Application of Random Matrix Theory to Future Wireless ... · Application of Random Matrix Theory...

Application of Random Matrix Theory to Future Wireless Networks

Romain Couillet1,2

1Alcatel-Lucent Chair on Flexible Radio, Supelec, Gif sur Yvette, FRANCE2ST-Ericsson, Sophia-Antipolis, FRANCE

[email protected]

PhD defenseSupelec, November 12th, 2010.

R. Couillet (Supelec, ST-Ericsson) Random Matrix Theory for Future Wireless Networks 12/11/2010 1 / 39

Outline

1 Cognitive Radios

2 Exploration: Spectrum Hole Detection

3 Exploration: User Detection and Power Inference

4 Exploitation: Optimal Ergodic Rate

5 Perspectives and Conclusion

Cognitive Radios

Outline

1 Cognitive Radios

Cognitive Radios

General system model

Figure: Cognitive radio setting.

Cognitive Radios

Exploration and Exploitation

Cognitive Radio Setting:The secondary network is entitled to reuse efficiently spectrum holes,

so to minimally interfere the primary network;so to maximise the throughput of secondary transmissions;with little or no feedback from the primary network, i.e. autonomously.

To this end, the secondary (or smart, cognitive) network mustlearn about the environment: this is the exploration phase.communicate within the secondary network: this is the exploitation phase.

Both scenarios are inherently multi-dimensional:efficient exploration requires large sensor networks;networks may be composed of multiple users with possibly multiple antennas.

We will discuss both exploration and exploitation phases as (possibly large) multivariate systems.

Cognitive Radios

Exploration: Spectrum Hole Detection

Outline

1 Cognitive Radios

Formulation of the exploration problem

We assume the scenario of a cognitive radio made ofa primary network of K transmitters, equipped with n1, . . . , nK antennas;a secondary network in sensing mode, equipped of N collocated sensors.

At time m, the secondary network receives y(m) ∈ CN as

y(m) =KX

pPk Hk x(m)

k + σw(m)

Depending on prior information, the secondary network willtry to infer the presence of a signal (we will assume a unique transmitter)try to infer the transmit powers of the K transmitters.

This information allows for:

the detection of spectrum holes;

the evaluation of the optimal secondary coverage;

ideally, the elaboration of a ‘map’ of space-frequency resources.

y(m) =KX

pPk Hk x(m)

k + σw(m)

y(m) =KX

pPk Hk x(m)

k + σw(m)

y(m) =√

PHx(m) + σw(m)

y(m) =KX

pPk Hk x(m)

k + σw(m)

y(m) =KX

pPk Hk x(m)

k + σw(m)

Problem formulation

We consider the model

y(m) =

σw(m) , (H0)√

PHx(m) + σw(m) , (H1)

We wish to confront the hypotheses H0 and H1 given the data matrixY , [y(1), . . . , y(M)] ∈ CN×M .

We consider, in a Bayesian framework, the Neyman-Pearson test ratio

C(Y)∆=

PH1|Y,I(Y)

PH0|Y,I(Y)

with prior information I on H, x(m), σ, . . ..

Problem formulation

y(m) =

σw(m) , (H0)√

PHx(m) + σw(m) , (H1)

C(Y)∆=

PH1|Y,I(Y)

PH0|Y,I(Y)

Problem formulation

y(m) =

σw(m) , (H0)√

PHx(m) + σw(m) , (H1)

C(Y)∆=

PH1|Y,I(Y)

PH0|Y,I(Y)

A Bayesian framework for cognitive radios

We assume prior statistical and deterministic knowledge I on H, σ, P

Using the maximum entropy principle (MaxEnt), a prior P(H,σ,P)(H, σ, P) can be derived

PY|Hi ,I(Y) =

Z(H,σ,P)

PY|Hi ,I,H,σ,P(Y)P(H,σ,P)(H, σ, P)d(H, σ, P)

In the following,we derive the case P = 1, σ known and the knowledge about H conveys unitary invariance

E[tr HHH] known: this is what we assume here;E[HHH] = Q unknown but such that E[tr Q] is known;rank(HHH) known.

we compare alternative methods when P = 1 and σ are unknown.

