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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

WCNC 2012, Paris, 1 April, 2012

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 1 / 101

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 WCNC12 Paris 2 / 101

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 WCNC12 Paris 3 / 101

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 WCNC12 Paris 4 / 101

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 WCNC12 Paris 5 / 101

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 WCNC12 Paris 6 / 101

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 WCNC12 Paris 7 / 101

Introduction Why do we need large random matrix theory?

A simple example: Network MIMO uplink channel

Received signal at the base stations:

y =Hx + n =

(g1,1

g1,2

g2,1 g2,2

)(x1x2

)+

(n1n2

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

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

I () = E[log det

(I + HHH

)]= ?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 8 / 101

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

= 0.5

(dB)

E[I()](nats/s/Hz)

SimulationApproximation

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

E [I ()] 2 log(

1 + 2(1 + )+

1 + 4(1 + )) 2 log(2e)

+4

1 +

1 + 4(1 + ).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 9 / 101

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 WCNC12 Paris 10 / 101

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 WCNC12 Paris 11 / 101

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 WCNC12 Paris 12 / 101

Theory

Theory

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 13 / 101

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 (,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 = R

12 WT

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

I H = [h1 hK ], where hj = R12j wj , Rj CNN , and wj CN (0, IN )

I H = W + A, where [W]i,j i.i.d., and is A is deterministicI ...

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 14 / 101

Theory Sequences of random matrices and convergence types

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().

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 15 / 101

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( : 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 .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 16 / 101

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, 2).

Matched filter: xk = hHk y

SINRMFk (2) =

|hHk hk |2hHk(2IN + Hk HHk

)hk

MMSE detector: xk = hHk

(HHH + 2IN

)1y

SINRMMSEk (2) = hHk

(Hk H

Hk +

2IN)1

hk

where Hk CN(K1) is H with its kth column removed.

Goal: Show that SINRka.s. SINRk , for N,K , NK c (0,).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 17 / 101

Theory Example: Asymptotic SINR with linear receivers

Two useful trace lemmas

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

Theory Example: Asymptotic SINR with linear receivers

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

SINRMFk (2) =

|hHk hk |2hHk(2IN + Hk HHk

)hk

(

NK

)22 N

K+ 1

K

j 6=k h

Hj hj

(

NK

)22 N

K+ K1

KNK

c2 + 1

, SINRMFk (2).

Similarly,

SINRMMSEk (2) = hHk

(Hk H

Hk +

2IN)1

hk

1K

tr(

Hk HHk +

2IN)1

Can we go further?

For two sequences (an)n1 and (bn)n1, the following notations are equivalent:

(i) an bn (ii) an bn a.s.n

0.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 19 / 101

Theory Example: Asymptotic SINR with linear receivers

Finite rank perturbations

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

Let z C \ R+, A CNN and B CNN with B Hermitian nonnegative definite, andx CN . Then, tr((B zIN )1 (B + xxH zIN)1)A Adist(z ,R+)where dist is the Euclidean distance. If z < 0 and lim supN A

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 + 2IN)1 a.s. c 1

2c2 1

2c+

(1 c + 2)2 + 4c2

2c2.

To this end, we need some preparation:

Definition (Stieltjes transform)

Let be a finite nonnegative measure with support supp() R, i.e., (R)

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

z dFA() =

1

N

Ni=1

1

i z =1

Ntr (A zIN )1 .

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

Properties

Let m(z) be the Stieltjes transform of a finite measure on R. Then,(i) (R) = limyiym(iy). If is a probability measure, we have (R) = 1.(ii) ([a, b]) = limy0+

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

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 WCNC12 Paris 22 / 101

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

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 IN a.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 WCNC12 Paris 23 / 101

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 WCNC12 Paris 24 / 101

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

Densityfc(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 WCNC12 Paris 25 / 101

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 Fcwhich 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 WCNC12 Paris 26 / 101

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

Kj=1

[(HHH zIN

)1]j,j

[Matrix inversion lemma] =K N

N

1

z+

1

N

Kj=1

1

z + hHj(

IN Hj(

HHj Hj zIK1)1

Hj

)hj

[Identity for matrix inverse] =K N

N

1

z+

1

N

Kj=1

1

z(

1 + hHj

(Hj HHj zIN

)1hj

)

