Showing papers on "Complex normal distribution published in 2008"
TL;DR: In this article, a power normal model is proposed for the analysis of data exhibiting a unimodal density having some skewness present, a structure often occurring in data analysis.
Abstract: The skew normal distribution, proposed by Azzalini (1985, A class of distributions which include the normal. Scand J Stat 12:171–178), can be a suitable model for the analysis of data exhibiting a unimodal density having some skewness present, a structure often occurring in data analysis. It has been observed that there are some practical problems in estimating the skewness parameter for small to moderate sample sizes. In this paper we point out those problems and propose another skewed model which we call “Power normal model”. The basic structural properties of the proposed model including its reliability properties are presented. The closeness of skew normal and power normal distributions is investigated. It is shown that the proposed model has some nice properties which make it feasible to study the estimation and testing of the skewness parameter. This can be achieved by transforming the data to the exponential model which has been studied extensively in the literature.
91 citations
TL;DR: This paper presents an analytical performance investigation of transmit beamforming (BF) systems in Rayleigh product multiple-input multiple-output (MIMO) channels, and derives new closed-form expressions for the cumulative distribution function, probability density function, and moments of a product of independent complex Gaussian matrices to provide a complete statistical characterization of the received signal-to-noise ratio (SNR).
Abstract: This paper presents an analytical performance investigation of transmit beamforming (BF) systems in Rayleigh product multiple-input multiple-output (MIMO) channels. We first derive new closed-form expressions for the cumulative distribution function, probability density function, and moments of the maximum eigenvalue of a product of independent complex Gaussian matrices, which are used to provide a complete statistical characterization of the received signal-to-noise ratio (SNR). We then derive a number of key performance metrics, including outage probability, symbol error rate, and ergodic capacity. We examine, in detail, three important special cases of the Rayleigh product MIMO channel: the degenerate keyhole scenario and the multiple-input single-output and single-input multiple-output scenarios, for which we derive insightful closed-form expressions for various exact and asymptotic measures (e.g., diversity order, array gain, and high SNR power offset, among others). We also compare the performance of transmit BF with orthogonal space-time block codes and quantify the benefit of exploiting transmitter channel knowledge in Rayleigh product MIMO channels. This is shown to be significant even for low-dimensional systems.
77 citations
TL;DR: In this article, the authors consider large complex random sample covariance matrices obtained from "spiked populations", that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to one.
Abstract: We consider large complex random sample covariance matrices obtained from "spiked populations", that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to one. We investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large. Under some conditions on moments of the sample distribution, we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex Gaussian sample with the same true covariance. The real setting is also considered.
42 citations
19 May 2008
TL;DR: Closed form expressions of the SEP are obtained for the cases of binary phase-shift keying, MPSK with high signal- to-noise ratio approximation, and MpsK with independent and identically distributed fading.
Abstract: Consider two independent complex Gaussian vectors having arbitrary mean vectors and covariance matrices which are scaled versions of the identity matrix. The joint characteristic function (c.f.) of the real and imaginary parts of the inner product of these two vectors is derived in closed form. This joint c.f. is applied to the analysis of the symbol error probability (ESP) of a multibranch diversity reception system in flat Rayleigh fading using M-ary phase-shift keying (MPSK). The receiver employs maximal- ratio combining with least squares channel estimation by means of pilot symbols. Closed form expressions of the SEP are obtained for the cases of binary phase-shift keying, MPSK with high signal- to-noise ratio approximation, and MPSK with independent and identically distributed fading.
21 citations
TL;DR: The Lyapunov exponent for the non-Hermitian Zakharov-Shabat eigenvalue problem corresponding to the attractive nonlinear Schrödinger equation with a Gaussian random pulse as an initial value function is calculated.
Abstract: We calculate the Lyapunov exponent for the non-Hermitian Zakharov-Shabat eigenvalue problem corresponding to the attractive nonlinear Schrodinger equation with a Gaussian random pulse as an initial value function. Using an extension of the Thouless formula to non-Hermitian random operators, we calculate the corresponding average density of states. We also calculate the distribution of a set of scattering data of the Zakharov-Shabat operator that determine the asymptotics of the eigenfunctions. We analyze two cases, one with circularly symmetric complex Gaussian pulses and the other with real Gaussian pulses. We discuss the implications in the context of information transmission through nonlinear optical fibers.