PY|Hi ,I(Y) =

Z(H,σ,P)

PY|Hi ,I(Y) =

Z(H,σ,P)

Evaluation of PY|Hi ,I(Y)

by MaxEnt, X, W are standard Gaussian matrix with Xij , Wij ∼ CN (0, 1).Under H0:

Y = σW

PY|H0,I(Y) =1

(πσ2)NMe−

1σ2 tr YYH

Under H1:

Y =ˆ√

PH σIN˜ »X

–PY|H1

ZΣ≥0

PY|Σ,H1(Y,Σ)PΣ(Σ)dΣ

with Σ = E[y(1)y(1)H] = HHH + σ2IN .From unitary invariance of H, denoting Σ = UGUH, diag(G) = (g1, . . . , gn, σ2, . . . , σ2)

PY|H1(Y) =

ZU(N)×(σ2,∞)n

PY|UGUH,H1(Y, U, G)PU(U)P(g1,...,gn)(g1, . . . , gn)dUdg1 . . . dgn

wherePY|UGUH,H1

is Gaussian with zero mean and variance UGUH;

PU is a constant (dU is a Haar measure);

if H is Gaussian, P(g1−σ2,...,gn−σ2)

is the joint eigenvalue distribution of a central Wishart;

Y = σW

PY|H0,I(Y) =1

(πσ2)NMe−

1σ2 tr YYH

Under H1:

Y =ˆ√

PH σIN˜ »X

–PY|H1

ZΣ≥0

PY|H1(Y) =

ZU(N)×(σ2,∞)n

wherePY|UGUH,H1

Y = σW

PY|H0,I(Y) =1

(πσ2)NMe−

1σ2 tr YYH

Under H1:

Y =ˆ√

PH σIN˜ »X

–PY|H1

ZΣ≥0

PY|H1(Y) =

ZU(N)×(σ2,∞)n

wherePY|UGUH,H1

Y = σW

PY|H0,I(Y) =1

(πσ2)NMe−

1σ2 tr YYH

Under H1:

Y =ˆ√

PH σIN˜ »X

–PY|H1

ZΣ≥0

PY|H1(Y) =

ZU(N)×(σ2,∞)n

wherePY|UGUH,H1

Y = σW

PY|H0,I(Y) =1

(πσ2)NMe−

1σ2 tr YYH

Under H1:

Y =ˆ√

PH σIN˜ »X

–PY|H1

ZΣ≥0

PY|H1(Y) =

ZU(N)×(σ2,∞)n

wherePY|UGUH,H1

Y = σW

PY|H0,I(Y) =1

(πσ2)NMe−

1σ2 tr YYH

Under H1:

Y =ˆ√

PH σIN˜ »X

–PY|H1

ZΣ≥0

PY|H1(Y) =

ZU(N)×(σ2,∞)n

wherePY|UGUH,H1

Y = σW

PY|H0,I(Y) =1

(πσ2)NMe−

1σ2 tr YYH

Under H1:

Y =ˆ√

PH σIN˜ »X

–PY|H1

ZΣ≥0

PY|H1(Y) =

ZU(N)×(σ2,∞)n

wherePY|UGUH,H1

Result in the Gaussian case, n = 1

R. Couillet, M. Debbah, “A Bayesian Framework for Collaborative Multi-Source Signal Sensing”,IEEE Transactions on Signal Processing, vol. 58, no. 10, pp. 5186-5195, 2010.

Theorem (Neyman-Pearson test)

The ratio C(Y) when the receiver knows n = 1, P = 1, E[ 1N tr HHH] = 1 and σ2, reads

C(Y) =1N

σ2(N+M−1)eσ2+λlσ2QN

i=1i 6=l

(λl − λi )JN−M−1(σ

2, λl )

with λ1, . . . , λN the eigenvalues of YYH and where

Jk (x , y) ,Z +∞

xtk e−t− y

t dt .

non trivial dependency on λ1, . . . , λN

contrary to energy detector,P

i λi is not a sufficient statistic;

integration over σ2 (or P when P 6= 1) is difficult.

C(Y) =1N

i=1i 6=l

2, λl )

Jk (x , y) ,Z +∞

xtk e−t− y

t dt .

C(Y) =1N

i=1i 6=l

2, λl )

Jk (x , y) ,Z +∞

xtk e−t− y

t dt .

Comparison to energy detector

1 · 10−3 5 · 10−3 1 · 10−2 2 · 10−20.2

False alarm rate

Energy detector

Neyman-Pearson test

Figure: ROC curve for single-source detection, K = 1, N = 4, M = 8, SNR = −3 dB, FAR range of practicalinterest, with signal power E = 0 dBm, either known or unknown at the receiver.

Comparison to energy detector

1 · 10−3 5 · 10−3 1 · 10−2 2 · 10−20.2

False alarm rate

Energy detector

Neyman-Pearson test

Neyman-Pearson test (E unknown)

Figure: ROC curve for single-source detection, K = 1, N = 4, M = 8, SNR = −3 dB, FAR range of practicalinterest, with signal power E = 0 dBm, either known or unknown at the receiver.