[Trace lem. & Rank-1 per.] K NN

1

z+

1

N

Kj=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 WCNC12 Paris 27 / 101

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 c2cz

12c

(1 c z)2 4cz2cz

where the branch of

(1cz)24cz

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 c1)+

(x) +1

2picx

(x a)+(b x)+

where a = (1c)2, b = (1 +c)2, (x)+ , max(0, x), and (x) is the Dirac delta.M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 28 / 101

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 (2)

a.s. SINRMFk (2) = c2 + 1

SINRMMSEk (2)

a.s. SINRMMSEk (2) = cmc (2).

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

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 29 / 101

Theory Limiting spectral distribution and the Marcenko-Pastur law

SINR with linear receivers: Numerical results

10 5 0 5 10 15 2010

0

10

20

N = 16, K = 8

SNR = 12 (dB)

SINRk(dB)

SINRMFk

SINRMMSEk

E[SINRMFk

]E[SINRMMSEk

]

10 5 0 5 10 15 2010

0

10

20

N = 64, K = 32

SNR = 12 (dB)

SINRMFk

SINRMMSEk

E[SINRMFk

]E[SINRMMSEk

]

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 WCNC12 Paris 30 / 101

Theory Asymptotic rates with linear receivers

Asymptotic rates with linear receivers

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

Let (Xn)n1 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 Xn

a.s. X , then f (Xn) a.s. f (X ).

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

Rk (2) , log

(1 + SINRk (

2))

a.s. Rk (2) , log(

1 + SINRk (2))

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

1

N

Kk=1

Rk (2) 1

N

Kk=1

Rk (2)

a.s. 0.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 31 / 101

Theory Asymptotic mutual information and its fluctuations

Asymptotic mutual informationF HH

H Fc , almost surely, implies that for any bounded continuous function g :g()dF HH

H

()a.s.

g()dFc ().

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

I (2) , 1N

log det

(IN +

1

2HHH

)=

log

(1 +

2

)dF HH

H

().

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

Denote by max and min the largest and smallest eigenvalue of HHH, respectively. Then,

(i) maxa.s. (1 +c)2, (ii) min a.s. (1

c)21{c 1}.

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

Theorem (Asymptotic mutual information)

I (2)a.s.

g()dFc () , I (2) =

1

clog(1 + cmc ) + log

(1 +

1

21

1 + cmc

) mc

1 + cmc

where mc = mc (2).M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 32 / 101

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 = 12 (dB)

(bits/s/Hz)

I(2

E[I(2)

]

Figure: Average normalized mutual information E[I (2)

]and its asymptotic approximation I (2)

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 WCNC12 Paris 33 / 101

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 (2) =

2

(1

xmHHH (x)

)dx

since

dI (2)

d2=

d

d2

(1

Nlog det

(IN +

1

2HHH

))=

d

d2

(1

Nlog det

(2IN + HH

H) log(2)

)=

1

Ntr(2IN + HH

H)1 12.

One can also verify that

I (2) =

2

(1

xmc (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 WCNC12 Paris 34 / 101

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

Nc

(I(2) I(2))

Frequency

HistogramN (0, 1) Example

Approximation of the outage probability:

Pout(r) = Pr(

NI (2) < r)

1 Q(

r NI (2)c

).

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

N

c

(I (2) I (2)

) N (0, 1)

where the asymptotic variance is given as 2c = log(

1 cmc (2)2(1+cmc (2))2

).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 35 / 101

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 b,

m(z) = 1z

k=0

Mkzk

where Mk =k dF () 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 ,

k dFc () =

k1i=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)kk! G (k)(0), where G (k)(z) isthe kth derivative of G(z).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 36 / 101

Theory Deterministic equivalents: Definition and overview of existing results

What happens if things do not converge?

