On the marginal distribution of the eigenvalues of wishart matrices

Abstract

Random matrices play a crucial role in the design and analysis of multiple-input multiple-output (MIMO) systems. In particular, performance of MIMO systems depends on the statistical properties of a subclass of random matrices known as Wishart when the propagation environment is characterized by Rayleigh or Rician fading. This paper focuses on the stochastic analysis of this class of matrices and proposes a general methodology to evaluate some multiple nested integrals of interest. With this methodology we obtain a closed-form expression for the joint probability density function of k consecutive ordered eigenvalues and, as a special case, the PDF of the th ordered eigenvalue of Wishart matrices. The distribution of the largest eigenvalue can be used to analyze the performance of MIMO maximal ratio combining systems. The PDF of the smallest eigenvalue can be used for MIMO antenna selection techniques. Finally, the PDF the k th largest eigenvalue finds applications in the performance analysis of MIMO singular value decomposition systems.

Tightness of Jensen’s Bounds and Applications to MIMO Communications

“…For a MU-MIMO MRC system modeled as in (11), whose spectral efficiency is given in (12), we note that the desired signal power and interference power are independent of each other, which makes it possible to derive them separately. We first notice that g (12), thus, we simply get [36, Lemma 4]…”

“…These results have been effectively applied to analyze the performance of SU-MIMO systems in different particular cases. More specifically, the joint probability density function (pdf) of ordered eigenvalues [9,10] was used to evaluate the information theoretical limits of SU-MIMO systems, while the pdfs of the smallest eigenvalue [11] and largest eigenvalue [12,13] have been utilized for designing antenna selection and multichannel beamforming schemes.…”

“…Rayleigh fading [1,2,14,15] to the case where correlation is present at one of the two sides [8,11,16,17] or at both sides [18] or even Ricean fading channels [6,19]- [21], exact, closedform capacity expressions are inherently hard to derive due to the complex distribution of the eigenvalues of HH † . In order to obtain more informative analytical insights into MIMO systems, a large number of works has focused on formulating upper and lower capacity bounds for both SU-MIMO (see [2,3,13,22]- [26]) and MU-MIMO systems [4,27] via Jensen's inequality 2 .…”

Opportunistic Interference Alignment Based on Dynamic Cell Selection

“…• 합 분포를 이용한다 [13] . Q    의 모든 고유값에 대한 결합 확률 밀도 함수(joint probability density function: joint pdf)는 다음과 같다.…”

Eigenvalue variation in MIMO OFDM systems

“…Now the marginal statistics, such as E [X] and var(X), can be computed using standard results [11], [12], [13], [14]. Thus, we can find the ACF from var(X − Y ) and vice-versa.…”