Unknown power and noise variances

Bayesian approaches:

PY|Hi ,I(Y) =

PY|Hi ,σ,P(Y)P(σ,P)(σ, P)d(σ, P)

limited by computational complexity (two-dimension numerical integration);inconsistence in MaxEnt uninformative priors on σ, P.

non-parametric GLRT approach:

CGLRT(Y)∆=

supH,σ PY|H1(Y)

supσ PY|H0(Y)

CGLRT(Y) expresses as a monotonic function of maxi λi1NP

i λi;

excludes prior information on H, σ, P.

ad-hoc methods, such as conditioning number:

Ccond(Y) ,maxi λi

mini λi

based on large random matrix considerations: underH0, as N/M → c

maxi λi

mini λi

a.s.−→(1 +

√c)2

(1−√

totally empirical.

PY|Hi ,I(Y) =

CGLRT(Y)∆=

supH,σ PY|H1(Y)

supσ PY|H0(Y)

i λi;

Ccond(Y) ,maxi λi

mini λi

maxi λi

mini λi

a.s.−→(1 +

√c)2

(1−√

totally empirical.

PY|Hi ,I(Y) =

CGLRT(Y)∆=

supH,σ PY|H1(Y)

supσ PY|H0(Y)

i λi;

Ccond(Y) ,maxi λi

mini λi

maxi λi

mini λi

a.s.−→(1 +

√c)2

(1−√

totally empirical.

Performance comparison for unknown σ2, P

1 · 10−3 5 · 10−3 1 · 10−2 2 · 10−20

False alarm rate

Neyman-Pearson, Jeffreys prior

Neyman-Pearson, uniform prior

Conditioning number test

Figure: ROC curve for a priori unknown σ2 of the Neyman-Pearson test, conditioning number method and GLRT,K = 1, N = 4, M = 8, SNR = 0 dB. For the Neyman-Pearson test, both uniform and Jeffreys prior, withexponent β = 1, are provided.

Exploration: User Detection and Power Inference

Outline

1 Cognitive Radios

Problem Statement

We now consider the model

y(m) =KX

pPk Hk x(m)

k + σw(m)

and wish to infer P1, . . . , PK .

With Y = [y(1), . . . , y(M)], this can be rewritten

pPk Hk Xk + σW =

ˆpP1H1 · · ·

pPK HK

˜| {z },HP

264X1...

375| {z }

+σW =hHP

12 σIN

i »XW

If H, (XT WT) are unitarily invariant, Y is unitarily invariant.

Most information about P1, . . . , PK is contained in the eigenvalues of BN , 1M YYH.

Problem Statement

y(m) =KX

pPk Hk x(m)

k + σw(m)

pPk Hk Xk + σW =

ˆpP1H1 · · ·

pPK HK

˜| {z },HP

264X1...

375| {z }

+σW =hHP

12 σIN

i »XW

Problem Statement

y(m) =KX

pPk Hk x(m)

k + σw(m)

pPk Hk Xk + σW =

ˆpP1H1 · · ·

pPK HK

˜| {z },HP

264X1...

375| {z }

+σW =hHP

12 σIN

i »XW

Problem Statement

y(m) =KX

pPk Hk x(m)

k + σw(m)

pPk Hk Xk + σW =

ˆpP1H1 · · ·

pPK HK

˜| {z },HP

264X1...

375| {z }

+σW =hHP

12 σIN

i »XW

Problem Statement

y(m) =KX

pPk Hk x(m)

k + σw(m)

pPk Hk Xk + σW =

ˆpP1H1 · · ·

pPK HK

˜| {z },HP

264X1...

375| {z }

+σW =hHP

12 σIN

i »XW

Problem Statement

y(m) =KX

pPk Hk x(m)

k + σw(m)

pPk Hk Xk + σW =

ˆpP1H1 · · ·

pPK HK

˜| {z },HP

264X1...

375| {z }

+σW =hHP

12 σIN

i »XW

From small to large system analysis

0.1 1 3 100

Eigenvalues of YYH

Eigenvalues of BN = 1M YYH

The classical approach requires to evaluate PP1,...,PK |Y

assuming Gaussian parameters, this is similar to previous calculus

leads to a sum of two-dimensional integrals

prohibitively expensive to evaluate even for small N, nk , M

From small to large system analysis

0.1 1 3 100

Eigenvalues of YYH

Limiting spectrum of BN

Assuming dimensions N, nk , M grow large, large dimensional random matrix theory providesa link between:

the “observation”: the limiting spectral distribution (l.s.d.) of BN ;the “hidden parameters”: the powers P1, . . . , PK , i.e. the l.s.d. of P.

consistent estimators of the hidden parameters.

Limiting spectrum of the sample covariance matrix

Definition. The Stieltjes transform mF (z), of a distribution function F is defined as

mF (z) =

ω − zdF (ω).

