4.2.4 Asymptotic Spectral Distributions for Hermitian Matrix Hence,we have ↓ 丽--u--P And finally,we have F(1-Pa)v (N-)-u 6
6 4.2.4 Asymptotic Spectral Distributions for Hermitian Matrix Hence，we have 2 2 1 fa N K F P v 2 1 2 1 fa N K F P v And finally, we have 1 2 2 1 F P v fa N K
Overview 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 -15 -10 5 0 5 10 15 SNR(dB) 5 users and 20 samples each user,P=0.005 7
7 Overview -15 -10 -5 0 5 10 15 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 SNR (dB) probability of detecting correctly 5 users and 20 samples each user, Pfa =0.005
4.3.Asymptotic Moment Method Definition:In asymptotic regime,H is an NXK matrix,we denote the empirical distribution of HHH as fx),and the i-th moment of fx) is defined as rar天] where [a,b]is domain of fx). Theorem 4.2.4.Consider an NXK matrix H whose entries are independent zero-mean complex (or real)random variables with variance 1/N and fourth moments of order O(1/N2 )As K,N->co with N./K→B,we have 安→以jr 8
8 4.3. Asymptotic Moment Method Theorem 4.2.4. Consider an N×K matrix H whose entries are independent zero-mean complex (or real) random variables with variance 1/N and fourth moments of order O( 1/N2 ). As K,N → ∞ with N/K→ β, we have Definition: In asymptotic regime, H is an N×K matrix, we denote the empirical distribution of HHH as f(x), and the i-th moment of f(x) is defined as 1 H ( )d lim b i i a K x f x x tr K H H 1 H 0 1 1 1 i i k k i i tr K i k k H H where [a,b] is domain of f(x)
4.3.Asymptotic Spectral Distributions for Hermitian Matrix Example 4.3.5:MMSE detector with low complexity Consider the following communication channel y=Hd+n with d=ddC"some transmitted vectorial data of dimension n,assumed to have zero mean and covariance matrix n,nCN is an additive Gaussian noise vector of zero mean and covarianceI,y=CNis the reccived signal,and HCNx"is the multi-dimensional communication channel. 9
9 4.3. Asymptotic Spectral Distributions for Hermitian Matrix Consider the following communication channel with some transmitted vectorial data of dimension , assumed to have zero mean and covariance matrix In , is an additive Gaussian noise vector of zero mean and covariance IN , is the received signal, and is the multi-dimensional communication channel. y Hd n T 1 ,..., n n d d d n N n T 1 ,..., N n y y y N n H Example 4.3.5：MMSE detector with low complexity 2
4.3.Asymptotic Spectral Distributions for Hermitian matrix Under these model assumptions,irrespective of the communication scenario under study(e.g.MIMO,CDMA),the minimum mean square error decoder output for the vector x reads: =(H"H+G2IH"y =H"(H+o21w)）厂y For practical applications,recovering therefore requires the inversion of the potentially large (HHH+021)matrix. 10
10 4.3. Asymptotic Spectral Distributions for Hermitian Matrix Under these model assumptions, irrespective of the communication scenario under study(e.g. MIMO, CDMA), the minimum mean square error decoder output for the vector x reads: For practical applications, recovering therefore requires the inversion of the potentially large (HHH+σ2 In ) -1 matrix. x ˆ 1 H 2 H 1 H H 2 ˆ n N x H Η I H y H ΗH I y x ˆ