Bn = HnTnHHn

where Hn Cnn, [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 WCNC12 Paris 37 / 101

Theory Deterministic equivalents: Definition and overview of existing results

Deterministic equivalents

Definition

Let (an)n1 be an infinite sequences of complex random variables and (bn)n1 an infinitesequence of complex numbers. We say that bn is a deterministic equivalent of an, iff

an bn a.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 asknowing an.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 38 / 101

Theory Deterministic equivalents: Definition and overview of existing results

Right-sided correlation model

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

Let X CNK , [X]i,j CN(1, 1

K

)i.i.d., and let T = diag (t1, . . . , tK ) RKK+ be

deterministic, satisfying maxk tk

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, z1e-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 WCNC12 Paris 40 / 101

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 CNK , [X]i,j CN (0,2i,jK

), and A CNK . Define B = (X + A) (X + A)H. Furtherdenote Dj = diag(

21,j , . . . ,

2N,j ) and Di = diag(

2i,1, . . . ,

2i,K ) i , j . Let N,K while

0 < lim inf NK lim sup N

K

Theory Deterministic equivalents: Definition and overview of existing results

Generalized variance profile

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

Let B = XXH + S, where X CNK is random and S CNN is Hermitian nonnegative definite.The jth column xj of X is given as xj = Rj zj , where zj CN (0, 1N IN ), and Rj CNN isdeterministic. Denote Rj = Rj R

Hj . Let D CNN be a deterministic Hermitian matrix. Then, as

N,K such that 0 < lim inf N/K lim sup N/K

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 WCNC12 Paris 43 / 101

Applications

Outline1 Introduction

4 Summary and perspectives

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 44 / 101

Applications

Applications

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 45 / 101

Applications Optimal channel training

Optimal channel training

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 46 / 101

Applications Optimal channel training

Optimal channel training

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

y =

KHx + n

where H CNK , [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 WCNC12 Paris 47 / 101

Applications Optimal channel training

Channel training

coherence time T

training data transmission

The receiver estimates the channel channel based on the observation

Y =

KH + N

where CK , H = 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,

+ K

), [H]i,j CN

(0,

K

+ K

).

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 48 / 101

Applications Optimal channel training

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

Rnet() ,(

1 T

)E[

1

Nlog det

(IN +

eff()

KHHH

)]and

eff() =2

K(1 + ) + .

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

Rnet() ,(

1 T

)I

(1

eff()

)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 = N

K, with SNR = 1

x.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 49 / 101

Applications Optimal channel training

Optimization of the large system approximation

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

d

dRnet() = 1

TI

(1

eff()

)(

1 T

) eff()eff()2

(mc

( 1eff()

) eff()

)!

= 0

where

eff() =2(1 + )K

(K(1 + ) + )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 a good approximation of for small N,K?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 50 / 101

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, = 0 dB, T = 100

Training length

Rnet()(bits/s/Hz)

SimulationApproximation

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

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 51 / 101

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 = (dB)

Optim

altraining

length

SimulationApproximation

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

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 52 / 101

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 WCNC12 Paris 53 / 101

Applications Large-scale MIMO systems

Large-scale MIMO systems

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 54 / 101

Applications Large-scale MIMO systems

Analysis of large-scale MIMO systems

yj =

Ll=1

Kk=1

hjlk xlk + nj =

Ll=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 CNN is deterministic and wjlk CN (0, IN ).M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 55 / 101

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

yjk = hjjk +l 6=j

hjlk +1

njk

where is the effective training SNR.

Computing the MMSE estimate hjjk = Rjjk Qjk yjk yields:

hjjk = hjjk + hjjk

hjjk CN (0,jjk ) , hjjk CN (0,Rjjk jjk )jlk = Rjjk Qjk Rjlk , Qjk =

(1 IN +

l

Rjlk)1

.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 56 / 101

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 + jm)]

with the effective SINR

jm =

rHjmhjjm2E[rHjm

(1

IN + hjjmhHjjm hjjmhHjjm +

l Hjl HHjl

)rjmHjj ] .

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 WCNC12 Paris 57 / 101

Applications Large-scale MIMO systems

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

1 Divide denominator and numerator by N2:

jm =

(1N

hHjjmhjjm)2

E[

1N2

hHjjm

(1

IN + hjjmhHjjm hjjmhHjjm +

l Hjl H

Hjl

)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

NhHjjmhjjm

)2(

1

Ntr jjm

)2a.s. 0.

