Showing papers on "Complex normal distribution published in 2006"

Polarimetric image segmentation via maximum-likelihood approximation and efficient multiphase level-sets

Abstract: This study investigates a level set method for complex polarimetric image segmentation. It consists of minimizing a functional containing an original observation term derived from maximum-likelihood approximation and a complex Wishart/Gaussian image representation and a classical boundary length prior. The minimization is carried out efficiently by a new multiphase method which embeds a simple partition constraint directly in curve evolution to guarantee a partition of the image domain from an arbitrary initial partition. Results are shown on both synthetic and real images. Quantitative performance evaluation and comparisons are also given

A rate of convergence result for the largest eigenvalue of complex white Wishart matrices

Abstract: It has been recently shown that if X is an n×N matrix whose entries are i.i.d. standard complex Gaussian and l1 is the largest eigenvalue of X*X, there exist sequences mn,N and sn,N such that (l1−mn,N)/sn,N converges in distribution to W2, the Tracy–Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of W2 is denoted F2. In this paper we show that, under the assumption that n/N→ γ∈(0, ∞), we can find a function M, continuous and nonincreasing, and sequences μn,N and σn,N such that, for all real s0, there exists an integer N(s0, γ) for which, if (n∧N)≥N(s0, γ), we have, with ln,N=(l1−μn,N)/σn,N, ∀ s≥s0 (n∧N)2/3|P(ln,N≤s)−F2(s)|≤M(s0)exp(−s). The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution W2 is a good approximation to the empirical distribution of ln,N in simulations, an important fact from the point of view of (e.g., statistical) applications.

Cramer-Rao bounds of DOA estimates for BPSK and QPSK Modulated signals

Abstract: This paper focuses on the stochastic Cramer-Rao bound (CRB) of direction of arrival (DOA) estimates for binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) modulated signals corrupted by additive circular complex Gaussian noise. Explicit expressions of the CRB for the DOA parameter alone in the case of a single signal waveform are given. These CRBs are compared, on the one hand, with those obtained with different a priori knowledge and, on the other hand, with CRBs under the noncircular and circular complex Gaussian distribution and with different deterministic CRBs. It is shown in particular that the CRBs under the noncircular [respectively, circular] complex Gaussian distribution are tight upper bounds on the CRBs under the BPSK [respectively, QPSK] distribution at very low and very high signal-to-noise ratios (SNRs) only. Finally, these results and comparisons are extended to the case of two independent BPSK or QPSK distributed sources where an explicit expression of the CRB for the DOA parameters alone is given for large SNR.

Probabilistic data association for symbol detection in MIMO systems

Abstract: The application of the probabilistic data association (PDA) algorithm to the symbol detection in narrowband spatial multiplexed multiple input multiple output systems with known channel state information is investigated. The performance of a new proposed complex formulation of PDA algorithm, which matches the full parameters of the complex Gaussian distribution (which are the mean vector, covariance matrix and the pseudo-covariance matrix), is compared with the real vector formulation of a generalised PDA algorithm and a PDA implementation, which employs complex Gaussian approximations with matched mean and covariance only.

Eigenvalue distributions for some correlated complex sample covariance matrices

Abstract: The distributions of the smallest and largest eigenvalues for the matrix product $Z^\dagger Z$, where $Z$ is an $n \times m$ complex Gaussian matrix with correlations both along rows and down columns, are expressed as $m \times m$ determinants. In the case of correlation along rows, these expressions are computationally more efficient than those involving sums over partitions and Schur polynomials reported recently for the same distributions.