Knowing the Stieltjes transform of F is equivalent to knowing F (similar to Fourier transform).

for simplicity, consider the sample covariance matrix model

Y∆=T

12 X ∈ CN×n, BN =

YYH ∈ CN×N , BN =1n

YHY ∈ Cn×n

where T ∈ CN×N has eigenvalues t1, . . . , tK , tk with multiplicity Nk and X ∈ CN×n is i.i.d. zeromean, variance 1.

If F T ⇒ T , then mFBN (z) = mBN (z)a.s.−→ mF (z) such that

mT`−1/mF (z)

´= −zmF (z)mF (z)

with mF (z) = cmF (z) + (c − 1) 1z and N/n → c.

mF (z) =

ω − zdF (ω).

Y∆=T

12 X ∈ CN×n, BN =

YHY ∈ Cn×n

mT`−1/mF (z)

mF (z) =

ω − zdF (ω).

Y∆=T

12 X ∈ CN×n, BN =

YHY ∈ Cn×n

mT`−1/mF (z)

Complex integration

From Cauchy integral formula, denoting Ck a contour enclosing only tk ,

ω − tkdω =

ω − tjdω =

N2πiNk

ωmT (ω)dω.

After the variable change ω = −1/mF (z),

tk =NNk

zmF (z)m′

m2F (z)

When the system dimensions are large,

mF (z) ' mBN (z)∆=

1λk − z

, with (λ1, . . . , λN) = eig(BN) = eig(YYH).

Dominated convergence arguments then show

tk − tka.s.−→ 0 with tk =

zmBN (z)m′

(z)dz =

Xm∈Nk

(λm − µm)

with Nk the indexes of cluster k and µ1 < . . . < µN are the ordered eigenvalues of the matrixdiag(λ)− 1

√λ√

λT, λ = (λ1, . . . , λN)T.

Complex integration

ω − tkdω =

ω − tjdω =

N2πiNk

ωmT (ω)dω.

tk =NNk

zmF (z)m′

m2F (z)

mF (z) ' mBN (z)∆=

1λk − z

zmBN (z)m′

(z)dz =

Xm∈Nk

(λm − µm)

√λ√

λT, λ = (λ1, . . . , λN)T.

Complex integration

ω − tkdω =

ω − tjdω =

N2πiNk

ωmT (ω)dω.

tk =NNk

zmF (z)m′

m2F (z)

mF (z) ' mBN (z)∆=

1λk − z

zmBN (z)m′

(z)dz =

Xm∈Nk

(λm − µm)

√λ√

λT, λ = (λ1, . . . , λN)T.

Complex integration

ω − tkdω =

ω − tjdω =

N2πiNk

ωmT (ω)dω.

tk =NNk

zmF (z)m′

m2F (z)

mF (z) ' mBN (z)∆=

1λk − z

zmBN (z)m′

(z)dz =

Xm∈Nk

(λm − µm)

√λ√

λT, λ = (λ1, . . . , λN)T.

Complex integration

ω − tkdω =

ω − tjdω =

N2πiNk

ωmT (ω)dω.

tk =NNk

zmF (z)m′

m2F (z)

mF (z) ' mBN (z)∆=

1λk − z

zmBN (z)m′

(z)dz =

Xm∈Nk

(λm − µm)

√λ√

λT, λ = (λ1, . . . , λN)T.

Complex integration

ω − tkdω =

ω − tjdω =

N2πiNk

ωmT (ω)dω.

tk =NNk

zmF (z)m′

m2F (z)

mF (z) ' mBN (z)∆=

1λk − z

zmBN (z)m′

(z)dz =

Xm∈Nk

(λm − µm)

√λ√

λT, λ = (λ1, . . . , λN)T.

Complex integration

ω − tkdω =

ω − tjdω =

N2πiNk

ωmT (ω)dω.

tk =NNk

zmF (z)m′

m2F (z)

mF (z) ' mBN (z)∆=

1λk − z

zmBN (z)m′

(z)dz =

Xm∈Nk

(λm − µm)

√λ√

λT, λ = (λ1, . . . , λN)T.

Application to the current model

R. Couillet, J. W. Silverstein, Z. Bai, M. Debbah, “Eigen-Inference for Energy Estimation of MultipleSources,” to appear in IEEE Transactions on Information Theory, 2010.

Extending Y with zeros, our model is a “double sample covariance matrix”

Y|{z}(N+n)×M

12 σIN

#| {z }(N+n)×(N+n)

–| {z }

(N+n)×M

Limiting distribution of 1M YYH

Theorem (l.s.d. of BN )

Let BN = 1M YYH with eigenvalues λ1, . . . , λN . Denote mBN

(z)∆= 1

λk−z , with λi = 0 fori > N. Then, for M/N → c, N/nk → ck , N/n → c0, for any z ∈ C+,

mBN(z)

a.s.−→ mF (z)

with mF (z) the unique solution in C+ of

1mF (z)

= −σ2 +1

»c0 − 1

c0+ mP

„−

1f (z)

«–, with f (z) = (c − 1)mF (z)− czmF (z)2.

