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Random Matrix Theory for Advanced Communication Systems erouane Debbah 1 and Jakob Hoydis 1,2 1 Alcatel-Lucent Chair on Flexible Radio 2 Department of Telecommunications Sup´ elec, Gif-sur-Yvette, France [email protected] [email protected] WCNC 2012, Paris, 1 April, 2012 M. Debbah and J. Hoydis (Sup´ elec) RMT for Advanced Communication Systems WCNC’12 Paris 1 / 101

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Page 1: RMT Software

Random Matrix Theory for Advanced Communication Systems

Merouane Debbah1 and Jakob Hoydis1,2

1Alcatel-Lucent Chair on Flexible Radio2Department of Telecommunications

Supelec, Gif-sur-Yvette, France

[email protected]

[email protected]

WCNC 2012, Paris, 1 April, 2012

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 1 / 101

Page 2: RMT Software

Outline1 Introduction

Advanced communication systems and related challengesWhy do we need large random matrix theory?Goals of this tutorial and take-away messages

2 TheorySequences of random matrices and convergence typesExample: Asymptotic SINR with linear receiversThe Stieltjes transform and its propertiesLimiting spectral distribution and the Marcenko-Pastur lawAsymptotic rates with linear receiversAsymptotic mutual information and its fluctuationsAsymptotic momentsDeterministic equivalents: Definition and overview of existing results

3 ApplicationsOptimal channel trainingLarge-scale MIMO systemsPolynomial expansion detectorsIterative deterministic equivalents: Multiphop relay channelRandom network topologies

4 Summary and perspectives

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 2 / 101

Page 3: RMT Software

Introduction

Outline1 Introduction

Advanced communication systems and related challengesWhy do we need large random matrix theory?Goals of this tutorial and take-away messages

2 TheorySequences of random matrices and convergence typesExample: Asymptotic SINR with linear receiversThe Stieltjes transform and its propertiesLimiting spectral distribution and the Marcenko-Pastur lawAsymptotic rates with linear receiversAsymptotic mutual information and its fluctuationsAsymptotic momentsDeterministic equivalents: Definition and overview of existing results

3 ApplicationsOptimal channel trainingLarge-scale MIMO systemsPolynomial expansion detectorsIterative deterministic equivalents: Multiphop relay channelRandom network topologies

4 Summary and perspectives

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 3 / 101

Page 4: RMT Software

Introduction Advanced communication systems and related challenges

Current challenges

Figure: Forecast of the global traffic [1] and carbon footprint [2] of cellular networks.

Mobile data traffic is exploding, expected yearly increase of almost 100 %.

ICT-related carbon emissions are expected to triple until 2020.

Soon, there will be more mobile-connected devices than humans on earth.

Current networks already reach their capacity limits in dense urban areas.

Sometimes even making voice calls can become a challenge.

How can the capacity of mobile networks be significantly increased?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 4 / 101

Page 5: RMT Software

Introduction Advanced communication systems and related challenges

Some possible solutions

More cells (small cells, multi-tier/heterogeneous networks)

More antennas (MIMO techniques, massive MIMO)

Cooperation and coordination (network MIMO, interference coordination, relays)

3D beamforming, antenna tilting, and smart antennas

Device-to-device communication, distributed caching

Cognitive radio (dynamic spectrum access)

More spectrum (millimeter waves)

Full-duplex transceivers

New coding and modulation schemes

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 5 / 101

Page 6: RMT Software

Introduction Advanced communication systems and related challenges

Theoretical challenges

Advanced communication systems become increasingly complex and are characterized bya dense deployment of different types of wireless access points. Any meaningful analysismust account for the most important characteristics of such networks, e.g.:

Fading and shadowing

Path loss

Interference

Imperfect channel state information (CSI)

Line-of-sight (LOS) channels

Antenna correlation

Limited backhaul capacity

Random network topologies

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 6 / 101

Page 7: RMT Software

Introduction Why do we need large random matrix theory?

Why do we need large random matrix theory?

1 Even some of these aspects taken alone are very difficult to model and to analyze.As a consequence, we are often unable to solve problems by exact analysis.

2 We need tools which reduce the system complexity and allow us to determine themost important system parameters.

3 The communication systems we study today are (very) large. Thus, asymptoticresults are not approximations anymore, but rather close to reality.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 7 / 101

Page 8: RMT Software

Introduction Why do we need large random matrix theory?

A simple example: Network MIMO uplink channel

Received signal at the base stations:

y =√ρHx + n =

√ρ

(g1,1

√α g1,2√

α g2,1 g2,2

)(x1

x2

)+

(n1

n2

)where gi,j , xi , ni ∼ CN (0, 1), ρ > 0, and the path loss is characterized by α ∈ [0, 1].

In general, no explicit expression of the ergodic mutual information is known:

I (ρ) = E[log det

(I + ρHHH

)]= ?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 8 / 101

Page 9: RMT Software

Introduction Why do we need large random matrix theory?

A simple example: Large-system approximation

−10 −5 0 5 10 15 200

2

4

6

8

α = 0.5

ρ (dB)

E[I(ρ)]

(nat

s/s/

Hz)

SimulationApproximation

Under the assumption that H is “very large”, we can find the approximation:

E [I (ρ)] ≈ 2 log(

1 + 2(1 + α)ρ+√

1 + 4(1 + α)ρ)− 2 log(2e)

+4

1 +√

1 + 4(1 + α)ρ.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 9 / 101

Page 10: RMT Software

Introduction Goals of this tutorial and take-away messages

Goal of this tutorial

1 Short introduction to the most important lemmas and proof-techniques necessary tounderstand and derive large-system approximations for a variety of channel models.

2 Focus on practical examples, applications, and implementation (e.g., Matlab).

3 State-of-the-art of random matrix research related to wireless communications(not exhaustive).

4 Outlook and perspectives for future research.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 10 / 101

Page 11: RMT Software

Introduction Goals of this tutorial and take-away messages

Take-away messages

Advanced communications system require novel tools for their theoretical analysis.

Under the assumption of a large system regime, random matrix theory allows us toobtain deterministic approximations of the system performance (e.g., mutualinformation, achievable rates, SINR, outage probability) for realistic channel models.

These approximations can help to identify the most important system parametersand to solve optimization problems which would be otherwise intractable (e.g.,precoding matrices, channel training, scheduling, base station placement).

Although all presented results are only exact in a large system regime, they providevery tight performance approximations for realistic system dimensions (i.e., smallnumbers of antennas, user terminals, etc.).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 11 / 101

Page 12: RMT Software

Theory

Outline1 Introduction

Advanced communication systems and related challengesWhy do we need large random matrix theory?Goals of this tutorial and take-away messages

2 TheorySequences of random matrices and convergence typesExample: Asymptotic SINR with linear receiversThe Stieltjes transform and its propertiesLimiting spectral distribution and the Marcenko-Pastur lawAsymptotic rates with linear receiversAsymptotic mutual information and its fluctuationsAsymptotic momentsDeterministic equivalents: Definition and overview of existing results

3 ApplicationsOptimal channel trainingLarge-scale MIMO systemsPolynomial expansion detectorsIterative deterministic equivalents: Multiphop relay channelRandom network topologies

4 Summary and perspectives

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 12 / 101

Page 13: RMT Software

Theory

Theory

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 13 / 101

Page 14: RMT Software

Theory Sequences of random matrices and convergence types

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 ...

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 14 / 101

Page 15: RMT Software

Theory Sequences of random matrices and convergence types

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).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 15 / 101

Page 16: RMT Software

Theory Sequences of random matrices and convergence types

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 .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 16 / 101

Page 17: RMT Software

Theory Example: Asymptotic SINR with linear receivers

Example: SINR with linear receiversAssume we want to estimate xk from the observation y ∈ CN :

y =K∑

j=1

hj xj + n = Hx + n

where hj ∼ CN (0, 1K

IN ), E[xxH]

= IK , and n ∼ CN (0, σ2).

Matched filter: xk = hHk y

SINRMFk (σ2) =

|hHk hk |2

hHk

(σ2IN + Hk HH

k

)hk

MMSE detector: xk = hHk

(HHH + σ2IN

)−1y

SINRMMSEk (σ2) = hH

k

(Hk HH

k + 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,∞).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 17 / 101

Page 18: RMT Software

Theory Example: Asymptotic SINR with linear receivers

Two useful trace lemmas

Lemma ([3, Lemma B.26], [4, 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

Ntr A

a.s.−→ 0.