ML-based symbol timing and frequency offset estimation for OFDM systems with noncircular transmissions

Abstract: This paper deals with the problem of joint symbol timing and carrier-frequency offset (CFO) estimation in orthogonal frequency-division multiplexing (OFDM) systems with noncircular (NC) transmissions. Maximum-likelihood (ML) estimators of symbol timing and CFO have been derived under the assumption of nondispersive channel and by modeling the OFDM signal vector as a circular complex Gaussian random vector (C-CGRV). The Gaussian assumption is reasonable when the number of subcarriers is sufficiently large. However, if the data symbols belong to an NC constellation, the received signal vector becomes an NC-CGRV, i.e., a CGRV whose relation matrix (defined as the statistical expectation of the product between the vector and its transpose) is not identically zero. Hence, in this case, previously mentioned estimators, termed MLC estimators, are not ML estimators. In this paper, by exploiting the joint probability density function for NC-CGRVs, ML estimators are derived. Moreover, since their implementation complexity is high, feasible computational algorithms are considered. Finally, refined symbol timing estimators, apt to counteract the degrading effects of intersymbol interference (ISI) in dispersive channels, are suggested. The performance of the derived estimators is assessed via computer simulation and compared with that of MLC estimators and that of modified MLC (MMLC) estimators exploiting only ISI-free samples of the cyclic prefix

The complex Bingham quartic distribution and shape analysis

Abstract: Summary. The complex Bingham distribution was introduced by Kent as a tractable model for landmark-based shape analysis. It forms an exponential family with a sufficient statistic which is quadratic in the data. However, the distribution has too much symmetry to be widely useful. In particular, under high concentration it behaves asymptotically as a normal distribution, but where the covariance matrix is constrained to have complex symmetry. To overcome this limitation and to provide a full range of asymptotic normal behaviour, we introduce a new ‘complex Bingham quartic distribution’ by adding a selection of quartic terms to the log-density. In the simplest case this new distribution corresponds to Kent's FB5-distribution. Asymptotic and saddlepoint methods are developed for the normalizing constant to facilitate maximum likelihood estimation. Examples are given to show the usefulness of this new distribution.

Statistics of the degree of polarization

Abstract: The degree of polarization of a wave, that depends on sensor and target parameters, provides information on the randomness of the scattering or emission from natural targets. It may be useful in the analysis and interpretation of remote sensing data. For that, precise statistical description of its estimate will be required. Probability density function, expected value and standard deviation of the maximum likelihood estimate of the degree of polarization are obtained assuming a complex Gaussian distribution of the field samples that appropriately describes radiometric data, and also, some radar data.

A Note on the Distribution of Linear Functions of Two Ordered Correlated Normal Random Variables

Abstract: We consider the problem of finding the distribution of linear functions of two ordered correlated normal random variables. We derive some distributional properties for these linear statistics and briefly discuss the use of them in location estimation. The connection of the subject with the skew normal distribution is also noted.

Spatially correlated multiple-antenna channel capacity distributions

Abstract: The capacity of multiple-input multiple-output (MIMO) wireless communication systems over spatially correlated Rayleigh distributed flat fading channels with complex Gaussian additive noise is investigated. Specifically, the probability density function of the mutual information between transmitted and received complex signals of MIMO systems is derived. Using this density the closed-form ergodic capacity (mean), delay-limited capacity, capacity variance and outage capacity formulas for spatially correlated channels are derived and then these formulas are evaluated numerically. Numerical results show how the channel correlation degrades the capacity of MIMO communication systems. It is also shown that the density of mutual information of correlated/uncorrelated MIMO systems can be approximated by a Gaussian density with derived mean and variance, even for a finite number of inputs and outputs.

An upper bound on the ergodic mutual information in Rician fading MIMO channels

Abstract: We consider Rician fading multiple-input multiple-output (MIMO) channels where the transmitted signal has complex Gaussian distribution, iid across the transmit antennas. Based on expected values of elementary symmetric functions of complex noncentral Wishart matrices, we derive an upper bound on the average (ergodic) mutual information for arbitrary SNR, arbitrary rank of the deterministic line-of-sight matrix, and arbitrary number of transmit/receive antennas. The Rayleigh fading signal component is allowed to have spatial correlation at one end of the link. Upper bounds for the cases of rank-1 line-of-sight component and pure Rayleigh fading emerge as special instances of the general result