Application to the current model (2)

R. Couillet, J. W. Silverstein, Z. Bai, M. Debbah, “Eigen-Inference for Energy Estimation of MultipleSources,” to appear in IEEE Trans. on Inf. Theory, 2010.

estimator calculus

Theorem (Estimator of P1, . . . , PK )

Let BN ∈ CN×N be defined as in Theorem 2, and λ = (λ1, . . . , λN), λ1 < . . . < λN . Assume thatasymptotic cluster separability condition is fulfilled for some k. Then, as N, n, M →∞,

Pk − Pka.s.−→ 0,

Pk =NM

nk (M − N)

Xi∈Nk

(ηi − µi )

with Nk the set indexing the eigenvalues in cluster k of F , η1 < . . . < ηN the eigenvalues ofdiag(λ)− 1

√λ√

and µ1 < . . . < µN the eigenvalues of diag(λ)− 1M

√λ√

Remarks

solution is computationally simple, explicit, and the final formula compact.cluster separability condition is fundamental. This requires

for all other parameters fixed, the Pk cannot be too close top one another: source separation problem.for all other parameters fixed, σ2 must be kept low: low SNR undecidability problem.for all other parameters fixed, M/N cannot be too low: sample deficiency issue (not such an issuethough).for all other parameters fixed, N/n cannot be too low: diversity issue.

exact spectrum separability is an essential ingredient (known for very few models to this day).

0.1 1 3 100

Eigenvalues of YYH

Remarks

solution is computationally simple, explicit, and the final formula compact.cluster separability condition is fundamental. This requires

for all other parameters fixed, the Pk cannot be too close top one another: source separation problem.for all other parameters fixed, σ2 must be kept low: low SNR undecidability problem.for all other parameters fixed, M/N cannot be too low: sample deficiency issue (not such an issuethough).for all other parameters fixed, N/n cannot be too low: diversity issue.

exact spectrum separability is an essential ingredient (known for very few models to this day).

0.1 1 3 100

Eigenvalues of YYH

Simulations

Cluster means

Estimated PkD

Figure: Histogram of the cluster-mean approach and of Pk for k ∈ {1, 2, 3}, P1 = 1/16, P2 = 1/4, P3 = 1,n1 = n2 = n3 = 4 antennas per user, N = 24 sensors, M = 128 samples and SNR = 20 dB.

Performance comparison

−5 0 5 10 15 20 25 30−20

SNR [dB]

Stieltjes transform estimator

Moment estimator

Cluster average estimator

Figure: Normalized mean square error of largest estimated power P3, P1 = 1/16, P2 = 1/4, P3 = 1,n1 = n2 = n3 = 4 ,N = 24, M = 128. Comparison between classical, moment and Stieltjes transformapproaches.

Exploitation: Optimal Ergodic Rate

Outline

1 Cognitive Radios

Problem statement

We assume, from the exploration phase, that power Qf can be transmitted in bandwidth Bf .The exploitation phase consists in optimally using these power resources.We consider the uplink scenario of

an N-antenna base station;K users equipped with n1, . . . , nK antennas;

Kronecker channels at all pairs Hk,f = R12k,f Xk,f T

12k,f ∈ CN×nk at frequency Bf ;

colored noise with covariance Σ.

Problem statement

Maximizing ergodic sum rate

Due to mobility, we wish to optimize the uplink ergodic sum rate (per antenna),

I(P1,1, . . . , PK ,F ) ,1N

|Bf |Pf ′ |Bf ′ |

24log det

0@IN +KX

Σ− 1

2f Hk,f Pk,f HH

k,f Σ− 1

1A35 .

Determine the sum rate maximizing precoders P?k,f

(P?1,1, . . . , P?

K ,F ) = arg max{Pk,f }PK

k=1 tr Pk,f≤Qf

I(P1,1, . . . , PK ,F ).

Simplifying assumptions:Problem can be treated for each f independently. We then assume F = 1.Taking Σ = σ2IN does not restrict generality.

I(P1,1, . . . , PK ,F ) ,1N

|Bf |Pf ′ |Bf ′ |

24log det

0@IN +KX

Σ− 1

2f Hk,f Pk,f HH

k,f Σ− 1

1A35 .

(P?1,1, . . . , P?

k=1 tr Pk,f≤Qf

I(P1,1, . . . , PK ,F ).

I(P1,1, . . . , PK ,F ) ,1N

|Bf |Pf ′ |Bf ′ |

24log det

0@IN +KX

Σ− 1

2f Hk,f Pk,f HH

k,f Σ− 1

1A35 .