18 citations
TL;DR: A new analytical framework provides a simple and accurate way to assess the effects of equal- and unequal-power cochannel interferers and thermal noise on the performance of OC.
Abstract: The performance of multiple-input-multiple-output systems with optimum combining (OC) is studied in a Rayleigh fading environment with arbitrary-power cochannel interference and thermal noise. Based on the joint eigenvalue distributions of quadratic functions of complex Gaussian matrices, a closed-form expression for the exact distribution of the output signal-to-interference-plus-noise ratio (SINR) is derived. A closed-form expression for the exact moment-generating function (MGF) of the output SINR of single-input-multiple-output (SIMO) systems is also derived. From the exact MGF, the moments of the output SINR and the symbol error rate of various M-ary modulation schemes are obtained. We verify the accuracy of our analytical results with numerical examples. The new analytical framework provides a simple and accurate way to assess the effects of equal- and unequal-power cochannel interferers and thermal noise on the performance of OC.
17 citations
12 May 2008
TL;DR: This paper proposes a new measure for noncircularity of complex random variables and derives the ML decision rule and its performance based on this measure, and shows that the ML detector performs pseudo correlation as well as conventional correlation to the signals-of-interest.
Abstract: In a wide range of communication systems, including DS-CDMA and OFDM systems, the signal-of-interest might be corrupted by an improper (F.D. Neeser et al.,1993) (also called non circularly symmetric (B. Picinbono, 1994)) interfering signal. This paper studies the maximum likelihood (ML) detection of binary signals in the presence of additive improper complex Gaussian noise. Proposing a new measure for noncircularity of complex random variables, we will derive the ML decision rule and its performance based on this measure. It will be shown that the ML detector performs pseudo correlation (F.D. Neeser et al.,1993) as well as conventional correlation of the observation to the signals-of-interest. As an alternative solution, we will propose a filter for converting improper signals to proper ones, called circularization filter, and will utilize it together with a conventional matched-filter (MF) to construct an ML detector.
17 citations
TL;DR: In this paper, the complex orbital exponents in Gaussian-type basis functions were optimized with the numerical Newton-Raphson method based on the variational principle for the frequency-dependent polarizabilities.
12 citations
TL;DR: The random input problem for a nonlinear system modeled by the integrable one-dimensional self-focusing nonlinear Schrödinger equation (NLSE) is considered and the properties obtained from the direct scattering problem associated with the NLSE are concentrated on.
Abstract: We consider the random input problem for a nonlinear system modeled by the integrable one-dimensional self-focusing nonlinear Schrodinger equation (NLSE). We concentrate on the properties obtained from the direct scattering problem associated with the NLSE. We discuss some general issues regarding soliton creation from random input. We also study the averaged spectral density of random quasilinear waves generated in the NLSE channel for two models of the disordered input field profile. The first model is symmetric complex Gaussian white noise and the second one is a real dichotomous (telegraph) process. For the former model, the closed-form expression for the averaged spectral density is obtained, while for the dichotomous real input we present the small noise perturbative expansion for the same quantity. In the case of the dichotomous input, we also obtain the distribution of minimal pulse width required for a soliton generation. The obtained results can be applied to a multitude of problems including random nonlinear Fraunhoffer diffraction, transmission properties of randomly apodized long period Fiber Bragg gratings, and the propagation of incoherent pulses in optical fibers.
12 citations
TL;DR: Within the framework of the complex basis function method, the photoionization cross sections of H 2+ and H2 were calculated based on the variational principle for the frequency‐dependent polarizabilities and implies the effectiveness of the optimization of orbital exponents to reduce the number of basis functions.
Abstract: Within the framework of the complex basis function method, the photoionization cross sections of H(2)(+) and H(2) were calculated based on the variational principle for the frequency-dependent polarizabilities. In these calculations, complex orbital exponents of Gaussian-type basis functions for the final state continuum wavefunctions were fully optimized for each photon energy with the numerical Newton-Raphson method. In most cases, the use of only one or two complex Gaussian-type basis functions was enough to obtain excellent agreement with previous high precision calculations and available experimental results. However, there were a few cases, in which the use of complex basis functions having various angular momentum quantum numbers was crucial to obtain the accurate results. The behavior of the complex orbital exponents as a function of photon energy was discussed in relation to the scaling relation and the effective charge for photoelectron. The success of this method implies the effectiveness of the optimization of orbital exponents to reduce the number of basis functions and shows the possibility to calculate photoionization cross sections of general molecules using only Gaussian-type basis functions.
