Massive MIMO: Performance Analysis Using Random Matrix Theory · Random Matrix Theory and Massive...
Transcript of Massive MIMO: Performance Analysis Using Random Matrix Theory · Random Matrix Theory and Massive...
Massive MIMO:Performance Analysis Using Random Matrix Theory
Jakob Hoydis
Alcatel-Lucent Bell Labs, [email protected]
ITG Fachgruppe “Angewandte Informationstheorie”Massive MIMO: Theory and Applications
Oct. 8, 2015, Stuttgart University, Germany
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 1 / 33
Outline
1 IntroductionSoftware-Defined Wireless NetworksPractical Challenges: Fronthaul
2 Random Matrix Theory and Massive MIMOThe Perfect ToolMathematical PreliminariesPerformance Analysis
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 2 / 33
Introduction
About myself
2008 Dipl. Ing. RWTH Aachen University, Germany
2012 Ph.D. Supelec, France
2012-13 Bell Labs, Germany
2014-15 Co-founded Spraed, France
Since 09/15 Bell Labs, France
Current interest
5G (beyond) research at the interface between thephysical layer and cloud computing
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 3 / 33
Introduction Software-Defined Wireless Networks
Software-Defined Wireless Networks
Essentially all components of the RAN can be virtualized on commodity hardware(RRH (SDR), Fronthaul (SDN), BBU (VM, Containers), Core (NFV))
Any component is instantiable/configurable on the fly
Benefit from resource pooling/sharing on all levels (only consume resources whenthey are needed (fronthaul capacity, CPUs, memory))
Data on all protocol layers accessible in real-time (analytics/optimization/learning)
Network components can be provided as (micro)-services (L1, L2, Core, eNBs, etc.)
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 4 / 33
Introduction Software-Defined Wireless Networks
SDWN Example: Massive MIMO for Antennas-as-a-Service
SDWN can even create antenna abstractions
Offer antennas/eNBs as a service to multiple operators
Antennas can be seen as a cloud resource similar to cpus/memory/storage
SDN enables bandwidth control/metering for different fronthaul traffic flows
Number of antennas/eNBs can be scaled according to the cell load
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 5 / 33
Introduction Practical Challenges: Fronthaul
Practical Challenges
The biggest challenge is the fronthaul
Why?
For plain I/Q samples, the required fronthaul capacity scales linearly with thenumber of antennas:
≈ 1.23Gbps/antenna (@20MHz BW)
Each RRH shares the fronthaul network with many other RRHs/services
Clock, latency, bandwidth and synchronous-transport requirements are hard to meetin packet-based networks
Possible solutions:
Adaptive split-processing between the RRH and the BBU
Low-resolution ADC/DACs (do not solve the fundamental scaling problem)
Compression through traffic inspection (e.g., only forward used resource blocks)
Clock-synchronization protocols (IEEE 1588-2008)
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 6 / 33
Random Matrix Theory and Massive MIMO
Random Matrix Theory and Massive MIMO
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 7 / 33
Random Matrix Theory and Massive MIMO The Perfect Tool
A simple uplink example
y = h1x1 + h2x2 + n
Assumptions
h1, h2 ∈ CN×1 have i.i.d. entries with zero mean and unit variance
h1, h2 perfectly known at the base station (BS)
E[|x1|2
]= E
[|x2|2
]= 1
n ∼ CN(0, σ2IN
)J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 8 / 33
Random Matrix Theory and Massive MIMO The Perfect Tool
Law of large numbers
The BS applies a simple matched filter to detect the symbol of UE 1:
1
NhH
1 y = x11
N
N∑i=1
|h1i |2︸ ︷︷ ︸useful signal
+ x21
N
N∑i=1
h∗1ih2i︸ ︷︷ ︸interference
+1
N
N∑i=1
h∗1ini︸ ︷︷ ︸noise
By the strong law of large numbers:
1
N
N∑i=1
h∗1ih2ia.s.−−−−→
N→∞E [h∗11h21] = 0 (interference vanishes)
1
N
N∑i=1
h∗1inia.s.−−−−→
N→∞E [h∗11n1] = 0 (noise vanishes)
Thus,
1
NhH
1 ya.s.−−−−→
N→∞x1E
[|h11|2
]= x1 (SNR can be made arbitrarily small)
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 9 / 33
Random Matrix Theory and Massive MIMO The Perfect Tool
Unfortunately, things are (a bit) more complicated
Let’s assume that there are K > 2 users.
