Showing papers on "Complex normal distribution published in 2013"

Statistical Characterization and Computationally Efficient Modeling of a Class of Underwater Acoustic Communication Channels

Abstract: Underwater acoustic channel models provide a tool for predicting the performance of communication systems before deployment, and are thus essential for system design. In this paper, we offer a statistical channel model which incorporates physical laws of acoustic propagation (frequency-dependent attenuation, bottom/surface reflections), as well as the effects of inevitable random local displacements. Specifically, we focus on random displacements on two scales: those that involve distances on the order of a few wavelengths, to which we refer as small-scale effects, and those that involve many wavelengths, to which we refer as large-scale effects. Small-scale effects include scattering and motion-induced Doppler shifting, and are responsible for fast variations of the instantaneous channel response, while large-scale effects describe the location uncertainty and changing environmental conditions, and affect the locally averaged received power. We model each propagation path by a large-scale gain and micromultipath components that cumulatively result in a complex Gaussian distortion. Time- and frequency-correlation properties of the path coefficients are assessed analytically, leading to a computationally efficient model for numerical channel simulation. Random motion of the surface and transmitter/receiver displacements introduce additional variation whose temporal correlation is described by Bessel-type functions. The total energy, or the gain contained in the channel, averaged over small scale, is modeled as log-normally distributed. The models are validated using real data obtained from four experiments. Specifically, experimental data are used to assess the distribution and the autocorrelation functions of the large-scale transmission loss and the short-term path gains. While the former indicates a log-normal distribution with an exponentially decaying autocorrelation, the latter indicates a conditional Ricean distribution with Bessel-type autocorrelation.

Multichannel Extensions of Non-Negative Matrix Factorization With Complex-Valued Data

Abstract: This paper presents new formulations and algorithms for multichannel extensions of non-negative matrix factorization (NMF). The formulations employ Hermitian positive semidefinite matrices to represent a multichannel version of non-negative elements. Multichannel Euclidean distance and multichannel Itakura-Saito (IS) divergence are defined based on appropriate statistical models utilizing multivariate complex Gaussian distributions. To minimize this distance/divergence, efficient optimization algorithms in the form of multiplicative updates are derived by using properly designed auxiliary functions. Two methods are proposed for clustering NMF bases according to the estimated spatial property. Convolutive blind source separation (BSS) is performed by the multichannel extensions of NMF with the clustering mechanism. Experimental results show that 1) the derived multiplicative update rules exhibited good convergence behavior, and 2) BSS tasks for several music sources with two microphones and three instrumental parts were evaluated successfully.

Transmit Optimization With Improper Gaussian Signaling for Interference Channels

Abstract: This paper studies the achievable rates of Gaussian interference channels with additive white Gaussian noise (AWGN), when improper or circularly asymmetric complex Gaussian signaling is applied. For the Gaussian multiple-input multiple-output interference channel (MIMO-IC) with the interference treated as Gaussian noise, we show that the user's achievable rate can be expressed as a summation of the rate achievable by the conventional proper or circularly symmetric complex Gaussian signaling in terms of the users' transmit covariance matrices, and an additional term, which is a function of both the users' transmit covariance and pseudo-covariance matrices. The additional degrees of freedom in the pseudo-covariance matrix, which is conventionally set to be zero for the case of proper Gaussian signaling, provide an opportunity to further improve the achievable rates of Gaussian MIMO-ICs by employing improper Gaussian signaling. To this end, this paper proposes widely linear precoding, which efficiently maps proper information-bearing signals to improper transmitted signals at each transmitter for any given pair of transmit covariance and pseudo-covariance matrices. In particular, for the case of two-user Gaussian single-input single-output interference channel (SISO-IC), we propose a joint covariance and pseudo-covariance optimization algorithm with improper Gaussian signaling to achieve the Pareto-optimal rates. By utilizing the separable structure of the achievable rate expression, an alternative algorithm with separate covariance and pseudo-covariance optimization is also proposed, which guarantees the rate improvement over conventional proper Gaussian signaling.

On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables

Abstract: The distribution of the ratio of two independent normal random variables X and Y is heavy tailed and has no moments. The shape of its density can be unimodal, bimodal, symmetric, asymmetric, and/or even similar to a normal distribution close to its mode. To our knowledge, conditions for a reasonable normal approximation to the distribution of Z = X/Y have been presented in scientific literature only through simulations and empirical results. A proof of the existence of a proposed normal approximation to the distribution of Z, in an interval I centered at β = E(X) /E(Y), is given here for the case where both X and Y are independent, have positive means, and their coefficients of variation fulfill some conditions. In addition, a graphical informative way of assessing the closeness of the distribution of a particular ratio X/Y to the proposed normal approximation is suggested by means of a receiver operating characteristic (ROC) curve.

Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits

Abstract: Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M=2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy-Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach—Part 2: The Under-Sampled Case

Abstract: In Abramovich [“Bounds on Maximum Likelihood Ratio-Part I: Application to Antenna Array Detection-Estimation With Perfect Wavefront Coherence,” IEEE Trans. Signal Process., vol. 52, pp. 1524-1536, June 2004], it was demonstrated, for multivariate complex Gaussian distribution, that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual covariance matrix R0 does not depend on this matrix and is fully specified by the matrix dimension M and the number of independent training samples T. This invariance property hence enables one to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically “as likely” as R0. This “expected likelihood” quality assessment allowed significant improvement of MUSIC DOA estimation performance in the so-called “threshold area,” and for diagonal loading and TVAR model order selection in adaptive detectors. Recently, the so-called complex elliptically symmetric (CES) distributions have been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix Σ0. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario (T ≥ M) while Part II deals with the undersampled scenario (T ≤ M).

Lyapunov Exponents for Products of Complex Gaussian Random Matrices

Abstract: The exact value of the Lyapunov exponents for the random matrix product P N =A N A N−1⋯A 1 with each $A_{i} = \varSigma^{1/2} G_{i}^{\mathrm{c}}$ , where Σ is a fixed d×d positive definite matrix and $G_{i}^{\mathrm{c}}$ a d×d complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.

Cramér-Rao Lower Bounds on Covariance Matrix Estimation for Complex Elliptically Symmetric Distributions

Abstract: This paper introduces the Cramer-Rao Lower Bounds (CRLBs) for the scatter matrix of Complex Elliptically Symmetric distributions and compares them to the performance of the (constrained-)ML estimators in the particular cases of complex Gaussian, Generalized Gaussian (GG) and t-distributed observation vectors. Numerical results confirm the goodness of the ML estimators and the advantage of taking into proper account a constraint on the matrix trace for small data size. The work is completed with the comparison with the performance of Tyler's matrix estimator that shows a very robust behavior in almost all the analyzed cases and with the CRLBs for the Complex Angular Elliptical distributions, whose Tyler's estimator is the ML one.

Cramér-Rao Bound for Circular and Noncircular Complex Independent Component Analysis

Abstract: Despite an increased interest in complex independent component analysis (ICA) during the last two decades, a closed form expression for the Cramer-Rao bound (CRB) for the demixing matrix is not known yet. In this paper, we fill this gap by deriving a closed-form expression for the CRB of the demixing matrix for instantaneous noncircular complex ICA. It contains the CRB for circular complex ICA and noncircular complex Gaussian ICA as two special cases. We also study the CRB numerically for the family of noncircular complex generalized Gaussian distributions and compare it to simulation results of two ICA estimators. Furthermore, we show how to extend the CRB to the case where the source signals are not temporally independent and identically distributed.

The complex statistics paradigm and the law of large numbers

Abstract: The five basic axioms of Kolmogorov define the probability in the real set R and do not take into cons ideration the imaginary part which takes place in the complex set C, a problem that we are facing in applied mathematics. Whatever the probability distribution of the random variable in R is, the corresponding probability in the whole set C equals always to one , so the outcome of the random experiment in C can be predicted totally. This is the consequence of the f act that the probability in C is got by subtracting the chaotic factor from the degree of our knowledge of the syst em. In this study, I will evaluate the complex rand om vectors and their resultant that represents the who le distribution and system in the complex space C. I will also define imaginary and complex expectations and variances and I will prove the law of large numbers usin g the concept of the resultant complex vector. In fact, a fter extending Kolmogorov’s system of axioms, the new axioms encompass the imaginary set of numbers and this by adding to the original five axioms of Kolmog orov an additional three axioms. Hence, the concept of c omplex random vector becomes clear, evident and it follows directly from the new axioms added. This re sult will be elaborated throughout this study using discrete probability distributions. Moreover, any experiment executed in the complex set C is the sum of the re al set R and the imaginary set M. Therefore, the whole probability distribution of random variables can be repr esented totally by the resultant complex random vector Z th at is used subsequently to prove the very well know n law of large numbers. In addition to my previous first paper, this second one elaborates the new field of “Complex Statistics” that considers random variables in the complex set C. Thus, the law of large numbers proves that this complex extension is successful and fruitful.

