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 nonasymptotic 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,N<M,and each entry of X is an independent complex random variable,for any 8>0,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 ( )