There are two ways to consider the asymptotic limit N →∞ :
1 K = const. (Tom Marzetta’s pioneering paper [1])→ The strong law of large numbers is enough.
2 K = K(N), such that lim infN→∞ K/N > 0→ We need other tools for the asymptotic analysis since
1
NhH
1
∑k>1
hkxk 6a.s.−−−−→
N→∞0
Remark
In general, the latter assumption leads to better approximations for finite N.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 10 / 33
Random Matrix Theory and Massive MIMO The Perfect Tool
The perfect tool
Large random matrix theory (RMT) deals with the asymptotic properties of randommatrices with growing dimensions.
Wireless communications with hundreds of antennas/users are becoming a reality.
Thus, RMT is the perfect tool to study the performance limits of massive MIMO.
Remark
For most of the asymptotic analysis to hold, the channel must be sufficiently rich, i.e.,< 6GHz carrier frequency. For mmWave-communications, this is rather unlikely.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 11 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
Mathematical Preliminaries
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 12 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
What is a random matrix?
A random matrix H is a matrix-valued random variable defined on a probabilityspace (Ω,F ,P) with entries in a measurable space (CN×K ,G).
We denote H(ω) the realization of H at sample point ω.
Examples:
I [H]i,j ∼ CN (0, 1), i.i.d.
I H = R12 WT
12 , where R ∈ CN×N , T ∈ CK×K , and [W]i,j i.i.d.
I H = [h1 · · · hK ], where hj = R12j wj , Rj ∈ CN×N , and wj ∼ CN (0, IN)
I H = W + A, where [W]i,j i.i.d., and is A is deterministic
I ...
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 13 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
Sequences of random matrices
We consider infinite sequences of random matrices (H(ω))n≥1 of growing dimensions:
H1(ω),H2(ω),H3(ω), . . .
where Hn(ω) ∈ CN(n)×K(n) and N(n),K(n)→∞ while
limn→∞
N(n)
K(n)= c ∈ (0,∞).
Keep in mind that:
Each ω creates an infinite sequence and not only a single random matrix.
All matrices/vectors considered in this tutorial must be understood as sequences ofgrowing matrices/vectors.
To simplify notations, we will write H instead of Hn(ω).
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 14 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
Convergence typesLet Xn = fn(Hn) ∈ R, where fn : CN(n)×K(n) 7→ R. Then, Xn has the distribution
Fn(x) = P(Xn ≤ x) = P(ω : Xn(ω) ≤ x).
Definition (Weak convergence)
The sequence of distribution functions (Fn)n≥1 converges weakly to the function F , if
limn→∞
Fn(x) = F (x)
for each x ∈ R at which F is continuous. This is denoted by Fn ⇒ F . If Xn and X havedistributions Fn and F , respectively, we also write Xn ⇒ X or Xn ⇒ F .
Definition (Almost sure convergence)
The sequence of random variables (Xn)n≥1 converges almost surely to X , if
P(ω : lim
nXn(ω) = X
)= 1.
This is denoted by Xna.s.−−→ X .
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 15 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
Two useful trace lemmas
Lemma ([2, Lemma B.26], [3, Lemma 14.2])
Let A ∈ CN×N and x = [x1 . . . xN ]T ∈ CN be a random vector of i.i.d. entries,independent of A. Assume E [xi ] = 0, E
[|xi |2
]= 1, E
[|xi |8
]<∞, and
lim supN‖A‖ <∞, almost surely. Then,
1
NxHAx− 1
NtrA
a.s.−→ 0.
Lemma ([3, Lemma 3.7])
Let y be another independent random vector with the same distribution as x. Then,
1
NxHAy
a.s.−→ 0.
