Supelec

Random Matrix Theoryfor

Wireless Communications

Merouane Debbahhttp://www.supelec.fr

merouane.debbah@supelec.fr

February, 2008

Presentation

MIMO Channel Modelling and random matrices

1

Where do we stand on Channel Modelling

Google search: ”MIMO Wireless Channel Modelling”

• Over 15 000 publications on channel modelling• At a rate of 10 papers per day, 1 500 days (nearly 4 years)!• The models are different and many validated by measurements!

Three conflicting schools

• Geometry based channel models.• Stochastic channel models based on channel statistics• Do not model, use test measurements

Not even within each school, all experts agree on fundamental issues.

2

MIMO System Model

TxRx

The channel is linear, noise is additive

y(t) =

√ρ

nt

∫Hnr×nt(τ)x(t− τ)dτ + n(t)

Y(f, t) =

√ρ

nt

Hnr×nt(f, t)X(f) + N(f)

3

Why do we need a channel model?

Our Vision

Step 1: Collection of informationThe user (or base station) download information on his environment (dense, number ofbuildings,...) through a localization service process

Step 2: Model generationA statistical channel model is automatically created (at the base station or the mobile unit)integrating only that information and not more!

Step 3: High speed connectionThe coding scheme is adapted to the (statistical not realization) modelExample

• Additive Gaussian: Euclidean distance coding• Rayleigh i.i.d: rank and determinant criteria

This scenario could be called ”User customized channel model coding service” and is aviable scenario from a Soft Defined Radio perspective.

4

Why do we need a statistical channel model?

Ergodic Channel Capacity: (The receiver knows the channel and the transmitter knowsthe statistics)

C = maxQE(C(Q)) with C(Q) = log2det(Int + ρ

ntHHQH

)

Q = E(XXH) = I only with i.i.d zero mean Gaussian MIMO model!

The need to model: Statistical channel models stimulate creativity (patents!):

• to optimize the codes• to estimate the channel

in order to achieve the information theoretic limits.

This can not be performed with simulation or measurement based models!

5

Types of questions channel modelling will answer

Multiple versus Single Antenna

SISO AWGN Channel: C = log2(1 + ρ) ∼ log2(ρ) at high SNR.

MIMO Channel:

• Suppose that the channel matrix is deterministic with equal entries 1 (Pure Line of Sightcase):

C = log2 det(Inr +ρ

nt

HHH).

=

nr∑

i=1

log2(1 +ρ

nt

λi)

= log2(1 + ρnr)

→ log2(ρ)at high SNR

• i.i.d Rayleigh fading: C = min(nr, nt) log2(ρ) at high SNR.

Is there a Contradiction?

6

Let us start...

Model Construction

7

The i.i.d Gaussian model

The modeler would like to attribute a joint probability distribution to:

H(f) =

h11(f) . . . . . . h1nt(f)... . . . . . . ...... . . . . . . ...

hnr1(f) . . . . . . hnrnt(f)

(1)

Assumption 1: The modeler has no knowledge where the transmission took place (thefrequency, the bandwidth, the type of room, the nature of the antennas...)

Assumption 2: The only things the modeler knows:For all {i, j},

E(∑

i,j | hij |2) = nrntE

What distribution P (H) should the modeler assign to the channel based only on thatspecific knowledge?

8

The i.i.d Gaussian model

Principle of maximum entropy

Maximize the following expression:

−∫

dHP (H)logP (H) + γ[nrntE −∫

dHnr∑

i=1

nt∑

j=1

| hij |2 P (H)]

+β

[1−

∫dHP (H)

]

Solution:

P (H) =1

(πE)nrntexp{−

nr∑

i=1

nt∑

j=1

| hij |2E

}

Contrary to past belief, the i.i.d Gaussian model is not an assumption but the result offinite energy knowledge.

This method can be extensively used whenever additional information is provided in termsof expected values.

9

Knowledge of the covariance structure

In the general case, under the constraint that:∫

CNhih

∗jPH|Q(H)dH = qi,j

for (i, j) ∈ [1, . . . , N ]2 (N = nrnt). Then using Lagrangian multipliers,

L(PH|Q) =

∫

CN− log(PH|Q(H))PH|Q(H)dH

+ β

[1−

∫

CNPH|Q(H)dH

]

+∑

αi,j

[∫

CNhih

∗jPH|Q(H)dH− qi,j

].

we obtain:

PH|Q(H) =1

det(πQ)exp

(−(vec(H)

HQ−1vec(H)))

.

