Massive MIMO: Performance Analysis Using Random Matrix Theory · Random Matrix Theory and Massive MIMO The Perfect Tool The perfect tool Large random matrix theory (RMT) deals with

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


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


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.)


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


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)


Random Matrix Theory and Massive MIMO

Random Matrix Theory and Massive MIMO


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


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)



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.



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.


Random Matrix Theory and Massive MIMO Mathematical Preliminaries

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 ...



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).



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 .



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.



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.



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?



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.



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 .



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.


Random Matrix Theory and Massive MIMO Performance Analysis

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,∞).



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.



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?



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.



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

.



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.



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));



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


http://www.massivemimo.eu/research-library


Thank you!



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.


http://www.eurecom.fr/people/cifre_wagner.en.htm


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.


http://arxiv.org/pdf/1310.1806.pdf

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Massive MIMO: Performance Analysis Using Random Matrix Theory · Random Matrix Theory and Massive MIMO The Perfect Tool The perfect tool Large random matrix theory (RMT) deals with