(P?1,1, . . . , P?

k=1 tr Pk,f≤Qf

I(P1,1, . . . , PK ,F ).

Deterministic Equivalents of the Sum Rate

The stochastic character of Hk,f makes things difficult.

We instead find a deterministic approximation I◦(P1, . . . , PK ) for I(P1, . . . , PK ) such that

I◦(P1, . . . , PK )− I(P1, . . . , PK ) → 0

as N, n1, . . . , nK →∞, and denote

(P◦1 , . . . , P◦K ) = arg max{Pk}

I◦(P1, . . . , PK ).

We can showI(P?

1 , . . . , P?K )− I(P◦1 , . . . , P◦K ) → 0.

Deterministic Equivalents of the Sum Rate

The stochastic character of Hk,f makes things difficult.

We instead find a deterministic approximation I◦(P1, . . . , PK ) for I(P1, . . . , PK ) such that

I◦(P1, . . . , PK )− I(P1, . . . , PK ) → 0

as N, n1, . . . , nK →∞, and denote

(P◦1 , . . . , P◦K ) = arg max{Pk}

I◦(P1, . . . , PK ).

We can showI(P?

1 , . . . , P?K )− I(P◦1 , . . . , P◦K ) → 0.

Deterministic Equivalents: Strategy

With BN ,PK

k=1 Hk Pk HHk (Hk = R

12k Xk T

12k ), notice that

I(P1, . . . , PK ) = E»

log det„

IN +1σ2

«–= E

»Z ∞

„1ω−

tr (BN + ωIN)−1«

–= E

»Z ∞

„1ω−mBN (−ω)

It suffices to find a deterministic equivalent for mBN (z).

Deterministic Equivalents: Strategy

With BN ,PK

k=1 Hk Pk HHk (Hk = R

12k Xk T

12k ), notice that

I(P1, . . . , PK ) = E»

log det„

IN +1σ2

«–= E

»Z ∞

„1ω−

tr (BN + ωIN)−1«

–= E

»Z ∞

„1ω−mBN (−ω)

It suffices to find a deterministic equivalent for mBN (z).

Final Results

R. Couillet, M. Debbah and J. W. Silverstein, “A Deterministic Equivalent for the Analysis ofCorrelated MIMO Multiple Access Channels”, to appear in IEEE Transactions on InformationTheory, arXiv Preprint 0906.3667.

Theorem (Deterministic equivalent of the Stieltjes transform)

Under some mild conditions on the Rk and Tk matrices, as N, n1, . . . , nK →∞

mBN (z)−mN(z)a.s.−→ 0

mN(z) =1N

0@−z

24 KXk=1

ek (z)Rk + IN

351A−1

and {ei (z)}, i ∈ {1, . . . , K}, form the unique solution to

ei (z) =1ni

0@−z

24 KXk=1

ek (z)Rk + IN

351A−1

ei (z) =1ni

tr T12i Pi T

„−z

»ei (z)T

12i Pi T

12i + Ini

–«−1

Final results (2)

Theorem (Deterministic equivalents of the sum rate)

Under similar conditions, with I(P1, . . . , PK ) = 1N E log det

“IN + 1

σ2 BN

”and z = −σ2,

I(P1, . . . , PK )− I◦(P1, . . . , PK ) → 0

I◦(P1, . . . , PK ) =1N

˛˛IN +

˛˛ +

log˛Ink + ek T

12k Pk T

˛− σ2

Nek ek .

It remains to maximize I◦(P1, . . . , PK ) over P1, . . . , PK withP

k tr Pk ≤ Q.This is obtained by iterative waterfilling

Pk and Tk have the same eigenspaceswith eig(P◦k ) = (p◦k,1, . . . , p◦k,nk

p◦k,i =

1e◦k tk,i

, e◦k = ek (P◦1 , . . . , P◦K ), µ set to satisfyX

tr Pk = Q.

Final results (2)

“IN + 1

σ2 BN

”and z = −σ2,

I(P1, . . . , PK )− I◦(P1, . . . , PK ) → 0

I◦(P1, . . . , PK ) =1N

˛˛IN +

˛˛ +

log˛Ink + ek T

12k Pk T

˛− σ2

Nek ek .

p◦k,i =

1e◦k tk,i

tr Pk = Q.

Final results (2)

“IN + 1

σ2 BN

”and z = −σ2,

I(P1, . . . , PK )− I◦(P1, . . . , PK ) → 0

I◦(P1, . . . , PK ) =1N

˛˛IN +

˛˛ +

log˛Ink + ek T

12k Pk T

˛− σ2

Nek ek .

p◦k,i =

1e◦k tk,i

tr Pk = Q.

Simulation results

−15 −10 −5 0 5 10 15 200

SNR [dB]

Sim., uni.

Det. eq., uni.

Sim., opt.