10

Existence of Correlation

Question

What to do if we know the existence of correlation but not its exact value?

Answer

P (H) =

∫P (H, Q)dQ =

∫P (H | Q)P (Q)dQ

1- Determine the a priori distribution of the covariance matrix based on limited informationat hand

2- Marginalize with respect to the a priori distribution

11

Construction of the a priori

Let us determine the a priori distribution of the covariance

Suppose that we only know that E(Trace(Q)) = nrntE (The covariance is not fixed butvaries due to mobility for example)

Result. The maximum entropy distribution for a covariance matrix Q under the constraintE(Trace(Q)) = nrntE is such as:

Q = UΛUH

where:

• U is haar unitary distributed matrix.• Λ = diag

(λ1, ..., λnrnt

)is diagonal matrix with independent Laplacian distributions.

P (Q)dQ =1

EnrntΠ

nrnr−1n=0 (n!(n + 1)!)e

−Trace(Q)E Πi>j(λi − λj)

2dUdΛ

Note that Q is nothing else than a Wishart matrix with nrnt degrees of freedom.

12

MIMO Channel distribution with correlation

What do we need to do?

P (H) =

∫P (H | Q)P (Q)dQ

=

∫ ∫1

π∏nrnt

i=1 λi

e−Trace(vec(H)vec(H)HUHΛ−1U)

1

EnrntΠ

nrnr−1n=0 (n!(n + 1)!)e

−Trace(Q)E Πi>j(λi − λj)

2dUdΛ

We need to integrate over U and Λ!

Difficult problem...but well known in statistical physics!

13

MIMO Channel distribution with correlation

Harish-Chandra, ”Differential Operator on a Semi-Simple Lie Algebra”, Amer. J. Math. 7987-120 (1957)

Harish-Chandra, 1923-1983

Harish-Chandra integral

∫

U∈U(m)

e−mTrace(Σ−1UΛU−1)

dU =det(e−σ−1

jλk)

∆(Σ−1)∆(Λ)

14

MIMO Channel distribution with correlation

”Maximum Entropy Analytical MIMO Channel Models”, M. Guillaud, M. Debbah and A.Moustakas, submitted to IEEE transactions on Information Theory, 2006.

Solution. P (H) is given by

P (H) =

nrnt∑n=1

2(Trace(HHH)

E)

n+nrnt−22 Kn+nrnt−2(2

√Trace(HHH)

E).

(−1)nnrnt

[(n− 1)!]2(nrnt − n)!

Kn(x) are bessel functions of order n.

We have therefore an explicit form that can be used for design when correlation exists in

the MIMO model but we are not aware of the explicit value of the correlation!

15

MIMO Channel distribution with correlation

What happens when we know of the existence of a channel covariance matrix with rank L?

Using the same methodology (and integration on a lower subspace), we obtain:

P(L)

(H) =2

Trace(HHH)

L∑

i=1

−L

√Trace(HHH)

E0

L+i

Ki+L−2

2L

√Trace(HHH)

E0

1

[(i− 1)!]2(L− i)!

.

Kn(x) are bessel functions of order n.

16

Some distributions

0 10 20 30 40 50 60−0.02

0

0.02

0.04

0.06

0.08

0.1

Energy x

PD

F o

f x=

||H|| F2

iid Gaussian (χ2)

MaxEnt P(16)x

(x)

MaxEnt P(12)x

(x)

MaxEnt P(8)x

(x)

MaxEnt P(4)x

(x)

MaxEnt P(2)x

(x)

Examples of the limited-rank covariance distribution for a 4× 4 MIMO matrix andL = 2, 4, 8, 12 and 16, and χ2 with 16 degrees of freedom, for E0 = 16.

17

Some distributions

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

mutual information (nats) at 15dB SNR, nr=n

t=4

CD

F

MaxEnt rank 1

MaxEnt rank 2

MaxEnt rank 3

MaxEnt rank 7

MaxEnt rank 16

iid Gaussian

CDF of the instantaneous mutual information of a 4× 4 flat-fading channel for the MaxEntmodel with various covariance ranks, at 15dB SNR.

