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Naples 2008 Free probability, random Vandermonde matrices, and applications Øyvind Ryan May 2008 Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Some important concepts from classical probability I Random variables are functions (i.e. they commute w.r.t. multiplication) with a given p.d.f. (denoted f ) I Expectation (denoted E ) is integration I Independence I Additive convolution (∗) and the logarithm of the Fourier transform I Multiplicative convolution I Central limit law, with special role of the Gaussian law ¢ ¢∗n ¡¡ Poisson distribution Pc : The limit of 1 − nc δ(0) + nc δ(1) as n → ∞. I I Divisibility: For a given a, nd i.i.d. b1 , ..., bn such that fa = fb1 +···+bn . Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Free probability I I I A more general theory, where the random variables are matrices (or more generally, elements in a unital ∗-algebra (denoted A), typically B (H)), with their eigenvalue distribution (spectrum) taking the role as the p.d.f. The above mentioned concepts have their analogues in this theory. For instance, the expectation (denoted φ) is a normalized linear functional on A. The pair (A, φ) is called a noncommutative probability space. For (random) matrices, φ will be the (expected) trace: φ(A) = trn (A) = n 1X aii (φ(A) = E (trn (A)). n i =1 I What should it mean that two random matrices are "free" (=analogue of independent, to be dened)? Think of as two independent random matrices, where eigenvectors of one point in all directions with equal probability (unitary invariance). Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati The semicircle law Free probability has a "Gaussian distribution counterpart": 35 30 25 20 15 10 5 0 −3 −2 −1 0 1 2 3 A = (1/sqrt(2000)) * (randn(1000,1000) + j*randn(1000,1000)); A = (sqrt(2)/2)*(A+A'); hist(eig(A),40) Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Motivation for free probability Assume that Xn , Yn are independent, Gaussian n × n-matrices. One can show that the limits ¡ ¢ ¡ ¢ φ X i1 Y j1 · · · X il Y jl := lim trn Xin1 Ynj1 · · · Xinl Ynjl n→∞ exist. If we linearly extend the linear functional φ to all polynomials in A and B, the following can be shown: Theorem If Pi , Qi are polynomials in X and Y respectively, with 1 ≤ i ≤ l , and φ(Pi (X )) = 0, φ(Qi (Y )) = 0 for all i, then φ (P1 (X )Q1 (Y ) · · · Pl (X )Ql (Y )) = 0. This motivates the denition of freeness (=analogue of independence): Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Denition of freeness Denition A family of unital ∗-subalgebras (Ai )i ∈I is called a free family if aj ∈ Aij ⇒ φ(a1 · · · an ) = 0. (1) i1 6= i2 , i2 6= i3 , · · · , in−1 6= in φ(a1 ) = φ(a2 ) = · · · = φ(an ) = 0 A family of random variables ai is called a free family if the algebras they generate form a free family. (1) is also called the freeness relation, and can be viewed as a rule for computing the mixed moments φ(a1 · · · an ) of the ai from their individual moments φ(aim ). In random matrix settings, it relates the moments (E (trn (Xm )) of random matrices. Recently, a theory called second order freeness has been developed, which also relates "higher order moments": Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Second order freeness For a random matrix ensemble X = {Xn }n , dene the second order limit moments ¡ ¢ ¡ ¢ ¡ ¢ αiX,j = lim E Tr (Xni )Tr (Xnj ) − E Tr (Xni ) E Tr (Xnj ) . n→∞ I I We say that {Xn }n has a second order limit distribution if the second order limit moments exist, and the higher order limit moments (not written down here) are all 0. Implies that tr (Xni ) − E (tr (Xni )) is asymptotically Gaussian of order n1 . It is known [1] that whenever An , Bn are independent, with second order limit distributions, one of them unitarily invariant, then An and Bn are asymptotically free of second p (A,B ) from the order [1] (a relation which determines all αi ,j A B individual αi ,j , αi ,j ). Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 I Free probability, random Vandermonde matrices, and applicati Second order freeness is an eective machinery for calculating p (A,B ) the αi ,j from αiA,j , αiB,j . Takes a particularly nice form for computation of αiA,j+B . I Gives alternative proofs of known reults. I More general than what we can do with the freeness relation, which only enables us to compute the (rst order) limit moments ¡ ¢ αiX = lim E tr (Xni ) n→∞ I from individual (rst order) limit moments. Used in the literature: I I Gaussian-type matrices have a second order limit distribution. In [2], the asymptotic Gaussianity of tr (Xni ) − E (tr (Xni )) is exploited with maximum likelihood estimation. Optimal weighting of moments [3]. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Additve and multiplicative free convolution I I I I I Additive/multiplicative free convolution (¢/£) corresponds to summing/multiplying free random variables. Can also be viewed as operations on measures, by associating the moments with a probability measure. In random matrix settings, additive free convolution cooresponds to estimating the eigenvalue distribution of the sum of two large, independent random matrices, where one is unitarily invariant. Alternative functional equations exist for computing additive/multiplicative free convolution. Uses the Stieltjes transform. One of the main questions in my papers: Let X and Y be random matrices. How can we make a good prediction of the eigenvalue distribution of X when one has the eigenvalue distribution of XY and Y (i.e. problem turned around to a deconvolution problem)? Simplest case is Y Gaussian. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Rectangular convolution Recently, additive free convolution has been extended to handle addition of rectangular matrices, with eigenvalue distributions replaced with singular value distributions: Assume Ak , Bk rectangular (n × N), with limiting singular laws µA , µB , and that limk →∞ Nn = λ. In many cases (when for instance the matrices are independent and unitarily invariant [4]), the limiting singular law of Ak + Bk (µA+B ) exists and can be computed from (and depends only on) µA , µB . We dene µA ¢λ µB = µA+B . I I I ¢λ is called rectangular free convolution with ratio λ. ¢λ can be extended to the set of all symmetric probability measures [4]. Expressible in terms of the free probability constructs additive (¢) and multiplicative (£) free convolution [5]. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Ways to compute free convolution I Using the fact that the Stieltjes transforms of the measures µ1 , µ2 , µ1 ¢ µ2 , µ1 £ µ2 satisfy certain functional equations: Discretize these equations, and turn the computational problem into a convex optimization problem [6]. I When the Stieltjes transforms of µ1 and µ2 satisfy certain polynomial equations, one can show that The Stieltjes transform of µ1 ¢ µ2 also satisfy a certain polynomial equation, and the polynomial of µ1 ¢ µ2 can be compute from those of µ1 and µ2 . This enables in turn to compute µ1 ¢ µ2 itself. This method is called the polynomial method of random matrices [7]. I Perform free convolution solely in terms of moments. Fast algorithms exist, both for rst and second order moments. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Application of free probability to channel capacity estimation [8] The capacity per receiving antenna of a channel with n × m channel matrix H and signal to noise ratio ρ = σ12 is given by µ ¶ n 1 1 1X 1 H C = log2 det In + HH = log2 (1 + 2 λl ) 2 n mσ n σ (2) l =1 where λl are the eigenvalues of m1 HHH . We would like to estimate C. To estimate C , we will use free probability tools to estimate the eigenvalues of m1 HHH based on some observations Ĥi Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Observation model The following is a much used observation model: Ĥi = H + σ Xi (3) where I The matrices are n × m (n is the number of receiving antennas, m is the number of transmitting antennas) I Ĥi is the measured MIMO matrix, Xi is the noise matrix with i.i.d standard complex Gaussian I entries. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Existing ways to estimate the channel capacity Several channel capacity estimators have been used in the literature: ³ ´ 1 PL 1 H Ĥ Ĥ C1 = nL log det I + n i =1 mσ 2 i i ´ ³2 1 1 PL C2 = n log2 det In + Lσ2 m i =1 Ĥi ĤH (4) i ³ ´ P P C3 = n1 log2 det In + σ21m ( L1 Li=1 Ĥi )( L1 Li=1 Ĥi )H ) Why not try to formulate an estimator based on free probability instead? Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati 1.6 1.4 Capacity 1.2 1 0.8 0.6 0.4 True capacity C1 C 2 C3 0.2 0 10 20 30 40 50 60 Number of observations 70 80 90 100 Comparison of the classical capacity estimators for various number of observations. σ 2 = 0.1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Form the compound observation matrix σ Ĥ1...L = H1...L + √ X1...L , where L i 1 h Ĥ1...L = √ Ĥ1 , Ĥ2 , ..., ĤL , L 1 H1...L = √ [H, H, ..., H] , L X1...L = [X1 , X2 , ..., XL ] . With free probability, moments of the observation matrix 1 Ĥ1...L ĤH1...L , m can be related with the moments of the channel matrix 1 1 H1...L HH1...L = HHH m m (one needs to perform additive free- and multiplicative free n ). From moments convolution with the Marchenko Pastur law (µ mL we can estimate eigenvalues, and then the channel capacity. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Free probability based estimator for the moments of the channel matrix Can also be written in the following way for the rst four moments: ĥ1 = h1 + σ 2 ĥ2 = h2 + 2σ 2 (1 + c )h1 + σ 4 (1 + c ) ĥ3 = h3 + 3¡σ 2 (1 + c )h2 ¢+ 3σ 2 ch12 +3σ 4¡ c 2 + 3c + 1¢ h1 +σ 6 c 2 + 3c + 1 ĥ4 = h4 + 4σ 2 (1 + c )h3 + 8σ 2 ch2 h1 +σ 4 (6c 2 + 16c + 6)h2 +14σ 4 c (1 + c )h12 +4σ 6¡(c 3 + 6c 2 + 6c + 1¢)h1 +σ 8 c 3 + 6c 2 + 6c + 1 , where ĥi are the moments of the observation matrix hi are the moments of m1 HHH . Øyvind Ryan (5) 1 H m Ĥ1...L Ĥ1...L , Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati 0.4 0.35 Capacity 0.3 0.25 0.2 0.15 0.1 0.05 True capacity Cf CG 0 0 10 20 30 40 50 60 Number of observations 70 80 90 100 Comparison of Cf and CG for various number of observations. σ 2 = 0.1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati 0.9 0.8 0.7 Capacity 0.6 0.5 0.4 0.3 True capacity, rank 3 Cf, rank 3 0.2 True capacity, rank 5 Cf, rank 5 0.1 True capacity, rank 6 C , rank 6 f 0 0 10 20 30 40 50 60 Number of observations 70 80 90 100 Cf for various number of observations. σ 2 = 0.1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3, 5 and 6. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Does other types of random matrices (i.e. non-unitarily invariant) t into a framework similar to free probability? We have investigated this for Vandermonde matrices [9, 10], which are widely used. They have the form 1 ··· 1 x1 · · · xL V= .. .. .. . . . N −1 N −1 x1 · · · xL It is straightforward to show that square Vandermonde matrices have determinant Y (xl − xk ). det(V) = 1≤k <l ≤N In particular, V is nonsingular if the xk are dierent. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Various results exist on the distribution of the determinant of Vandermonde matrices (Gaussian entries (Metha), entries with B-distribution (Selberg)), but there are many open problems (below, VH V is used since V is rectangular in general): I How can³we nd the ¡ H ¢k ´ moments of the Vandermonde matrices (i.e. trL V V ) (not the determinant itself)? I I I Deconvolution problem: How to estimate the moments of D from mixed moments DVH V? Mixed moments of independent Vandermonde matrices? Asymptotic results? If X is an N × N standard, complex, Gaussian matrix, then ¡ ¡1 ¢¢ H limN →∞ N1 log det I + ρ XX = ´ 2 N ³ ¢2 ¢2 ¡ ¡√ 1 √ 2 log2 1 + ρ − 4 4ρ + 1 − 1 − log4ρ2 e 4ρ + 1 − 1 . (which is the expression for the capacity). We are not aware of similar asymptotic expressions for the determinant/capacity of Vandermonde matrices. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Random Vandermonde matrices We will consider Vandermonde matrices V of dimension N × L of the form 1 ··· 1 −j ω · · · e −j ωL 1 e 1 V= √ (6) .. . . . . . . N . e −j (N −1)ω1 · · · e −j (N −1)ωL (i.e. we assume that the xi lie on the unit circle). The ωi are called phase distributions. We will limit the study of Vandermonde matrices to cases where I I The phase distributions are i.i.d. The asymptotic case N , L → ∞ with limN →∞ NL = c. The normalizing factor √1 is included to ensure limiting N asymptotic behaviour. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Where can such Vandermonde matrices appear? Consider a multi-path channel of the form: h(τ ) = L X αi g (τ − τi ) i =1 αi are i.d. Gaussian random variables with power Pi , I τi are uniformly distributed delays over [0, T ], I g is the low pass transmit lter. I L is the number of paths In the frequency domain, the channel is given by: I c (f ) = L X αi G (f )e −j2πf τi i =1 We suppose the transmit lter to be ideal (G (f ) = 1). Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Sampling the continuous frequency signal at fi = i W N (N is the number of frequency samples) where W is the bandwidth, our model becomes α1 n1 1 . . r = VP 2 (7) .. + .. , αL nN where V is a random Vandermonde matrix of the type (6), and I P is the L × L diagonal power matrix, I ni is independent, additive, white, zero mean Gaussian noise of variance √σ . N Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Main result Denition Dene Kρ,ω,N = RN 1 n+1−|ρ| × (0,2π)|ρ| Qn jN (ω −ω ) b(k −1) b(k ) 1−e k =1 1−e j (ωb(k −1) −ωb(k ) ) , (8) d ω1 · · · d ω|ρ| , where ωρ1 , ..., ωρ|ρ| are i.i.d. (indexed by the blocks of ρ), all with the same distribution as ω , and where b(k ) is the block of ρ which contains k (where notation is cyclic, i.e. b(−1) = b(n)). If the limit Kρ,ω = lim Kρ,ω,N N →∞ exists, then Kρ,ω is called a Vandermonde mixed moment expansion coecient. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Main result 2 Assume that I {Dr (N )}1≤r ≤n are diagonal L × L matrices which have a joint limit distribution as N → ∞, I L N → c. We would like to express the limits Mn = lim E [trL (D1 (N )VH VD2 (N )VH V · · · × Dn (N )VH V)]. (9) N →∞ It turns out that this is feasible when all Vandermonde mixed moment expansion coecients Kρ,ω exist. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati For convenience, dene £ ¡¡ ¢n ¢¤ mn = (cM )n = c limN →∞ E trL D(N )VH V , dn = (cD )n = c limN →∞ trL (Dn (N )) , (10) Theorem Assume D1 (N ) = D2 (N ) = · · · = Dn (N ). When ω = u, m1 = d1 m2 = d2 + d12 m3 = d3 + 3d2 d1 + d13 m4 = d4 + 4d3 d1 + 8/3d22 + 6d2 d12 + d14 m5 = d5 + 5d4 d1 + 25/3d3 d2 + 10d3 d12 + 40/3d22 d1 + 10d2 d13 + d15 m6 = d6 + 6d5 d1 + 12d4 d2 + 15d4 d12 + 151/20d32 + 50d3 d2 d1 +20d3 d13 + 11d23 + 40d22 d12 + 15d2 d14 + d16 m7 = d7 + 7d6 d1 + 49/3d5 d2 + 21d5 d12 + 497/20d4 d3 + 84d4 d2 d1 +35d4 d13 + 1057/20d32 d1 + 693/10d3 d22 + 175d3 d2 d12 +35d3 d14 + 77d23 d1 + 280/3d22 d13 + 21d2 d15 + d17 . Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Comparison The Gaussian equivalent of this is m1 m2 m3 m4 m5 m6 d1 d2 + d12 d3 + 3d2 d1 + d13 d4 + 4d3 d1 + 3d22 + 6d2 d12 + d14 d5 + 5d4 d1 + 5d3 d2 + 10d3 d12 + 10d22 d1 + 10d2 d13 + d15 d6 + 6d5 d1 + 6d4 d2 + 15d4 d12 + 3d32 + 30d3 d2 d1 +20d3 d13 + 5d23 + 10d22 d12 + 15d2 d14 + d16 m7 = d7 + 7d6 d1 + 7d5 d2 + 21d5 d12 + 7d4 d3 + 42d4 d2 d1 +35d4 d13 + 21d32 d1 + 21d3 d22 + 105d3 d2 d12 +35d3 d14 + 35d23 d1 + 70d22 d13 + 21d2 d15 + d17 , (11) 1 H H when we replace V V with N XX , with X an L × N complex, standard, Gaussian matrix. = = = = = = Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Sketch of proof We can write h ³ ´i E trL D1 (N )VH VD2 (N )VH V · · · Dn (N )VH V as L−1 P i1 ,...,in j1 ,...,jn (12) E ( D1 (N )(j1 , j1 )VH (j1 , i2 )V(i2 , j2 ) D2 (N )(j2 , j2 )VH (j2 , i3 )V(i3 , j3 ) .. . (13) Dn (N )(jn , jn )VH (jn , i1 )V(i1 , j1 )) The (j1 , ..., jn ) give rise to a partition ρ of {1, ..., n}, where each block ρj consists of equal values, i.e. ρj = {k |jk = j }. This ρ will actually represent the ρ used in the denition of Kρ,ω,n . The rest of the proof goes by carefully computing this limit quantity using much combinatorics. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Comparisons Denote by µµ ¶ ¶∗n λ λ ν(λ, α) = lim 1− δ − 0 + δα n→∞ n n the (classical) Poisson distribution of rate λ and jump size α. Denote also by µµ ¶ ¶¢n λ λ 1− δ − 0 + δα n→∞ n n µ(λ, α) = lim the free Poisson distribution of rate λ and jump size α (also called the Marchenko Pastur law). Denote also µc = µ( c1 , c ), νc = ν(c , 1). Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Comparisons 2 Corollary Assume that V has uniformly distributed phases. Then the limit moment h ³³ ´n ´i Mn = lim E trL VH V N →∞ satsies the inequality φ(a1n ) ≤ Mn ≤ 1 E (a2n ), c where a1 ∼ µc , a2 ∼ νc . In particular, equality occurs for m = 1, 2, 3 and c = 1. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Comparisons 3 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 2 3 4 5 6 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10 H 1 N XX , H with X an 800 × 1600 com(a) V V, with V a 1600 × 800 Van- (b) dermonde matrix with uniformly dis- plex, standard, Gaussian matrix. tributed phases. Figure: Histogram of mean eigenvalue distributions. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Other results I Mixed moments of (more than one) independent Vandermonde matrices. I Generalized ¢Vandermonde matrices: These have the form ¡ V = e j αk βl 1≤k ≤N ,1≤l ≤L . It is known that V is nonsingular i all αk are dierent, and all βl are dierent. The papers also contain results on the asymptotics of generalized Vandermonde matrices. I Exact moments of lower order Vandermonde matrices. Reveals slower convergence. I Computation of the asymptotic moments when the phase distribution is not uniform. Phase distributions with continous density, and phase distributions with singularities. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Application 1: Estimation of the number of paths Return to the multi-path channel model (7). For simplicity, set W = T = 1, so that the phase distribution of the Vandermonde matrix is uniform. We take K observations of (7) and form the observation matrix Y = [r1 · · · rK ] (1) (K ) α1 1 . = VP 2 .. ··· .. . α1 .. . αL ··· αL (1) (K ) (1) (K ) n1 .. + . ··· .. . n1 .. . nN ··· nN (1) , (K ) (14) It is now possible to combine the deconvolution result for Vandermonde matrices with known deconvolution results for Gaussian matrices to estimate L from a number of observations (assuming P is known). All values of L are tried, and the one which "best matches" the observed values is chosen: Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Estimation of the number of paths 2 Proposition Assume that V has uniformly distributed phases, and let mPi be the ^i ) the moments of the sample moments of P, and mR̂i = trN (R covariance matrix ^ = 1 YYH . R K N , c2 = NL , and c3 = KL . Then Dene also c1 = K £ ¤ E mR̂ = c2 mP1 + σ 2 µ ¶ h i 1 2 E mR̂ = c2 1 − mP2 + c2 (c2 + c3 )(mP1 )2 N +2σ 2 (c2 + c3 )mP1 + σ 4 (1 + c1 ) h i E mR̂3 = ··· Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Estimation of the number of paths 3 70 70 Estimate of L Actual value of L 50 50 40 40 Estimate of L Actual value of L L 60 L 60 30 30 20 20 10 10 0 0 10 20 30 40 50 60 Number of observations 70 80 90 (a) K = 1 100 0 0 10 20 30 40 50 60 Number of observations 70 80 90 100 (b) K = 10 Figure: √ Estimate for the number of paths. Actual value of L is 36. Also, σ = 0.1, N = 100. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Application 2: Wireless capacity Analysis For a general matrix W, the mean capacity is dened as ¡ ¡ ¢¢ CN = N1 E log2 det IN + σ12 WWH ¡ ¡ ¡ ¢¢¢ R ¡ P 1 H = N1 N = log2 1 + k =1 E log2 1 + σ 2 λk WW ¢ t µ(dt ) (15) H where µ is the mean empirical eigenvalue distribution of WW . P k +1 t k , Substituting the Taylor series log2 (1 + t ) = ln12 ∞ k =1 (−1) k we obtain P (−1)k +1 mk (µ)ρk (16) , CN = ln12 ∞ k =1 k where ρ is SNR, and where mk (µ) = 1 σ2 Z t k d µ(t ) for k ∈ Z+ However, many more moments are required for precise estimation of capacity than we can provide with the formulas for the rst 7 moments. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati Wireless capacity Analysis 2 3 Asymptotic capacity sample capacity 2.5 Capacity 2 1.5 1 0.5 0 0 1 2 3 4 5 ρ 6 7 8 9 10 ¡ ¢ Figure: Several realizations of the capacity N1 log2 det I + ρ N1 XXH when X is standard, complex, Gaussian. Matrices of size 36 × 36 were used. The known expression for the asymptotic capacity is also shown. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati 3 3 2.5 2.5 2 2 Capacity Capacity Wireless capacity Analysis 3 1.5 1.5 1 1 0.5 0.5 0 0 1 2 3 4 5 ρ 6 7 8 (a) ¡Realizations H¢ 1 when N log2 det I + ρVV has uniform phase distribution. 9 10 0 0 1 2 3 4 5 ρ 6 7 8 9 10 of (b) of ¡Realizations ¢ ω N1 log2 det I + ρVVH when ω has a certain non-uniform phase distribution. Figure: Several realizations of the capacity for Vandermonde matrices for two dierent phase distributions. Matrices of size 36 × 36 were used. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati I This talk is available at http://heim.i.uio.no/∼oyvindry/talks.shtml. I My publications are listed at http://heim.i.uio.no/∼oyvindry/publications.shtml THANK YOU! Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati B. Collins, J. A. Mingo, P. niady, and R. Speicher, Second order freeness and uctuations of random matrices: III. higher order freeness and free cumulants, Documenta Math., vol. 12, pp. 170, 2007. N. R. Rao, J. Mingo, R. Speicher, and A. Edelman, Statistical eigen-inference from large Wishart matrices, 2007, arxiv.org/abs/math.ST/0701314. Ø. Ryan and M. Debbah, Free deconvolution for signal processing applications, Submitted to IEEE Trans. on Information Theory, 2007, http://arxiv.org/abs/cs.IT/0701025. F. Benaych-Georges, Rectangular random matrices. related convolution, 2008, arxiv.org/abs/math.PR/0507336. Ø. Ryan and M. Debbah, Multiplicative free convolution and information-plus-noise type matrices, 2007, http://arxiv.org/abs/math.PR/0702342. Øyvind Ryan Free probability, random Vandermonde matrices, and applica Naples 2008 Free probability, random Vandermonde matrices, and applicati N. E. Karoui, Spectrum estimation for large dimensional covariance matrices using random matrix theory, 2006, arxiv.org/abs/math/0609418. N. R. Rao and A. Edelman, The polynomial method for random matrices, 2007, arxiv.org/abs/math.PR/0601389. Ø. Ryan and M. Debbah, Channel capacity estimation using free probability theory, To appear in IEEE Trans. Signal Process., 2007, http://arxiv.org/abs/0707.3095. , Random Vandermonde matrices-part I: Fundamental results, Submitted to IEEE Trans. on Information Theory, 2008. , Random Vandermonde matrices-part II: Applications, Submitted to IEEE Trans. on Information Theory, 2008. Øyvind Ryan Free probability, random Vandermonde matrices, and applica