Remark
For A = IN , these results are simple consequences of the strong law of large numbers:
1
NxHINx =
1
N
N∑i=1
|x2i |
a.s.−→ E[|xi |2
]= 1,
1
NxHINy =
1
N
N∑i=1
x∗i yia.s.−→ E [x∗i yi ] = 0.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 16 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
Finite rank perturbations
Lemma (Rank-1 perturbation lemma [4, Lemma 2.1])
Let z ∈ C \ R+, A ∈ CN×N and B ∈ CN×N with B Hermitian nonnegative definite, andx ∈ CN . Then,∣∣∣∣trA (B− zIN)−1 − trA
(B + xxH − zIN
)−1∣∣∣∣ ≤ ‖A‖
dist(z ,R+)
where dist is the Euclidean distance. If z < 0 and lim supN ‖A‖ <∞, this implies∣∣∣∣ 1
NtrA (B− zIN)−1 − 1
NtrA
(B + xxH − zIN
)−1∣∣∣∣ ≤ ‖A‖N|z |
N→∞−−−−→ 0.
Remark
By iteration of this lemma, we can see that finite rank perturbations of B areasymptotically negligible.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 17 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
On the empirical spectral distribution of large random matrices
Assume hj ∼ CN(0, 1
KIN)
i.i.d., for j = 1, . . . ,K .
What we expect from the strong law of large numbers:
For K →∞ and while N = const., we have
HHH =K∑j=1
hjhHj =
1
K
K∑j=1
hj hjH a.s.−−→ E
[h1h
H1
]= IN , h ∼ CN (0, IN).
But what happens if also N →∞, while N/K → c ∈ (0,∞)?
We can still say that[HHH
]i,i
a.s.−−→ 1 and[HHH
]i,j
a.s.−−→ 0 for j 6= i .
However, it is not true that HHH − INa.s.−−→ 0!
What happens to the eigenvalues of HHH?
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 18 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
Empirical and limiting spectral distribution
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
Eigenvalues of HHH
Den
sity
Empirical eigenvaluesMarcenko-Pastur density
Figure: Histogram of the eigenvalues of a single realization of HHH for N = 500, K = 2000.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 19 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
The Marcenko-Pastur law
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
x
Den
sity
fc(x
)
c = 0.1
c = 0.2
c = 0.5
Figure: Marcenko-Pastur density fc for different limit ratios c = limN/K .
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 20 / 33
Random Matrix Theory and Massive MIMO Mathematical Preliminaries
The Marcenko-Pastur lawIt was shown in [5]:
1
Ntr(HHH + σ2IN
)−1 a.s.−−→ mc(σ2) =c − 1
2cσ2− 1
2c+
√(1− c + σ2)2 + 4cσ2
2cσ2.
Remark
The function mN(z) = 1N
tr(HHH − zIN
)−1is known as the Stieltjes-Transform of the
eigenvalue distribution FHHH
of the matrix HHH, where
FHHH
(x) =1
N
N∑i=1
1λi ≤ x.
Remark
The convergence mN(z)a.s.−−→ mc(z) implies by [2, Theorem B.9]
FHHH
⇒ Fc
where Fc is Marcenko-Pastur law.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 21 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
Performance analysis
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 22 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
Example: SINR with linear receiversAssume we want to estimate xk from the observation y ∈ CN :
y =K∑j=1
hjxj + n = Hx + n
where hj ∼ CN (0, 1KIN), E
[xxH]
= IK , and n ∼ CN (0, σ2).
Matched filter: xk = hHk y
SINRMFk (σ2) =
|hHk hk |2
hHk
(σ2IN + HkHH
k
)hk
MMSE detector: xk = hHk
(HHH + σ2IN
)−1y
SINRMMSEk (σ2) = hH
k
(HkH
Hk + σ2IN
)−1
hk
where Hk ∈ CN×(K−1) is H with its kth column removed.
Goal: Show that SINRka.s.−−→ SINRk , for N,K →∞, N
K→ c ∈ (0,∞).
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 23 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
Example: SINR with linear receivers (cont.)
SINRMFk (σ2) =
|hHk hk |2
hHk
(σ2IN + HkHH
k
)hk
(trace lemma) (NK
)2
σ2 NK
+ 1K
∑j 6=k h
Hj hj
(trace lemma) (NK
)2
σ2 NK
+ K−1K
NK
c
σ2 + 1
, SINRMFk (σ2).
For two sequences (an)n≥1 and (bn)n≥1, the following notations are equivalent:
(i) an bn (ii) an − bna.s.−−−→
n→∞0.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 24 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
Example: SINR with linear receivers (cont.)
