IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 4, NO. 6, DECEMBER 2015
653
Size and Array Shape for Massive MIMO Khawla A. Alnajjar, Peter J. Smith, Philip Whiting, and Graeme K. Woodward
Abstract—With massive multi-input multi-output, it may be the case that large numbers of antennas are closely packed to fit in some available space. Here, channel correlations become important and it is of interest to investigate the space requirements of different array shapes. We focus on uniform square and linear arrays and consider a range of correlation models. We show that the benefits of two-dimensional arrays are dependent on the type of correlation. When the correlation decays slowly over small antenna separations then square arrays can be far more compact than linear arrays or they can offer substantial sum rate enhancements. When the correlation decays more quickly, then the main benefit is compactness. Index Terms—Correlation, massive MIMO.
I. I NTRODUCTION
M
ASSIVE multi-input multi-output (MIMO) is one of the key technologies proposed for 5th generation wireless systems (5G) [1], [2]. In this paper, we look at the correlations present in massive uniform square arrays (USAs) and linear arrays (ULAs) and evaluate the impact of correlation on sum rate. The advantages and disadvantages of different array configurations have been discussed in [1]–[6]. One-dimensional arrays, such as the ULA, have been shown to have some disadvantages such as an inability to resolve signals in azimuth and elevation [5], [6]. In contrast, two and three-dimensional arrays, such as the USA, offer extra degrees of freedom due to the variation in elevation angle [3] and a compact layout [4]. This paper focuses on the particular issue of how compact a two-dimensional array can be made relative to a ULA. We consider an uplink massive MIMO deployment with a co-located base station (BS) array and distributed single antenna users. The performance of these systems is evaluated
Manuscript received May 3, 2015; accepted August 26, 2015. Date of publication September 9, 2015; date of current version December 15, 2015. The associate editor coordinating the review of this paper and approving it for publication was V. Raghavan. K. A. Alnajjar was with the Department Electrical & Computer Engineering, Wireless Research Center, University Canterbury, Christchurch, 8140, New Zealand. She is now with the Department of Electrical & Computer Engineering, University of Sharjah, Sharjah, United Arab Emirates (e-mail:
[email protected]). P. J. Smith was with the Department Electrical & Computer Engineering, Wireless Research Center, University Canterbury, Christchurch, 8140, New Zealand. He is now with the School of Mathematics and Statistics, Cotton Building, Gate 7 Kelburn Pde, Kelburn Campus, Victoria University of Wellington, New Zealand (e-mail:
[email protected]). P. Whiting is with the Department Electrical & Electronic Engineering, Macquarie University, North Ryde NSW 2109, Australia (e-mail: philip.
[email protected]). G. K. Woodward is with the Department Electrical & Computer Engineering, Wireless Research Center, University Canterbury, Christchurch, 8140, New Zealand (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LWC.2015.2477513
by ergodic sum rate and sum rate outage. Square arrays offer two types of benefits relative to linear arrays. One benefit is compactness since a ULA of length bULA has approximately the same antenna spacing as a√USA with√the same number of antennas and with dimension bULA × bULA. If size is not an issue, then a bULA × bULA USA will deliver increased sum rate compared to a ULA with the same number of antennas and length bULA. In general, the compactness of the array can be traded off for sum rate improvements. The main contributions of this paper are: • We evaluate these benefits for a range of correlation models and show that USAs offer considerable space savings,1 usually requiring only 30%−50% of the ULA width. When the correlations decay slowly at small distances, which is the case for traditional channel models, then a USA also offers a considerable increase in sum rate compared to a ULA of the same width. • We prove the validity of a family of correlation models previously used in the literature [7]. II. U PLINK S YSTEM M ODEL Consider Nt single antenna user equipments (UEs) and a BS with Nr receive antennas. The transmit vector, x, is x = [x1 , x2 , . . . xNt ]T , [.]T is the transpose operator, and the channel matrix of size Nr × Nt is H = [h1 · · · hNt ]. The noise vector, n, of size Nr × 1 has independent and identically distributed (i.i.d.) complex Gaussian entries, i.e., ni ∼ CN (0, σ 2 ). The received signal, y, is given by y = Hx + n =
Nt
hj xj + n.
