电子科技大学：《贝叶斯学习与随机矩阵及在无线通信中的应用 BI-RM-AWC》课程教学资源（课件讲稿）06 Non-asymptotic Analysis for Large Random Matrix

6.Non-asymptotic Analysis The importance of the asymptotic theory is that it often makes possible to carry out the analysis and state many results which cannot be obtained within the standard "finite-sample theory" In practical applications,asymptotic theory is applied by treating the asymptotic results as approximately valid for finite sample sizes as well.But,such approach is often criticized for not having any mathematical grounds behind it. In some other cases where we want to know the how the performance metric (for example,channel capacity in MIMO vary with the dimension of system,the asymptotic result may not be very useful. 2

2 The importance of the asymptotic theory is that it often makes possible to carry out the analysis and state many results which cannot be obtained within the standard “finite-sample theory” 6. Non-asymptotic Analysis In practical applications, asymptotic theory is applied by treating the asymptotic results as approximately valid for finite sample sizes as well. But, such approach is often criticized for not having any mathematical grounds behind it. In some other cases where we want to know the how the performance metric (for example, channel capacity in MIMO ) vary with the dimension of system, the asymptotic result may not be very useful

6.Non-asymptotic Analysis In Chapter 2,we have some results also named non-asymptotic results such as a m x m complex Wishart matrix A=YY f,the PDF of the smallest eigenvalue is fn()=C之之(-1)+m元N-M+m-2e入det(M,) n=1m=1 T(g,D,i<nand j<m where C=I(N-m)!(M-m]and M,(ij)=T(q+2,1),izn and jzm T(q+1,), others T(g.x)is incomplete Gamma function defined as r(g,x)=ed It can be noted that,the complexity can not be accepted in high dimension.Here,in this Chapter,we will introduce the non- asymptotic results in high dimension. 3

3 It can be noted that, the complexity can not be accepted in high dimension. Here, in this Chapter, we will introduce the non￾asymptotic results in high dimension. 6. Non-asymptotic Analysis In Chapter 2, we have some results also named non-asymptotic results, such as A m × m complex Wishart matrix A=YY † , the PDF of the smallest eigenvalue is 2 min 1 1 ( ) C ( 1) det( ) M M n m N M n m n m f e + − + + − −  = =  = −    M 1 1 C [( !( !] M m N - m) M - m) − = = ( ), and ( ) ( 2 ), and ( 1, ), others q,l i < n j < m i, j q+ ,l i n j m q l    =        + M 1 ( , ) q t x q x t e dt  − −  =  where and ( , ) q x is incomplete Gamma function defined as

6.1 Non-asymptotic Analysis:Concentration inequality Concentration inequality can describe the probability that function of random matrix deviates its ergodic value X is an N X Mmatrix,N0,the following holds valid: l恨]n小s"t2ans 2(N+M)3 2.D where R is a non-negative diagonal matrix,A is the spectral radius of R and L is the Lipschitz norm of g(x)=f(x2)as follow L=sup g(x)-g(y) x≠y x-y 4

4 X is an N × M matrix, N ≤ M, and each entry of X is an independent complex random variable, for any δ > 0, the following holds valid: 6.1 Non-asymptotic Analysis: Concentration inequality Concentration inequality can describe the probability that function of random matrix deviates its ergodic value † † 2 2 2 1 1 ( ) Tr ( ) Tr ( ) 2exp 2 N M N M p f f N N N L   +  +             −    −       XRX XRX where R is a non-negative diagonal matrix, λ is the spectral radius of R and L is the Lipschitz norm of as follow 2 g x f x ( ) ( ) = ( ) ( ) sup x y g x g y L  x y − = −

6.1 Non-asymptotic Analysis:Concentration inequality We can analyze the MIMO capacity through concentration inequality System Model:A point-to-point MIMO transmission is considered, which consists of a transmitter and a receiver,equipped with n,and n, antennas,respectively.The received signal at receiver can be expressed as y=Hx+n where x∈CXI denotes the transmit signal and H C mrxm represents the MIMO channel.Assume that each entry of channel matrix H is independent random variable with zero-mean and unit variance.n is the additive white Gaussian noise vectors and its covariance matrix Enn'=(1/p)I where p is the signal-to-noise ratio (SNR) 5

5 6.1 Non-asymptotic Analysis: Concentration inequality We can analyze the MIMO capacity through concentration inequality System Model: A point-to-point MIMO transmission is considered, which consists of a transmitter and a receiver, equipped with nt and nr antennas, respectively. The received signal at receiver can be expressed as y Hx n = + where x ∈ C nt×1 denotes the transmit signal and H ∈ C nr×nt represents the MIMO channel. Assume that each entry of channel matrix H is independent random variable with zero-mean and unit variance. n is the additive white Gaussian noise vectors and its covariance matrix †   =  (1/ )   nn I where ρ is the signal-to-noise ratio (SNR)