Lemma ([4, 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

NxHIN x =

1

N

N∑i=1

|x2i | a.s.−→ E

[|xi |2

]= 1,

1

NxHIN y =

1

N

N∑i=1

x∗i yia.s.−→ E [x∗i yi ] = 0.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 18 / 101

Page 19: RMT Software

Theory Example: Asymptotic SINR with linear receivers

Example: SINR with linear receivers (cont.)As a direct application of the trace lemma, we obtain

SINRMFk (σ2) =

|hHk hk |2

hHk

(σ2IN + Hk HH

k

)hk

(

NK

)2

σ2 NK

+ 1K

∑j 6=k hH

j hj

(

NK

)2

σ2 NK

+ K−1K

NK

→ c

σ2 + 1, SINR

MFk (σ2).

Similarly,

SINRMMSEk (σ2) = hH

k

(Hk HH

k + σ2IN

)−1

hk

1

Ktr(

Hk HHk + σ2IN

)−1

→ Can we go further?

For two sequences (an)n≥1 and (bn)n≥1, the following notations are equivalent:

(i) an bn (ii) an − bna.s.−−−→

n→∞0.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 19 / 101

Page 20: RMT Software

Theory Example: Asymptotic SINR with linear receivers

Finite rank perturbations

Lemma (Rank-1 perturbation lemma [5, Lemma 2.1])

Let z ∈ C \ R+, A ∈ CN×N and B ∈ CN×N with B Hermitian nonnegative definite, andx ∈ CN . Then, ∣∣∣∣tr

((B− zIN )−1 −

(B + xxH − zIN

)−1)

A

∣∣∣∣ ≤ ‖A‖dist(z ,R+)

where dist is the Euclidean distance. If z < 0 and lim supN ‖A‖ <∞, this implies∣∣∣∣ 1

Ntr

((B− zIN )−1 −

(B + xxH − zIN

)−1)

A

∣∣∣∣ ≤ ‖A‖N|z |N→∞−−−−→ 0.

Remark

By iteration of this lemma, we can see that finite rank perturbations of B areasymptotically negligible.

Coming back to our example, this implies

SINRMMSEk (σ2) 1

Ktr(

Hk HHk + σ2IN

)−1

c1

Ntr(

HHH + σ2IN

)−1

.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 20 / 101

Page 21: RMT Software

Theory The Stieltjes transform and its properties

How can we go further: The Stieltjes transformIn the next slides, we will proof the following result [6]:

1

Ntr(

HHH + σ2IN

)−1 a.s.−−→ c − 1

2cσ2− 1

2c+

√(1− c + σ2)2 + 4cσ2

2cσ2.

To this end, we need some preparation:

Definition (Stieltjes transform)

Let µ be a finite nonnegative measure with support supp(µ) ⊂ R, i.e., µ(R) <∞. TheStieltjes transform m(z) of µ is defined for z ∈ C \ supp(µ) as

m(z) =

∫R

1

λ− zdµ(λ).

Definition (Empirical spectral distribution (e.s.d.))

Let A ∈ CN×N be a Hermitian matrix with eigenvalues λ1, . . . , λN . The e.s.d. F A(x) ofthe eigenvalues of A is defined as

F A(x) =1

N

N∑i=1

1λi ≤ x.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 21 / 101

Page 22: RMT Software

Theory The Stieltjes transform and its properties

Properties of the Stieltjes transformThe Stieltjes transform mA(z) of the of the e.s.d. F A of some Hermitian matrix A can bewritten in several ways:

mA(z) =

∫R

1

λ− zdF A(λ) =

1

N

N∑i=1

1

λi − z=

1

Ntr (A− zIN )−1 .

One can recover µ when only its Stieltjes transform m(z) is known:

Properties

Let m(z) be the Stieltjes transform of a finite measure µ on R. Then,

(i) µ(R) = limy→∞−iym(iy). If µ is a probability measure, we have µ(R) = 1.

(ii) µ([a, b]) = limy→0+

∫ b

a=m(x + iy)dx , if a, b are continuity points of µ.

Moreover, let m(z) be an analytic function over C+ such that m(z) ∈ C+ if z ∈ C+. Iflim supy→∞ |iym(iy)| = 1, then m(z) is the Stieltjes transform of a probability measureon R.

Remember that:

Knowing the Stieltjes transform of a matrix is as good as knowing its eigenvalues.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 22 / 101

Page 23: RMT Software

Theory The Stieltjes transform and its properties

On the empirical spectral distribution of large random matricesRemember that 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

hj hHj =

1

K

K∑j=1

hj hjH a.s.−−→ E

[h1hH

1

]= 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?

Remark (Wishart matrices)

The matrix HHH is called a “Wishart matrix” with K degrees of freedom. For finiteN,K, its exact eigenvalue distribution is known, e.g., [7].

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 23 / 101

Page 24: RMT Software

Theory Limiting spectral distribution and the Marcenko-Pastur law

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.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 24 / 101

Page 25: RMT Software

Theory Limiting spectral distribution and the Marcenko-Pastur law

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

sityfc(x

)

c = 0.1

c = 0.2

c = 0.5

Figure: Marcenko-Pastur density fc for different limit ratios c = lim N/K .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 25 / 101

Page 26: RMT Software

Theory Limiting spectral distribution and the Marcenko-Pastur law

Deriving the The Marcenko-Pastur lawWe want to show that, almost surely,

F HHH ⇒ Fc , where Fc is Marcenko-Pastur law.

Outline: (for a detailed proof, see, e.g., [4])1 Starting from the Stieltjes transform of HHH, show that

mHHH (z)− 1− c

cz+

1

cz (1 + cmHHH (z))a.s.−−→ 0.

2 Verify that there is a unique Stieltjes transform mc (z) of a probability distribution Fc

which satisfies

mc (z) =1− c

cz− 1

cz (1 + cmc (z)).

3 One can then show thatmHHH (z)

a.s.−−→ mc (z).

4 By [3, Theorem B.9], the almost sure convergence of the Stieltjes transforms implies

the almost sure weak convergence of F HHH

to Fc .5 From the formula for the inverse Stieltjes transform one can recover Fc and its

density fc .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 26 / 101

Page 27: RMT Software

Theory Limiting spectral distribution and the Marcenko-Pastur law

Deriving the The Marcenko-Pastur law: Step 1

1

Ntr(

HHH − zIN

)−1=

K − N

N

1

z+

1

Ntr(

HHH− zIN

)−1

=K − N

N

1

z+

1

N

K∑j=1

[(HHH− zIN

)−1]

j,j

[Matrix inversion lemma] =K − N

N

1

z+

1

N

K∑j=1

1

−z + hHj

(IN −Hj

(HH

j Hj − zIK−1

)−1Hj

)hj

[Identity for matrix inverse] =K − N

N

1

z+

1

N

K∑j=1

1

−z

(1 + hH

j

(Hj HH

j − zIN

)−1hj

)

[Trace lem. & Rank-1 per.] K − N

N

1

z+

1

N

K∑j=1

1

−z(

1 + NK

1N

tr(HHH − zIN

)−1)

Thus,

mHHH (z)− 1− c

cz+

1

cz (1 + cmHHH (z))a.s.−−→ 0 .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 27 / 101

Page 28: RMT Software

Theory Limiting spectral distribution and the Marcenko-Pastur law

Deriving the The Marcenko-Pastur law: Steps 2-5The equation

mc (z) =1− c

cz− 1

cz (1 + cmc (z))

has a unique Stieltjes transform solution which satisfies

Stieltjes transform of the Marcenko-Pastur law

mc (z) =1− c

2cz− 1

2c−√

(1− c − z)2 − 4cz

2cz

where the branch of

√(1−c−z)2−4cz

2czis chosen such that mc (z) ∈ C+ for z ∈ C+ and

mc (z) > 0 for z < 0.

Taking the inverse Stieltjes transform of mc (z) leads to the distribution Fc with density

Density of the Marcenko-Pastur law

fc (x) =(

1− c−1)+

δ(x) +1

2πcx

√(x − a)+(b − x)+

where a = (1−√c)2, b = (1 +√

c)2, (x)+ , max(0, x), and δ(x) is the Dirac delta.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 28 / 101

Page 29: RMT Software

Theory Limiting spectral distribution and the Marcenko-Pastur law

Example: SINR with linear receivers (cont.)