10 citations
TL;DR: The analytical expressions of the Wigner distribution for a Gaussian beam passing through a spatial filtering optical system with an internal hard aperture are obtained and are compared with the numerical integral results to show that the analytical results are proper and ascendant.
Abstract: The effect of an apertured optical system on Wigner distribution can be expressed as a superposition integral of the input Wigner distribution function and the double Wigner distribution function of the apertured optical system. By introducing a hard aperture function into a finite sum of complex Gaussian functions, the double Wigner distribution functions of a first-order optical system with a hard aperture outside and inside it are derived. As an example of application, the analytical expressions of the Wigner distribution for a Gaussian beam passing through a spatial filtering optical system with an internal hard aperture are obtained. The analytical results are also compared with the numerical integral results, and they show that the analytical results are proper and ascendant.
12 May 2008
TL;DR: An explicit characterization of the (ergodic) rate-optimal input covariance for systems that operate at low signal-to-noise ratios (SNRs) is derived and a closed-form expression for the matrix whose principal eigenvector yields the optimal beamforming direction is obtained.
Abstract: We consider a multiple-input multiple-output (MIMO) wireless communication scenario in which the channel follows a general spatially-correlated complex Gaussian distribution with non-zero mean. We derive an explicit characterization of the optimal input covariance from an ergodic rate perspective for systems that operate at low SNRs. This characterization is in terms of the eigen decomposition of a matrix that depends on the mean and the covariance of the channel, and typically results in a beamforming strategy along the principal eigenvector of that matrix. Simulation results show the potential impact of (jointly) exploiting the mean and the covariance of the channel on the ergodic achievable rate at both low and moderate- to-high SNRs.
Posted Content•
TL;DR: In this article, lower convex order bounds for sums of dependent log-skew normal random variables have been derived for modeling dependent random variables, based on the class of multivariate closed skew-normal.
Abstract: When it comes to modeling dependent random variables, not surprisingly, the multivariate normal distribution has received the most attention because of its many appealing properties. However, when it comes to practical implementation, the same family of distribution is often rejected for modeling nancial and insurance data because they do not apparently behave in the multivariate normal sense. In this paper, we consider the construction of lower convex order bounds, in the sense of Kaas et al. (2000), to approximate sums of dependent log-skew normal random variables. The dependence structure of these random variables is based on the class of multivariate closed skew-normal (CSN) distribution that appears in Gonzalez-Farias et al. (2004b) and which carries several interesting properties of the normal distribution apart from allowing additional parameters to regulate skewness. The bounds that we present in this paper are therefore natural extensions to the results presented in Dhaene et al. (2002b) and Dhaene et al. (2002a), where bounds for sums of log-normal random variables have been derived. These lower bound approximations are constructed based on the additional information provided by a conditioning variable which when optimally chosen can provide an accurate approximation. We exploit inherent properties of this family of skew-normal distributions in order to choose the optimal conditioning variable. Results of our simulations provide an indication of the performance of these approximations.
12 Dec 2008
TL;DR: A generalized likelihood ratio test is derived for detecting target patterns in multi-band spectral images with improper complex Gaussian noise and shows that the test achieves a constant false alarm rate.
Abstract: A generalized likelihood ratio test is derived for detecting target patterns in multi-band spectral images with improper complex Gaussian noise. The GLR test generalizes previously derived tests, which were restricted to proper complex Gaussian noise. However, this generalization comes for the price of increased computational time. The GLR test is derived based on the joint multivariate pdf of the magnitude and phase of correlated complex Gaussian variables. An expression for the probability of false alarm is obtained which shows that the test achieves a constant false alarm rate. The GLR test was applied to simulation and experimental data and showed satisfying results.
TL;DR: The corrected versions for these expressions for the distribution of a linear combination of Normal and Laplace random variables, Z, given in formulae (3, Theorem 1), (6), and (7) are presented here.
Abstract: In the above mentioned paper, some errors were found in the expressions given for the distribution of a linear combination of Normal and Laplace random variables, Z, given in formulae (3, Theorem 1), (6), and (7) that can lead to obtaining negative values for the mentioned distribution. The corrected versions for these expressions are presented here. In addition, the density function of Z is also provided.
06 Jul 2008
TL;DR: A closed-form expression for distribution of the smallest eigenvalue of the Wishart matrix, which is generated from the square Gaussian matrix, and a complicated equality is proved due to the fact that the distribution is mathematically equivalent to those reported.