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

1

N2hHjjmhjjmh

Hjjmhjjm

a.s. 0

1

N2hHjjmhjjmh

Hjjmhjjm

(1

Ntr jjm

)2a.s. 0.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 58 / 101

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

Ll=1

hHjjm(

Hjl HHjl hjlmhHjlm

)hjjm 1

N2

Ll=1

k 6=m

hHjlk jjmhjlka.s. 0

1

N2

Ll=1

k 6=m

hHjlk jjmhjlk 1NL

l=1

Kk=1

1

Ntr Rjlk jjm

a.s. 0.

6 Lastly notice, that hjjm = RjjmQjm(

l hjlm +1

njm)

. Using this relation and

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

1

N2

Ll=1

hHjjmhjlmhHjlmhjjm

Ll=1

1N tr jlm2 a.s. 0.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 59 / 101

Applications Large-scale MIMO systems

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

jm jm a.s. 0where

jm =

(1N

tr jjm)2

1

N2tr jjm

noise

+1

N

l,k

1

Ntr Rjlk jjm

interference

+l 6=j

1N tr jlm2

pilot contamination

.

Remarks:

As N , the performance becomes independent of .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 jlm = tr Rjlk RjjmQjm,

1

Ntr Rjlk jjm =

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 WCNC12 Paris 60 / 101

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

Average

rate

peruser

(bits/s/Hz)

SimulationDeterministic equivalent

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

Remark: The channel correlation matrices were generated as

Rjlk = [ (j l)( 1)]Vjlk VHjlkwhere Vjlk CNN/2 are standard complex Gaussian with variance 1/N and = 0.1.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 61 / 101

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 WCNC12 Paris 62 / 101

Applications Polynomial expansion detectors

Polynomial expansion detectors

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 63 / 101

Applications Polynomial expansion detectors

Problem setting and motivation

Consider the N K MIMO channel

y = Hx + n =K

k=1

hk xk + n

where x CN (0, IK ), n CN (0, IN ) and H CNK random but known to the receiver.

Several linear detectors require the matrix inversion ( 0):

(HHH + IN

)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 WCNC12 Paris 64 / 101

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 i ) =N

i=0

i zi

where i are the eigenvalues of A and i depend on the eigenvalues of A.

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

Ni=0

i Ai = 0 A1 =

N1l=0

l+10

Al .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 65 / 101

Applications Polynomial expansion detectors

Polynomial expansion detectors

Assume we want to estimate x with a linear detector:

x = HH(

HHH + IN)1

y HHL1l=0

wl(

HHH)l

y, L < rank(H).

The weights w = [w0, . . . ,wL1]T can be chosen to minimize:

w = arg minwE

xHH(

L1l=0

wl(

HHH)l)

y

2

2

= 1where RLL+ and RL+ are defined as

[]ij =1

Ntr(

HHH)i+j

+ 21

Ntr(

HHH)i+j1

[]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 WCNC12 Paris 66 / 101

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 variance1K

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 WCNC12 Paris 67 / 101

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]

Average

received

SINR[dB]

Matched FilterL = 2

L = 3

L = 6

LMMSE

0 2 4 6 8 10 12 14104

103

102

101 Matched Filter

L = 2

L = 3

L = 6

LMMSE

Transmit SNR [dB]

Average

biterrorrate

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 Jakes 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 WCNC12 Paris 68 / 101

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 WCNC12 Paris 69 / 101

Applications Iterative deterministic equivalents: Multiphop relay channel

Iterative deterministic equivalents:Multiphop relay channel

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 70 / 101

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 respect

to 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...

fK1 (HK ) fK a.s. 0.The same idea has been applied to different channel models, e.g.,I Double-scattering channel: H = R

12 W1S

12

W2T12 , 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 WCNC12 Paris 71 / 101

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 WCNC12 Paris 72 / 101

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 WCNC12 Paris 73 / 101

Applications Iterative deterministic equivalents: Multiphop relay channel

Channel model

Received signal vector yk Cnk at node k:

yk =k Hk

k1nk1

yk1 Signal from node k 1

+ nk

Hk Cnknk1 is a standard complex Gaussian matrix, i.e., [Hk ]i,j CN (0, 1).y0 = x CN (0, In0 ) is the channel input vectornk CN (0, Ink ) noise at node kEach nodes scales the transmit signal according to its power constraint k :

k =k

1nk

trE[yk yHk

]Large system limit: n0 , such that

0 < lim infn

ck ,nk1

nk lim sup ck

Applications Iterative deterministic equivalents: Multiphop relay channel

Mutual information

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

Ik (k ) = Jk (1,k ) Jk (1,k )

where

Jk (x ,k ) =1

nklog det

(Ink + x

kk1nk1

Hk Rk1HHk

)with

R0 = E[xxH]

= In

Rk = E[yk y

Hk

]= Ink +

kk1nk1

Hk Rk1HHk , k = 1, . . . ,K

and k = [0, , k1], k = [0, 1, , k1].

