elements are given by [D]kk = βk , while H models the physical channel which will be described in more detail in the next section. A. Virtual Channel Representation

C. Mask Matrix Modeling

The virtual channel representation characterizes the physical channel matrix H via coupling between spatial beams in fixed virtual receive directions at the BS and MSs [12], [15]. Due to its tractability, we use it for our analysis henceforth. Note that this model was originally proposed for point-to-point singleuser MIMO systems. Yet, it can be easily extended to the considered MU-SIMO setup provided that a ULA is deployed at the BS. For an N -dimensional ULA at the BS, the receiver response is associated with the N × 1 array steering vector aN (θ) [14] iT 1 h −j2πθ 1, e · · · e−j2π(N −1)θ (4) aN (θ) = √ N

The entries of the mask matrix M can be modeled as Bernoulli random variables ( 1, pk [M]nk = 0, 1 − pk . That is, with probability 1 − pk , the virtual path [HV ]nk between the kth MS and the angle-of-arrival (AoA) θn at the BS is vanished. The assumption of equal probability of a nonvanishing path for each MS is reasonable, since the distances between the MSs and the BS are much larger than the distance between the antennas. Moreover, we define fki as

where θ is the spatial frequency, which is related to the physical angle (in the plane of arrays) φ ∈ [−π/2, π/2] as θ = d sin(φ)/λ, with d the antenna spacing and λ the carrier wavelength [15]. Considering K single-antenna MSs and a N -antenna BS, the physical MIMO channel matrix H can be mapped onto the virtual channel matrix, HV , through the following transformation H = AR H V (5)

fki =

[M]nk [M]ni

(8)

n=1

as the number of common non-vanishing AoAs between the kth and ith MSs. III. ACHIEVABLE R ATES & A SYMPTOTIC A NALYSIS Considering the full CSI scenario, we now derive analytical approximations of the uplink achievable rates for both linear receivers, namely MRC and ZF receivers. Furthermore, the asymptotic behavior of the achievable rate is studied as N, K → ∞, while N/K → c ≥ 1.

where AR = [aN (θn )]n∈I(N ) and the set of indices I(N ) is defined as I(N ) , {n − (N − 1)/2 : n = 0, 1, ..., N − 1}.

N X

(6)

22


A. Maximum-Ratio Combining

B. Zero-forcing Receiver

Proposition 1: The exact achievable rate of the kth MS for MRC receivers, Rkmrc , is approximately given by

The case of ZF receivers imposes mathematical challenges. Thus, we assume, without significant loss of generality that p = pk , ∀k. Proposition 2: The exact achievable rate of the kth MS for ZF receivers, Rkzf , for p = pk , ∀k, is approximately given by:

˜ kmrc = log2 1 + g¯−1 Rkmrc ≈ R k where g¯k ,

ρu (fk − 1)

PK

i=1,i6=k

(9)

˜ zf = log2 (1 + ρu p(N − K)βk ) . Rkzf ≈ R k

fki βi + (fk − 2)(fk − 3)

ρu (fk − 1)(fk − 2)(fk − 3)βk

Proof: See Appendix II. Although, this expression is asymptotic, it is shown by simulations that is quite accurate, even for a small number of antennas. Corollary 2: When N, K → ∞, while N/K → c ≥ 1, then (15) grows logarithmically as

. (10)

Proof: See Appendix I. Note that the expression in (9) is an accurate large-antenna approximation, which depends on the transmit power, largescale fading coefficients and the sparsity parameters fk and fki . We now turn our attention to the asymptotic regime and provide the following simplified expression: Corollary 1: When N, K → ∞, while N/K → c ≥ 1, then (9) converges asymptotically to cβk pk mrc ˜ Rk → log2 1 + (11) E{βk }¯ p(k)

˜ kzf ≈ log2 (1 + ρu p(c − 1)Kβk ) . R

µ 1−α

1−α 1−α RM − Rm RM − Rm

.