18

Presentation

Asymptotic Analysis of MIMO systems

19

Representation of a multiple-antenna system: example

Rx Tx

Φnr×sr Ψst×ntΘsr×st

• Ψst×nt: matrix of direction of departure of size st × nt.• Φnr×sr: matrix of direction of arrival of size nr × sr.• Θsr×st i.i.d Gaussian matrix of size sr × st.

This model is known as the Maxent model (the Kronecker model, Sayeed’s virtual

representation and the key-hole model can be shown to be particular cases).

20

CLT for MIMO systems

M. Debbah and R. Muller, ”MIMO Channel Modelling and the Principle of MaximumEntropy,” IEEE Transactions on Information Theory, Vol. 51 , pp. 1667 - 1690, May, No.5-2005

y =

√ρ

nt

Hs + n

=

√ρ

nt

1√srst

Φnr×srΘsr×stΨst×nts + n

For many types of random matrices:

limnt→∞,nr

nt=α

log2det(

Inr +ρ

nt

HHH

)− ntµ → N (0, σ

2).

21

General result for H = 1√srst

ΦΘΨ

Result: Let η et ξ be the eigenvalues of matrices 1sr

ΦΦH and 1st

ΨHΨ respectively (Θ isan sr × st i.i.d zero mean Gaussian matrix):

µ =

nt∑

i=1

log2(1 + ρξir) +

nr∑

i=1

log2(1 + ρηiq)− nrqr

σ2= −2 log(1− g(r, q))

g(r, q) =

[1

nt

nt∑

i=1

(ρηi

1 + ηiρq)2

] [1

nt

nr∑

i=1

(ρξi

1 + ξiρr)2

]

r =1

nt

nt∑

i=1

ρηi

1 + ηiρq

q =1

nt

nr∑

i=1

ρξi

1 + ξiρr

22

Elements of Proof

µ =1

nt

log det(

Inr +ρ

nt

HHH

)

=1

nt

nr∑

i=1

log(1 + ρnr

nt

λi)

=nr

nt

1

nr

nr∑

i=1

log(1 + ρnr

nt

λi)

→ nr

nt

∫log(1 + ρ

nr

nt

λ)dF 1nrHHH(λ)

23

Elements of Proof

More specifically, let1

sr

ΦHΦ = VφΛφVHφ

1

st

ΨΨH= VψΛψVH

ψ

Vψ and Vφ are unitary matrices while Λφ and Λψ are diagonal matrices representing

respectively the eigenvalues of matrices 1sr

Pr12ΦHΦPr1

2 and 1st

Pt12ΨΨHPt

12 . The

non-zero eigenvalues of matrix 1nr

HHH = 1nrsrst

ΦPr12ΘPt

12ΨΨHPt

12ΘHPr1

2ΦH are the

same as Θ1ΘH1 = 1

nr[Λ

12Φ(VΦ

HΘVΨ)Λ12Ψ][Λ

12Ψ(V H

Ψ ΘHVΦ)Λ12Φ]. Without loss of generality,

we will suppose that sr ≤ nr. Therefore, the spectra of 1nr

HHH and Θ1ΘH1 are related

by:fHHH(x) = (1− sr

nr

)δ(x− 0) +sr

nr

fΘ1ΘH1(x)

and their Stieltjes transforms are related as:

mHHH(z) = (1− sr

nr

)1

z+

sr

nr

mΘ1ΘH1(z)

24

Elements of Proof

Matrix VφΘVΨ is an i.i.d zero mean Gaussian matrix with unit variance (only unitary

transforms are applied). Therefore, matrix Θ1 = 1√nr

[Λ12Φ(VΦΘVΨ)Λ

12Ψ] is a sr × st

random matrix composed of independent entries with zero mean and variances1

nrλi

φλjψ = 1

sr

λiΦλj

Ψ

γ . The weak convergence of the empirical eigenvalue distribution of

Θ1ΘH1 to a limiting distribution holds under certain assumptions and is an application of

the theorem of Girko.

