SupelecRandom Matrix TheoryforWireless CommunicationsM´erouane Debbahhttp://[email protected], 2008PresentationThe role of the Cauchy-Stieltjes Transform in Communications1General Multiple Input Multiple Output ModelModel representing multiple-antennas, CDMA, OFDM, ad-hoc networks withcooperation,...y = W s + nReceived signal MIMO matrix emitted signal AWGNN × 1 N × K K × 1 ∼ N(0, σ2IN)LetW =u U, s =·s1x¸The goal is to detect s.2Communications Notations UsedReceiving vector• r as ”Received”.• y as ”Y do we care?”.Transmitting vector• s as ”Signal”.• x as ”Xciting”.Noise vector• n as ”Noise” (english).• b as ”Bruit” (french).• z as ”Zzzzzzz...” (disturbance)MIMO matrix• W when the matrix is non-isometric.• Θ when the matrix is isometric• H when we consider multiple-antennas.3Shannon CapacityMutual information M between input and output:M(s; (y, W)) = M(s; W) + M(s; y | W)= M(s; y | W)= H(y | W) − H(y | s, W)= H(y | W) − H(n)The differential entropy of a complex Gaussian vector x with covariance Q is given bylog2det(πeQ).4Shannon CapacityIn the case of Gaussian independent entries, sinceE(yyH) = σ2IN+ WQWHE(nnH) = σ2INThe mutual information per dimension is:CN=1N(H(y | W) − H(n))=1N³log2det(πe(σ2IN+ WQWH)) − log2det(πeσ2IN)´=1Nµlog2det(IN+1σ2WQWH)¶5Shannon CapacityConsider the random variableCN=1Nlog detµIN+1σ2WWH¶=1NNXk=1logµ1 +1σ2λk³WWH´¶When N → ∞ and K/N → α,CN→Zlogµ1 +1σ2t¶µ(dt) a.s.dCNd1σ2→ σ2− σ4Gµ(−σ2) a.s.The capacity is strongly related to the Cauchy-Stieltjes transform.6Some numerical factsW i.i.d. zero mean with variance1N: CN=1Nlog det³IN+1σ2WWH´0 5 10 152.72.752.82.852.92.95Number of antennasb/s/HzMean for variable matrix size at 10dBSimulationsTheoretical formula• With N= 6, the empirical mean is at 0.02% of the asymptotic value.• With N= 3, the empirical mean is at 0.6% of the asymptotic value.7MMSE ReceiverModel example :y = Ws + n= us1+ Ux + n= us1+ n0E(n0n0H) = (UUH+ σ2I) = QΛQHWhitening filter:˜y = Λ−12QHy = Λ−12QHus1+ Λ−12QHn0= g s1+ bb is a white Gaussian noise.8MMSE Receiver˜y = Λ−12QHus1+ bDefine g = Λ−12QHuThe output SINR is maximized with:gH˜y = gHgs1+ gHbAs a consequence, the receiver is:gHΛ−12QH= uH³QΛ−1QH´= uH³UUH+ σ2IN´−1Remark: The usual MMSE receiver is the unbiased one:uH³WWH+ σ2IN´−1=11 + uH¡UWUH+ σ2IN¢−1uuH³UUH+ σ2IN´−19MMSE ReceiverAfter MMSE filtering, we obtain:gH˜y = gHgs1+ gHbwith g = Λ−12QHuSignal to Interference plus Noise Ratio (SINR):βN=(gHg)2E(| s1|2)gHg= gHg = uH³UUH+ σ2IN´−1uDepends strongly on the choice of U.10Example: the i.i.d. model caseβN= uH³UUH+ σ2IN´−1uSuppose the matrix W = [Wij] has i.i.d. elements, E [Wij] = 0, EhW2iji= 1/N.Example : IS95.You remember that important lemma?u vector N × 1 with i.i.d elements. Each element : zero mean and variance 1/N.A matrix N × N independent of u. Then, under some assumptions,uHAu −1Ntrace (A) → 0 a.s.when N → ∞.Application : u et U independent, soβN−1Ntraceµ³UUH+ σ2IN´−1¶→ 0 a.s.11Example: the i.i.d. model case1Ntraceµ³UUH+ σ2IN´−1¶=1Ntrace³f³UUH´´where f(t) = 1/(t + σ2).Since UUHhas a limiting