Example

I2(2) =1

n2log det

(In2 +

21n1

H2R1HH2

) 1

n2log det

(In2 +

21n1

H2HH2

)

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 75 / 101

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:

ka.s.

nk =

k1 + kk1

, k = 1, . . . ,K 1

where 0 = 0 = 0.

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

Rk = Ink +kk1

nk1Hk Rk1H

Hk

to find iterative deterministic equivalents of

Jk (x ,k ) =1

nklog det

(Ink + x

kk1nk1

Hk Rk1HHk

).

Note that Hk Rk1HHk can be seen as a right-sided correlation model with randomcorrelation. In each step of the analysis, we derive a deterministic equivalent of Jk withrespect to Hk while Rk1 is assumed to be deterministic.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 76 / 101

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 , k1) Jk (x , k1) a.s. 0, k 1where

Jk(x , k1

)=

1

nklog det

([ck + ek1] Ink1 + xk k1Rk1

)+ (1 ck ) log (ck + ek1)

ek1ck + ek1

log (ck )

and ek1 is given as the unique positive solution to

ek1 =1

nktrk k1Rk1

(k k1Rk1

ck + ek1+

1

xInk1

)1.

2 Using the recursion Rk1 = Ink1 +k1k2

nk2Hk1Rk2HHk1, we obtain

Jk (x , k1) = ckJk1(

xk k1ck + xk k1 + ek1

, k2

)+ ck log

(1 +

xk k1ck + ek1

)+ log

(1 +

ek1ck

) ek1

ck + ek1.

3 Similarly, we can find recursive expressions for ek .

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 77 / 101

Applications Iterative deterministic equivalents: Multiphop relay channel

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

Jk (x ,k ) Jk(x , k

) a.s.n

0

where Jk(x , k

)is recursively defined for k 2 as

Jk(x , k

)= ck Jk1

(xk k1

ck + xk k1 + ek1(x , k1

) , k1)

+ ck log

(1 +

xkk1ck + ek1

(x , k1

))

+ log

(1 +

ek1(x , k1

)ck

) ek1

(x , k1

)1 + ek1

(x , k1

)and ek

(x , k1

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

J1(x , 0

)= c1 log

(1 +

x10

c1 + e0(x , 0

))+ log(1 + e0 (x , 0)c1

) e0

(x , 0

)1 + e0

(x , 0

) .M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 78 / 101

Applications Iterative deterministic equivalents: Multiphop relay channel

Theorem (Recursive definition of ek(x , k

)[53])

ek(x , k

)is given as the unique positive solution to

ek(x , k

)= ck+1

(ck+1 + ek

(x , k

)) ck+1

(ck+1 + ek

(x , k

))2xk+1k

mk1

(xk+1k

ck+1 + xk+1k + ek(x , k

) , k1)

where

mk(x , k

)=

xck+1

ck+1 + ek(x , k

) .The initial values m0(x , 0) and e0(x , 0) are given in closed-form:

m0(x , 0) =c1

10c1+e0(x,0)

+ 1x

+ (1 c1)x

e0(x , 0

)= x10(1 c1) + c1

2+

(x10(1 c1) + c1

)2+ 4x10c21

2.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 79 / 101

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

0 (dB)

E[ n k n 0I

k(

k)] (b

its/s/Hz)

k = 1

k = 2

k = 3

k = 4

1 = 3 = 0.70, 2 = 0.50, 1 = 1, 2 = 4 = 0.7, 3 = 0.5

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 80 / 101

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 WCNC12 Paris 81 / 101

Applications Random network topologies

Random network topologies

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 82 / 101

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)

for some path loss exponent [0, 5].What is the average mutual information of this channel?