We have shown that for N,K →∞ while N/K → c, the following holds:

SINRMFk (σ2)

a.s.−−→ SINRMFk (σ2) =

c

σ2 + 1

SINRMMSEk (σ2)

a.s.−−→ SINRMMSEk (σ2) = cmc (−σ2).

But are these asymptotic results good approximations for realistic values of N,K?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 29 / 101

Page 30: RMT Software

Theory Limiting spectral distribution and the Marcenko-Pastur law

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.

The asymptotic results are quite accurate for reasonable values of N,K .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 30 / 101

Page 31: RMT Software

Theory Asymptotic rates with linear receivers

Asymptotic rates with linear receivers

Theorem (Continuous mapping theorem [8, Theorem 2.3])

Let (Xn)n≥1 be a sequence of real random variables and let f : R 7→ R be continuous atevery point of a set A such that P(X ∈ A) = 1, for some random variable X .

(i) If Xn ⇒ X , then f (Xn)⇒ f (X );

(ii) If Xna.s.−−→ X , then f (Xn)

a.s.−−→ f (X ).

The direct application of this theorem provides the asymptotic rate with linear detection:

Rk (σ2) , log(

1 + SINRk (σ2))

a.s.−−→ Rk (σ2) , log(

1 + SINRk (σ2))

Remark : The same holds also for the normalized sum-rate:

1

N

K∑k=1

Rk (σ2)− 1

N

K∑k=1

Rk (σ2)a.s.−−→ 0.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 31 / 101

Page 32: RMT Software

Theory Asymptotic mutual information and its fluctuations

Asymptotic mutual informationF HHH ⇒ Fc , almost surely, implies that for any bounded continuous function g :∫

g(λ)dF HHH

(λ)a.s.−−→

∫g(λ)dFc (λ).

The normalized mutual information between y and x is not bounded, since

I (σ2) ,1

Nlog det

(IN +

1

σ2HHH

)=

∫log

(1 +

λ

σ2

)dF HHH

(λ).

Theorem (No eigenvalues outside the support [9, Theorem 1.1], [10, Theorem 3])

Denote by λmax and λmin the largest and smallest eigenvalue of HHH, respectively. Then,

(i) λmaxa.s.−→ (1 +

√c)2, (ii) λmin

a.s.−→ (1−√c)21c ≤ 1.

Thus, we have (see, e.g., [11])

Theorem (Asymptotic mutual information)

I (σ2)a.s.−−→

∫g(λ)dFc (λ) , I (σ2) =

1

clog(1 + cmc ) + log

(1 +

1

σ2

1

1 + cmc

)−

mc

1 + cmc

where mc = mc (−σ2).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 32 / 101

Page 33: RMT Software

Theory Asymptotic mutual information and its fluctuations

Asymptotic mutual information: Numerical results

−10 −5 0 5 10 15 200

1

2

3

4

N = 6, K = 3

SNR = 1σ2 (dB)

(bits

/s/H

z)

I(σ2

E[I(σ2)

]

Figure: Average normalized mutual information E[I (σ2)

]and its asymptotic approximation I (σ2)

vs. SNR for N = 6 and K = 3. Errorbars correspond to one standard deviation in each direction.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 33 / 101

Page 34: RMT Software

Theory Asymptotic mutual information and its fluctuations

Relation between the mutual information and the Stieltjes transform

Interestingly, the mutual information and the Stieltjes transform are related by

I (σ2) =

∫ ∞σ2

(1

x−mHHH (−x)

)dx

since

dI (σ2)

dσ2=

d

dσ2

(1

Nlog det

(IN +

1

σ2HHH

))=

d

dσ2

(1

Nlog det

(σ2IN + HHH

)− log(σ2)

)=

1

Ntr(σ2IN + HHH

)−1

− 1

σ2.

One can also verify that

I (σ2) =

∫ ∞σ2

(1

x−mc (−x)

)dx .

Remark : The Stieltjes transform of the e.s.d. of a matrix is sufficient to calculate themutual information. The exact eigenvalue distribution is not needed.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 34 / 101

Page 35: RMT Software

Theory Asymptotic mutual information and its fluctuations

Fluctuations of the mutual information

−4 −3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

N = 6, K = 3

SNR = 2 dB

NΘc

(I(σ2)− I(σ2))

Freq

uenc

y

HistogramN (0, 1) Example

Approximation of the outage probability:

Pout(r) = Pr(

NI (σ2) < r)

≈ 1− Q

(r − NI (σ2)

Θc

).

Theorem (Central limit theorem of the mutual information [4, Theorem 3.18])

N

Θc

(I (σ2)− I (σ2)

)⇒ N (0, 1)

where the asymptotic variance is given as Θ2c = − log

(1− cmc (−σ2)2

(1+cmc (−σ2))2

).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 35 / 101

Page 36: RMT Software

Theory Asymptotic moments

What about moments?

Theorem ([4, Theorem 3.3])

Let F be a probability distribution and denote by m(z) its Stieltjes transform. Assumethat supp(F ) ⊂ [a, b] for 0 < a < b <∞. Then, for z ∈ C \ R, |z | > b,

m(z) = −1

z

∞∑k=0

Mk

zk

where Mk =∫λk dF (λ) are the moments of F .

Theorem (Moments of the Marcenko-Pastur law [12])

Denote by Mk = 1N

tr(HHH

)kthe kth moment of H. Then,

Mka.s.−→ Mk ,

∫λk dFc (λ) =

k−1∑i=0

(k

i

)(k

i + 1

)c i

k.

Remark : We can recover the moments of F through differentiation of the function

G(z) = 1z

m(−1/z) =∑∞

k=0(−1)k zk Mk . Note that Mk = (−1)k

k!G (k)(0), where G (k)(z) is

the kth derivative of G(z).M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 36 / 101

Page 37: RMT Software

Theory Deterministic equivalents: Definition and overview of existing results

What happens if things do not converge?

Bn = HnTnHHn

where Hn ∈ Cn×n, [Hn]i,j ∼ CN (0, 1/n), and

Tn =

In , for n odd(

In/2 0

0 0

), for n even

.

Question: F Bn ⇒ ?

F Bn does not converge to a limiting distribution.

F B2n+1 ⇒ F1 , F B2n ⇒ F2, where Fc is the Marcenko-Pastur law,

However, F Bn − Fn ⇒ 0

where Fn =

F1 , for n odd

F2 , for n even.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 37 / 101

Page 38: RMT Software

Theory Deterministic equivalents: Definition and overview of existing results

Deterministic equivalents

Definition

Let (an)n≥1 be an infinite sequences of complex random variables and (bn)n≥1 an infinitesequence of complex numbers. We say that bn is a deterministic equivalent of an, iff

an − bna.s.−−−→

n→∞0.

Remarks:

Remember that we consider functionals of random matrices with growingdimensions, e.g., an is the mutual information of a N(n)× K(n) MIMO channel.

Deterministic equivalents (DEs) are a “natural” way of thinking about large randommatrices: “if n is large, an ≈ bn”.

Often a DE of an exists, although an does not converge.

bn can be seen as a deterministic approximation of an for each n.