Det. eq., opt.

Figure: Ergodic MAC sum rate for an N = 4 antenna receiver and K = 4 single-antenna transmitters under sumpower constraint. Every user transmit signal has different correlation patterns at the receiver, and different pathlosses. Deterministic equivalents (det. eq.) against simulation (sim.), with uniform (uni.) or optimal (opt.) powerallocation.

Perspectives and Conclusion

Outline

1 Cognitive Radios

The road ahead

signal sensing:optimal hypothesis tests require symmetry, are often computationally prohibitive;situations with more side information demand simpler tests, based on eigen-structure;more realistic scenarios with cooperation will demand a further improvement of such tests.

statistical inference:many methods have been proposed recently (moments, direct inversion, Stieltjes transform . . . )

Stieltjes transform approach seems the most powerful;Stieltjes transform approach suffers when separability is lost;estimating number of sources remains.

test performance must be better evaluated;there is room for extension to more realistic/involved models.

overlaid communications:capacity evaluation in multi-dimensional networks is progressing fast;optimal feedback and cooperation needs to be developed with similar random matrix tools.

Future cognitive radio communications with

open-access communications,

secondary networks coordination,

will demand

a characterization of what information to share,

a merger between random matrix theory and game theory,

a stronger effort on graph-oriented random matrix theory.

The road ahead

will demand

The road ahead

will demand

The road ahead

will demand

List of Publications (1)

Publications in Journals: (7 accepted / 2 submitted)R. Couillet, J. Hoydis, M. Debbah, “A deterministic equivalent approach to the performance analysisof isometric random precoded systems,” submitted to IEEE Transactions on Information Theory, arXivPreprint XXXX.XXXX.S. Wagner, R. Couillet, M. Debbah, D. T. M. Slock, “Large System Analysis of Linear Precoding inMISO Broadcast Channels with Limited Feedback”, submitted to IEEE Transactions on InformationTheory, arXiv Preprint 0906.3682.R. Couillet, J. W. Silverstein, Z. Bai, M. Debbah, “Eigen-Inference for Energy Estimation of MultipleSources”, to appear in IEEE Transactions on Information Theory, arXiv Preprint 1001.3934.R. Couillet, M. Debbah, J. W. Silverstein, “A Deterministic Equivalent for the Analysis of CorrelatedMIMO Multiple Access Channels”, to appear in IEEE Transactions on Information Theory, arXivPreprint 0906.3667.R. Couillet, M. Debbah, “A Bayesian Framework for Collaborative Multi-Source Signal Sensing”, IEEETransactions on Signal Processing, vol. 58, no. 10, pp. 5186-5195, 2010, arXiv Preprint 0811.0764.R. Couillet, A. Ancora, M. Debbah, “Bayesian Foundations of Channel Estimation for CognitiveRadios”, Advances in Electronics and Telecommunications, vol. 1, no. 1, pp. 41-49, 2010.R. Couillet, M. Debbah, “Le telephone du futur : plus intelligent pour une exploitation optimale desfrequences” Revue de l’Electricite et de l’Electronique, no. 6, pp. 71-83, 2010.R. Couillet, M. Debbah, “Mathematical foundations of cognitive radios”, Journal ofTelecommunications and Information Technologies, no. 4, 2009.R. Couillet, M. Debbah, “Outage performance of flexible OFDM schemes in packet-switchedtransmissions”, Eurasip Journal on Advances on Signal Processing, Volume 2009, Article ID 698417,2009.

List of Publications (2)Publications in International Conferences: (16 accepted / 3 submitted)