Similarly,
SINRMMSEk (σ2) = hH
k
(HkH
Hk + σ2IN
)−1
hk
(trace lemma) 1
Ktr(HkH
Hk + σ2IN
)−1
(rank-1 perturbation) 1
Ktr(HHH + σ2IN
)−1
(MP law) cmc(−σ2)
, SINRMMSEk (σ2).
Are these asymptotic results good approximations for realistic values of N,K?
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 25 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
SINR with linear receivers: Numerical results
−10 −5 0 5 10 15 20−10
0
10
20
N = 16, K = 8
SNR = 1σ2 (dB)
SIN
Rk(d
B)
SINRMFk
SINRMMSEk
E[SINRMF
k
]
E[SINRMMSE
k
]
−10 −5 0 5 10 15 20−10
0
10
20
N = 64, K = 32
SNR = 1σ2 (dB)
SINRMFk
SINRMMSEk
E[SINRMF
k
]
E[SINRMMSE
k
]
Figure: Errorbars correspond to one standard deviation in each direction.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 26 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
Analysis for more complex channel models
The trace and rank-1 perturbation lemmas are essential tools which directly apply tomore complex channel models.
The results are generally not given in closed form, but can be computed by quicklyconverging fixed-point algorithms.
We can even tackle more realistic scenarios with channel estimation, antennacorrelation and pilot contamination (e.g. [6]).
Example (Individual user antenna correlation – Generalized variance profile)
Let hj ∼ CN (0, 1NRj) for j = 1, . . . ,K , where ‖Rj‖ is bounded for all N. By the trace
lemma,
hHj hj
1
NtrRj
hHj hkh
Hk hj
1
N2trRjRk
hHj
(HjHj + σ2IN
)−1
hj 1
NtrRj
(HjHj + σ2IN
)−1
.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 27 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
Deterministic equivalent for generalized variance profile
Theorem ([7],[8, Theorem 2.3])
Let H ∈ CN×K be random and S ∈ CN×N Hermitian nonnegative definite. The jthcolumn hj of H is distributed as CN (0, 1
NR). Let D ∈ CN×N be a deterministic Hermitian
matrix. Then, as N,K →∞ such that 0 < lim inf N/K ≤ lim supN/K <∞ and undersome mild technical conditions, the following holds:
1
NtrD
(HHH + S− zIN
)−1
− 1
NtrDT(z)
a.s.−→ 0, for z ∈ C \ R+
where δ1(z), . . . , δK (z) are the unique Stieltjes transform solutions to
δj(z) =1
NtrRjT(z) =
1
NtrRj
(1
N
K∑k=1
Rk
1 + δk(z)+ S− zIN
)−1
, 1 ≤ j ≤ K .
Remark
The values of δj(z) can be computed by a classical fixed-point algorithm which normallyconverges in a few number of iterations (< 20) and does not pose any computationalchallenge.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 28 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
Matlab implementation
Example
function [m,delta,T] = stieltjes(rho,R,D,S)
% R is a cell of size 1xK, where each cell contains a NxN matrix
% D is NxN, S is NxN, rho>0
N = size(D,1); K = size(R,2);
delta = ones(K,1); delta old = zeros(K,1);
INIT = rho*eye(N)+S;
while (max(abs(delta-delta old))>1e-6)
delta old = delta;
TMP = INIT;
for k=1:K
TMP = TMP + 1/N*Rk/(1+delta old(k));
end
for k=1:K
delta(k) = real(1/N*trace(Rk/TMP));end
end
T = eye(N)/(TMP);
m = real(1/N*trace(D*T));
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 29 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
Some applications of RMT to Massive MIMO
Effects of hardware impairments: [9]
Exploitation of antenna correlation diversity (JSDM): [10]
Pilot contamination mitigation: [11]
Low-complexity receivers/precoders: [12, 13, 14]
Mobility analysis: [15, 16, 17, 18]
Checkout the rapidly growing massive-MIMO portal:
http://www.massivemimo.eu/research-library
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 30 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
Thank you!
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 31 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
References I
[1] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. WirelessCommun., vol. 9, no. 11, pp. 3590–3600, Nov. 2010.
[2] Z. D. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer Series inStatistics, New York, NY, USA, 2009.
[3] R. Couillet and M. Debbah, Random matrix methods for wireless communications, 1st ed. New York, NY, USA:Cambridge University Press, 2011.
