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  • Massive MIMO:Performance Analysis Using Random Matrix Theory

    Jakob Hoydis

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

    ITG Fachgruppe Angewandte InformationstheorieMassive 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 CN1 have i.i.d. entries with zero mean and unit varianceh1, h2 perfectly known at the base station (BS)

    E[|x1|2

    ]= E

    [|x2|2

    ]= 1

    n CN(0, 2IN

    )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

    NhH1 y = x1

    1

    N

    Ni=1

    |h1i |2 useful signal

    + x21

    N

    Ni=1

    h1ih2i interference

    +1

    N

    Ni=1

    h1ini noise

    By the strong law of large numbers:

    1

    N

    Ni=1

    h1ih2ia.s.

    NE [h11h21] = 0 (interference vanishes)

    1

    N

    Ni=1

    h1inia.s.

    NE [h11n1] = 0 (noise vanishes)

    Thus,

    1

    NhH1 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

    Lets assume that there are K > 2 users.

    There are two ways to consider the asymptotic limit N :

    1 K = const. (Tom Marzettas 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

    NhH1k>1

    hkxk 6a.s.

    N0

    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 (,F ,P) with entries in a measurable space (CNK ,G).

    We denote H() the realization of H at sample point .

    Examples:

    I [H]i,j CN (0, 1), i.i.d.

    I H = R12 WT

    12 , where R CNN , T CKK , and [W]i,j i.i.d.

    I H = [h1 hK ], where hj = R12j wj , Rj C

    NN , 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())n1 of growing dimensions:

    H1(),H2(),H3(), . . .

    where Hn() CN(n)K(n) and N(n),K(n) while

    limn

    N(n)

    K(n)= c (0,).

    Keep in mind that:

    Each 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().

    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( : Xn() x).

    Definition (Weak convergence)

    The sequence of distribution functions (Fn)n1 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)n1 converges almost surely to X , if

    P( : lim

    nXn() = 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 CNN 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 CNN and B CNN with B Hermitian nonnegative definite, andx CN . Then,trA (B zIN)1 trA(B + xxH zIN)1 Adist(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

    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 =Kj=1

    hjhHj =

    1

    K

    Kj=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. H

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