Marchenko-Pastur distribution law µ, we haveβN→ β =Z1t + σ2µ(dt) = Gµ(−σ2) a.s.Solution :β =1 − α2σ2−12+s(1 − α)24σ4+(1 + α)2σ2+14The SINR at the output of the MMSE receiver is exactly the Cauchy-Stieltjes transform.12MMSE and CapacitySINRMMSE= Gµ(−σ2)dCd1σ2= σ2− σ4Gµ(−σ2)dCσ2= SINRMMSE−1σ2The derivative of the capacity is strongly related to the performance of the MMSE receiver!13When Wiener meets ShannonD. Guo, S. Shamai, and S. Verd´u, “Mutual Information and Minimum Mean-Square Error inGaussian Channels,” IEEE Trans. Information Theory, vol. 51, no. 4, pp. 1261-1283, Apr.2005.D. P. Palomar and S. Verd´u, “Gradient of Mutual Information in Linear Vector GaussianChannels,” IEEE Trans. Information Theory , vol. 52, no. 1, pp. 141-154, Jan. 2006.Norbert Wiener, 1894-1964Claude Shannon, 1916-200114The Universality of Wiener’s solutionStill, no explanation for why an MMSE estimator, which is seemingly an artifact from theworld of analog signals, plays such a key role in achieving channel capacity.Perhaps, best stated by Shannon himself in 1948 paper acknowledging Wiener:”Credit should also be given to Prof. N. Wiener whose elegant solution of the problem offiltering and prediction of stationary ensembles has considerably influenced the writer’sthinking in this field.”15Outage performanceYou remember the CLT result?limK→∞,KN=αN (C − µ) → N(0, σ2)C =1Nlog det(I +1σ2WWH) is the mutual information per dimension and R = NC.• To optimize the network (CDMA for example), only the mean and the variance isneeded!• Application: How to derive the outage mutual information? Let q be the outageprobability and Rqbe the :q = P (R ≤ Rq) =ZTq−∞p(R)dRRq= Nµ + σQ−1(1 − q)with Q(x) =1√2πR∞xe−t22dt16PresentationAsymptotic Analysis of (MC)-CDMA systems17Uplink CDMA• K users in the cell.• Code Division Multiple Access (CDMA): Simultaneous communication of all the usersto the base station using different codes.18Uplink CDMA: General fading modelThe N × 1 received signal vector y at the base station has the form:y = H1w1√P1s1+ H2w2√P2s2+ ... + HKwK√PksK+ n• s = (s1, . . . , sK) is the emitted symbol vector.• wkis the N × 1 k-th user code.• Hkis the N × N channel matrix of user k.• N is the spreading length.•√Pkis the amplitude of user k.• n is a N × 1 white complex gaussian noise vector of variance σ2.19Uplink CDMA: Flat fading modelD. N Tse and S. Hanly, ”Linear Multiuser Receivers: Effective Interference, EffectiveBandwidth and User Capacity”, IEEE Transactions on Information Theory, Vol.45, No. 2,Mar. 1999S. Verdu and S. Shamai, ”The Impact of Frequency-Flat Fading on the Spectral Efficiencyof CDMA”, IEEE Transactions on Information Theory, Vol. 45, No. 2, Mar. 1999The N × 1 received signal vector y at the base station has the form:y = h1w1√P1s1+ h2w2√P2s2+ ... + hKwK√PKsK+ n= WHP12s + n• s = (s1, . . . , sK) is the emitted symbol vector.• wkis the N × 1 k-th user code.• H = diag[h1, ..., hK] an K × K matrix with hkflat fading of user k.• P = diag[P1, ..., PK] an K × K matrix with Pkpower of user k.• N is the spreading