- Home
- Documents
*Massive MIMO: Performance Analysis Using Random Matrix Theory Random Matrix Theory and Massive MIMO*

prev

next

out of 33

View

2Download

1

Embed Size (px)

Massive MIMO: Performance Analysis Using Random Matrix Theory

Jakob Hoydis

Alcatel-Lucent Bell Labs, France jakob.hoydis@alcatel-lucent.com

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 Introduction Software-Defined Wireless Networks Practical Challenges: Fronthaul

2 Random Matrix Theory and Massive MIMO The Perfect Tool Mathematical Preliminaries Performance 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. Supéléc, 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 the physical 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 when they 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 the number 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 meet in 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

N hH1 y = x1

1

N

N∑ i=1

|h1i |2︸ ︷︷ ︸ useful signal

+ x2 1

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∗1ih2i a.s.−−−−→

N→∞ E [h∗11h21] = 0 (interference vanishes)

1

N

N∑ i=1

h∗1ini a.s.−−−−→

N→∞ E [h∗11n1] = 0 (noise vanishes)

Thus,

1

N hH1 y

a.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

N hH1 ∑ k>1

hkxk 6 a.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 random matrices 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 probability space (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 = R 1 2 WT

1 2 , where R ∈ CN×N , T ∈ CK×K , and [W]i,j i.i.d.

I H = [h1 · · · hK ], where hj = R 1 2 j wj , Rj ∈ C

N×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

lim n→∞

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 of growing 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 types Let 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

lim n→∞

Fn(x) = F (x)

for each x ∈ R at which F is continuous. This is denoted by Fn ⇒ F . If Xn and X have distributions 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

n Xn(ω) = X

) = 1.

This is denoted by Xn a.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

]

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, and x ∈ 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‖

Random Matrix Theory and Massive MIMO Mathematical Preliminaries

On the empirical spectral distribution of large random matrices

Assume hj ∼ CN ( 0, 1

K IN )