In many cases, an ≈ bn for very small n. Thus, knowing bn is often “as good as”knowing an.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 38 / 101

Page 39: RMT Software

Theory Deterministic equivalents: Definition and overview of existing results

Right-sided correlation model

Theorem (Right-sided correlation model [13],[14])

Let X ∈ CN×K , [X]i,j ∼ CN(1, 1

K

)i.i.d., and let T = diag (t1, . . . , tK ) ∈ RK×K

+ be

deterministic, satisfying maxk tk <∞. Denote B = XTXH and c = NK

and let N,K →∞while 0 < lim inf c ≤ lim sup c <∞. Then,

(i)1

Ntr (B− zIN )−1 − m(z)

a.s.−→ 0, for z ∈ C \ R+

(ii)1

Nlog det

(IN +

1

σ2B

)− I (σ2)

a.s.−→ 0, for σ2 ∈ R+

where m(z) is the unique Stieltjes transform which satisfies

m(z) =

(1

K

K∑k=1

tk

1 + ctk m(z)− z

)−1

and

I (σ2) =1

N

K∑k=1

log(

1 + ctk m(−σ2))− log

(σ2m(−σ2)

)− 1

K

K∑k=1

tk m(−σ2)

1 + ctk m(−σ2).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 39 / 101

Page 40: RMT Software

Theory Deterministic equivalents: Definition and overview of existing results

Right-sided correlation model: Matlab code

Example (Stieltjes transform of the right-sided correlation model)

function m = stieltjes(z,c,T)

% T is a KxK diagonal matrix, z<0 and c=N/K

m = 0;

m old = 1;

t = diag(T);

while abs(m-m old)>1e-9 % change desired accuracy here

m old = m;

m = 1/(mean(t./(1+c*t*m old)) - z);

end

end

Remark

This classical fixed-point algorithm normally converges in a few number of iterations(< 20) and does not pose any computational challenge. All other implicit equations (seenext slides) can be solved in a similar manner.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 40 / 101

Page 41: RMT Software

Theory Deterministic equivalents: Definition and overview of existing results

Rician channels with a variance profile

Theorem ([15, Theorems 2.4, 2.5, 4.1])

Let X ∈ CN×K , [X]i,j ∼ CN (0,σ2

i,j

K), and A ∈ CN×K . Define B = (X + A) (X + A)H. Further

denote Dj = diag(σ21,j , . . . , σ

2N,j ) and Di = diag(σ2

i,1, . . . , σ2i,K ) ∀i , j . Let N,K →∞ while

0 < lim inf NK≤ lim sup N

K<∞. Then, under some mild technical assumptions,

E[

1

Nlog det

(IN +

1

σ2B

)]− I (σ2)→ 0, for σ2 ∈ R+

where

I (σ2) =1

Nlog det

(Ψ−1

σ2+ AΨAH

)+

1

Nlog det

(Ψ−1

σ2

)−

σ2

NK

∑i,j

σ2ij Tii Tjj

and where

T =(

Ψ−1 − zAΨAH)−1

, T =(

Ψ−1 − zAHΨA

)−1

and Ψ = diag (ψ1, . . . , ψN ) , Ψ = diag(ψ1, . . . , ψK

)are the unique positive solutions to the

following set of N + K fixed-point equations

ψi =1

σ2(

1 + 1K

tr Di T) , 1 ≤ i ≤ N, ψj =

1

σ2(1 + 1

Ktr Dj T

) , 1 ≤ j ≤ K .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 41 / 101

Page 42: RMT Software

Theory Deterministic equivalents: Definition and overview of existing results

Generalized variance profile

Theorem ([16],[17, Theorem 2.3])

Let B = XXH + S, where X ∈ CN×K is random and S ∈ CN×N is Hermitian nonnegative definite.The jth column xj of X is given as xj = Rj zj , where zj ∼ CN (0, 1

NIN ), and Rj ∈ CN×N is

deterministic. Denote Rj = Rj RHj . Let D ∈ CN×N be a deterministic Hermitian matrix. Then, as

N,K →∞ such that 0 < lim inf N/K ≤ lim sup N/K <∞ and under some mild technicalconditions, the following holds:

(i)1

Ntr D (B− zIN )−1 −

1

Ntr DT(z)

a.s.−→ 0, for z ∈ C \ R+

(ii) E[

1

Nlog det

(IN +

1

σ2B

)]− I (σ2)→ 0, for σ2 ∈ R+

where δ1(z), . . . , δK (z) are the unique Stieltjes transform solutions to

δj (z) =1

Ntr Rj T(z) =

1

Ntr Rj

(1

N

K∑k=1

Rk

1 + δk (z)+ S− zIN

)−1

, 1 ≤ j ≤ K

and

I (σ2) =1

Nlog det

(IN +

1

σ2S +

1

σ2

1

N

K∑k=1

Rk

1 + δk

)+

1

N

K∑k=1

log (1 + δk )−δk

1 + δk

where δk = δk

(−σ2

), ∀k.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 42 / 101

Page 43: RMT Software

Theory Deterministic equivalents: Definition and overview of existing results

Some remarks and literature overview

Remark

Most of the presented deterministic equivalents also hold for non-Gaussian randommatrices. It is often sufficient to assume zero mean, variance 1/K and finite 4th or 8thorder moment of the matrix entries. The convergence for these distributions is in generalslower than for Gaussian matrices.

Limiting e.s.d. and deterministic equivalents of the mutual information:

[14] : B =∑L

l=1 R12l Xl Tl Xl R

12l , where Rl ,Tl deterministic, Xl Gaussian

[18] : B =

(∑Ll=1 R

12l Xl T

12l

)(∑Ll=1 R

12l Xl T

12l

)H

, where Rl ,Tl deterministic, Xl Gaussian

Central limit theorems:

[19] : Mutual information for channels with a variance profile

[20] : SINR of the MMSE detector for channels with a variance profile

[21] : Mutual information of a Kronecker channel with interference from anotherKronecker channel

...and many more. See, e.g., [4] for an extensive overview.M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 43 / 101

Page 44: RMT Software

Applications

Outline1 Introduction

Advanced communication systems and related challengesWhy do we need large random matrix theory?Goals of this tutorial and take-away messages

2 TheorySequences of random matrices and convergence typesExample: Asymptotic SINR with linear receiversThe Stieltjes transform and its propertiesLimiting spectral distribution and the Marcenko-Pastur lawAsymptotic rates with linear receiversAsymptotic mutual information and its fluctuationsAsymptotic momentsDeterministic equivalents: Definition and overview of existing results

3 ApplicationsOptimal channel trainingLarge-scale MIMO systemsPolynomial expansion detectorsIterative deterministic equivalents: Multiphop relay channelRandom network topologies

4 Summary and perspectives

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 44 / 101

Page 45: RMT Software

Applications

Applications

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 45 / 101

Page 46: RMT Software

Applications Optimal channel training

Optimal channel training

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 46 / 101

Page 47: RMT Software

Applications Optimal channel training

Optimal channel training

Consider a N × K MIMO point-to-point link with output

y =

√ρ

KHx + n

where H ∈ CN×K , [H]i,j ∼ CN (0, 1) i.i.d., x ∼ CN (0, IK ), and n ∼ CN (0, IN ).

Problem setting:

Blockfading channel model with coherence time T .

The channel is unknown and must be estimated at the receiver.

Which fraction of T should be used for channel training?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 47 / 101

Page 48: RMT Software

Applications Optimal channel training

Channel training

coherence time T

training τ data transmission

The receiver estimates the channel channel based on the observation

Yτ =

√τρ

KHΦ + N

where Φ ∈ CK×τ , ΦHΦ = IK , is a matrix with known pilot tones.

The receiver computes the MMSE estimate H of H, such that

H = H + H

where H and the estimation error H are independent and distributed as

[H]i,j ∼ CN(

0,ρτ

ρτ + K

), [H]i,j ∼ CN

(0,

K

ρτ + K

).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 48 / 101

Page 49: RMT Software

Applications Optimal channel training

Net ergodic achievable rateAs in [22], our goal is to find τ∗ which maximizes the net ergodic achievable rate, i.e.,τ∗ = arg maxτ∈[K ,T ] Rnet(τ), where

Rnet(τ) ,(

1− τ

T

)E[

1

Nlog det

(IN +

ρeff(τ)

KHHH

)]and

ρeff(τ) =ρ2τ

K(1 + ρ) + ρτ.

As this problem is difficult to solve exactly, we consider the large-system approximationand find τ∗ = arg maxτ∈[K ,T ] Rnet(τ), where

Rnet(τ) ,(

1− τ

T

)I

(1

ρeff(τ)

)and

I (x) =1

clog(1 + cmc (−x)) + log

(1 +

1

x

1

1 + cmc (−x)

)− mc (−x)

1 + cmc (−x)

is the asymptotic mutual information of a N ×K MIMO channel, c = NK

, with SNR = 1x

.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 49 / 101

Page 50: RMT Software

Applications Optimal channel training

Optimization of the large system approximation

The asymptotically optimal training length τ∗ can be computed by a simple line search:

d

dτRnet(τ) = − 1

TI

(1

ρeff(τ)

)−(

1− τ

T

) ρ′eff(τ)

ρeff(τ)2

(mc

(− 1

ρeff(τ)

)− ρeff(τ)

)!

= 0

where

ρ′eff(τ) =ρ2(1 + ρ)K

(K(1 + ρ) + ρτ)2 .