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 83 / 101

Applications Random network topologies

Channel model and mutual information

The received signal at the BS is given as

y = Hs + n = XT12 s + n

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 (2) = E[

1

Nlog det

(IN +

1

2XTXH

)]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 WCNC12 Paris 84 / 101

Applications Random network topologies

Asymptotic analysis

Recall the deterministic equivalent for fixed T:

I (2) =1

N

Kk=1

log (1 + ctk m) log(2m

) 1

K

Kk=1

tk m

1 + ctk m

where m is the unique solution to

m =

(1

K

Kk=1

tk1 + ctk m(z)

+ 2)1

.

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

ma.s.

( R0

`(d)

1 + c`(d)mdF (d) + 2

)1I (2)

a.s. 1c

R0

log (1 + c`(d)m) dF (d) log(2m

) 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 WCNC12 Paris 85 / 101

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 = 12 (dB)

I( 2)

(bits/s/Hz)

SimulationApproximation

Simulation assumptions:

Path loss function

f (d) =1

(1 + d), = 3.7

Uniform user distribution

dF (d) =2d

R2, d [0,R]

N = 8, K = 4, R = 1

Corollary

I (2) 1c

R0

log (1 + c`(d)m) dF (d) log (2m)+ R0

`(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) + 2

)1.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 86 / 101

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 WCNC12 Paris 87 / 101

Summary and perspectives

Outline1 Introduction

4 Summary and perspectives

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 88 / 101

Summary and perspectives

Summary and perspectives

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 89 / 101

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 WCNC12 Paris 90 / 101

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 adeterministic 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 multihop

relay 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 WCNC12 Paris 91 / 101

Summary and perspectives

Available in bookstores!

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 92 / 101

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 WCNC12 Paris 93 / 101

Summary and perspectives

Thank you!

www.flexible-radio.com

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 94 / 101

Summary and perspectives

References I

[1] Cisco, Cisco visual networking index: Global mobile data traffic forecast update, 20102015, 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. 5562, 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. 81101, 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. 457483, 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. 316345, 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. 24202439, Apr. 2011.

[11] S. Verdu and S. Shamai, Spectral efficiency of CDMA with random spreading, IEEE Trans. Inf. Theory, vol. 45, no. 2, pp.622640, Mar. 1999.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 95 / 101

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. 138, 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. 175192, 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. 34933514, 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. 875930, 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. 57375753, 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. 20712130, 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. 50485063, 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. 25452561, Oct.2003.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 96 / 101

Summary and perspectives

References III

[22] B. Hassibi and B. M. Hochwald, How much training is needed in multiple-antenna wireless links? IEEE Trans. Inf.Theory, vol. 49, no. 4, pp. 951963, Apr. 2003.

[23] J. Hoydis, M. Kobayashi, and M. Debbah, Optimal channel training in uplink network MIMO systems, IEEE Trans. SignalProcess., vol. 59, no. 6, pp. 28242833, Jun. 2011.

[24] A. L. Moustakas and S. H. Simon, On the outage capacity of correlated multiple-path MIMO channels, IEEE Trans. Inf.Theory, vol. 53, no. 11, pp. 38873903, Nov. 2007.

[25] G. Taricco, Asymptotic mutual information statistics of separately correlated rician fading MIMO channels, IEEE Trans.Inf. Theory, vol. 54, no. 8, pp. 34903504, Aug. 2008.

[26] J. Dumont, W. Hachem, S. Lasaulce, P. Loubaton, and J. Najim, On the capacity achieving covariance matrix for RicianMIMO channels: An asymptotic approach, IEEE Trans. Inf. Theory, vol. 56, no. 3, pp. 10481069, Mar. 2010.

[27] B. Hochwald and S. Vishwanath, Space-time multiple access: Linear growth in the sum rate, in Proc. IEEE AnnualAllerton Conference on Communication, Control, and Computing, Monticello, Illinois, US, Oct. 2002, pp. 387396.

[28] C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst, A vector-perturbation technique for near-capacity multiantennamultiuser communication-part I: channel inversion and regularization, IEEE Trans. Commun., vol. 53, no. 1, pp. 195 202,Jan. 2005.