IV. N UMERICAL R ESULTS The analytical results are now evaluated in a single-cell simulation scenario. We consider a circular cell with a radius of 1000m and assuming uniform MSs distribution as well as no user is closer to the BS than 100m. The lognormal random variable ζk has parameters ln N (2.8, 8dB) and the path loss exponent is α = 3.8. Finally, for the ease of exposition we consider that p = pk , ∀k. In Fig. 1, a comparison between the Monte-Carlo simulations and our analytical approximations is provided for the MRC and ZF receivers, according to the expressions in (9) and (15), respectively. In the case of p = 1, the channel experiences rich scattering and the best performance is achieved for both receivers. However, when p = 0.5 half DoF are expected to be vanished, and thus, the channel can be considered sparse. In this case, MRC receivers experience less inter-user interference but the useful received signal is also weaker and, thus, the thermal noise dominates it. In order to elaborate on the impact of sparsity, we can rewrite (13) in the following way: # ! ρ N β u k ˜ kmrc ≈ log2 1 + . (17) R PK ρu i=1,i6=k βi + p1

(12)

Proof: Note that by increasing the number of BS antennas, the DoF can be assumed to be fk ≫ 3; hence, (9) can be simplified as ! # ρ p N β u k k ˜ kmrc ≈ log2 1 + R (13) PK ρu i=1,i6=k pi βi + 1

where for a sufficiently large number of antennas fk ≈ N pk while fki ≈ N pi pk . In the limit, N converges to cK and using the Kolmogorov’s strong law, w , PK K→∞ 1 −−−→ E{w}, such that (13) converges i=1,i6=k pi βi − K−1 asymptotically to (11). By noting that the random variables n o ζi and ri are independent we have E{βi } = E {ζi } E r1α , i where E {ζi } = µ. Thus, E {βi } =

RM

µ − Rm

Z

RM Rm

1 dri . riα

(16)

Proof: Asymptotically, N can be replaced by cK in (15), the proof follows trivially We see that, in contrast to MRC receivers, the achievable rate of ZF receivers could still benefit from a very large antenna array, since they can successfully eliminate all inter-user interference, to yield a logarithmic scaling of the achievable rate.

PK where p¯(k) , i=1,i6=k pi /(K − 1). Considering a circular cell with a radius of RM meters and assuming uniform MSs distribution as well as no user is closer to the BS than Rm meters and pathloss α > 1, we have that E{βk } =

(15)

Now, we can see from (17), that highly sparse channels (low values on p) magnify the impact of the thermal noise on MRC receivers. For ZF receivers, we can directly see from (15) that more sparse channels reduce the power of the useful received signal at the BS. Moreover, note that the difference between the ZF and MRC curves increases with higher values of p, which implies that for higher sparsity the impact of inter-user interference is not so pronounced.

(14)

Due to the structure of D, E {βi } = E {βk } which concludes the proof. From the expression in (11), we see that, in the asymptotic limit, the achievable rate of MRC receivers reaches a saturation point.

23


Finally, Fig. 2 depicts the achievable rate per MS for MRC receivers versus different number of BS antennas under the condition N/K = 10. We see that both the channel with rich scattering (p = 1) and a sparse channel with p = 0.5 converge to the asymptotic limit in (11), for pk /¯ p(k) = 1.

and ZF receivers, when both the number of the antennas at the BS and the number of MSs increase without bound. Our results demonstrated that the achievable rate of MRC receivers reaches a saturation point, whereas the rate of ZF receivers increases logarithmically with the number of MSs. A PPENDIX I

Achievable Rate per MS (bits/s/Hz)

6

For MRC receivers, the exact achievable uplink rate of the kth user, Rkmrc , can be lower bounded as [5, Eq. (15)]

p=1

5

4

where

p=1 p = 0.5 3

1

MRC simulation ZF simulation Analytical approximation 200

300 400 500 Number of BS antennas

600

700

(19)

g¯k = E {gk (yk , yi )} ) ( PK H 2 1 i=1,i6=k |yk yi | + E . =E ||yk ||4 ρu ||yk ||2 Now, let tik , |ykH yi |2 /||yk ||4 . Then, we have PN ∗ yim |2 | m=1 ykm tik = 4 ||yk || N N X X y∗ yim ykl y∗ km il = 4 ||y || k m=1

4.4

4.2 Achievable Rate per MS (bits/s/Hz)

ˆ kmrc , log2 1 + (E {gk (yk , yi )})−1 R

The expectation, g¯k , of gk (yk , yi ) can be evaluated as

Fig. 1. Achievable rate per MS of MRC and ZF receivers. In this example, K = 10, ρu = 10dB and the propagation channel parameters are σ = 8dB and pathloss α = 3.8.

