A closed-form upper-bound for the distribution of the weighted sum of Rayleigh variates

The problem of finding the distribution of the sum of more than two Rayleigh fading envelopes has never been solved in terms of tabulated functions. In this letter, we present a closed-form union upper-bound for the cumulative distribution function of the weighted sum of N independent Rayleigh fading envelopes. Computer simulation results verify the tightness of the proposed bound for several values of N. The proposed bound can be efficiently applied in various wireless applications, such as satellite communications, equal-gain receivers, and radars.     

On the PDF of the sum of random vectors

There are various cases in physics and engineering sciences (especially communications) where one requires the envelope probability density function (PDF) of the sum of several random sinusoidal signals. According to the correspondence between a random sinusoidal signal and a random vector, the sum of random vectors can be considered as an abstract mathematical model for the above sum. Now it is desired to obtain the PDF of the length of the resulting vector. Considering the common and reasonable assumption of uniform distributions for the angles of the vectors, many researchers have obtained the PDF of the length of the resulting vector only for special cases. However in this paper, the PDF is obtained for the most general case in which the lengths of vectors are arbitrary dependent random variables. This PDF is in the form of a definite integral, which may be inappropriate for analytic manipulations and numerical computations. So an appropriate infinite Laguerre expansion is also derived. Finally, the results are applied to solve a typical example in computing the scattering cross section of random scatterers     

Information inequalities for the sum of independent random vectors (Corresp.)

Lower bounds are given for the rate-distortion and distortion-rate functions of the sums of linear transformations of independent random vectorsXandY. These lower bounds are sharp in the sense that they are exactly achieved when the random vectorsXandYare Gaussian and identically distributed.     

Nakagami-m approximation to the sum of M non-identical independent Nakagami-m variates

A Nakagami-m approximate distribution to the sum of M non-identical independent Nakagami-m variates is derived. Comparisons between exact and approximate sum distributions show excellent agreement. This new approach finds applicability in important communications issues such as equal-gain combining, signal detection, linear equalisers, outage probability, intersymbol interference and phase jitter.     

Closed-form statistics for the sum of squared Nakagami-m variates and its applications

We present closed-form expressions for the probability density function (PDF) and the cumulative distribution function (CDF) of the sum of non-identical squared Nakagami-m random variables (RVs) with integer-order fading parameters. As it is shown, they can be written as a weighted sum of Erlang PDFs and CDFs, respectively, while the analysis includes both independent and correlated sums of RVs. The proposed formulation significantly improves previously published results, which are either in the form of infinite sums or higher order derivatives of the fading parameter m. The obtained formulas can be applied to the performance analysis of diversity combining receivers operating over Nakagami-m fading channels.     

A simulation-based approach to estimate the correlation parameters of multivariate Nakagami-m distribution

A Bayesian estimator for the correlation parameters of the multivariate Nakagami-m distribution is presented in this letter. The proposed estimator is a generalized Gibbs sampler, called group-based sampler. Numerical simulation results demonstrate that our new estimator offers reasonably small relative error and it also provides an acceptable performance.     

On the multivariate Nakagami-m distribution with exponential correlation

Capitalizing on the proof of a theorem presented by L.E. Blumenson and K.S. Miller (see Ann. Math. Statist., vol.34, p.903-10, 1963), we propose a useful closed formula for the exponentially correlated n-variate Nakagami-m probability density function. Moreover, an infinite series approach for the corresponding cumulative distribution function is presented. Bounds on the error resulting from the truncation of the infinite series are also derived. Finally, in order to check the accuracy of the proposed formulation, numerical results are presented.     

On higher order statistics of the Nakagami-m distribution

This paper proposes a fading model that leads to a formal, but simple method to obtain the exact formula of the Nakagami-m published in 1960 distribution for m=n/2, with n a nonzero integer number. Based on such a model, the joint distribution of the envelope and its time derivative are accomplished, and exact formulas for the level crossing rate (closed-form formula) and for the average fade duration are derived. Simulation curves and exact formulas are checked against each other and a very good agreement between them is attained. Although derived for discrete values of m (m being an integer multiple of 1/2), there are no mathematical constraints for these expressions to be used for any real value of m⩾1/2     

A closed-form solution of vertical dipole antennas above adielectric half-space

This paper derives a convenient closed-form expression for the input impedance of a vertical antenna above a dielectric half-space. The expression is obtained from the induced electromotive force (EMF) method using a complex-image spatial Green's function. It is found that the effect of the dielectric half-space can be modeled by a short image array of three to five complex image antennas. Numerical results verify the accuracy and convenience of the method for a vertical antenna at any height above a dielectric half-space with and without loss     

Cooperative localization for next-generation converged networks with closed-form solution

In this article, a novel cooperative wireless localization scheme based on information fusion is proposed. The scheme combines large-scale arrival time and small-scale distance measurements obtained from the next-generation converged networks. The maximum likelihood (ML) estimate of the terminai's position is derived with closed-form solution, and the Cramtr-Rao lower bound (CRLB) of the estimate error is investigated. Both theoretical analysis and simulation results verify that the proposed localization scheme can significantly enhance the location precision. Moreover, the mean square error of position estimate approximates the CRLB when the number of reference stations increases, which indicates that the proposed ML estimator is asymptotically efficient.     