On Some Properties of the Unified Skew Normal Distribution

Abstract: Like some other multivariate skew normal distributions, the unified skew normal (SUN) distribution preserves important properties of the normal distribution. In this article, we show that for a random vector with a unified multivariate skew normal distribution all column (row) full rank linear transformations are in the same family of distributions. Using this property, we provide a characterization for the SUN distribution. In addition, we show that the joint distribution of the independent SUN random vectors is again a SUN distributed random vector. With this property and closure under the linear transformation, we finally show the closure of sums of independent SUN random vectors, and as expected it belongs to the same family.

Method for computing harmonic impedance of system based on maximum likelihood estimation theory

Abstract: The invention discloses a method for computing harmonic impedance of a system based on maximum likelihood estimation theory. The method is characterized by comprising the following steps: collecting bus voltage instantaneous value of common coupling point and current instantaneous value of user access system, and establishing a relation between harmonic voltage phasor and harmonic current phasor; on the basis of defining a complex covariance, deriving to obtain a probability density function of unary complex normal distribution so as to obtain a maximum likelihood estimation function; establishing the maximum likelihood estimation theory of complex field estimated by the harmonic impedance of system; utilizing an extreme value theory to solve the maximum likelihood estimation function so as to obtain the estimated value of harmonic impedance of the system finally. The method has the beneficial effects that the method for computing harmonic impedance of the system based on maximum likelihood estimation theory is capable of relatively accurately computing equivalent harmonic impedance of the system and has important meanings for further solving the problem of harmonic pollution and improving the management level of electric energy quality.

The Augmented Complex Particle Filter

Abstract: Complex Gaussian signals in engineering applications are typically noncircular, that is, they have rotation dependent distributions. However, current complex valued particle filters (PFs) have assumed (implicitly or explicitly) circular signal distributions, which for noncircular signals leads to suboptimal performance. In this correspondence, we employ a recent development in the so-called augmented complex statistics and propose the augmented complex PF (ACPF) and the augmented complex Gaussian PF (ACGPF) for the sequential estimation of complex states in both circular and noncircular noise. The analysis and simulations illuminate theoretical and practical advantages for the proposed solutions.

Complex Gaussian multiplicative chaos

Abstract: In this article, we study complex Gaussian multiplicative chaos. More precisely, we study the renormalization theory and the limit of the exponential of a complex log-correlated Gaussian field in all dimensions (including Gaussian Free Fields in dimension 2). Our main working assumption is that the real part and the imaginary part are independent. We also discuss applications in 2D string theory; in particular we give a rigorous mathematical definition of the so-called Tachyon fields, the conformally invariant operators in critical Liouville Quantum Gravity with a c=1 central charge, and derive the original KPZ formula for these fields.

On the Expected Value and Higher-Order Moments of the Euclidean Norm for Elliptical Normal Variates

Abstract: We consider the Euclidean norm of a bivariate normal vector, equivalently, the amplitude or envelope of a complex Gaussian variable with correlated real and imaginary parts - known also as Hoyt or Nakagami-q distribution. Such a model, popular in fading and other communication contexts, is also relevant for the analysis of spatial errors in positioning-related applications, namely localization. In this paper an efficient expression in terms of complete elliptic integrals is given for the expected value, and good approximations in simple algebraic form are derived. Results are generalized to moments of any order.

Modeling multilook polarimetric SAR images with heavy-tailed rayleigh distribution and novel estimation based on matrix log-cumulants

Abstract: With the improvements in modern radar resolution, the Gaussian-fluctuation model based on the central limit theorem does not accurately describe the scattering echo from targets. In contrast, the heavytailed Rayleigh distribution, based on the generalized central limit theorem, performs well in modeling the synthetic aperture radar (SAR) images, whereas its application to multi-look image processing is difficult. We describe successful modeling of multilook polarimetric SAR images with the heavy-tailed Rayleigh distribution and present novel parameter estimators based on matrix log-cumulants for the heavy-tailed Rayleigh distribution including the equivalent number of looks (ENL). First, a compound variable of heavy-tailed Rayleigh distribution is divided into a product of a positive alpha-stable variable and a complex Gaussian variable. The parameter estimations of the characteristic exponent and scale parameter based on log-cumulants in a single polarization channel are then derived. Second, the matrix log-cumulants (MLCs) for full polarization in multilook images are obtained, which can be applied to estimate model parameters. Therefore, a novel ENL estimator based on MLC is presented that describes the model more precisely. Extended to all other multivariable product models, this estimator performs better than existing methods. Finally, calculations on both simulated and real data are performed that give good fits with theoretical results. Multilook processing in one image with a fixed pixel number can improve parameter estimations over single-look processing. Our heavy-tailed Rayleigh model with its parameter estimation provides a new method to analyze the multilook polarimetric SAR images for target detection and classification.