A. Kammoun, R. Couillet, J. Najim, M. Debbah, “A G-estimator for rate adaption in cognitive radios,” submitted to IEEE Dynamic SpectrumAccess Networks, Aachen, Germany, 2011.J. Yao, R. Couillet, J. Najim, E. Moulines, M. Debbah, “CLT for eigen-inference methods in cognitive radios,” submitted to IEEE InternationalConference on Acoustics, Speech and Signal Processing, Prague, Czech Republic, 2011.J. Hoydis, R. Couillet, M. Debbah, “Deterministic Equivalents for the Performance Analysis of Isometric Random Precoded Systems,”submitted to IEEE International Conference on Communications, Kyoto, Japan, 2011.J. Hoydis, J. Najim, R. Couillet, M. Debbah, “Fluctuations of the Mutual Information in Large Distributed Antenna Systems with ColoredNoise,” Forty-Eighth Annual Allerton Conference on Communication, Control, and Computing, Allerton, IL, USA, 2010.R. Couillet, H. V. Poor, M. Debbah, “Self-organized spectrum sharing in large MIMO multiple-access channels,” IEEE InternationalSymposium on Information Theory, Austin TX, USA, 2010.R. Couillet, J. W. Silverstein, M. Debbah, “Eigen-inference for multi-source power estimation,” IEEE International Symposium on InformationTheory, Austin TX, USA, 2010.S. Wagner, R. Couillet, D. T. M. Slock, M. Debbah, “Optimal Training in Large TDD Multi-user Downlink Systems under Zero-forcing andRegularized Zero-forcing Precoding,” submitted to IEEE Global Communication Conference, Miami, 2010.S. Wagner, R. Couillet, D. T. M. Slock, M. Debbah, “Large System Analysis of Zero-Forcing Precoding in MISO Broadcast Channels withLimited Feedback” IEEE International Workshop on Signal Processing Advances for Wireless Communications, Marrakech, Morocco, 2010.R. Couillet, M. Debbah, “Information theoretic approach to synchronization: the OFDM carrier frequency offset example”, AdvancedInternational Conference on Telecommunications, Barcelona, Spain, 2010.R. Couillet, M. Debbah, “Uplink capacity of self-organizing clustered orthogonal CDMA networks in flat fading channels” IEEE InformationTheory Workshop Fall’09, Taormina, Sicily, 2009.R. Couillet, M. Debbah, J. W. Silverstein, “Asymptotic Capacity of Multi-User MIMO Communications” IEEE Information Theory WorkshopFall’09, Taormina, Sicily, 2009.R. Couillet, M. Debbah, J. W. Silverstein, “Rate region of correlated MIMO multiple access channel and broadcast channel” IEEE Workshopon Statistical Signal Processing, Cardiff, Wales, UK, 2009.R. Couillet, M. Debbah, “Mathematical foundations of cognitive radios” U.R.S.I.’09, Warsaw, Poland, 2009.R. Couillet, M. Debbah, “A maximum entropy approach to OFDM channel estimation”, IEEE International Workshop on Signal ProcessingAdvances for Wireless Communications, Perugia, Italy, 2009.R. Couillet, M. Debbah, “Bayesian inference for multiple antenna cognitive receivers”, IEEE Wireless Communications & NetworkingConference, Budapest, Hungary, 2009.R. Couillet, M. Debbah, “Flexible OFDM schemes for bursty transmissions”, IEEE Wireless Communications & Networking Conference,Budapest, Hungary, 2009.R. Couillet, S. Wagner, M. Debbah, “Asymptotic Analysis of Correlated Multi-Antenna Broadcast Channels”, IEEE Wireless Communications& Networking Conference, Budapest, Hungary, 2009.R. Couillet, S. Wagner, M. Debbah, A. Silva, “The Space Frontier: Physical Limits of Multiple Antenna Information Transfer”, ValueTools,Inter-Perf Workshop, Athens, Greece, 2008. BEST STUDENT PAPER AWARD

R. Couillet, M. Debbah, “Free deconvolution for OFDM multicell SNR detection”, IEEE Personal, Indoor and Mobile Radio Communications

Symposium, Cognitive Radio Workshop, Cannes, France, 2008.

Patents: (4 owned by ST-Ericsson)R. Couillet, M. Debbah, Application no. 08368028.0 “Process and apparatus for performing initialcarrier frequency offset in an OFDM communication system”R. Couillet, M. Debbah, Application no. 08368023.1 “Method for short-time OFDM transmission andapparatus for performing flexible OFDM modulation”R. Couillet, S. Wagner, Application no. 09368025.4 “Precoding process for a transmitter of aMU-MIMO communication system”R. Couillet, Application no. 09368030.4 “Process for estimating the channel in an OFDMcommunication system, and receiver for doing the same”

Books and book chapters: (1 book / 2 book chapters)Random Matrix Methods for Wireless Communications [book]Theoretical random matrix tools (finite dimensional analysis, limiting spectral laws, free probability,deterministic equivalents, statistical inference) and applications to wireless communications(SU-MIMO, MU-MIMO, CDMA, detection, estimation, channel modelling).

Authors: R. Couillet and M. DebbahPublisher: Cambridge University PressYear: 2011 (to appear)

Mathematical Foundations for Signal Processing, Communications and Networking [bookchapter XX]Chapter “Random matrix theory” on reminders of random matrix theory and especially statisticalinference methods.

Chapter authors: R. Couillet and M. DebbahPublisher: CRC Press, Taylor & and Francis GroupYear: 2011 (to appear)

Orthogonal Frequency Division Multiple Access Fundamentals and Applications [book chapter13]Chapter “Fundamentals of OFDMA Synchronization” on theoretical considerations and applicationtools for time offset and frequency offset regulation in OFDM and OFDMA systems.

Chapter authors: R. Couillet and M. DebbahPublisher: Auerbach Publications, CRC Press, Taylor & and Francis GroupISBN: 978-1-4200-8824-3Year: 2010

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