Remark : For the derivation, remember from the relation between the mutualinformation and the Stieltjes transform that

dI (x)

dx= mc (−x)− 1

x.

Is τ∗ a good approximation of τ∗ for small N,K?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 50 / 101

Page 51: RMT Software

Applications Optimal channel training

Optimal channel training: Numerical results

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

N = 4, K = 2, ρ = 0 dB, T = 100

Training length τ

Rne

t(τ)

(bits

/s/H

z)

SimulationApproximation

Figure: Net ergodic achievable rate and its asymptotic approximation versus training length τ .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 51 / 101

Page 52: RMT Software

Applications Optimal channel training

Optimal channel training: Numerical results

−10 −5 0 5 10 15 200

10

20

30

N = 4, K = 2, T = 100

SNR = ρ (dB)

Opt

imal

trai

ning

leng

thτ∗

SimulationApproximation

Figure: Optimal training length and its asymptotic approximation versus SNR.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 52 / 101

Page 53: RMT Software

Applications Optimal channel training

Remarks & Extensions

Surprisingly, the asymptotic results are already optimal for a 4× 2 MIMO channel!

The same idea can be applied to more involved channel models(see, e.g., [23] for a channel model with a variance profile).

The general idea of optimizing a large-system approximation with respect to certainparameters has been extensively applied in the literature, e.g.:

I Precoding: [24, 25, 26, 14, 18]

I Beamforming: [27, 28, 29, 30, 31, 32, 33, 16]

I Channel training: [23, 16]

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 53 / 101

Page 54: RMT Software

Applications Large-scale MIMO systems

Large-scale MIMO systems

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 54 / 101

Page 55: RMT Software

Applications Large-scale MIMO systems

Analysis of large-scale MIMO systems

yj =√ρ

L∑l=1

K∑k=1

hjlk xlk + nj =√ρ

L∑l=1

Hjl xl + nj

where xl ∼ CN (0, IK ), nj ∼ CN (0, IN ).

Each channel can have a different correlation structure:

hjlk = R12jlk wjlk

where Rjlk ∈ CN×N is deterministic and wjlk ∼ CN (0, IN ).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 55 / 101

Page 56: RMT Software

Applications Large-scale MIMO systems

Channel estimation – Pilot contamination

The same set of K orthogonal pilot sequences is reused in every cell.

BS j estimates the channel hjjk based on the observation

yτjk = hjjk +∑l 6=j

hjlk +1√ρτ

njk

where ρτ is the effective training SNR.

Computing the MMSE estimate hjjk = Rjjk Qjk yτjk yields:

hjjk = hjjk + hjjk

hjjk ∼ CN (0,Φjjk ) , hjjk ∼ CN (0,Rjjk −Φjjk )

Φjlk = Rjjk Qjk Rjlk , Qjk =(ρ−1τ IN +

∑l

Rjlk

)−1

.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 56 / 101

Page 57: RMT Software

Applications Large-scale MIMO systems

Ergodic achievable rate with linear detectorsWith a linear detector rjm, the following rate of UT m in cell j is achievable [22]:

Rjm = EHjj[log2 (1 + γjm)]

with the effective SINR

γjm =

∣∣∣rHjmhjjm

∣∣∣2E[rH

jm

(1ρ

IN + hjjmhHjjm − hjjmhH

jjm +∑

l Hjl HHjl

)rjm

∣∣∣Hjj

] .Remarks:

Computing Rjm for typical receivers (MMSE detector, matched filter) seemsintractable.

Assuming N,K to be very large allows us to compute deterministic equivalents ofthe SINR and the achievable rates.

We consider the achievable rates with a matched filter, i.e.:

rjm = hjjm

A more involved analysis for MMSE detection and the downlink with different linearprecoders can be found in [34, 35, 36].

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 57 / 101

Page 58: RMT Software

Applications Large-scale MIMO systems

Deriving a deterministic equivalent for the matched filter performance (1)

1 Divide denominator and numerator by N2:

γjm =

(1N

hHjjmhjjm

)2

E[

1N2 hH

jjm

(1ρ

IN + hjjmhHjjm − hjjmhH

jjm +∑

l Hjl HHjl

)hjjm

∣∣∣Hjj

] .2 We can circumvent the evaluation of the expectation by computing deterministic

equivalents of all terms. (Notice that this step requires dominated convergence arguments[37] as we exchange the order of the limit and the expectation.)

3 As a direct application of the trace lemma and the continuous mapping theorem, we obtain(1

NhH

jjmhjjm

)2

−(

1

Ntr Φjjm

)2a.s.−−→ 0.

4 Using the trace lemma for independent vectors, we arrive at

1

N2hH

jjmhjjmhHjjmhjjm

a.s.−→ 0

1

N2hH

jjmhjjmhHjjmhjjm −

(1

Ntr Φjjm

)2a.s.−→ 0.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 58 / 101

Page 59: RMT Software

Applications Large-scale MIMO systems

Deriving a deterministic equivalent for the matched filter performance (2)

5 Similarly, we obtain for the remaining terms:

1

N2

L∑l=1

hHjjm

(Hjl H

Hjl − hjlmhH

jlm

)hjjm − 1

N2

L∑l=1

∑k 6=m

hHjlk Φjjmhjlk

a.s.−→ 0

1

N2

L∑l=1

∑k 6=m

hHjlk Φjjmhjlk − 1

N

L∑l=1

K∑k=1

1

Ntr Rjlk Φjjm

a.s.−→ 0.

6 Lastly notice, that hjjm = RjjmQjm

(∑l hjlm + 1√

ρτnjm

). Using this relation and

applying the trace lemmas to each of the remaining individual terms yields:

1

N2

L∑l=1

hHjjmhjlmhH

jlmhjjm −L∑

l=1

∣∣∣∣ 1

Ntr Φjlm

∣∣∣∣2 a.s.−→ 0.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 59 / 101

Page 60: RMT Software

Applications Large-scale MIMO systems

Deriving a deterministic equivalent for the matched filter performance (3)Putting all pieces together, we finally arrive at

γjm − γjma.s.−→ 0

where

γjm =

(1N

tr Φjjm

)2

1

ρN2tr Φjjm︸ ︷︷ ︸

noise

+1

N

∑l,k

1

Ntr Rjlk Φjjm︸ ︷︷ ︸

interference

+∑l 6=j

∣∣∣∣ 1

Ntr Φjlm

∣∣∣∣2︸ ︷︷ ︸pilot contamination

.

Remarks:

As N →∞, the performance becomes independent of ρ.

If K/N → 0, the interference vanishes and pilot contamination remains as the onlyperformance limitation.

Antenna correlation can have a positive effect since

1

Ntr Φjlm = tr Rjlk RjjmQjm,

1

Ntr Rjlk Φjjm =

1

Ntr Rjlk RjjmQjmRjjm

can be small if the subspaces spanned by Rjlk and Rjjm are almost orthogonal.

Smart user scheduling and/or pilot assignment based on statistical CSI could reduceinterference and pilot contamination.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 60 / 101

Page 61: RMT Software

Applications Large-scale MIMO systems

Large-scale MIMO: Numerical results

50 100 150 2000

1

2

3

N = 16, K = 8

Number of antennas N

Ave

rage

rate

per

user

(bits

/s/H

z)

SimulationDeterministic equivalent

Figure: Average rate per user vs. N for L = 4, K = 10, ρ = 0 dB, ρτ = 10 dB.

Remark: The channel correlation matrices were generated as

Rjlk = [α− δ(j − l)(α− 1)]Vjlk VHjlk

where Vjlk ∈ CN×N/2 are standard complex Gaussian with variance 1/N and α = 0.1.M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 61 / 101

Page 62: RMT Software

Applications Large-scale MIMO systems

Large-scale MIMO: Some conclusions

Large random matrix theory allows us to provide easy computable approximations ofthe SINR and sum-rate with linear receivers/precoders.

This technique can treat very general channel models with user specific antennacorrelation.

Since the dimensions of the channel are from the beginning assumed to be verylarge, RMT seems to be the appropriate tool.

The asymptotic expressions are simple, depend only on the channel statistics, andallow us to draw far-reaching conclusions (e.g., impact of correlation, noise vanishes,pilot contamination).

These results could be further leveraged for optimization on higher layers(scheduling, resource allocation, etc.).