25

Remarks on capacity of the double directional model

Proposition: In the high SNR regime, the mean mutual information of the doubledirectional model converges to:

min(

nt, nr, sr

∫

λ>0

dSdoa(λ), st

∫

λ>0

dSdod(λ)

)log2(ρ)

The factor min(nt, nr, sr

∫λ>0

dSdoa(λ), st

∫λ>0

dSdod(λ))

denotes the multiplexing gain.

∫λ>0

dSdoa(λ) and∫

λ>0dSdod(λ)) express the correlation factor of the sr and st

scatterers respectively.

The MIMO mutual information is limited by the scattering environment!

26

Mutual information compliance

For a given frequency f , a model will be called capacity complying if it minimizes:

∫ ∞

0

| F (C, f)− Fempirical(C, f) |2 dC (2)

Here Fempirical(C, f) is the empirical cdf given by measurements.

• How to derive the theoretical cdf F (C) of the capacity?

F (C) = 1−Q(C − ntµ

σ)

with

Q(x) =1√2π

∫ ∞

x

dte−t2

2

since C = log2det(Int + ρ

ntHHH

)is asymptotically Gaussian.

27

Measurements of channels

Measurements have been performed:

• at 2.1 GHz• at 5.2 GHz

In the following scenarios:

• Indoor• Atrium• Urban Open Place• Urban Regular High Antenna• Urban Regular Low Antenna

28

Sounder characteristics at 2.1 GHz

Measurement frequency 2.1 GHzMeasurement bandwidth 100 MHz

Delay resolution 10nsSounding signal linear frequency chirp

Transmitter antenna 8 element ULAElement spacing 71.4 mm (0.5 λ)Receiver antenna 32 (8× 4) elementElement spacing 73.0 mm (0.51 λ)

29

Sounder characteristics at 5.2 GHz

Measurement frequency 5.255 GHzMeasurement bandwidth 100 MHz

Sounding signal linear frequency chirpTransmitter antenna 8 element ULA

Element spacing 28.54 mm (0.5 λ)Receiver antenna 8 element ULA

30

Antennas

The receiver is an 8× 4 antenna whereas the transmitter is an 8 elements ULA.

31

Example: Urban Open Place at 2.1 GHz

32

Is mutual information Gaussian at 2.1 GHz

15 16 17 18 19 20 21 22 23 24 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

b/s/Hz

CD

F

Are the measured mutual information Gaussian?

MeasuredGaussian approximation

Urban RegularHIgh Antenna

Indoor

Urban Open Place

Atrium

Urban regular low Antenna

Mutual information has a Gaussian behavior for an 8× 8 system at 2.1Ghz!

33

Is mutual information Gaussian at 5.2 GHz

15 16 17 18 19 20 21 22 23 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

b/s/Hz

CD

F

Are the measured mutual information Gaussian?

MeasuredGaussian approximation

Urban Open Place

Indoor

Atrium

Urban Low Antenna

• Mutual information has a Gaussian behavior for 8× 8 system at 5.2Ghz!

34

Rice Model Case

Single Antenna/Multiple Antennas

SISO Channel: Capacity of the AWGN channel is higher than the ergodic capacity of aRayleigh fading channel (Jensen’s inequality).

Rice MIMO Channel: Suppose that the matrix representing the channel has deterministicentries equal to 1:

C = log2det(Inr +ρ

nt

HHH).

=

nr∑

i=1

log2(1 +ρ

nt

λi)

= log2(1 + ρnr)

→ log2(ρ)at high SNR

MIMO i.i.d zero mean Gaussian : C = min(nr, nt)log2(ρ) at high SNR.Is there a contradiction?

35

Rice Model Case

Diversity versus path loss trade-off

Free Space Propagation:ρ ∼ ρmax

r2

Propagation with reflections:ρ ∼ ρmax

r2n+2

(n is the number of reflections)

There is a trade-off between the gain in SNR due to the line of sight component and themultiplexing gain due to multiple reflections.

The analysis of MIMO Rice channels is important to determine the interval of interest.

36

Rice Model Case

Rice versus Rayleigh

Rayleigh fading:

• Rayleigh channels have been studied extensively in the case of non-line of sight.• In the case of i.i.d entries, ergodic capacity increase is min(nr, nt) bits per second per

hertz for every 3dB increase at high SNR.• Highly scattered environnements increase ergodic capacity.