[4] Z. D. Bai and J. W. Silverstein, “On the signal-to-interference ratio of CDMA systems in wireless communications,” TheAnnals of Applied Probability, vol. 17, no. 1, pp. 81–101, 2007.
[5] V. A. Marcenko and L. A. Pastur, “Distributions of eigenvalues for some sets of random matrices,” Math USSR-Sbornik,vol. 1, no. 4, pp. 457–483, April 1967.
[6] J. Hoydis, S. Ten Brink, and M. Debbah, “Massive MIMO in the UL/DL of cellular networks: How many antennas do weneed?” IEEE J. Sel. Areas Commun., vol. 31, no. 2, pp. 160–171, Feb. 2013.
[7] S. Wagner, R. Couillet, M. Debbah, and D. T. M. Slock, “Large system analysis of linear precoding in MISO broadcastchannels with limited feedback,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4509–4537, Jul. 2012.
[8] S. Wagner, “MU-MIMO Transmission and Reception Techniques for the Next Generation of Cellular Wireless Standards(LTE-A),” Ph.D. dissertation, EURECOM, 2229 route des cretes, BP 193 F-06560 Sophia-Antipolis cedex, 2011. [Online].Available: http://www.eurecom.fr/people/cifre wagner.en.htm
[9] E. Bjorson, J. Hoydis, M. Kountouris, and M. Debbah, “Massive MIMO Systems With Non-Ideal Hardware: EnergyEfficiency, Estimation, and Capacity Limits,” IEEE Trans. Inf. Theory, vol. 60, no. 11, pp. 7112–7139, Nov. 2014.
[10] A. Adhikary, J. Nam, J.-Y. Ahn, and G. Caire, “Joint spatial division and multiplexing—the large-scale array regime,” IEEETrans. Inf. Theory, vol. 59, no. 10, pp. 6441–6463, Sep. 2013.
[11] R. Muller, L. Cottatellucci, and M. Vehkapera, “Blind pilot decontamination,” IEEE J. Sel. Topics Signal Process., vol. 8,no. 5, pp. 773–786, Oct. 2014.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 32 / 33
Random Matrix Theory and Massive MIMO Performance Analysis
References II
[12] J. Hoydis, M. Debbah, and M. Kobayashi, “Asymptotic moments for interference mitigation in correlated fading channels,”in IEEE International Symposium on Information Theory (ISIT), 2011, pp. 2796–2800.
[13] A. Kammoun, E. Bjorson, and M. Debbah, “Linear Precoding Based on Polynomial Expansion: Large-Scale Multi-CellMIMO Systems,” Selected Topics in Signal Processing, IEEE Journal of, vol. 8, no. 5, pp. 861–875, Oct. 2014.
[14] A. Muller, A. Kammoun, E. Bjornson, and M. Debbah, “Linear Precoding Based on Polynomial Expansion: ReducingComplexity in Massive MIMO,” IEEE Trans. Inf. Theory, 2014, submitted. [Online]. Available:http://arxiv.org/pdf/1310.1806.pdf
[15] J. Hoydis, A. Muller, R. Couillet, and M. Debbah, “Analysis of multicell cooperation with random user locations viadeterministic equivalents,” in Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), 2012 10thInternational Symposium on, May 2012, pp. 374–379.
[16] L. Sanguinetti, A. L. Moustakas, E. Bjorson, and M. Debbah, “Large system analysis of the energy consumption distributionin multi-user MIMO systems with mobility,” IEEE Trans. Wireless Commun., vol. 14, no. 3, pp. 1730–1745, Mar. 2015.
[17] M. Girnyk, A. Muller, M. Vehkapera, L. Rasmussen, and M. Debbah, “On the Asymptotic Sum Rate of Downlink CellularSystems With Random User Locations,” Wireless Communications Letters, IEEE, vol. 4, no. 3, pp. 333–336, Jun. 2015.
[18] A. Muller, E. Bjorson, R. Couillet, and M. Debbah, “Analysis and management of heterogeneous user mobility in large-scaledownlink systems,” in Signals, Systems and Computers, 2013 Asilomar Conference on, Nov. 2013, pp. 773–777.
J. Hoydis (Bell Labs) RMT for Massive MIMO ITG Fachtagung 33 / 33