For more literature on this topic, see, e.g., [38, 39, 34, 35, 36].

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 62 / 101

Page 63: RMT Software

Applications Polynomial expansion detectors

Polynomial expansion detectors

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 63 / 101

Page 64: RMT Software

Applications Polynomial expansion detectors

Problem setting and motivation

Consider the N × K MIMO channel

y = Hx + σn =K∑

k=1

hk xk + σn

where x ∼ CN (0, IK ), n ∼ CN (0, IN ) and H ∈ CN×K random but known to the receiver.

Several linear detectors require the matrix inversion (α ≥ 0):

(HHH + αIN

)−1

This operation is expensive (complexity and energy) : O(N2) [40].

In particular, for large (distributed) antenna arrays for multi-user communications.

N ≈ 100, K ≈ 50 (or even more)

Asymptotic moments can be used calculate approximations of the matrix inverse.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 64 / 101

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Applications Polynomial expansion detectors

A note on matrix inversion

Let A be a non-singular N × N matrix with characteristic polynomial:

det (zIN − A) =N∏

i=1

(z − λi ) =N∑

i=0

αi zi

where λi are the eigenvalues of A and αi depend on the eigenvalues of A.

Caley-Hamilton Theorem (“Every matrix satisfies its own characteristic polynomial”):

N∑i=0

αi Ai = 0 ⇐⇒ A−1 =

N−1∑l=0

−αl+1

α0Al .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 65 / 101

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Applications Polynomial expansion detectors

Polynomial expansion detectors

Assume we want to estimate x with a linear detector:

x = HH(

HHH + αIN

)−1

y ≈ HHL−1∑l=0

wl

(HHH

)l

y, L < rank(H).

The weights w = [w0, . . . ,wL−1]T can be chosen to minimize:

w = arg minw

E

∥∥∥∥∥x−HH

(L−1∑l=0

wl

(HHH

)l)

y

∥∥∥∥∥2

2

= Φ−1ϕ

where Φ ∈ RL×L+ and ϕ ∈ RL

+ are defined as

[Φ]ij =1

Ntr(

HHH)i+j

+ σ2 1

Ntr(

HHH)i+j−1

[ϕ]i =1

Ntr(

HHH)i

.

Idea: Replace the moments 1N

tr(HHH

)kby their deterministic approximations Mk .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 66 / 101

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Applications Polynomial expansion detectors

Related works and existing moment results

Asymptotic moment results:

[12] : H as i.i.d. entries with zero mean and variance 1/K .

[41] : Hij have zero mean and variance 1K

Vij .

[41] : H = [A1h1, . . . ,AK hK ] B, where hj are vectors of i.i.d. entries with zero mean andvariance 1/K , B is diagonal and Aj are absolutely summable Toeplitz matrices.

[42] : H = TWP12 , where T is Toeplitz, P is diagonal and W has either i.i.d. elements

with zero mean and variance 1/K or is created by taking K ≤ N columns from arandom Haar (unitary) matrix.

[43] : H = [h1, . . . , hK ] where hk = R12k wk and wk are vectors of i.i.d. with zero mean

and variance 1/K .

[44] : H is a random Vandermonde matrix.

Related works on polynomial expansion receivers:

[45, 46, 47, 48, 40, 49, 50, 43, 51]

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 67 / 101

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Applications Polynomial expansion detectors

Polynomial expansion receivers : Numerical results

0 2 4 6 8 10 12 140

5

10

15

Matched Filter

L = 2

L = 3

L = 6

LMMSE

Transmit SNR [dB]

Ave

rage

rece

ived

SIN

R[d

B]

Matched FilterL = 2

L = 3

L = 6

LMMSE

0 2 4 6 8 10 12 1410−4

10−3

10−2

10−1 Matched Filter

L = 2

L = 3

L = 6

LMMSE

Transmit SNR [dB]

Ave

rage

bit

erro

rra

te

Matched FilterL = 2

L = 3

L = 6

LMMSE

N = 100, K = 40

Channel with a generalized variance profile: Correlation matrices Rk randomlycreated (extended version of Jake’s model) (see [43] for details)

Uncoded BER = E[Q(√

SINR)]

One can also derive a deterministic equivalent of the SINR for a given approximationorder L, as shown by solid lines in the left plot (see, e.g., [40, 43]).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 68 / 101

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Applications Polynomial expansion detectors

Some remarks

Polynomial expansion detectors are a low complexity receiver architecture for largeMIMO channels (also for CDMA).

Polynomial expansion detectors allow to trade-off complexity against performancewith a very fine granularity (L = 1: matched filter, L = rank(H): MMSE detector).

If the channel coherence time is large and the channel statistics are known, theasymptotic moments can be precomputed and only the matrices(HHH)l , l = 1, . . . , L, must be calculated.

The asymptotic moments for a wide range of channel models are known and couldbe used.

Practical implementations?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 69 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Iterative deterministic equivalents:Multiphop relay channel

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 70 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Iterative deterministic equivalents

So far: Random matrix models composed of sums of independent random matrices.

But what about products?Key idea:

1 Consider a functional f (H1, . . . ,HK ) of K independent random matrices H1, . . . ,HK .

2 Treat H2, . . . ,HK as constant and derive a deterministic equivalent of f with respectto H1, i.e.,

f (H1, . . . ,HK )− f1 (H2, . . . ,HK )a.s.−−→ 0.

3 Repeat this procedure by iteratively removing the “randomness” related to all matrices:

f1 (H2, . . . ,HK )− f2 (H3, . . . ,HK )a.s.−−→ 0

...

fK−1 (HK )− fKa.s.−−→ 0.

The same idea has been applied to different channel models, e.g.,I Double-scattering channel: H = R

12 W1S 1

2W2T

12 , where W1,W2 are standard complex

Gaussian and R,S,T are deterministic [52, 53].

I Random beamforming: H = WUP12 , where W has a generalized variance profile, U is

a random unitary matrix, and P is diagonal and deterministic [54].

Today: Multihop relay channel

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 71 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Multihop MIMO amplify-and-forward relay channel

A source communicates its message x to a destination via K − 2 relay nodes.

Each node receives a signal only from its predecessor.

The relays amplify-and-forward their received signal to the next node.

We do not account for the transmission delay(e.g., K − 1 time slots with a TDMA protocol).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 72 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

About the literature

Asymptotic analysis of the dual-hop relay channel, mutual information via numericalintegration and inversion of the Stieltjes transform [55]

Asymptotic analysis of our model, recursive expression of the Stieltjes transform, noclosed-form results of the mutual information [56]

Asymptotic analysis of the dual-hop relay channel with antenna correlation,cumulants of the mutual information (Replica method) [57]

K -hop AF channel with antenna correlation, but noise only at destination, optimalprecoders and mutual information [58]

AF-relay channel without noise (K →∞) [59]

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 73 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Channel model

Received signal vector yk ∈ Cnk at node k:

yk =√αk Hk

√βk−1

nk−1yk−1︸ ︷︷ ︸

Signal from node k − 1

+ nk

Hk ∈ Cnk×nk−1 is a standard complex Gaussian matrix, i.e., [Hk ]i,j ∼ CN (0, 1).

y0 = x ∼ CN (0, In0 ) is the channel input vector

nk ∼ CN (0, Ink ) noise at node k

Each nodes scales the transmit signal according to its power constraint ρk :

βk =ρk

1nk

trE[yk yH

k

]Large system limit: n0 →∞, such that

0 < lim infn

ck ,nk−1

nk≤ lim sup ck <∞, k = 1, . . . ,K .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 74 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Mutual information

The normalized mutual information Ik (βk ) between yk and x can be written as:

Ik (βk ) = Jk (1,βk )− Jk (1,β′k )

where

Jk (x ,βk ) =1

nklog det

(Ink + x

αkβk−1

nk−1Hk Rk−1HH

k

)with

R0 = E[xxH]

= In

Rk = E[yk yH

k

]= Ink +

αkβk−1

nk−1Hk Rk−1HH

k , k = 1, . . . ,K

and βk = [β0, · · · , βk−1], β′k = [0, β1, · · · , βk−1].

Example

I2(β2) =1

n2log det

(In2 +

α2β1

n1H2R1HH

2

)− 1

n2log det

(In2 +

α2β1

n1H2HH

2

)

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 75 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Asymptotic analysis

By the trace Lemma and the fact that the matrices Rk have a.s. bounded spectral norm,it is straight-forward to prove the following result:

βka.s.−−−→

n→∞βk =

ρk

1 + αkρk−1, k = 1, . . . ,K − 1

where β0 = β0 = ρ0.

