Real Gaussian Research Articles

Spectrum Sensing in Cognitive Radio using Frequency Domain

An efficient bandwidth allocation and dynamic bandwidth access away from its previous limits is referred as cognitive radio (CR).The limited spectrum with inefficient usage requires the advances of dynamic spectrum access approach, where the secondary users are authorized to utilize the unused temporary licensed spectrum. For this reason it is essential to analyze the absence/presence of primary users for spectrum usage. So spectrum sensing is the main requirement and developed to sense the absence/ presence of a licensed user. This paper shows the design model of energy detection based spectrum sensing in frequency domain utilizing Binary Symmetric Channel (BSC) ,Additive white real Gaussian channel (AWGN), Rayleigh fading channel users for 16-Quadrature Amplitude Modulation(QAM) which is utilized for the wide band sensing applications at low Signal to noise Ratio(SNR) level to reduce the false error identification. The spectrum sensing techniques has least computational complexity. Simulink model for the energy detection based spectrum sensing using frequency domain in MATLAB 2014a.

Gaussian Immature Bayes Classifier and Uneven Lay down Conception for Information Flow Group in Frequent Concept Drift

The requirement of acting class in streaming environments, investigators have exuberance a range of circulation classification procedures by way of managing idea flow. However, routine idea waft is a tough disaster in data flow as the span of the statistics isn't always fixed more than the point in time frame. As a result of taking into consideration the habitual concept flow, this manuscript propose a novel classifier representation, referred to as Real Gaussian naïve Bayes classifier for the information circulate categorization. To carry out the confront of habitual concept float, the first part is to construct the use of the difficult set idea intended for detect the idea waft. After that, Gaussian naïve classifier is tailored accurately to replace active information devoid of the usage of the museum facts. Also, the class is featured using the posterior possibility and the goal characteristic which focus the couple of criterion. The predicted RGNBC version is experiment through two big datasets and the effects are proven from corner to corner the existent MReC-DFS set of rules via sen, spe and corr. from the results, we ought to find out that the projected version attain the utmost accurateness of 63.97 % while evaluated with the aid of the available set of rules.

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure [Formula: see text], where each [Formula: see text] is a standard real Gaussian matrix, and [Formula: see text] is a real anti-symmetric matrix can be determined. For [Formula: see text] and [Formula: see text] the bidiagonal anti-symmetric matrix with 1’s above the diagonal, this reclaims results of Defosseux. For general [Formula: see text], and this choice of [Formula: see text], or [Formula: see text] itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

AbstractThe singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product {X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each {X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the {X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.

Propagation of the Measurement Error and the Measured Mean of a Physical Quantity for Elementary Functions ax and loga x

Rules have been obtained for the propagation of the error and the mean value for a measured physical quantity onto another one with a functional relation of the type ax or loga x between them. In essence, these rules are inherently based on the Gaussian weight scheme. Therefore, they should be valid in the framework of a real Gaussian weight scheme applied to discrete data of a real physical experiment (a sample). An analytical form that was used to present the rules concerned (“analytical propagation rules”) and their character allow the processing and the analysis of experimental data to be simplified and accelerated.

New Approximation to Distribution of Positive RVs Applied to Gaussian Quadratic Forms

This letter introduces a new approach to the problem of approximating the probability density function (PDF) and the cumulative distribution function (CDF) of a positive random variable. The novel approximation strategy is based on the analysis of a suitably defined sequence of auxiliary variables which converges in distribution to the target variable. By leveraging such convergence, simple approximations for both the CDF and PDF of the target variable are given in terms of the derivatives of its moment generating function (MGF). In contrast to classical approximation methods based on truncated series of moments or cumulants, our approximations always represent a valid distribution and the relative error between variables is independent of the variable under analysis. The derived results are then used to approximate the statistics of positive-definite real Gaussian quadratic forms, comparing our proposed approach with other existing approximations in the literature.

Quasi-Concavity for Gaussian Multicast Relay Channels.

Standard upper and lower bounds on the capacity of relay channels are cut-set (CS), decode-forward (DF), and quantize-forward (QF) rates. For real additive white Gaussian noise (AWGN) multicast relay channels with one source node and one relay node, these bounds are shown to be quasi-concave in the receiver signal-to-noise ratios and the squared source-relay correlation coefficient. Furthermore, the CS rates are shown to be quasi-concave in the relay position for a fixed correlation coefficient, and the DF rates are shown to be quasi-concave in the relay position. The latter property characterizes the optimal relay position when using DF. The results extend to complex AWGN channels with random phase variations.

Floating Orbitals Reconsidered: The Difference an Imaginary Part Can Make

Floating orbitals for valence electrons have made cameo appearances at several stages in the history of quantum chemistry. Most often, they were considered as potentially useful basis functions and, more recently, also as muses for the development of subatomistic force fields. To facilitate computation, these orbitals are generally taken to be real spherical Gaussians. However, the computational advantages carry over to complex Gaussians. Here, we explore the potential utility of an imaginary part. Analytical equations for two mobile electrons show that an imaginary part shifts the balance between contributions to the exchange energy that favor parallel versus antiparallel electron spins. However, an imaginary part also carries a large kinetic energy penalty. The imaginary part is therefore negligible for two valence electrons, except in the case of strong core–valence exchange interactions. This consideration allows a self-consistent model for the nd2 triplet ground states of transition metal ions versus the ns2 singlet ground states of main group ions.