(1)
j=1
The channel coefficient, hij , from UE j to receive antenna i has link gain E[|hij|2 ] = Pj . The transmit signal power is E[|xj |2 ] = Es = 1 and the noise power is E[|ni |2 ] = σ 2 . The signal-tonoise ratio (SNR) is ρ = E[|xj |2 ]/E[|ni|2 ] = 1/σ 2 . The one-sided Kronecker correlated Rayleigh channel matrix is defined as 1
1
H = Rr2 UP 2 ,
(2)
where the entries of U are i.i.d. CN (0, 1), Rr is the correlation matrix at the receiver side, assumed constant for all UEs for simplicity, P = diag(P1 , P2 , . . . , PNt ) and Pj is the link gain for user j modeled by Pj = Aβ j−1 , j = {1, 2, . . . , Nt } [8], where A is the link gain of the strongest UE,2 β controls the rate of 1 Space saving is defined as the ratio of the width of a USA to the length of a ULA where the arrays have the same sum rate. 2 Without loss of generality, we index the UEs so that they are ordered from strongest to weakest.
2162-2337 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
654
IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 4, NO. 6, DECEMBER 2015
decay of the link gains and 0 < β ≤ 1. The benefit of this link gain model is that it controls the decay rate deterministically so that the effects of unequal link gains are not confounded by statistical variation. As β → 1 the link gains become equal and as β → 0 the link gains are dominated by a strong UE.
Gaussian variable), the square root model (when c = 12 , the Sqrt correlation is the CF of a Levy variable) and the exponential model (when c = 1, the Exp correlation is the CF of a Cauchy variable). To understand why the model is invalid for c > 2, consider three closely spaced antennas in a ULA with antenna separation > 0. The correlation matrix, using e−z ≈ 1 − z for small z > 0, is given by ⎡ ⎤ 1 − a2c c 1 1 − a c 1 1 − a c ⎦ . Rr ≈ ⎣ 1 − a c (3) c c c 1 − a2 1 − a 1
III. C ORRELATION M ODELS Let Rr = (Rij ), then Rij could be defined by a wide variety of different models. We consider four common models from the literature [7]. These models are chosen due to their simplicity and their ability to model a wide range of correlation behavior with a single real parameter μ, 0 < μ < 1, as in [7]. • The Jakes model (Jakes): Rij = J0 (2πμdij), where J0 (.) is the zeroth order Bessel function and dij is the physical distance between two antennas measured in terms of the wavelength [7]. 2 • A Gaussian decay model (Gauss): Rij = μdij [7]. √ • A square root model (Sqrt): Rij = μ dij [7]. • The exponential correlation model (Exp): Rij = μdij [9]. The Jakes model is certainly a valid correlation model since it was constructed from a physical model. However, to the best of our knowledge, the other three models have not been validated. Hence, we prove their validity below. Definition 1: A correlation model for the channel vector, hi , is valid if Rr = Rij satisfies Rii = 1 for all i, |Rij | ≤ 1 for all i = j and if Rr is positive semidefinite [10]. A more convenient approach to checking the validity of the Gauss, Sqrt and Exp models is given in the Theorem below. ¯c Theorem 1: The correlation model, Rij = e−ad , for two antennas i and j separated by distance d¯ is valid for all a > 0 and 0 ≤ c ≤ 2. Proof: The spectral theorem for autocorrelation functions (ACF) [11, page 82] states that a given function is the ACF of a complex weakly stationary mean square continˇ is a positive integer, if and uous process in RDˇ , where D only if it is the characteristic function (CF) of a certain random variable. Let h(ˇx, yˇ ) refer to a channel coefficient from a given source to an antenna located at position (ˇx, yˇ ). The Rayleigh channel assumptions imply that E[h(ˇx, yˇ )] = 0 and E[|h(ˇx, yˇ )|2 ] = 1 for all (ˇx, yˇ ). Furthermore, the ACF is defined as E[h(ˇx1 , yˇ 1 )h∗ (ˇx2 , yˇ 2 )] = exp(−ad¯ c ), where d¯ = (ˇx1 − xˇ 2 )2 + (ˇy1 − yˇ 2 )2 is the distance between (ˇx1 , yˇ 1 ) and (ˇx2 , yˇ 2 ), and (.)