In order to proceed, we will exploit the recursive definition of the matrices

Rk = Ink +αkβk−1

nk−1Hk Rk−1HH

k

to find iterative deterministic equivalents of

Jk (x ,βk ) =1

nklog det

(Ink + x

αkβk−1

nk−1Hk Rk−1HH

k

).

Note that Hk Rk−1HHk can be seen as a right-sided correlation model with random

correlation. In each step of the analysis, we derive a deterministic equivalent of Jk withrespect to Hk while Rk−1 is assumed to be deterministic.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 76 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Proof sketch1 It follows from the deterministic equivalent of the mutual information for the right-sided

correlation model [13, 14] that

Jk (x , βk−1)− Jk (x , βk−1)a.s.−→ 0, k ≥ 1

where

Jk

(x , βk−1

)=

1

nklog det

([ck + ek−1] Ink−1 + xαk βk−1Rk−1

)+ (1− ck ) log (ck + ek−1)−

ek−1

ck + ek−1− log (ck )

and ek−1 is given as the unique positive solution to

ek−1 =1

nktrαk βk−1Rk−1

(αk βk−1Rk−1

ck + ek−1+

1

xInk−1

)−1

.

2 Using the recursion Rk−1 = Ink−1 +αk−1βk−2

nk−2Hk−1Rk−2HH

k−1, we obtain

Jk (x , βk−1) = ckJk−1

(xαk βk−1

ck + xαk βk−1 + ek−1

, βk−2

)+ ck log

(1 +

xαk βk−1

ck + ek−1

)+ log

(1 +

ek−1

ck

)−

ek−1

ck + ek−1.

3 Similarly, we can find recursive expressions for ek .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 77 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Theorem (Deterministic equivalent of Jk (x ,βk ) [53])

Jk (x ,βk )− Jk

(x , βk

) a.s.−−−→n→∞

0

where Jk

(x , βk

)is recursively defined for k ≥ 2 as

Jk

(x , βk

)= ck Jk−1

(xαk βk−1

ck + xαk βk−1 + ek−1

(x , βk−1

) , βk−1

)

+ ck log

(1 +

xαkβk−1

ck + ek−1

(x , βk−1

))

+ log

(1 +

ek−1

(x , βk−1

)ck

)− ek−1

(x , βk−1

)1 + ek−1

(x , βk−1

)and ek

(x , βk−1

)for k ≥ 0 is given on the next slide. Moreover,

J1

(x , β0

)= c1 log

(1 +

xα1β0

c1 + e0

(x , β0

))+ log

(1 +

e0

(x , β0

)c1

)− e0

(x , β0

)1 + e0

(x , β0

) .M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 78 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Theorem (Recursive definition of ek

(x , βk

)[53])

ek

(x , βk

)is given as the unique positive solution to

ek

(x , βk

)= ck+1

(ck+1 + ek

(x , βk

))− ck+1

(ck+1 + ek

(x , βk

))2

xαk+1βk

mk−1

(xαk+1βk

ck+1 + xαk+1βk + ek

(x , βk

) , βk−1

)where

mk

(x , βk

)=

xck+1

ck+1 + ek

(x , βk

) .The initial values m0(x , β0) and e0(x , β0) are given in closed-form:

m0(x , β0) =c1

α1β0

c1+e0(x,β0)+ 1

x

+ (1− c1)x

e0

(x , β0

)= −xα1β0(1− c1) + c1

2+

√(xα1β0(1− c1) + c1

)2+ 4xα1β0c2

1

2.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 79 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Numerical example: Amplify-and-forward Multi-hop relay channel

−10 0 10 20 300

2

4

6

8

10n0 = n4 = 4, n1 = n3 = 8, n2 = 12

ρ0 (dB)

E[ n

k

n0I k

(βk)]

(bits

/s/H

z)

k = 1

k = 2

k = 3

k = 4

ρ1 = ρ3 = 0.7ρ0, ρ2 = 0.5ρ0, α1 = 1, α2 = α4 = 0.7, α3 = 0.5

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 80 / 101

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Applications Iterative deterministic equivalents: Multiphop relay channel

Some conclusions

Iterative deterministic equivalents are a natural way to extend deterministicequivalents to products (or other functionals) of independent matrices.

Key idea: Iteratively remove “randomness” related to one/some of the randommatrices until none is left.

One can easily derive new deterministic equivalents for involved matrix models byrelying of the large pool of existing results (see [53, 54] for some examples).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 81 / 101

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Applications Random network topologies

Random network topologies

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 82 / 101

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Applications Random network topologies

Moving to random network topologies

Consider a single cell uplink channel from K UTs to a BS with N antennas.

The UTs are randomly distributed over the cell of radius R. The distancesd1, . . . , dK from the UTs to the BS are i.i.d. random variables with distribution F .

The path loss from UT k to the BS depends on its distance via the path lossfunction `(dk ), e.g.,

`(d) =1

(1 + d)β

for some path loss exponent β ∈ [0, 5].

What is the average mutual information of this channel?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 83 / 101

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Applications Random network topologies

Channel model and mutual information

The received signal at the BS is given as

y = Hs + σn = XT12 s + σn

where s ∼ CN (0, IK ), n ∼ CN (0, IN ), [X]i,j ∼ CN(0, 1

KIN

), and

T = diag (`(d1), . . . , `(dK )).

We are interested in the quantity

I (σ2) = E[

1

Nlog det

(IN +

1

σ2XTXH

)]where the expectation is with respect to H and T.

Key idea:

Notice H = XT12 is a right-sided correlation model with random correlation matrix T.

Since the distances dk are i.i.d., the e.s.d. of T converges weakly to a limitingdistribution.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 84 / 101

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Applications Random network topologies

Asymptotic analysis

Recall the deterministic equivalent for fixed T:

I (σ2) =1

N

K∑k=1

log (1 + ctk m)− log(σ2m

)− 1

K

K∑k=1

tk m

1 + ctk m

where m is the unique solution to

m =

(1

K

K∑k=1

tk

1 + ctk m(z)+ σ2

)−1

.

As N,K →∞ while N/K → c ∈ (0,∞), we have from the strong law of large numbers:

ma.s.−→

(∫ R

0

`(d)

1 + c`(d)mdF (d) + σ2

)−1

I (σ2)a.s.−→ 1

c

∫ R

0

log (1 + c`(d)m) dF (d)− log(σ2m

)−∫ R

0

`(d)m

1 + c`(d)mdF (d).

Based on this observation, we can prove the following result:

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 85 / 101

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Applications Random network topologies

Numerical results

−10 −5 0 5 10 15 200

0.5

1

1.5

2

2.5

N = 8, K = 4

SNR = 1σ2 (dB)

I( σ

2)

(bits

/s/H

z)

SimulationApproximation

Simulation assumptions:

Path loss function

f (d) =1

(1 + d)β, β = 3.7

Uniform user distribution

dF (d) =2d

R2, d ∈ [0,R]

N = 8, K = 4, R = 1

Corollary

I (σ2)→1

c

∫ R

0log (1 + c`(d)m) dF (d)− log

(σ2m

)+

∫ R

0

`(d)m

1 + c`(d)mdF (d)

where m is the unique solution to the following fixed-point equation

m =

(∫ R

0

`(d)

1 + c`(d)mdF (d) + σ2

)−1

.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 86 / 101

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Applications Random network topologies

Some remarks and conclusions

Large random matrix theory can be also used to derive asymptotic approximations ofthe performance of networks with random topologies.

Surprisingly, these results are also very accurate for small system dimensions.

Many extensions are possible : random BS locations, multi-cell systems with/withoutcooperation, more complex channel models with directional antennas (see, e.g., [60])

These results allow one to optimize certain system parameters in order to maximizethe average performance for a given channel model and user distribution. This is,e.g., important for optimal cell planning.