4

(21) (22)

(23) (24)

l=1

MRC simulation p=1 MRC approximation p=1 MRC approximation p=0.5 MRC simulation p=0.5 Asymptotic limit (11)

3.8

where ykm is the mth element of the column vector yk . Given, that AR is unitary and noticing that the kth and ith columns hV,k and hV,i of HV are independent, the expected value of (24) simplifies to fki X |hV,km |2 βi E E{tik } = . (25) ||hV,k ||4 m=1 2 ˜ (m) ||2 = Pfki We define ||h i=1,i6=m |hV,ki | . For a sufficiently V,k (m) 4 ˜ || ≈ ||hV,k ||4 and then large number of antennas ||h V,k ˜ (m) ||4 can be considered independent. Thus, |hV,km |2 and ||h V,k the expected value of the first term in (21) can be approximated as ( ) fki X |hV,km |2 βi E E {tik } ≈ (26) ˜ (m) ||4 ||h m=1 V,k ) ( fki X 1 . (27) β i βk E = ˜ (m) ||4 ||h m=1

3.6

3.4

3.2 50

(18)

where gk (yk , yi ) is a function of the channel vectors yk and yi PK ρu i=1,i6=k |ykH yi |2 + ||yk ||2 gk (yk , yi ) , . (20) ρu ||yk ||4

p = 0.1

2

0 100

ˆ mrc Rkmrc ≥ R k

p = 0.5

100

150

200 250 Number of BS antennas

300

350

400

Fig. 2. Asymptotic achievable rate per MS of MRC receivers. In this example, N/K = 10, ρu = 10dB and the propagation channel parameters are σ = 8dB and pathloss α = 3.8.

V. C ONCLUSIONS This paper analyzed the effect of sparse channels on a single-cell massive MIMO system. In particular, we have considered a stochastic model for the sparse channel where the virtual paths vanish randomly. Furthermore, we derived analytical approximations for the achievable uplink rate of MRC and ZF receivers. We show that sparsity reduces considerably the performance compared to Rayleigh fading conditions. We also observed that ZF receivers can offer a higher achievable rate compared to MRC receivers when the sparsity is low. Moreover, we studied the asymptotic performance of MRC

V,k

The expectation in (27) can be written as a scalar central complex Wishart matrix with fk − 1 DoF [16, Eq. (2.9)] ) ( 1 1 . (28) = E (m) ˜ ||4 (fk − 2)(fk − 3)βk2 ||h V,k

24


The entries of HV are i.i.d. zero-mean with variance p. Therefore, as N and K grow without bound, WV converges to the Marchenko-Pastur distribution, and similarly with Wiid , its eigenvalues converge to a deterministic constant ˜ which depends on the variance of HV . Furthermore, we λ ˜ Furthermore have that tr(WV ) = ||HV ||2F ≈ N Kp ≈ K λ. 1 −1 −1 ˜ ˜ λ ≈ N p ≈ λp, and so λ ≈ p λ . The latter leads to −1 ). Substituting it into (35) we can tr(WV−1 ) ≈ p1 tr(Wiid conclude the proof.

Therefore, the first term of the expression in (22) becomes ( PK ) K H 2 X i=1,i6=k |yk yi | E = E {tik } (29) ||yk ||4 i=1,i6=k

=

K X

i=1,i6=k

fki βi . (fk − 2)(fk − 3)βk

(30)

The expectation of the second term in (22) can be found similarly with (28) and it is given as 1 1 . (31) = E 2 ρu ||yk || ρu (fk − 1)βk

ACKNOWLEDGMENTS This work has been supported in part by the Swedish Governmental Agency for Innovation Systems (VINNOVA) within the VINN Excellence Center Chase.

Substituting (30) and (31) into (22), we can easily derive the expression in (9).