Abstract: The complex central semi-correlated Wishart matrix is the product of a zero-mean row-wise correlated (or columnwise correlated) complex Gaussian matrix and its Hermitian transposition. In this paper, we investigate the distribution of the smallest eigenvalue of the Wishart matrix, which is generated from the square Gaussian matrix. A closed-form expression for distribution of the smallest eigenvalue is derived. Compared with previous results, our expression is very concise, since the distribution has been known in terms of matrix determinants. Numerical results that confirm the expression are also presented. And a complicated equality is proved due to the fact that our distribution is mathematically equivalent to those reported.
08 Dec 2008
TL;DR: The previous attempt for capacity analysis of full correlated MIMO channels is correct for square channel matrices only and the approach from square matrices to rectangular matrices is modified to obtain the correct joint eigenvalue distribution of the correlated Wishart matrix.
Abstract: Multiple-input multiple-output (MIMO) channels have been studied from various aspects including the average of the mutual information between the transmitter and receiver (ergodic capacity) when the channel gains are known to the receiver only. A common approach for capacity analysis is to find the moment generating function (MGF) of the mutual information and by direct differentiation, the mean of the mutual information (capacity) is calculated. Recently, character expansions of groups have been used for integration over unitary matrices to obtain the joint eigenvalue distribution of the correlated Wishart matrix i.e. HH* where H is the zero mean full correlated complex Gaussian random MIMO channel matrix. In this paper, we show that the previous attempt for capacity analysis of full correlated MIMO channels is correct for square channel matrices only. We modify the approach from square matrices to rectangular matrices to obtain the correct joint eigenvalue distribution of the correlated Wishart matrix. The result can be used to obtain the MGF and the capacity of full correlated MIMO channels.
TL;DR: The statistical characterisation of the output of a memory less nonlinear device having, as an input, a zero mean proper complex Gaussian random process is studied.
Abstract: The statistical characterisation of the output of a memory less nonlinear device having, as an input, a zero mean proper complex Gaussian random process is studied. The closed form expression obtained for the high-order moments involved allowed for neat mathematical expressions for the autocorrelation function and the power spectral density of the signal at the nonlinearity output. Parts of the autocorrelation function and of the power spectral density that correspond to the desired signal and to the intermodulation products of different orders are easily identified.
TL;DR: In this paper, a family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of matrices with iid centered complex Gaussian entries is considered, and scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum.
Abstract: A family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian entries is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order $n^{-1/3}$. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.
TL;DR: Zhao and Idota as discussed by the authors corrected the description of the definition of normalization according to their indication, and also corrected related to the correlation coefficient ρ 0,12 and the equation ƒ(χ 1, χ2).
Abstract: The authors thank Yan-Gang Zhao and Hideki Idota for their discussion. The answers are as follows;(1) We corrected the description of the definition of normalization according to their indication. And also we corrected the description related to the correlation coefficient ρ0,12 and the equation ƒ(χ1, χ2)(2) The cause of other indications seems to be the normalization. Joint probability density function for non-normal random variables was derived by bivariate normal distribution function. In the process of derivation, we used the normalization at any point of non-normal random variables. By this normalization the equivalent normal distribution can be gotten according to the point. Then the mean and standard deviation of the equivalent normal distribution are different at each point. But in the process of derivation, normal variate by normalization is transformed to standard normal variate, whose mean and standard deviation is zero and one respectively, even though the points are different. Finally the random variables included in the derived equation are standard normal variates and the original distribution function. So we think the derivation is correct.
Patent•
10 Jun 2008
TL;DR: In this article, the authors proposed a method for detecting the information in the OFDM system using the probability density function of the multi-dimensional complex Gaussian distribution (MDCG).
Abstract: The method for detecting the information in the OFDM system includes the following steps. A) The vector including several received signals is inputted into the receiver, when one symbol of several symbols to be detected is equal to one of candidate values, the vector would obey the multi-dimensional complex Gaussian distribution when the conditions about the channels and the other data are unknown. B) The posterior probability of every data that is equal to every candidate values can be calculated with the probability density function of the multi-dimensional complex Gaussian distribution. So the posterior probability of the data can be calculated accurately when the channel conditions would not be estimated. The performance of the detection can be close to the optimum results, and the complexity is in direct ratio with the number of carriers.