Variance of the number of zeroes of shift-invariant Gaussian analytic functions

Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle [−T,T] × [a, b] is asymptotically between cT and CT2, with positive constants c and C. We also supply with conditions (in terms of the spectral measure) under which the variance grows asymptotically linearly with T, as a quadratic function of T, or has intermediate growth. The results are compared with known results for real stationary Gaussian processes and other models.

A note on the characterizations of the distributions of the condition numbers of real Gaussian matrices

Abstract Many researchers and authors have studied the distributions of the condition numbers of real Gaussian matrices, which appear in many fields of pure and applied sciences, such as, probability, statistics, multivariate statistics, linear algebra, operator algebra theory, actuarial science, physics, wireless communications, and polarimetric synthetic aperture radar (PolSAR). Motivated by this, in this paper, we first present several new distributional properties of the distributions of the condition numbers of real Gaussian matrices. Since it is important to know the percentage points of a given distribution for any statistical application, we have also computed percentiles of the said distributions of the condition numbers. Before a particular probability distribution model is applied to fit the real world data, it is necessary to confirm whether the given continuous probability distribution satisfies the underlying requirements by its characterizations. Also, the truncated distributions arise in practical statisticswhere the ability of record observations is limited to a given threshold or within a specified range. In view of these facts, some characterizations by truncated first moment are also presented. We hope that the findings of this paper will be quite useful to the researchers in various fields of pure and applied sciences as stated above.

Phase Retrieval via Reweighted Amplitude Flow

This paper deals with finding an $n$ -dimensional solution $\boldsymbol {x}$ to a system of quadratic equations of the form $y_i=|\langle \boldsymbol {a}_i,\boldsymbol {x}\rangle |^2$ for $1\leq i \leq m$ , which is also known as the generalized phase retrieval problem. For this NP-hard problem, a novel approach is developed for minimizing the amplitude-based least-squares empirical loss, which starts with a weighted maximal correlation initialization obtainable through a few power or Lanczos iterations, followed by successive refinements based on a sequence of iteratively reweighted gradient iterations. The two stages (initialization and gradient flow) distinguish themselves from prior contributions by the inclusion of a fresh (re)weighting regularization procedure. For certain random measurement models, the novel scheme is shown to be able to recover the true solution $\boldsymbol {x}$ in time proportional to reading the data $\lbrace (\boldsymbol {a}_i;y_i)\rbrace _{1\leq i \leq m}$ . This holds with high probability and without extra assumption on the signal vector $\boldsymbol {x}$ to be recovered, provided that the number $m$ of equations is some constant $c>0$ times the number $n$ of unknowns in the signal vector, namely $m>cn$ . Empirically, the upshots of this contribution are: first, (almost) $\text{100}{\%}$ perfect signal recovery in the high-dimensional (say $n\geq 2000$ ) regime given only an information-theoretic limit number of noiseless equations, namely $m=2n-1$ , in the real Gaussian case; and second, (nearly) optimal statistical accuracy in the presence of additive noise of bounded support. Finally, substantial numerical tests using both synthetic data and real images corroborate markedly improved recovery performance and computational efficiency of the novel scheme relative to the state-of-the-art approaches.

Symbol Error Rate Performance of Box-Relaxation Decoders in Massive MIMO

The maximum-likelihood (ML) decoder for symbol detection in large multiple-input multiple-output wireless communication systems is typically computationally prohibitive. In this paper, we study a popular and practical alternative, namely the Box-relaxation optimization (BRO) decoder, which is a natural convex relaxation of the ML. For iid real Gaussian channels with additive Gaussian noise, we obtain exact asymptotic expressions for the symbol error rate (SER) of the BRO. The formulas are particularly simple, they yield useful insights, and they allow accurate comparisons to the matched-filter bound (MFB) and to the zero-forcing decoder. For BPSK signals the SER performance of the BRO is within 3dB of the MFB for square systems, and it approaches the MFB as the number of receive antennas grows large compared to the number of transmit antennas. Our analysis further characterizes the empirical density function of the solution of the BRO, and shows that error events for any fixed number of symbols are asymptotically independent. The fundamental tool behind the analysis is the convex Gaussian min-max theorem.