Rice fading:

• Many propagation scenarios have line of sight components (Inter-base stationtransmissions,..).

• Capacity studies limited to rank 1 Rice fading or based on very loose bounds.• The impact of line sight with scattering is still an issue!

37

Rice Model Case

General Model

H =

√K

K + 1A +

√1

K + 1B

A is the deterministic line of sight component part of the matrix such as ‖A‖2F = ntnr

B is a gaussian unit variance zero mean i.i.d matrix.

The average energy of the channel is normalized according to: E(tr(HHH))nrnt

= 1.

Assumption: The matrix size 1nt

AAH grows large with β = nrnt

remaining fixed such asthe empirical eigenvalue distribution of 1

ntAAH converges in distribution to a deterministic

limit function F A√nt

.

Remark: The rank of A increases at the same rate as the number of antennas.

38

Rice Model Case

Remember the useful result

Let mH(z) be the Stieljes transform of the limiting eigenvalues FH(λ) of HHHnt

:

mH(z) =

∫dFH(λ)

λ− z

As nr = βnt →∞,

mH(z) =

∫d FA(λ)

KλβmH(z)+K+1 − z

(βmH(z)

K+1 + 1)

+ 1−βK+1

The asymptotic eigenvalue distribution of HHHnt

is completely determined knowing onlyFA(λ), β and K and not the particular fluctuations of the fading!

39

Rice Model Case

Perfect Channel Knowledge at the transmitter

The channel capacity per receiving antenna converges almost surely, as nr = βnt →∞,to:

C1 =1

ln 2

∫ +∞

1µ∗

ln λµ∗d FH(λ)

β

∫ +∞

1µ∗

(µ∗ − 1

λ

)d FH(λ) = ρ

Remark:

• The capacity is achieved through waterfilling power allocation.• The channel should change slowly enough in order to have complete channel

knowledge at the transmitter.

40

Rice Model Case

Channel knowledge of the limiting singular value distribution of A

L. Cottatellucci, M. Debbah, ”The Effect of Line of Sight Components on the AsymptoticCapacity of MIMO Systems,” 2004 IEEE International Symposium on InformationTheory,Chicago, June 27 - July 2, 2004, USA

Since the transmitter has no knowledge of the eigenvector structure, the transmittedpower is equally distributed among the antennas.

The channel mutual information per receiving antenna converges almost surely, asnr = βnt →∞, to:

C =1

ln 2

∫ ρ

0

1

x

(1− 1

xmH

(−1

x

))d x

Remark:

• If K → 0, mH(z) = 1−z(βmH(z)+1)+(1−β) (Marchenko Pastur Distribution)

• If K →∞, mH(z) =∫ d FA(λ)

λ−z (distribution of the mean)

41

Rice Model Case

Perfect Knowledge of line of sight matrix A

D. Hosli and A. Lapidoth, The Capacity of a MIMO Ricean Channel is Monotonic in theSingular Values of the Mean, 5th International ITG Conference on Source and ChannelCoding (SCC), Erlangen, Nuremberg, jan, 2004

The optimal covariance matrix Q is shown to have the same eigenvectors as AHA.

42

Rice Model Case

Perfect Knowledge of line of sight matrix A

The capacity is then given by:

J(H, UQUH) =

1

nr

log2 det(

Inr +ρ

nt

HUQUHHH)

=1

nr

log2 det(

Inr +ρ

nt

VHQHHVH)

=1

nr

log2 det(

Inr +ρ

nt

HQHH

)

H = VHHU =√

KK+1A +

√1

K+1VHBU

VHBU is a random matrix with i.i.d zero mean Gaussian entries

The line of sight matrix√

KK+1A is diagonal.

43

Rice Model Case

Perfect Knowledge of line of sight matrix A

The asymptotic channel capacity per receive antenna converges almost surely to:

C2(H) = maxQ(λ,FA(λ),K,β)

J(H, UQUH)

J(H, UQUH) =

1

ln 2

∫ ρ

0

1

x

(1− 1

xm2

(−1

x

))d x

m1(z) =∫ β(m1(z)− z(K + 1))q(K + 1) d FA(λ)

(K + 1 + βm2(z)q)(m1(z)− z(K + 1)) + (K + 1)Kλq+

(1− β)q0(K + 1)

(K + 1 + βm2(z)q0)

m2(z) =∫ (K + 1 + qβm2(z))(K + 1) d FA(λ)

(K + 1 + βm2(z)q)(m1(z)− z(K + 1)) + λq(K + 1)K

q = q(λ, F (λ), K, β) denotes the diagonal entries of Q and q0 = q(0, F (λ), K, β).