Frequency response estimation for NCFs of an MIMO plant from closed-loop time-domain experimental data

Abstract: Frequency response estimation from closed-loop time-domain experimental data is investigated in this paper for normalized coprime factors (NCF) of a multiple-input-multiple-output (MIMO) plant. Based on a linear fractional transformation (LFT) representation for all the NCFs of plants internally stabilizable by a known controller, this estimation problem is converted into the open-loop nonparametric estimation of an inner transfer function matrix (TFM). An estimate is derived through constrained data-matching. It is proved that when the probing signals are periodic and the NCFs of the auxiliary plant are appropriately selected, the estimate is asymptotically unbiased and with a normal/complex normal distribution. It has been made clear that the estimation bias and the estimation variance are always finite. Computationally tractable procedures are suggested for choosing the desirable auxiliary TFMs.

Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrödinger Equation Driven by the Square of a Gaussian Field

Abstract: Solutions to the equation Open image in new window are investigated, where S(x, t) is a complex Gaussian field with zero mean and specified covariance, and m≠0 is a complex mass with Im(m) ≥ 0. For real m this equation describes the backscattering of a smoothed laser beam by an optically active medium. Assuming that S(x, t) is the sum of a finite number of independent complex Gaussian random variables, we obtain an expression for the value of λ at which the qth moment of Open image in new window w.r.t. the Gaussian field S diverges. This value is found to be less or equal for all m ≠ 0, Im(m) ≥ 0 and |m|<+∞ than for |m| = +∞, i.e. when the Open image in new window term is absent. Our solution is based on a distributional formulation of the Feynman path-integral and the Paley-Wiener theorem.

Quantiles for finite mixtures of normal distributions

Abstract: Quantiles for finite mixtures of normal distributions are computed. The difference between a linear combination of independent normal random variables and a linear combination of independent normal densities is emphasized.

On the linear combination of normal and Laplace random variables

Abstract: The exact distribution of the linear combination ?X+βY is derived when X and Y are normal and Laplace random variables distributed independently of each other. A program in MAPLE is provided to compute the associated percentage points.

An OFDM Rayleigh Fading Channel Simulator

Abstract: Several models have been proposed for the simulation of Rayleigh fading channels. However, existing simulators lack to properly consider the correlation between the subchannels of an OFDM system. We use a recently developed cross-correlation function that describes both temporal and frequency correlation in order to generate channel parameters. This correlation function is decomposable into multiplication of two correlation functions. The first term characterizes only the temporal correlation and the other characterizes the correlation between subchannels. Using this property, the proposed simulator is implemented in cascade of two steps. In the first step, we propose an improved IFFT method for generation of multiple independent temporally correlated complex Gaussian processes following the given temporal correlation. We then transform these processes into a vector random processes by a transformation which is obtained by factorization of the frequency-correlation matrix. Our results reveal that the proposed technique accurately generates the desired statistical properties. This method is efficient in terms of computation complexity and runtime cost.

JMASM22: A Convenient Way Of Generating Normal Random Variables Using Generalized Exponential Distribution

Abstract: A convenient method to generate normal random variable using a generalized exponential distribution is proposed. The new method is compared with the other existing methods and it is observed that the proposed method is quite competitive with most of the existing methods in terms of the KS − distances and the corresponding p-values.

Optimal Combining with Arbitrary Power Interferers and Thermal Noise on Rayleigh Fading Channels

Abstract: The performance of optimum combining is studied in a Rayleigh fading environment with arbitrary-power cochannel interferers and thermal noise. Based on the joint eigenvalue distributions of quadratic forms in complex Gaussian matrices, closed form expressions for exact moment generating function (MGF) of the output signal-to-interference-plus-noise ratio are derived. From the exact MGF, the moments of the output SINR and the symbol error rate of various M-ary moulation schemes are obtained. We verify the accuracy of our analytical results by 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 optimum combining.

