Let \(\mathbf{W}_1\) and \(\mathbf{W}_2\) be independent \(n\times n\) complex central Wishart matrices with \(m_1\) and \(m_2\) degrees of freedom respectively. This paper is concerned with the ...extreme eigenvalue distributions of double-Wishart matrices \((\mathbf{W}_1+\mathbf{W}_2)^{-1}\mathbf{W}_1\), which are analogous to those of F matrices \({\bf W}_1 {\bf W}_2^{-1}\) and those of the Jacobi unitary ensemble (JUE). Defining \(\alpha_1=m_1-n\) and \(\alpha_2=m_2-n\), we derive new exact distribution formulas in terms of \((\alpha_1+\alpha_2)\)-dimensional matrix determinants, with elements involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-\(n\) analysis with \(\alpha_1\) and \(\alpha_2\) fixed (i.e., under the so-called "hard-edge" scaling limit); the analysis is based on new asymptotic properties of Legendre polynomials and their relation with Bessel functions that are here established. Specifically, we present limiting formulas for the smallest and largest eigenvalue distributions as \(n \to \infty\) in terms of \(\alpha_1\)- and \(\alpha_2\)-dimensional determinants respectively, which agrees with expectations from known universality results involving the JUE and the Laguerre unitary ensemble (LUE). We also derive finite-\(n\) corrections for the asymptotic extreme eigenvalue distributions under hard-edge scaling, giving new insights on universality by comparing with corresponding correction terms derived recently for the LUE. Our derivations are based on elementary algebraic manipulations, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painlevé differential equations, or hypergeometric functions of matrix arguments.
This paper shows that the proposed Rician shadowed model for multi-antenna communications allows for the unification of a wide set of models, both for multiple-input multiple output (MIMO) and ...single-input single output (SISO) communications. The MIMO Rayleigh and MIMO Rician can be deduced from the MIMO Rician shadowed, and so their SISO counterparts. Other SISO models, besides the Rician shadowed proposed by Abdi et. al., are included in the model, such as the \(\kappa\)-\(\mu\) defined by Yacoub, and its recent generalization, the \mbox{\(\kappa\)-\(\mu\)} shadowed model. Moreover, the SISO \mbox{\(\eta\)-\(\mu\)} and \mbox{Nakagami-\(q\)} models can be seen as particular cases of the MIMO Rician shadowed. The literature already presents the probability density function (pdf) of the Rician shadowed Gram channel matrix in terms of the well-known gamma-Wishart distribution. We here derive its moment generating function in a tractable form. Closed-form expressions for the cumulative distribution function and the pdf of the maximum eigenvalue are also carried out.
This paper proposes a novel approach to the statistical characterization of non-central complex Gaussian quadratic forms (CGQFs). Its key strategy is the generation of an auxiliary random variable ...(RV) that converges in distribution to the original CGQF. Since the mean squared error between both is given in a simple closed-form formulation, the auxiliary RV can be particularized to achieve the required accuracy. The technique is valid for both definite and indefinite CGQFs and yields simple expressions of the probability density function (PDF) and the cumulative distribution function (CDF) that involve only elementary functions. This overcomes a major limitation of previous approaches, in which the complexity of the resulting PDF and CDF prevents from using them for subsequent calculations. To illustrate this end, the proposed method is applied to maximal ratio combining systems over correlated Rician channels, for which the outage probability and the average bit error rate are derived.
We present a general and tractable fading model for line-of-sight (LOS) scenarios, which is based on the product of two independent and non-identically distributed \(\kappa\)-\(\mu\) shadowed random ...variables. Simple closed-form expressions for the probability density function, cumulative distribution function and moment-generating function are derived, which are as tractable as the corresponding expressions derived from a product of Nakagami-\(m\) random variables. This model simplifies the challenging characterization of LOS product channels, as well as combinations of LOS channels with non-LOS ones. We leverage these results to analyze performance measures of interest in the contexts of wireless powered and backscatter communications, where both forward and reverse links are inherently of LOS nature, as well as in device-to-device communications subject to composite fading. In these contexts, the model shows a higher flexibility when fitting field measurements with respect to conventional approaches based on product distributions with deterministic LOS, together with a more complete physical interpretation of the underlying propagation characteristics.
This paper shows that the recently proposed $\kappa$-$\mu$ shadowed fading
model includes, besides the $\kappa$-$\mu$ model, the $\eta$-$\mu$ fading model
as a particular case. This has important ...relevance in practice, as it allows
for the unification of these popular fading distributions through a more
general, yet equally tractable, model. The convenience of new underlying
physical models is discussed. Then, we derive simple and novel closed-form
expressions for the asymptotic ergodic capacity in $\kappa$-$\mu$ shadowed
fading channels, which illustrate the effects of the different fading
parameters on the system performance. By exploiting the unification here
unveiled, the asymptotic capacity expressions for the $\kappa$-$\mu$ and
$\eta$-$\mu$ fading models are also obtained in closed-form as special cases.
This paper shows that the recently proposed \(\kappa\)-\(\mu\) shadowed fading model includes, besides the \(\kappa\)-\(\mu\) model, the \(\eta\)-\(\mu\) fading model as a particular case. This has ...important relevance in practice, as it allows for the unification of these popular fading distributions through a more general, yet equally tractable, model. The convenience of new underlying physical models is discussed. Then, we derive simple and novel closed-form expressions for the asymptotic ergodic capacity in \(\kappa\)-\(\mu\) shadowed fading channels, which illustrate the effects of the different fading parameters on the system performance. By exploiting the unification here unveiled, the asymptotic capacity expressions for the \(\kappa\)-\(\mu\) and \(\eta\)-\(\mu\) fading models are also obtained in closed-form as special cases.
The ion exchange reactions of n-butylamine intercalates of tin(IV) hydrogen phosphate and hydrogen uranyl phosphate towards carbonatotetraamminecobalt(III), chloropentaamminecobalt(III), and ...hexaamminecobalt(III) have been investigated. Independent of the complex cation charges, the amounts of Co(III) complex exchanged by the n-butylamine intercalate of tin(IV) hydrogen phosphate are practically the same. With the n-butylamine intercalate of hydrogen uranyl phosphate, the ionic exchange was completed and the composition was fixed by the exchanged Co(III) complex. The layer charge densities of these phosphates justify the different ionic exchange behaviour observed towards the large complex cations. All the products were characterized by chemical analysis, X-ray diffractometry, infrared spectroscopy, diffuse reflectance spectroscopy, and thermal analysis.
The solids derived from HUP by substituting metallic ions for protons, take up Lewis base molecules (like NH
3
and C
4
H
9
NH
2
) in the interlamellar space. The diffractograms of these inclusion ...compounds indicate that the host crystallinity was preserved.The infrared spectroscopy of the Ni, Co, Cu, Zn, and Cd intercalates revealed that part of the sorbate was protonated in the interlaminar space while the other part displaced the water in their coordination spheres. However, in the Mn derivative, no coordinated or free Lewis base could be detected: it was all in the protonated form.