Scalable Solvers of Random Quadratic Equations via Stochastic Truncated Amplitude Flow

A novel approach termed \emph{stochastic truncated amplitude flow} (STAF) is developed to reconstruct an unknown $n$-dimensional real-/complex-valued signal $\bm{x}$ from $m$ `phaseless' quadratic equations of the form $\psi_i=|\langle\bm{a}_i,\bm{x}\rangle|$. This problem, also known as phase retrieval from magnitude-only information, is \emph{NP-hard} in general. Adopting an amplitude-based nonconvex formulation, STAF leads to an iterative solver comprising two stages: s1) Orthogonality-promoting initialization through a stochastic variance reduced gradient algorithm; and, s2) A series of iterative refinements of the initialization using stochastic truncated gradient iterations. Both stages involve a single equation per iteration, thus rendering STAF a simple, scalable, and fast approach amenable to large-scale implementations that is useful when $n$ is large. When $\{\bm{a}_i\}_{i=1}^m$ are independent Gaussian, STAF provably recovers exactly any $\bm{x}\in\mathbb{R}^n$ exponentially fast based on order of $n$ quadratic equations. STAF is also robust in the presence of additive noise of bounded support. Simulated tests involving real Gaussian $\{\bm{a}_i\}$ vectors demonstrate that STAF empirically reconstructs any $\bm{x}\in\mathbb{R}^n$ exactly from about $2.3n$ magnitude-only measurements, outperforming state-of-the-art approaches and narrowing the gap from the information-theoretic number of equations $m=2n-1$. Extensive experiments using synthetic data and real images corroborate markedly improved performance of STAF over existing alternatives.

A central limit theorem for Lipschitz–Killing curvatures of Gaussian excursions

This paper studies the excursion set of a real stationary isotropic Gaussian random field above a fixed level. We show that the standardized Lipschitz–Killing curvatures of the intersection of the excursion set with a window converges in distribution to a normal distribution as the window grows to the d-dimensional Euclidean space. Moreover a lower bound for the asymptotic variance is derived.

Exact Distribution for the Product of Two Correlated Gaussian Random Variables

This letter considers the distribution of product for two correlated real Gaussian random variables with nonzero means and arbitrary variances, which arises widely in radar and communication societies. We determine the exact probability density function (PDF) in terms of an infinite sum of modified Bessel functions of second kind, which includes some existent results, i.e., zero-means and/or independent variables, as special cases. Then, we study the approximation error and convergence rate when finite summations are exploited in practice. Finally, we evaluate the PDF behaviors of the derived expression as well as the Monte Carlo simulations.

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly k real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these — the eigenvalue density of the real eigenvalues — gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.

Joint Independent Subspace Analysis Using Second-Order Statistics

This paper deals with a novel generalization of classical blind source separation (BSS) in two directions. First, relaxing the constraint that the latent sources must be statistically independent. This generalization is well-known and sometimes termed independent subspace analysis (ISA). Second, jointly analyzing several ISA problems, where the link is due to statistical dependence among corresponding sources in different mixtures. When the data are one-dimensional, i.e., multiple classical BSS problems, this model, known as independent vector analysis (IVA), has already been studied. In this paper, we combine IVA with ISA and term this new model joint independent subspace analysis (JISA). We provide full performance analysis of JISA, including closed-form expressions for minimal mean square error (MSE), Fisher information and Cramer–Rao lower bound, in the separation of Gaussian data. The derived MSE applies also for non-Gaussian data, when only second-order statistics are used. We generalize previously known results on IVA, including its ability to uniquely resolve instantaneous mixtures of real Gaussian stationary data, and having the same arbitrary permutation at all mixtures. Numerical experiments validate our theoretical results and show the gain with respect to two competing approaches that either use a finer block partition or a different norm.

Equidistribution of zeros of random holomorphic sections

We study asymptotic distribution of zeros of random holomorphic sections of high powers of positive line bundles defined over projective homogenous manifolds. We work with a wide class of distributions that includes real and complex Gaussians. As a special case, we obtain asymptotic zero distribution of multivariate complex polynomials given by linear combinations of orthogonal polynomials with i.i.d. random coefficients. Namely, we prove that normalized zero measures of m i.i.d random polynomials, orthonormalized on a regular compact set $K\subset \Bbb{C}^m,$ are almost surely asymptotic to the equilibrium measure of $K$.

Distribution of the largest root of a matrix for Roy’s test in multivariate analysis of variance

Let ${\bf X, Y} $ denote two independent real Gaussian $\mathsf{p} \times \mathsf{m}$ and $\mathsf{p} \times \mathsf{n}$ matrices with $\mathsf{m}, \mathsf{n} \geq \mathsf{p}$, each constituted by zero mean i.i.d. columns with common covariance. The Roy's largest root criterion, used in multivariate analysis of variance (MANOVA), is based on the statistic of the largest eigenvalue, $\Theta_1$, of ${\bf{(A+B)}}^{-1} \bf{B}$, where ${\bf A =X X}^T$ and ${\bf B =Y Y}^T$ are independent central Wishart matrices. We derive a new expression and efficient recursive formulas for the exact distribution of $\Theta_1$. The expression can be easily calculated even for large parameters, eliminating the need of pre-calculated tables for the application of the Roy's test.

Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for the product of two Gaussian matrices

We provide a proof to a recent conjecture by Forrester (2014 J. Phys. A: Math. Theor. 47 065202) regarding the algebraic and arithmetic structure of Meijer G-functions which appear in the expression for probability of all eigenvalues real for the product of two real Gaussian matrices. In the process we come across several interesting identities involving Meijer G-functions.