[29] V. K. Nguyen and J. S. Evans, Multiuser transmit beamforming via regularized channel inversion: A large system analysis,in Proc. IEEE Global Telecommunications Conference (GLOBECOM08), Dec. 2008, pp. 14.

[30] R. Zakhour and S. V. Hanly, Large system analysis of base station cooperation on the downlink, in Allerton Conf. onCommun., Control, and Computing, Urbana-Champaign, IL, US, Oct. 2010, pp. 270277.

[31] R. Muharar and J. Evans, Downlink beamforming with transmit-side channel correlation: A large system analysis, in Proc.IEEE International Conference on Communications (ICC11), Jun. 2011, pp. 15.

[32] H. Huh, G. Caire, S.-H. Moon, and I. Lee, Multi-cell MIMO downlink with fairness criteria: The large-system limit, inProc. IEEE Int. Symp. on Inf. Theory (ISIT), Austin, TX, US, Jun. 2010, pp. 20582062.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 97 / 101

Summary and perspectives

References IV

[33] H. Huh, G. Caire, H. C. Papadopoulos, and S. A. Ramprashad, Achieving massive MIMO spectral efficiency with anot-so-large number of antennas, IEEE Trans. Wireless Commun., 2011, submitted. [Online]. Available:http://arxiv.org/abs/1107.3862

[34] J. Hoydis, S. ten Brink, and M. Debbah, Massive MIMO: How many antennas do we need? in Proc. IEEE AllertonConference on Communication, Control, and Computing (ALLERTON), Urbana-Champaign, IL, US, Sep. 2011.

[35] , Comparison of linear precoding schemes for the massive MIMO downlink, in Proc. IEEE International Conferenceon Communications (ICC), Ottawa, Canada, Jun. 2012.

[36] , Massive MIMO in the UL/DL of cellular networks: How many antennas do we need? IEEE J. Sel. Areas Commun.,Dec. 2012, submitted.

[37] P. Billingsley, Probability and Measure, 3rd ed. John Wiley & Sons, Inc., 1995.

[38] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta, O. Edfors, and F. Tufvesson, Scaling up MIMO:Opportunities and challenges with very large arrays, IEEE Signal Process. Mag., 2012, to appear. [Online]. Available:http://liu.diva-portal.org/smash/get/diva2:450781/FULLTEXT01

[39] T. L. Marzetta, Noncooperative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. WirelessCommun., vol. 9, no. 11, pp. 35903600, Nov. 2010.

[40] L. Cottatellucci and R. R. Muller, A systematic approach to multistage detectors in multipath fading channels, IEEETrans. Inf. Theory, vol. 51, no. 9, pp. 31463158, Sep. 2005.

[41] L. Li, A. M. Tulino, and S. Verdu, Design of reduced-rank MMSE multiuser detectors using random matrix methods,IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 9861008, Jun. 2004.

[42] W. Hachem, Simple polynomial detectors for CDMA downlink transmissions on frequency-selective channels, IEEE Trans.Inf. Theory, vol. 50, no. 1, pp. 164171, Jan. 2004.

[43] J. Hoydis, M. Debbah, and M. Kobayashi, Asymptotic moments for interference mitigation in correlated fading channels,in Proc. IEEE International Symposium on Information Theory (ISIT), Saint Petersburg, Russia, Aug. 2011, pp. 27962800.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 98 / 101

Summary and perspectives

References V

[44] O. Ryan and M. Debbah, Asymptotic behavior of random Vandermonde matrices with entries on the unit circle, IEEETrans. Inf. Theory, vol. 55, no. 7, pp. 31153147, Jul. 2009.

[45] S. Moshavi, E. G. Kanterakis, and D. L. Schilling, Multistage linear receivers for DS-CDMA systems, International Journalof Wireless Information Networks, vol. 3, no. 1, pp. 117, Jan. 1996.

[46] R. R. Mueller and S. Verdu, Design and analysis of low-complexity interference mitigation on vector channels, vol. 19,no. 8, pp. 14291441, Aug. 2001.

[47] M. L. Honig and W. Xiao, Performance of reduced-rank linear interference suppression, IEEE Trans. Inf. Theory, vol. 47,no. 5, pp. 19281946, Jul. 2001.