A note on the ratio of normal and Laplace random variables

Abstract: The normal and Laplace are the two earliest known continuous distributions in statistics and the two most popular models for analyzing symmetric data. In this note, the exact distribution of the ratio | X / Y | is derived when X and Y are respectively normal and Laplace random variables distributed independently of each other. A MAPLE program is provided for computing the associated percentage points. An application of the derived distribution is provided to a discriminant problem.

Simple error probability derivation for binary DPSK over fast Rician channels with diversity

Abstract: Bit error probability (BEP) performance of binary differential phase shift keying (DPSK) with differential detection over the nonselective, fast Rician fading channels with combining diversity reception is analysed. The analytical approach that exists in previously published literature for computing the BEP relied on a special case of the derivation given by Proakis that was concerned with the probability that a general quadratic form in complex Gaussian random variables is less than zero. However, evaluating the various coefficients required in the derivation leads to a computationally intensive solution. A simple derivation is presented which leads to a new, alternative BEP expression.

Truncations of random unitary matrices and Young tableaux

Abstract: Let $U$ be a matrix chosen randomly, with respect to Haar measure, from the unitary group $U(d).$ We express the moments of the trace of any submatrix of $U$ as a sum over partitions whose terms count certain standard and semistandard Young tableaux. Using this combinatorial interpretation, we obtain a simple closed form for the moments of an individual entry of a random unitary matrix and use this to deduce that the entries converge in moments to standard complex Gaussian random variables. In addition, we recover a well-known theorem of E. Rains which shows that the moments of the trace of a random unitary matrix enumerate permutations with restricted increasing subsequence length.

A Novel Algorithm for the Simulation of Cross-correlated Rayleigh Fading Channels

Abstract: A novel and efficient algorithm for the simulation of cross-correlated Rayleigh fading channels is proposed by investigating the channel characteristics. In the proposed algorithm, we lead the solution of coloring matrix to an optimization problem. And then, the proximal matrix Ktilde for correlation coefficient matrix K is obtained by unitary similarity transform in Frobenius norm. After the achievement of coloring matrix, linear transform to independent and identically distributed complex Gaussian random processes is implemented to simulate the correlated Rayleigh fading channels with desired cross correlation properties. The proposed algorithm has a good generality for relaxing the restriction of K to be positive definite or positive semi-definite. Also the simulation results are approximately consistent with the theoretic analysis

Asymptotic Analysis in MIMOMRT/MRC Systems

Abstract: Through the analysis of the probability density function of the largest squared singular value of a complex Gaussian matrix at the origin and tail, we obtain two asymptotic results related to the multi-input multi-output (MIMO) maximum-ratiotransmission/maximum-ratio-combining (MRT/MRC) systems. One is the asymptotic error performance (in terms of SNR) in a single-user system, and the other is the asymptotic system capacity (in terms of the number of users) in the multiuser scenario when multiuser diversity is exploited. Similar results are also obtained for two other MIMO diversity schemes, space-time block coding and selection combining. Our results reveal a simple connection with system parameters, providing good insights for the design of MIMO diversity systems.

A Correlation Based Double Directional Stochastic Model for MIMO-UWB Propagation Channels

Abstract: This paper proposes a correlation based double directional stochastic model (CBDDSM) for indoor Multiple Input Multiple Output (MIMO) Ultra Wideband (UWB) propagation channels in the passband domain. Both angle of arrival (AoA) and time of arrival statistics are taken into account in the modelling procedure. Spatial correlations are introduced into the amplitude matrices and delay matrices of the channel model. Under the assumption of Rayleigh amplitude fading statistics, each amplitude matrix is obtained from the underlying correlated complex Gaussian matrix, while each delay matrix is obtained as the sum of a reference matrix and a difference matrix. Model parameters are determined based on well-known measurement campaigns. Simulation results show that the characteristics of the proposed model are compatible with those of the IEEE 802.15.3a standard model. In addition, our initial analysis indicates that this model yields desirable spatial correlation properties in both the time and frequency domains.