Remark: The solution is not waterfilling on the mean matrix!

44

Rice Model Case

Simulations: general resultsFA = 1

4δ(λ− 3) + 34δ(λ− 1

3), β = 1, ρ = 5dB.

10−2

10−1

100

101

102

103

104

1.6

1.65

1.7

1.75

1.8

1.85

Rice channel capacity for FA

(λ)=δ(λ−3)/4+3 δ(λ−1/3)/4, β=1, ρ=5 dB

Rice factor K

Cap

acity

(bi

t/s/H

z)

C1: Asymptotic evaluationC1: Simulation with 8 × 8 MIMO systemC2: Asymptotic evaluationC2: Simulation with 8 × 8 MIMO systemC3: Asymptotic evaluationC3: Simulation with 8 × 8 MIMO system

Rayleigh channel

• Close match with ergodic capacities for a system with 8× 8 antennas.• Mean feedback is sufficient for values of K > 10.

45

Rice Model Case

Simulations: Low and high SNR regimeFA = 1

4δ(λ− 3) + 34δ(λ− 1

3), β = 1.

10−2

10−1

100

101

102

103

104

0

0.05

0.1

0.15

0.2

0.25

Rice channel with FA(λ)=1/4 δ(λ−3)+3/4 δ(λ−1/3), β=1.

Rice factor K

(C1−

C3)/

C3

ρ=0 dB

1 dB

2 dB

4 dB

6 dB

8 dB

10 dB

10−2

10−1

100

101

102

103

104

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Rice channel with FA(λ)=1/4 δ(λ−3)+3/4 δ(λ−1/3), β=1.

Rice factor K

(C2−

C3)/

C3

ρ=0 dB

1 dB

2 dB

4 dB

6 dB 8 dB 10 dB

• At high SNR, the effect of feedback on the capacity diminishes.• For low SNR, the effect of feedback is quite important and depends mainly on the Ricean

factor K.

46

Rice Model Case

Simulations: reduced rank, mean matrix known at the transmitter.FA =

nt8 δ(λ) +

8−ntnt

δ(λ− 88−nt

), β = 1, ρ = 10.

10−2

10−1

100

101

102

103

104

1

1.5

2

2.5

3

3.5

Rice factor K

Cap

acity

C2, (

bits

/s/H

z)

Rice channel capacity C2 for F

A(λ)=p

0 δ(λ)+p

1δ(λ−1/p

1), β=1, ρ=10.

p0= 0

1/8

2/8

3/8

4/8

5/8

6/8

7/8

• Rank reduction yields a dramatric effect on the capacity.• Transition phase with respect to K around [1,10].

47

Rice Model Case

Simulations: rank 1, perfect channel knowledge at the transmitter.FA =

nt−1nt

δ(λ) + 1nt

δ(λ− nt), β = 1, ρ = 10.

10−2

10−1

100

101

102

103

104

0.5

1

1.5

2

2.5

3

Rice factor K

Cap

acity

C1, (

bits

/s/H

z)

Rice channel capacity C1 for F

A(λ)=(n

t−1)/n

t δ(λ)+δ(λ−n

t)/n

t, β=1, ρ=10.

4

5

678

10

16

23

nt= 2

3

• Asymptotic analysis able to capture the rank 1 case.• Close match with a small number of antenna elements.

48

Rice Model Case

Rice or Rayleigh?

Asymptotic analysis of Ricean MIMO channels show that Rice fading is not always betterthan Rayleigh fading.

The result depends in fact on:

• the limiting behavior of the mean matrix.• the ratio β = nr

nt.

• the SNR ρ.• the Ricean factor K.• through various fixed point equations depending on the channel state of knowledge

available at the transmitter.