This line of research is in its infancy; not many results are known.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 87 / 101

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Summary and perspectives

Outline1 Introduction

Advanced communication systems and related challengesWhy do we need large random matrix theory?Goals of this tutorial and take-away messages

2 TheorySequences of random matrices and convergence typesExample: Asymptotic SINR with linear receiversThe Stieltjes transform and its propertiesLimiting spectral distribution and the Marcenko-Pastur lawAsymptotic rates with linear receiversAsymptotic mutual information and its fluctuationsAsymptotic momentsDeterministic equivalents: Definition and overview of existing results

3 ApplicationsOptimal channel trainingLarge-scale MIMO systemsPolynomial expansion detectorsIterative deterministic equivalents: Multiphop relay channelRandom network topologies

4 Summary and perspectives

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 88 / 101

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Summary and perspectives

Summary and perspectives

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 89 / 101

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Summary and perspectives

Summary

Large random matrix theory can be used to obtain deterministic approximations ofseveral performance measures of advanced communication systems:

I Mutual information/achievable ratesI SINR with linear receiversI Outage probability

In many cases, the asymptotic results provide very tight approximations for smallsystem dimensions (even for 2× 2 MIMO channels!).

With the tools presented, one can easily account for many important characteristicsof modern communication systems, e.g.:

I Path lossI Imperfect channel state informationI InterferenceI Antenna correlationI Line-of-sight channelsI Random user locations/network topologies

These results can be used, e.g., to:

I Gain insights about the most important system parameters (e.g., antenna correlation)I Simplify optimization problems (e.g., channel training, precoding matrices)I Design reduced complexity detectors

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 90 / 101

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Summary and perspectives

Perspectives

Large random matrix theory is a highly developed tool for the theoretical analysis ofcommunication systems.

Asymptotic results for most relevant channel models exist.

Some promising avenues for future research are in the areas of:

I Combination of random matrix theory and stochastic geometry for the analysis of largerandom networks.

I Cross-layer optimization, e.g., user scheduling, pilot assignment based on a“deterministic abstraction of the physical layer”.

I Game theory and random matrix theory for distributed resource allocation.

I Analysis of “new” channel models, e.g., full-duplex radios, more involved multihoprelay channels, 3D beamforming.

I Signal processing, e.g., multi-source power estimation, capacity estimation, failuredetection in large networks [61].

To dig deeper, we recommend the textbooks [4, 62] for applications of randommatrix theory to problems in wireless communications and signal processing and [3]for a more theoretical introduction.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 91 / 101

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Summary and perspectives

Available in bookstores!

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 92 / 101

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Summary and perspectives

Detailed outlineRomain Couillet, Merouane Debbah, Random Matrix Methods for WirelessCommunications.

1 Theoretical aspects1 Preliminary2 Tools for random matrix theory3 Deterministic equivalents4 Central limit theorems5 Spectrum analysis6 Eigen-inference7 Extreme eigenvalues

2 Applications to wireless communications1 Introduction2 System performance: capacity and rate-regions

1 Introduction2 Performance of CDMA technologies3 Performance of multiple antenna systems4 Multi-user communications, rate regions and sum-rate5 Design of multi-user receivers6 Analysis of multi-cellular networks7 Communications in ad-hoc networks

3 Detection4 Estimation5 Modelling6 Random matrix theory and self-organizing networks7 Perspectives8 Conclusion

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 93 / 101

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Summary and perspectives

Thank you!

www.flexible-radio.com

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 94 / 101

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Summary and perspectives

References I

[1] Cisco, “Cisco visual networking index: Global mobile data traffic forecast update, 2010–2015,” Feb. 2011, White paper.[Online]. Available:http://www.cisco.com/en/US/solutions/collateral/ns341/ns525/ns537/ns705/ns827/white paper c11-520862.pdf

[2] A. Fehske, G. Fettweis, J. Malmodin, and G. Biczok, “The global footprint of mobile communications: The ecological andeconomic perspective,” IEEE Commun. Mag., vol. 49, no. 8, pp. 55–62, Aug. 2011.

[3] Z. D. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer Series inStatistics, New York, NY, USA, 2009.

[4] R. Couillet and M. Debbah, Random matrix methods for wireless communications, 1st ed. New York, NY, USA:Cambridge University Press, 2011.

[5] 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.

[6] 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.

[7] A. Edelman, “Eigenvalues and condition numbers of random matrices,” Ph.D. dissertation, Massachusetts Institute ofTechnology, Department of Mathematics, Cambridge, MA, US, 1989.

[8] A. W. van der Vaart, Asymptotic Statistics (Cambridge Series in Statistical and Probabilistic Mathematics). CambridgeUniversity Press, New York, 2000.

[9] Z. D. Bai and J. Silverstein, “No eigenvalues outside the support of the limiting spectral distribution of large-dimensionalrandom matrices,” Ann. Probab., vol. 26, pp. 316–345, 1998.

[10] R. Couillet, J. W. Silverstein, Z. D. Bai, and M. Debbah, “Eigen-inference for energy estimation of multiple sources,” IEEETrans. Inf. Theory, vol. 57, no. 4, pp. 2420–2439, Apr. 2011.

[11] S. Verdu and S. Shamai, “Spectral efficiency of CDMA with random spreading,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp.622–640, Mar. 1999.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 95 / 101

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Summary and perspectives

References II

[12] D. Jonsson, “Some limit theorems for the eigenvalues of a sample covariance matrix,” J. Multivariate Anal., vol. 12, no. 1,pp. 1–38, 1982.

[13] J. W. Silverstein and Z. D. Bai, “On the empirical distribution of eigenvalues of a class of large dimensional randommatrices,” Journal of Multivariate Analysis, vol. 54, no. 2, pp. 175–192, 1995.

[14] R. Couillet, M. Debbah, and J. W. Silverstein, “A deterministic equivalent for the analysis of correlated MIMO multipleaccess channels,” IEEE Trans. Inf. Theory, vol. 57, no. 6, pp. 3493–3514, Jun. 2011.

[15] W. Hachem, P. Loubaton, and J. Najim, “Deterministic equivalents for certain functionals of large random matrices,”Annals of Applied Probability, vol. 17, no. 3, pp. 875–930, 2007.

[16] 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, 2012, to appear. [Online]. Available:http://arxiv.org/abs/0906.3682

[17] 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

[18] F. Dupuy and P. Loubaton, “On the capacity achieving covariance matrix for frequency selective MIMO channels using theasymptotic approach,” IEEE Trans. Inf. Theory, vol. 57, no. 9, pp. 5737–5753, Sep. 2011. [Online]. Available:http://arxiv.org/abs/1007.0875

[19] W. Hachem, P. Loubaton, and J. Najim, “A CLT for information theoretic statistics of Gram random matrices with a givenvariance profile,” The Annals of Probability, vol. 18, no. 6, pp. 2071–2130, Dec. 2008.

[20] A. Kammoun, M. Kharouf, W. Hachem, and J. Najim, “A central limit theorem for the SINR at the LMMSE estimatoroutput for large dimensional systems,” IEEE Trans. Inf. Theory, vol. 55, pp. 5048–5063, Nov. 2009.

[21] A. L. Moustakas, S. H. Simon, and A. M. Sengupta, “MIMO capacity through correlated channels in the presence ofcorrelated interferers and noise: A (not so) large n analysis,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2545–2561, Oct.2003.

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References III

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[23] J. Hoydis, M. Kobayashi, and M. Debbah, “Optimal channel training in uplink network MIMO systems,” IEEE Trans. SignalProcess., vol. 59, no. 6, pp. 2824–2833, Jun. 2011.

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References IV

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References V

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References VI

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Some useful lemmas

Matrix inversion lemma

Let A ∈ CN×N , D ∈ Cn×n be invertible, and B ∈ CN×n, C ∈ Cn×N . Then,(A BC D

)−1

=

( (A− BD−1C

)−1 −A−1B(D− CA−1B

)−1

−(A− BD−1C

)−1CA−1

(D− CA−1B

)−1

).

Identity for the matrix inverse

Let A ∈ CN×n and B ∈ Cn×N such that (IN + AB) is invertible. Then

IN − A (In + BA)−1 B = (IN + AB)−1 .

Resolvent identity

For invertible matrices A and B, we have the following identity:

A−1 − B−1 = A−1(B− A)B−1.

Matrix inversion lemma [13, Eq. (2.2)]

Let A ∈ CN×N be invertible. Then, for any vector x ∈ CN and any scalar c ∈ C such thatA + cxxH is invertible,

xH(

A + cxxH)−1

=xHA−1

1 + cxHA−1x.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC’12 Paris 101 / 101