[48] P. Loubaton and W. Hachem, Asymptotic analysis of reduced rank Wiener filters, in Proc. of Information TheoryWorkshop (ITW03), Paris, France, Mar. 31 - Apr. 4, 2003, pp. 328331.

[49] L. Cottatellucci, R. Muller, and M. Debbah, Asynchronous CDMA systems with random spreading - part i:Fundamentallimits, IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 14771497, Apr. 2010.

[50] L. Cottatellucci, R. R. Muller, and M. Debbah, Asynchronous CDMA systems with random spreading - part ii: Designcriteria, IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 14981520, Apr. 2010.

[51] A. M. Masucci, O. Ryan, and M. Debbah, Polynomial expansion detector for uniform linear arrays, in Proc. IEEE Int.Conf. Acoustics, Speech and Signal Proc. (ICASSP11), May 2011, pp. 33923395.

[52] J. Hoydis, R. Couillet, and M. Debbah, Asymptotic analysis of double-scattering channels, in Proc. IEEE AsilomarConference on Signals, Systems, and Computers (ASILOMAR), Pacific Grove, CA, USA, Nov. 2011.

[53] , Iterative deterministic equivalents for the capacity analysis of communication systems, IEEE Trans. Inf. Theory,Dec. 2011, submitted. [Online]. Available: http://arxiv.org/abs/1112.4167

[54] R. Couillet, J. Hoydis, and M. Debbah, Random beamforming over quasi-static and fading channels: A deterministicequivalent approach, IEEE Trans. Inf. Theory, Nov. 2011, accepted. [Online]. Available: http://arxiv.org/abs/1011.3717

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 99 / 101

Summary and perspectives

References VI

[55] V. I. Morgenshtern and H. Bolcskei, Random matrix analysis of large relay networks, in Proc. 44th Allerton AnnualConference on Communications, Control and Computing, Urbana-Champaign, IL, US, Sep. 2006, pp. 106112.

[56] S.-P. Yeh and O. Leveque, Asymptotic capacity of multi-level amplify-and-forward relay networks, in Proc. IEEEInternational Symposium on Information Theory (ISIT07), Jun. 2007, pp. 14361440.

[57] J. Wagner, B. Rankov, and A. Wittneben, Large n analysis of amplify-and-forward MIMO relay channels with correlatedRayleigh fading, IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 57355746, Dec. 2008.

[58] N. Fawaz, K. Zarifi, M. Debbah, and D. Gesbert, Asymptotic capacity and optimal precoding in MIMO multi-hop relaynetworks, IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 20502069, Apr. 2011.

[59] G. H. Tucci, Spectral analysis of the amplify and forward relay network as the number of relay layers increases, in Proc.8th International Symposium on Wireless Communication Systems (ISWCS11), Aachen, Germany, Nov. 2011.

[60] J. Hoydis, A. Muller, R. Couillet, and M. Debbah, Analysis of multicell cooperation with random user locations viadeterministic equivalents, in Proc. Workshop on Spatial Stochastic Models for Wireless Networks (SPASWIN), Paderborn,Germany, May 2012.

[61] R. Couillet and M. Debbah, Signal processing in large systems: A new paradigm, IEEE Signal Process. Mag., 2012, toappear. [Online]. Available: http://arxiv.org/abs/1105.0060

[62] A. M. Tulino and S. Verdu, Random matrix theory and wireless communications, Foundations and Trends in Commun.Inf. Theory, vol. 1, no. 1, pp. 1182, Jun. 2004.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 100 / 101

Summary and perspectives

Some useful lemmas

Matrix inversion lemma

Let A CNN , D Cnn be invertible, and B CNn, C CnN . Then,(A BC D

)1=

( (A BD1C)1 A1B (D CA1B)1

(A BD1C)1 CA1 (D CA1B)1).

Identity for the matrix inverse

Let A CNn and B CnN such that (IN + AB) is invertible. ThenIN A (In + BA)1 B = (IN + AB)1 .

Resolvent identity

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

A1 B1 = A1(B A)B1.

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

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

xH(

A + cxxH)1

=xHA1

1 + cxHA1x.

M. Debbah and J. Hoydis (Supelec) RMT for Advanced Communication Systems WCNC12 Paris 101 / 101

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

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

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

Summary and perspectives