R EFERENCES A PPENDIX II

[1] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities and challenges with very large arrays,” IEEE Signal Process. Mag., vol. 30, no. 1, pp. 40–46, Jan. 2013. [2] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3590–3600, Nov. 2010. [3] H. Yang and T. L. Marzetta, “Performance of conjugate and zeroforcing beamforming in large-scale antenna systems,” IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 172–179, Feb. 2013. [4] E. G. Larsson, F. Tufvesson, O. Edfors, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” IEEE Commun. Mag., vol. 52, no. 2, pp. 186–195, Feb. 2014. [5] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy and spectral efficiency of very large multiuser MIMO systems,” IEEE Trans. Commun., vol. 61, no. 4, pp. 1436–1449, Apr. 2013. [6] J. Hoydis, S. ten Brink, and M. Debbah, “Massive MIMO in the UL/DL of cellular networks: How many antennas do we need?” IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 160–171, Feb. 2013. [7] H. Yin, D. Gesbert, M. Filippou, and Y. Liu, “A coordinated approach to channel estimation in large-scale multiple-antenna systems,” IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 264–273, Feb. 2013. [8] J. Chen, “When does asymptotic orthogonality exist for very large arays?” in Proc. IEEE GLOBECOM, Dec. 2013. [9] M. Matthaiou, A. M. Sayeed, and J. A. Nossek, “Sparse multipath MIMO channels: Performance implications based on measurement data,” in Proc. IEEE SPAWC, June 2009, pp. 364–368. [10] L. Wood and W. S. Hodgkiss, “A reduced-rank eigenbasis MIMO channel model,” in Proc. IEEE WTS, Apr. 2008, pp. 78–83. [11] S. Wyne, A. Molisch, P. Almers, G. Eriksson, J. Karedal, and F. Tufvesson, “Statistical evaluation of outdoor-to-indoor office MIMO measurements at 5.2 GHz,” in Proc. IEEE VTC, May 2005, pp. 146–150. [12] A. M. Sayeed, “Deconstructing multiantenna fading channels,” IEEE Trans. Signal Process., vol. 50, no. 10, pp. 2563–2579, Oct. 2002. [13] T. Datta and A. Chockalingam, “Increasing system capacity in uplink multiuser MIMO using sparsity exploiting receiver,” in Proc. IEEE PIMRC, Sep. 2010. [14] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “The multicell multiuser MIMO uplink with very large antenna arrays and a finite-dimensional channel,” IEEE Trans. Commun., vol. 61, no. 6, pp. 2350–2361, June 2013. [15] A. M. Sayeed and V. Raghavan, “Maximizing MIMO capacity in sparse multipath with reconfigurable antenna arrays,” IEEE J. Sel. Topics Signal Process., vol. 1, no. 1, pp. 156–166, June 2007. [16] A. M. Tulino and S. Verd´u, “Random matrix theory and wireless communications,” Foundations and Trends in Communications and Information Theory, vol. 1, no. 1, pp. 1–182, 2004. [Online]. Available: http://dx.doi.org/10.1561/0100000001 [17] Z. D. Bai, “Methodologies in spectral analysis of large dimensional random matrices, a review,” Statistica Sinica, vol. 9, pp. 611–677, 1999.

For ZF receivers the exact uplink rate for the kth user, Rkzf , can be lower bounded according to [5, Eq. (19)] ˆ kzf Rkzf ≥ R

(32)

where 

ˆ kzf , log2 1 + R

ρu E

nh

(YH Y)

−1



i o .

(33)

kk

Exploiting the fact that the matrix AR is unitary, we have nh −1 i o −1 i o 1 nh H YH Y E = HV HV E (34) βk kk kk n o 1 −1 E tr (WV ) (35) = βk K

where WV , HH V HV . From (34) to (35), we have used the 1 fact that [WV−1 ]kk = det(W [C]kk , where C is the cofactor V) matrix of WV . We see that [C]kk has the same distribution for all k = 1, ..., K; thus, the diagonal elements of HV follow the same distribution. Now, let Wiid , HH iid Hiid and c , N/K. It is well known that as K and N grow large, the eigenvalues of N1 Wiid converge to a fixed deterministic distribution known as the Marchenko-Pastur distribution. The largest and the smallest eigenvalues of N1 Wiid converge to [17] 2 2 1 1 2 2 √ √ λmax → γ 1 + , λmin → γ 1 − (36) c c where γ 2 is the variance of the entries of Hiid . Thus, the condition number of Wiid , κ(Wiid ), converges to 2 √ c+1 . (37) κ(Wiid ) → √ c−1 If c ≫ 1, then κ(Wiid ) ≈ 1 and therefore, the eigenvalues of Wiid become identical, i.e., λ1 ≈ λ2 ≈ ... ≈ λk , λ. As a result, we have that tr(Wiid ) = ||Hiid ||2F ≈ N K ≈ Kλ, where || · ||2F is the Frobenius norm. Thus, we conjecture that the eigenvalues of Wiid are approximately λ ≈ N .

25