6.1 Non-asymptotic Analysis:Concentration inequality we suppose that equal power is allocated at each transmit antennas, thus the capacity is given by C-oe,d1+号m】 In asymptotic analysis,the number of transmit antennas s goes to infinity,the row vectors of H become orthogonal,hence we can get HH" N Therefore,the capacity in can be written as C=n,log2 (1+p) 6

6 6.1 Non-asymptotic Analysis: Concentration inequality we suppose that equal power is allocated at each transmit antennas, thus the capacity is given by 2 † log det t C n    = +     I HH In asymptotic analysis, the number of transmit antennas goes to infinity, the row vectors of H become orthogonal, hence we can get † Nr Nt → HH I Therefore, the capacity in can be written as C n = +  r log 1 2 ( )

6.1 Non-asymptotic Analysis:Concentration inequality We will analyze the MIMO capacity in Concentration inequality such as P传w.k是n时iy小&sa(s) where R-PI and元-P and n n, f(x)=log2(1+x) g(x)=l1og2(1+x2) 7

7 6.1 Non-asymptotic Analysis: Concentration inequality We will analyze the MIMO capacity in Concentration inequality such as 2 2 2 † † 1 1 ( ) Tr ( ) Tr ( ) 2exp 2 r t r t r r r n n n n p n f n n L f     +  +             −    −         HRH HRH t n  where R I = and t n   = and ( ) ( ) 2 2 2 ( ) log 1 ( ) log 1 f x x g x x   = + = +

6.1 Non-asymptotic Analysis:Concentration inequality The Lipschitz norm of g(x) L=sup g(x)-8(y)-d2g(x)-1 x≠y x-y In 2 And we can get Tr[f(XRX')]=Tr[/(HRH)]->[(HRH)]=I0g.det(I+HRH)=C. The Concentration inequality of MIMO capacity can be written as δ2(n,+n,)尸n plC-[C]>δ(n,+n,)}≤2e2osd 8

8 6.1 Non-asymptotic Analysis: Concentration inequality The Lipschitz norm of g x( ) 2 2 ( ) ( ) ( ) 1 sup x y ln 2 g x g y d g x L  x y dx − = = = − ( ) ( ) ( ) 2 ( ) 1 † † † Tr =Tr log det . r n i i f f f C =       =  = + =     XRX HRH HRH I HRH    And we can get The Concentration inequality of MIMO capacity can be written as    ( ) 2 2 2 ( ) 2 2 t r t LS n n n c L t r p C C n n e  + −  −   + 

6.1 Non-asymptotic Analysis:Concentration inequality Sett=δ(n,+n,),asδis an arbitrary positive constant,t is also an arbitrary positive constant.Replacing t=5(n,+n)with t,then we can get 1'n, pC-E[C]>2e 2ot →plc-[C]>ts2er where im and B- n This result demonstrates the probability that the instantaneous MIMO capacity deviates from its ergodic capacity.Next,we will exploit this result to derive the deterministic bounds on ergodic capacity

6.1 Non-asymptotic Analysis: Concentration inequality Set , as δ is an arbitrary positive constant, t is also an arbitrary positive constant. Replacing with t, then we can get t n n =  + ( t r ) t n n =  + ( t r )         2 2 2 2 2 2 t t n L t p C C t e p C C t e −  −  −    −   where and 1 ln 2 L = 2 2 t n L  =  This result demonstrates the probability that the instantaneous MIMO capacity deviates from its ergodic capacity. Next, we will exploit this result to derive the deterministic bounds on ergodic capacity

6.1 Non-asymptotic Analysis:Concentration inequality Due to C-E[C]=C-E[C],and based on the concentration inequality, we can get pC-[C]>≤plC-[C]>≤2exp(-Bt2) For convenience,let T denote C-E[C],yields [e-]=∫e'd1-p(T>]=e'[1-pT>]P-∫e1-pT>]d s[-e'xo]ex2ene'p(T>1di sfn7>=28 10

10 6.1 Non-asymptotic Analysis: Concentration inequality Due to C C C C −  −     , and based on the concentration inequality, we can get     2 p C C t p C C t t { } { } 2exp( ) −   −   − For convenience, let T denote C C −   , yields             2 1 4 1 + 1 1 0 2 2 C C t t t t t t t t t t e e d p T t e e p T t dt e e e e p T t dt e p p T t d T e t t   − − −  −  =  =− −  −  −      =  −   =  −          −     −   −     +         =     

6.1 Non-asymptotic Analysis:Concentration inequality Based on Taylor series expansion,we arrive at -]小 DeoB=,henee wen%≥8p，叱2作1，his 5.学 Hnce ege[ew]s2辰e-2a 11

11 Based on Taylor series expansion, we arrive at 6.1 Non-asymptotic Analysis: Concentration inequality ( ) ( ) 1 2 1 1/4 1/4 4 2 2 1 1! 2! e e e        = + + +         Due to 2 2 t n L  =  , hence when , 2 8 t n L   2 1    , thus   1/4 1+2 ( 1) C C e e  −      − =    Hence we get ( ) ( ) 1 2 1 1/4 1/4 4 1/4 2 1 2 1+2 ( 1) 1! 2! e e e e           + + + = −         