∗ is the conjugate operator. Hence, the process is a weakly stationary two dimensional process. Since the ACF, exp(−ad¯ c ), is continuous at d¯ = 0 it follows from [11, page 81] that h(ˇx, yˇ ) is mean square continuous. Hence, exp(−ad¯ c ) is valid if and only if it is the CF of a certain random variable. A stable random variable with parameters α (0 < α ≤ 2), σˇ (σˇ ≥ 0), βˇ (−1 ≤ βˇ ≤ 1), μˇ (−∞ < μˇ < ∞) is deˇ μ) ˇ [12, page 5] and the particular stanoted Sα (σˇ , β, ble variable Sα (σˇ , 0, 0) has the CF ϕ(t) = exp(−σˇ α |t|α ) [12, page 9]. Hence, the ACF is the CF of a Sc (a1/c , 0, 0) variable and the ACF is valid for any c ∈ (0, 2]. The theorem demonstrates the validity of the Gaussian model (when c = 2, the Gauss correlation is the CF of a zero-mean
Taking the determinant of both sides of (3) and keeping dominant terms gives |Rr | ≈ a2 2c 2c (4 − 2c ).
(4)
From (4) we see that c > 2 causes |Rr | to be negative for small spacings and violates the positive semi-definite property of Rr . The result also identifies the allowable order of “roll off” of the correlation function at the origin, with the Gaussian model giving the slowest valid decay at small spacings. Specifically, the correlation at a small spacing, , can take the form 1 − a c + o( c ) only for c ≤ 2 and a > 0. Hence, a quadratic roll-off is possible but not a cubic. In fact, quadratic roll-off follows naturally from standard channel modeling approaches. Let R(d) represent the spatial correlation between two receive antennas separated by a distance, d. A reasonably general representation for R(d) in a 3D channel model is given by [13] as g(θ, φ) exp (jdf (θ, φ))p(θ, φ)dθ dφ (5) R(d) = φ
θ
≈ 1 + jdI1 − d2 /2I2 , where I1 and I2 are defined by g(θ, φ)f r (θ, φ)p(θ, φ)dθ dφ, Ir = φ
θ
and we have used a small separation expansion for the exponential in (5). Here, θ and φ are angles of arrival in elevation and azimuth, g(θ, φ) is the antenna pattern, d is separation, df (θ, φ) gives the phase shift between the two antennas and p(θ, φ) is the joint probability density function of the elevation and azimuth angles. The main assumptions behind this representation are independent scatterers located at a distance which is large compared to the spacing. Note that this becomes increasingly reasonable for the small spacings considered here. This model gives the magnitude of R(d) as
1 I2 − I12 d2 . (6) |R(d)| ≈ 1 − 2 Since I2 > I12 , we see that any channel model of this type will give a quadratic roll-off. This observation makes the use of the exponential and square root models questionable at small spacings and raises the general question of what type of channel model can produce a roll off quicker than quadratic? IV. S UM R ATE M ETRICS The ergodic sum rate of the Nt UEs is given by E [R] ≤ E log2 det(I + ρ HH H) ,
(7)
ALNAJJAR et al.: SIZE AND ARRAY SHAPE FOR MASSIVE MIMO
Fig. 1. Sum rate metrics vs Rtarget for the Jakes model. The metrics are ergodic sum rate (given by lines) and Rout with p = 10% (given by lines with points). β = 0.9.