Improper Gaussian signaling for the K-user SISO interference channel

Abstract: This paper studies the transmit optimization for the K-user Gaussian single-input single-output interference channel (SISO-IC), with the interference treated as Gaussian noise and by applying improper or circularly asymmetric complex Gaussian signaling. The transmit optimization with improper Gaussian signaling involves not only the signal covariance as in the conventional proper or circularly symmetric complex Gaussian signaling, but also the signal pseudo-covariance, which is conventionally set to zero in proper Gaussian signaling. By utilizing the rate-profile method, the achievable rate region of the K-user SISO-IC is characterized by solving a sequence of minimum-weighted-rate maximization (MinWR-Max) problems, which are non-convex and thus difficult to be solved globally optimally. By applying the semidefinite relaxation (SDR) technique, we propose an efficient approximate solution, which jointly optimizes the covariance and pseudo-covariance of the transmitted signals. Simulation results demonstrate the effectiveness of the proposed algorithm for the K-user SISO-IC with improper Gaussian signaling.

A recursive Bayesian beamforming for steering vector uncertainties

Abstract: A recursive Bayesian approach to narrowband beamforming for an uncertain steering vector of interest signal is presented. In this paper, the interference-plus-noise covariance matrix and signal power are assumed to be known. The steering vector is modeled as a complex Gaussian random vector that characterizes the level of steering vector uncertainty. Applying the Bayesian model, a recursive algorithm for minimum mean square error (MMSE) estimation is developed. It can be viewed as a mixture of conditional MMSE estimates weighted by the posterior probability density function of the random steering vector given the observed data. The proposed recursive Bayesian beamformer can make use of the information about the steering vector brought by all the observed data until the current short-term integration window and can estimate the mean and covariance of the steering vector recursively. Numerical simulations show that the proposed beamformer with the known signal power and interference-plus-noise covariance matrix outperforms the linearly constrained minimum variance, subspace projection, and other three Bayesian beamformers. After convergence, it has similar performance to the optimal Max-SINR beamformer with the true steering vector.

Introducing a Novel Bivariate Generalized Skew-Symmetric Normal Distribution

Abstract: We introduce a generalization of the bivariate generalized skew-symmetric normal distribution [5]. We denote this distribution by

Approximating the outage capacity of asymmetric 2×2 dual-polarized MIMO at high SNR

Abstract: We present an outage probability analysis for asymmetric dual-polarized channels, where the elements of the channel gain matric are represented by independent nonidentical complex Gaussian distributions. We apply a moment generating function method to derive statistics associated with the determinant of the desired asymmetric random matrices. Using a two-step distribution model to characterize the mutual information we provide outage capacity approximates for various asymmetric channel realizations that are parameterized by the co-polarized power ratio, the cross-polarization discrimination ratio, and sub-channel Rician K-factors. Deriving an exact outage capacity formulation is made difficult by the inherent asymmetry of the model. The Lognormal and the Gamma distributions are assumed for mutual information exponent and the Weibull and Normal distributions are used to represent the mutual information. Contrary to current notions regarding the use of Gaussian approximations for mutual information, more accurate results are obtained using the Weibull distribution. We also show that partially-correlated channel gains typical of dual-polarized antennas negligibly impact the outage probabilities reported for the uncorrelated cases.

Propagation equation of Hermite-Gauss beams through a complex optical system with apertures and its application to focal shift

Abstract: Based on the generalized Huygens-Fresnel diffraction integral (Collins' formula), the propagation equation of Hermite-Gauss beams through a complex optical system with a limiting aperture is derived. The elements of the optical system may be all those characterized by an ABCD ray-transfer matrix, as well as any kind of apertures represented by complex transmittance functions. To obtain the analytical expression, we expand the aperture transmittance function into a finite sum of complex Gaussian functions. Thus the limiting aperture is expressed as a superposition of a series of Gaussian-shaped limiting apertures. The advantage of this treatment is that we can treat almost all kinds of apertures in theory. As application, we define the width of the beam and the focal plane using an encircled-energy criterion and calculate the intensity distribution of Hermite-Gauss beams at the actual focus of an aperture lens. 2013 Optical Society of America.