49

Rice Model Case

More Reading

J. Dumont, P. Loubaton, S. Lasaulce and M. Debbah, ”On the Asymptotic Performance ofMIMO Correlated Ricean Channels,” in Proc. IEEE Int. Conf. on Acoustics, Speech andSig. Proc. (ICASSP), Montreal, Canada, Mar. 2005, pp. 813-816.

J. Dumont, W. Hachem, P. Loubaton and J. Najim, ”On the asymptotic Analysis of mutualinformation of MIMO Rician correlated channels”, IEEE-EURASIP Int. Sym. on Control,Commun. and Sig. Proc. (ISCCSP 2006), Marrakech, Morocco, Mar. 2006

A. L. Moustakas and S.H. Simon, ”Random matrix theory of multi-antennacommunications: the ricean case”, J. Phys. A: Math. Gen. 38 (2005), 10859-10872.

J. Dumont, P. Loubaton and S. Lasaulce, ”On the capacity Achieving Transmit CovarianceMatrices of MIMO Rician Channels: A Large System Approach”, GlobeCom 2006,San-Francisco, USA, FIRST PATENT DUE TO FRANCE TELECOM based on Randommatrices.

50

Presentation

Asymptotic design of receivers

51

Model

y = W P12 s + n

Received signal code matrix Amplitude Gains emitted signal AWGN

N × 1 N ×K K ×K K × 1 ∼ N (0, σ2IN)

The goal is to detect s.

Linear MMSE Filter:LMMSE = P

12

H

WH(R + σ

2I)−1

R = WPWH

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Matrix Inversion

Let R be non-singular.Let λi be the eigenvalues of R.

Then, ∏Kk=1 (R− λkI) = 0 ⇒ −I +

∑Kk=1 αkRk−1 = 0

Cayley − Hamilton Theoreom with appropriate αk′s.

Solution to matrix inversion problem given the eigenvalues

R−1=

K∑

k=1

αkRk

53

Linear Multi-Stage Detection

M. Honig and W. Xiao, ”Performance of Reduced Rank Linear Interference Suppression”,IEEE Transactions on Information Theory, vol. 47, No.5, July 2001.

Linear MMSE Filter: LMMSE =(R + σ2I

)−1

Approximation by Power Series:

Cayley-Hamilton Theorem yields:

(R + σ

2I)−1

=

K−1∑

i=0

θiRi

'D−1∑

i=0

θiRi for D < K

For a N ×K matrix W with i.i.d random spreading and R = WWH, the o¯ptimum

weights converge almost surely as K, N →∞ with α = KN , and can be given in c

¯losed

form.

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Semi-Universal Weights

R. Muller and S. Verdu, ”Design and Analysis of Low-Complexity Interference Mitigationon Vector Channels”, IEEE Journal on Selected Areas in Communications, vol. 19, no. 8,pp. 1429-1441, August 2001.

Filter shall be independent from the realization of the random matrix W, but may use itsstatistics.

For most large random matrices, as K = αN →∞, many finite dimensional functions ofthe eigenvalues, e.g, the filter coefficients free.

The asymptotic limits depend only on parts of the statistics of the random matrix.

The weights can be calculated off-line with the help of random matrix theory and freeprobability theory.

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Weight Design

is given by the Yule-Walker equations:

m1

m2...

mD+1

=

m2 + σ2m1 m3 + σ2m2 . . . mD+2 + σ2mD+1

m3 + σ2m2 m4 + σ2m3 . . . mD+3 + σ2mD+2... ... . . . ...

mD+2 + σ2mD+1 mD+3 + σ2mD+2 . . . m2D+2 + σ2m2D+1

w0

w1...

wD+1

with the total moments:mn = E(λn

) = Trace(WWH)n

.

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Example for Weight Design

Random with i.i.d entries

D=2 w0 = −σ2m1 + 2 + 2β

w1 = −1

D=3 w0 = −σ2m1 + 3 + 4β + 3β2

w1 = −σ2m2 − 3− 3β

w2 = 1

D=4 w0 = −σ2m1 + 4 + 6β + 6β2 + 4β3

w1 = −σ2m2 − 6− 9β − 6β2

w2 = −σ2m3 + 4 + 4β

w3 = −1

mn =1

n

n∑

i=1

Cni C

ni−1β

i

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Rate of Convergence

Ph. Loubaton and W. Hachem, ”Asymptotic Analysis of Reduced Rank Wiener Filters”,Information Theory Workshop 2003, Paris, France

Theorem: Suppose that the elements of the matrix are i.i.d zero-mean random variablewith finite fourth moment. Then the multi-user efficiencies η of all users c

¯onverge almost

surely as N, K →∞ but KN fixed to:

ηD+1 =1

1 + β

σ2+ηD

with η0 = 0 for optimally chosen weights.