655
Fig. 2. Ergodic sum rate vs Rtarget for Jakes (lines only) and Gauss (lines with points) models. β = 0.9.
where det(.) is the determinant of a matrix and [.]H is the Hermitian operator. It is well-known that E [R] tends to grow linearly in the uncorrelated case with min(Nr , Nt ) whereas the growth is logarithmic in the perfectly correlated case. This difference is more noticeable for massive MIMO and so the effects of correlation can be extremely large. The p-percentage outage sum rate, Rout , is the rate that can be supported by (100-p)% of the channel realizations, i.e., P(R < Rout) = p. Since no exact analytical results are known for Rout, simulations are used for the sum rate metrics. V. R ESULTS We assume the system size is Nr = 100, Nt = 10 and the link gains are defined by A = 1, β ∈ {0.1, 0.9}. The scale factor A, is unity and β = 0.1, β = 0.9 correspond to a dominant user and several similar strength users, respectively. Due to space limitations a single SNR value of ρ = 0 dB is used. The length of the ULA, (bULA), is given by bULA = 1 without loss of generality. The USA has width bUSA expressed as a fraction of bULA = 1, e.g., bUSA = 0.3 bULA. Varying levels of correlation are introduced by defining Rtarget as the correlation between the first and last ULA antennas, i.e., at separation 1 and the value of Rtarget gives the value of μ required to define the correlation models in Section II. In Fig. 1, we show ergodic sum rate and the 10% sum rate outage vs Rtarget for the Jakes model with β = 0.9. Results are shown for a ULA with bULA = 1 and a USA with 30% (bUSA = 0.3 bULA) and 50% (bUSA = 0.5 bULA) of the width of the ULA. We observe that the ULA performs similarly to a USA of 30% the width whereas 50% width gives a noticeable sum rate improvement. Due to the stability of massive MIMO, i.e., channel hardening, the sum rate distribution is not highly variable and the Rout results are simply shifted versions of the ergodic results. Since no further insights are found from Rout, we restrict the following results to ergodic sum rate, and include the outage results in Fig. 1 for completeness. Figs. 2 and 3 show ergodic sum rate vs Rtarget for the Jakes and Gauss models. Clearly, the ergodic sum rate increases with β since the link gains grow with β. The results also show that
Fig. 3. Ergodic {sum rate} vs Rtarget for Jakes (lines only) and Gauss (lines with points) models. β = 0.1.
ULA performance is usually bracketed by bUSA = 0.3 bULA and bUSA = 0.5 bULA and that a larger USA is needed for β = 0.1 compared to β = 0.9. Figs. 4 and 5 repeat Figs. 2 and 3 for the Sqrt and Exp models. A comparison of Figs. 2 and 3 with Figs. 4 and 5 shows that the Jakes and Gauss models are more sensitive to correlation as shown by the substantial variation in sum rate between the three USAs of varying widths. This difference is caused by the slow, quadratic roll-off of the correlation at small distances provided by Jakes and Gauss. Also shown in Figs. 4 and 5 is the convergence of ergodic sum rate to the Independent case, shown explicitly in Fig. 5, which gives the plateau as Rtarget becomes small. The Independent and Full Corr. results in Fig. 5 show the potential variation in ergodic sum rate. The Independent limit is approached faster by the Sqrt and Exp models as they decay more quickly. The Jakes model alone does not converge to the Independent limit since J0 (2πμdij) is not monotonic in dij. Hence, equating the Bessel function to Rtarget at dij = 1 does not provide small correlations as Rtarget → 0 since the first lobe of the Bessel function does not diminish. Figs. 1–5 show that ULA performance is achieved for bUSA ∈ (0.3 bULA, 0.5 bULA) for virtually all cases. Factors which require a larger equivalent USA are higher correlation and slowly decaying correlation.
656
IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 4, NO. 6, DECEMBER 2015
Fig. 4. Ergodic sum rate vs Rtarget for Sqrt (lines only) and Exp (lines with points) models. β = 0.9.