An infinite series for the computation of the complementaryprobability distribution function of a sum of independent randomvariables and its application to the sum of Rayleigh random variables

The properties of the series are studied for both bounded and unbounded random variables. The technique is used to find efficient series for computation of the distributions of sums of uniform random variables and sums of Rayleigh random variables. A useful closed-form expression for the characteristic function of a Rayleigh random variable is presented, and an efficient method for computing a confluent hypergeometric function is given. An infinite series for the probability density function of a sum of independent random variables is also derived. The inversion of characteristic functions, a trapezoidal rule for numerical integration, and the sampling theorem in the frequency domain are related to, and interpreted in terms of, the results     

AN IMPROVEMENT OF THE RECURSIVE CLOSED-FORM ALGORITHM FOR NON-MINIMUM PHASE FIR SYSTEM IDENTIFICATION

An improved algorithm which is based on the recursive closed-form algorithm fornon-minimum phase FIR system identification via higher order statistics is presented.In order toincrease the parametric estimation accuracy,the improved algorithm uses the optimal iterativemethod to seek the nonlinear least-square solution.Finally,the simulation examples are alsogiven.     

Directivity optimization of a reflector antenna with cluster feeds: A closed-form solution

The directivity of a reflector antenna deteriorates as the feed moves away from the focal point for beam scanning. This deterioration can be substantially reduced if a cluster feed instead of a single feed is used to control a beam. A closed-form solution is presented for the cluster excitation to achieve the optimum directivity. For an offset108 lambdaparabolic reflector scanning 10 beamwidths, the optimum-directivity achieved by a 19-element (seven-element) cluster is 12 dB (8 dB) higher than that of a single element. Comparison of the optimum-directivity design and the popular conjugate field matching design is made. When the cluster spacingdis greater than1 lambda, it is found that the optimum directivity is higher than that of conjugate field matching (CFM) scheme by an insignificant amount, although the excitations of two designs can be quite different. Ford < 0.5 lambda, the optimum design may exhibit the supergain phenomenon, namely, extremely high directivities achieved by an oscillatory cluster excitation.     

A robust approach for the derivation of closed-form Green'sfunctions

Spatial-domain Green's functions for multilayer, planar geometries are cast into closed forms with two-level approximation of the spectral-domain representation of the Green's functions. This approach is very robust and much faster compared to the original one-level approximation. Moreover, it does not require the investigation of the spectral-domain behavior of the Green's functions in advance to decide on the parameters of the approximation technique, and it can be applied to any component of the dyadic Green's function with the same ease     

Error probability of binary phase shift keying in Nakagami-m fadingchannel with phase noise

The error performance of coherent detection of binary phase shift keying (BPSK) signals with noisy phase reference is analysed for a flat Nakagami-m fading channel and in the presence of additive white Gaussian noise (AWGN). By assuming Gaussian and Tikhonov probability density functions (PDFs) for the phase error, closed-form expressions for the average bit error rate (BER) are derived     

Closed-form expressions for distribution of sum of exponentialrandom variables

In many systems which are composed of components with exponentially distributed lifetimes, the system failure time can be expressed as a sum of exponentially distributed random variables. A previous paper mentions that there seems to be no convenient closed-form expression for all cases of this problem. This is because in one case the expression involves high-order derivatives of products of multiple functions. The authors prove a simple intuitive multi-function generalization of the Leibnitz rule for high-order derivatives of products of two functions and use this to simplify this expression, thus giving a closed form solution for this problem. They similarly simplify the state-occupancy probabilities in general Markov models     

Efficient closed-form solution for target localization using TDOA measurements in the presence of sensor position errors

An efficient solution for locating a target was proposed, which by using time difference of arrival (TDOA) measurements in the presence of random sensor position errors to increase the accuracy of estimation. The cause of position estimation errors in two-stage weighted least squares (TSWLS) method is analyzed to develop a simple and effective method for improving the localization performance. Specifically, the reference sensor is selected again and the coordinate system is rotated according to preliminary estimated target position by using TSWLS method, and the final position estimation of the target is obtained by using weighted least squares (WLS). The proposed approach exhibits a closed-form and is as efficient as TSWLS method. Simulation results show that the proposed approach yields low estimation bias and improved robustness with increasing sensor position errors and thus can easily achieve the Cramer-Rao lower bound (CRLB) easily and effectively improve the localization accuracy.     

General expression for pdf of a sum of independent exponential random variables

In order to evaluate exactly the performance of some diversity schemes, the probability density function (pdf) of a sum of independent exponential random variables (r.v.'s) must be known. This paper proposes a simple method to find it by using characteristic function., The resultant pdf is successfully applied to formulate the closed-form BER expression of 2 Tx-J Rx transmit diversity as well as the outage probability of repetition coding.     

On the maximum entropy of the sum of two dependent random variables

Investigates the maximization of the differential entropy h(X+Y) of arbitrary dependent random variables X and Y under the constraints of fixed equal marginal densities for X and Y. We show that max[h(X+Y)]=h(2X), under the constraints that X and Y have the same fixed marginal density f, if and only if f is log-concave. The maximum is achieved when X=Y. If f is not log-concave, the maximum is strictly greater than h(2X). As an example, identically distributed Gaussian random variables have log-concave densities and satisfy max[h(X+Y)]=h(2X) with X=Y. More general inequalities in this direction should lead to capacity bounds for additive noise channels with feedback     

Importance sampling for the random phase Gaussian channel

Importance sampling (IS) is developed as a variance reduction technique for Monte Carlo simulation of data communications over random phase additive white Gaussian noise channels. The binary problem (with known performance) is examined initially to determine parameter values and estimate the performance gain of IS. These results can then be applied to intractable m-ary signaling problems through composite IS. An example compares the performance of linear, square-law, and optimum receivers for binary block coded data