SupelecRandom Matrix TheoryforWireless CommunicationsM´erouane Debbahhttp://[email protected], 2008PresentationMIMO Channel Modelling and random matrices1Where do we stand on Channel ModellingGoogle 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 measurementsNot even within each school, all experts agree on fundamental issues.2MIMO System ModelTxRxThe channel is linear, noise is additivey(t) =rρntZHnr×nt(τ)x(t − τ)dτ + n(t)Y(f, t) =rρntHnr×nt(f, t)X(f) + N(f)3Why do we need a channel model?Our VisionStep 1: Collection of informationThe user (or base station) download information on his environment (dense, number ofbuildings,...) through a localization service processStep 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 criteriaThis scenario could be called ”User customized channel model coding service” and is aviable scenario from a Soft Defined Radio perspective.4Why 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) = log2det³Int+ρntHHQH´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 channelin order to achieve the information theoretic limits.This can not be performed with simulation or measurement based models!5Types of questions channel modelling will answerMultiple versus Single AntennaSISO 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 = log2det(Inr+ρntHHH).=nrXi=1log2(1 +ρntλi)= 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?6Let us start...Model Construction7The i.i.d Gaussian modelThe 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(Pi,j| hij|2) = nrntEWhat distribution P (H) should the modeler assign to the channel based˚only on thatspecific knowledge?8The i.i.d Gaussian modelPrinciple of maximum entropyMaximize the following expression:−ZdHP (H)logP (H) + γ[nrntE −ZdHnrXi=1ntXj=1| hij|2P (H)]+β·1 −ZdHP (H)¸Solution:P (H) =1(πE)nrntexp{−nrXi=1ntXj=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.9Knowledge of the covariance structureIn the general case, under the constraint that:ZCNhih∗jPH|Q(H)dH = qi,jfor (i, j) ∈ [1, . . . , N ]2(N = nrnt). Then using Lagrangian multipliers,L(PH|Q) =ZCN−log(PH|Q(H))PH|Q(H)dH+ β·1 −ZCNPH|Q(H)dH¸+Xαi,j·ZCNhih∗jPH|Q(H)dH − qi,j¸.we obtain:PH|Q(H) =1det(πQ)exp³−(vec(H)HQ−1vec(H))´.10Existence of CorrelationQuestionWhat to do if we know the existence of correlation but not its exact value?AnswerP (H) =ZP (H, Q)dQ =ZP (H | Q)P (Q)dQ1- Determine the a priori distribution of the covariance matrix based on limited informationat hand2- Marginalize with respect to the a priori distribution11Construction of the a prioriLet us determine the a priori distribution of the covarianceSuppose 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 = UΛUHwhere:• U is haar unitary distributed matrix.• Λ = diag¡λ1, ..., λnrnt¢is diagonal matrix with independent Laplacian distributions.P (Q)dQ =1EnrntΠnrnr−1n=0(n!(n + 1)!)e−Trace(Q)EΠi>j(λi− λj)2dUdΛNote that Q is nothing else than a Wishart matrix with nrntdegrees of freedom.12MIMO Channel distribution with correlationWhat do we need to do?P (H) =ZP (H | Q)P (Q)dQ=Z Z1πQnrnti=1λie−Trace(vec(H)vec(H)HUHΛ−1U)1EnrntΠnrnr−1n=0(n!(n + 1)!)e−Trace(Q)EΠi>j(λi− λj)2dUdΛWe need to integrate over U and Λ!Difficult problem...but well known in statistical physics!13MIMO Channel distribution with correlationHarish-Chandra, ”Differential Operator on a Semi-Simple Lie Algebra”, Amer. J. Math. 7987-120 (1957)Harish-Chandra, 1923-1983Harish-Chandra integralZU∈U(m)e−mTrace(Σ−1UΛU−1)dU =det(e−σ−1jλk)∆(Σ−1)∆(Λ)14MIMO 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 byP (H) =nrntXn=12(Trace(HHH)E)n+nrnt−22Kn+nrnt−2(2sTrace(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 inthe MIMO model but we are not aware of the explicit value of the correlation!15MIMO Channel distribution with correlationWhat 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) =2Trace(HHH)LXi=1−LsTrace(HHH)E0L+iKi+L−22LsTrace(HHH)E01[(i − 1)!]2(L − i)!.Kn(x) are bessel functions of order n.16Some distributions0 10 20 30 40 50 60−0.0200.020.040.060.080.1Energy xPDF of x=||H||F2iid 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