MISO interference channel with improper Gaussian signaling

Abstract: This paper studies the achievable rate region of the K-user Gaussian multiple-input single-output interference channel (MISO-IC) with interference treated as noise, when improper or circularly asymmetric complex Gaussian signaling is applied. By exploiting the separable rate expression with improper Gaussian signaling, we propose a separate covariance and pseudo-covariance optimization algorithm, which is guaranteed to improve the users' rates over the conventional proper or circularly symmetric complex Gaussian signaling. In particular, for the pseudo-covariance optimization, the semidefinite relaxation (SDR) technique is applied to provide a high-quality approximate solution. For the special case of two-user MISO-IC, the SDR technique yields the optimal pseudo-covariance solution.

Spectrum estimation using new efficient 2-D AR lattice modeling of random fields

Abstract: In this paper, the new efficient 2-D autoregressive (AR) lattice modeling technique of random fields is applied to predict the reflection parameters of the 2-D AR data field and the spectrum estimation is carried out. In addition, the AR coefficients of the data field generated as the sum of 2-D complex sinusoids corrupted by additive complex Gaussian noise are predicted and its spectrum is estimated. The results are compared with the classical 2-D FFT-based estimates. The new efficient 2-D (AR) lattice modeling technique is newly presented and based on employing the auxiliary vertical and horizontal prediction error fields. The aim of this method is to obtain simultaneously all possible types of causal and noncausal 2-D models for an arbitrary rectangular shape of the prediction support region. The obtained causal quarter-plain models are mostly found stable.

The glassy phase of complex branching Brownian motion

Abstract: In this paper, we study complex valued branching Brownian motion in the so-called glassy phase, or also called phase II. In this context, we prove a limit theorem for the complex partition function hence confirming a conjecture formulated by Lacoin and the last two authors in a previous paper on complex Gaussian multiplicative chaos. We will show that the limiting partition function can be expressed as a product of a Gaussian random variable, mainly due to the windings of the phase, and a stable transform of the so called derivative martingale, mainly due to the clustering of the modulus. The proof relies on the fine description of the extremal process available in the branching Brownian motion context.

Pre-coding method under non-accurate channel information of multi-user multiple-input-multiple-output system

Abstract: The invention discloses a pre-coding method under non-accurate channel information of a multi-user multiple-input-multiple-output system. The pre-coding method is characterized in that errors between non-accurate estimation channel state matrixes obtained by a practical channel state matrix and a base station is modeled to an additive complex gaussian random process, the structure of a pre-coded matrix is formed by pseudo-inverse multiplying regularization of a total estimation channel matrix from the base station to all users with a block diagonal matrix, the block diagonal matrix is calculated according to the minimum mean square error, the obtained pre-coded matrix pre-codes data flow and transmits the data flow out from an antenna of the base station, all the users multiplies self receiving signals with corresponding decoding matrixes of the users to convert a channel to a plurality of single-input single-output sub channels, and all the sub channels are subjected to data detection to estimate original data. The pre-coding method considers influence of channel estimation errors during calculation of the pre-coded matrix and the decoding matrix, solves the problem that the channel errors generate the plateau effect in a pre-coding system, and ensures transmission effectiveness of the pre-coding system.

Some Complex Statistical Distributions in Multichannel Complex-Valued Signal Detection Theory

Abstract: The probability density functions(PDF's)and cumulative distribution functions(CDF's) of complex random variables RVA's)are basis for calculating the probabilities of false alarm(PFA's)and probabilities of detection(PD's)in complex-valued signal detection theory.In this paper,some relationship between the complex statistical distributions and the real ones is investigated. The PDF's and CDF's of the complex F and complex Beta distributions,both central and noncentral,and the PDF of the central complex t distribution are derived.

Asymptotic distributions of estimated cyclic autocorrelations of DSSS signals and the applications

Abstract: Asymptotic distributions of estimated cyclic autocorrelations (CA) of direct sequence spread spectrum (DSSS) signals are derived in this paper. The estimation follows a zero-mean complex normal distribution in which the variance exhibits a cyclic thumbtack form, and the cyclic period equals the symbol period. This property of the estimated CA can be used in the detection and recognition problem of DSSS signals. The asymptotic performances of detection and recognition are carried out, and the simulations also verify the theoretical analysis.