The approximation converges to the exact MMSE performance as a continued fraction.For optimally weights wi, the approximation error ε decreases exponentially with thenumber of stages D.

ε < Const(1 + SNR)−D

58

Individual Weight Design

Allow for different weight for different users:

(R + σ

2I)−1

=

K−1∑

i=1

wiRi

≈D−1∑

i=0

WiRi for D < K and all Wi diagonal

Weight design by the same Yule-Walker equations, but with the k-partial moments:

mkn =

[(SSH

)n]

kk

For users with different powers, individual weight design is better.

Do the k-partial moments converge asymptotically?

59

Convergence of partial moments

Let the random matrix S fulfill the same conditions as before. Let A be an K ×K

diagonal matrix such as its singalur value distribution converges almost surely asK →∞ to a non-random limit distribution. Let R = ASSHAH

Then Rlkk, the k-th diagonal element of Rl converges, conditioned on akk, the k-th

diagonal element of A, almost surely, as K = βN →∞ to

Rlkk =| akk |2 β

∑lq=1 Rq−1

kk mRl−q with mR

q = Trace(Rq)

The total moments of R are conveniently given by the recursion:

mRl = β

l∑q=1

mRl−q lim

K→∞1

K

K∑

k=1

| akk |2 Rq−1kk

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Model

The system is described by a virtual NR×K spreading matrix

S =

h11s1 h12s2 . . . h1KsK

h21s1 h22s2 . . . h2KsK... ... . . . ...

hR1s1 hR2s2 . . . hRKsK

Note that with the Kronecker product ⊗:

sk = hk ⊗ sk

Note also that the entries of S are not jointly independent even if those ones of S and H

are.

61

Ressource Pooling Result

S. Hanly and D. Tse, ”Resource Pooling and Effective Bandwidths in CDMA Systems withMultiuser Receivers and Spatial Diversity”, IEEE Transactions on Information Theory, vol.47(4), May 2001, pp. 1328-1351

Theorem Let the chips of any user be i.i.d zero-mean random variables with finite 6thmoment and the antenna array channel hrk follow the i.i.d Complex Gaussian Model.Then the multiuser efficiency ηMMSE of the linear MMSE detector c

¯onverges for all users

almost surely as N, K →∞ but β = KN and R fixed to the deterministic unique positive

solution of the fixed point equation:

1

ηMMSE= 1 +

β

R

∫x

σ2 + ηMMSExdPA2(x)

if the power of the users converge weakly to the limit distribution PA2 with:

| Ak |2=| Ak |2R∑

r=1

| hrk |2

62

Ressource Pooling result for correlated MIMO Channels

L. Cottatellucci, R. Muller, ”A Generalized Resource Pooling Result for CorrelatedAntennas with Applications to Asynchronous CDMA” International Symposium onInformation Theory and its Applications, Parma, Italy, October 2004

Theorem Let the chips of any user be i.i.d zero-mean random variables with finite 6thmoment and the empirical distribution of the channel gains hrk across the usersconverge, jointly for all receive antennas r to an R-dimensional joint limit distribution PH.Then, with linear MMSE detection, the SINR converges as N, K →∞ but β = K

N and Rfixed, conditioned on the channel gains of user k to:

hHk Ahk

σ2

where A is the deterministic unique positive definite solution of the matrix valued fixedpoint equation

A−1

= I + β

∫xxH

σ2 + xHAxdPH(x)

Asymptotic Performance is characterized by an R× R matrix.

63

Multi-stage Detection for Correlated Ressource Pooling

As the dimension of S grows, the following result hold:

the k partial moments conditioned on hk converge

Explicit Recursive expressions for them are now available

The proof follow along the same lines as before.

The system is sensitive to the correlation at the receiving side.

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Last Slide

THANK YOU!

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