Fig. 6. Ergodic sum rate vs Rtarget for different array types with β = 0.9. Lines represents bULA = 1, lines and circles represents bUSA = 0.5 bULA , lines and squares represent bUSA = 0.3 bULA .
Powers greater than two are invalid and so a quadratic roll-off is the slowest possible. We have also shown that this roll off is a natural result of standard channel modeling assumptions. Finally, we have demonstrated that the sum rate benefits of USAs are highly dependent on the correlation models. When the spatial correlation decays very slowly at the origin, then USAs offer substantial benefits in sum rate. However, for more rapidly decaying correlations, the space savings offered by USAs are more important.
R EFERENCES Fig. 5. Ergodic sum rate vs R for Sqrt (lines only) and Exp (lines with points) models. β = 0.1.
Figs. 1–5 also show that the benefits of USAs are dependent on the correlation models. For models with a quadratic rolloff, the high correlations at small spacing impact strongly on ULAs and a substantial sum rate improvement is possible using USAs. For example, in Fig. 2, 50% improvements are possible (comparing bULA = 1 with bUSA = 1bULA). In contrast, for Sqrt and Exp the sum rate improvements are much smaller. This contrasts to the space saving benefits which are between 30% and 50% for all correlation types. For different system sizes, the effects are more prominent as Nt increases for a fixed Nr . This is shown in Fig. 6, where we use the Exp model with β = 0.9 and we vary Nr ∈ {100, 200} and Nt ∈ {10, 20, 40}. The space saving benefits for the cases Nt ∈ {20, 40} are greater than for Nt = 10 and here the USA gives space savings less than 30%. VI. C ONCLUSION In this paper, we have analytically justified a range of correlation models that have previously been used in the literature without verification. This analysis has also shown that correlation models must decay away from unity at zero separation with a power law behavior where the power is at most quadratic.
[1] L. Lu, G. Li, A. Swindlehurst, A. Ashikhmin, and R. Zhang, “An overview of massive MIMO: Benefits and challenges,” IEEE Sel. Topics Signal Process., vol. 8, no. 5, pp. 742–758, Oct. 2014. [2] 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. [3] Y. Nam et al., “Full-dimension MIMO (FD-MIMO) for next generation cellular technology,” IEEE Commun. Mag., vol. 51, no. 6, pp. 172–179, Jun. 2013. [4] K. Zheng, S. Ou, and X. Yin, “Massive MIMO channel models: A survey,” Int. J. Antennas and Propag., vol. 2014, 2014, Art. ID. 848071. [5] J. Li et al., “Codebook design for uniform rectangular arrays of massive antennas,” in Proc. IEEE VTC, 2013, pp. 1–5. [6] X. Su et al., “Limited feedback precoding for massive MIMO,” Int. J. Antennas Propag., vol. 2013, 2013, Art. ID. 416352. [7] P. Smith, P. Dmochowski, M. Chiani, and A. Giorgetti, “On the number of independent channels in multi-antenna systems,” IEEE Trans. Wireless Commun., vol. 13, no. 1, pp. 75–85, Jan. 2014. [8] H. Gao, P. J. Smith, and M. V. Clark, “Theoretical reliability of MMSE linear diversity combining in rayleigh-fading additive interference channels,” IEEE Trans. Commun., vol. 46, no. 5, pp. 666–672, May 1998. [9] S. L. Loyka, “Channel capacity of MIMO architecture using the exponential correlation matrix,” Commun. Lett., vol. 5, no. 9, pp. 369–371, 2001. [10] R. Horn and C. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [11] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. Cambridge, MA, USA: MIT Press, 2006. [12] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, vol. 1. Boca Raton, FL, USA: CRC Press, 1994. [13] Q.-U.-A. Nadeem, A. Kammoun, M. Debbah, and M.-S. Alouini, “Spatial correlation in 3D MIMO channels using fourier coefficients of power spectrums,” in Proc. IEEE WCNC, 2015, pp. 165–170.
