probability of error for equal gain combiners over rayleigh channels some closed form solutions the probability of error of a general equal gain combiner eg in closed form is not available in the literature by formulating the problem in the framework of signal detection and evaluating the probability of error directly from the characteristic function cf of the signal at the combiner output we obtain closed form solutions for the probability of error of the eg combiner with two or three branches operating over a rayleigh fading channel

decomposition formulas of exponential operators and lie exponentials with some applications to quantum mechanics and statistical physics decomposition formulas of general exponential operators in a banach algebra and in a lie algebra are presented that yield a basis of monte carlo simulation of quantum systems they are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena a global approximation method of transient phenomena is also formulated on the basis of decomposition formulas it is applied to the laser model as a simple example

new array codes for multiple phased burst correction an optimal family of array codes over gf q for correcting multiple phased burst errors and erasures where each phased burst corresponds to an erroneous or erased column in a code array is introduced as for erasures these array codes have an efficient decoding algorithm which avoids multiplications or divisions over extension fields replacing these operations with cyclic shifts of vectors over gf q the erasure decoding algorithm can be adapted easily to handle single column errors as well the codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array thus generalizing a previously known construction for the special case of two erasures algebraically they can be interpreted as reed solomon codes when q is primitive in gf q the resulting codes become conventional reed solomon codes of length p over gf q sup p 1 in which case the new erasure decoding technique can be incorporated into the berlekamp massey algorithm yielding a faster way to compute the values of any prescribed number of errors

x code mds array codes with optimal encoding we present a new class of mds maximum distance separable array codes of size n spl times n n a prime number called x code the x codes are of minimum column distance 3 namely they can correct either one column error or two column erasures the key novelty in x code is that it has a simple geometrical construction which achieves encoding update optimal complexity i e a change of any single information bit affects exactly two parity bits the key idea in our constructions is that all parity symbols are placed in rows rather than columns

analytical level crossing rates and average fade durations for diversity techniques in nakagami fading channels the level crossing rates lcrs and average fade durations afds of a fading channel find diverse applications in the evaluation and design of wireless communication systems analytical expressions for these quantities are available in the literature for certain diversity reception techniques but are generally limited to the rayleigh fading channel with few exceptions moreover the methods employed are usually specific to a certain channel diversity pair and thus cannot be applied to all cases of interest using a unified methodology we derive analytical expressions for the lcrs and afds for three diversity reception techniques and a general nakagami 1960 fading channel we provide novel analytical expressions for selection combining sc and equal gain combining egc and rederive in a more general manner the case of maximal ratio combining mrc it is shown that our general results reduce to some specific cases previously published these results are used to examine the effects of the diversity technique the number of receiving branches and severity of the fading on the concerned quantities it is observed that as the nakagami m parameter and the diversity order increase the behavior of the combined received envelope for egc follows closely the one for mrc and distances itself from sc

symmetry and classes of transport equations abstract a method for classification and determination of local transport equations is presented based on the use of lie symmetries the starting point is the choice of a specific symmetry group and the definition of a class of partial differential equations results associated with the fokker planck equation in 1 1 and 2 1 dimensions are given

microdiversity on rician fading channels formats coherent phase shift keying cpsk and noncoherent frequency shift keying ncfsk results using both maximal ratio combining mrc and selection diversity combining sc are presented for comparison our results show that from a feasibility and practical tradeoffs point of view the performance of an eg combiner may be as good as that of a mr combiner the effects of gain unbalance between branches of the eg combiner on the probability distribution of snr and on the bit error rates are also investigated the rician fading model may be used to model both the microcellular environment and the mobile satellite fading channel hence the results of this paper may be useful in both of these areas furthermore in the development of the analysis we present an efficient method for computing the distribution of sums of rich random variables this may be useful for other problems involving rician fading the suitability of modeling a rician fading environment by a properly chosen nakagami model is examined a formula for determining the corresponding values of rician parameter k and nakagami parameter m is also assessed

new exact closed form pdf of the sum of nakagami m random variables with applications an exact closed form expression is developed for the probability density function pdf of the sum of independent but not necessarily identically distributed nakagami m random variables by utilizing the newly developed pdf and the associated developments new exact expressions of the bit error probability are presented for predetection equal gain combining diversity

linear diversity combining on nakagami 0 5 fading channels the nakagami 0 5 distribution is an important special case of the nakagami distribution as it represents a worst case fading scenario this scenario is of particular significance for wireless applications with high quality of service requirements the bit error probability bep performances of three basic diversity combining schemes in nakagami 0 5 fading are considered it is shown that the error rate performances of dual branch equal gain and selection combining sc diversity are identical regardless of modulation format it is also shown that l branch maximal ratio combining diversity in nakagami 0 5 fading has the same bep performance as single branch reception in nakagami l 2 fading with l times the transmission power of each branch a useful upper bound for the bep of l branch sc diversity in nakagami 0 5 fading is also derived the performance degradation for the worst case nakagami 0 5 fading relative to rayleigh fading can be as much as 24 6 db for a dual diversity receiver at target bep of 10 6 the additional resources required to restore a specified performance level as increased channel signal to noise ratio and as number of additional diversity branches are quantified it is concluded that worst case nakagami channels can result in dramatically poorer performance than rayleigh channels making proper system design and evaluation essential to achieve a required transmission quality

a novel extension to craig s q function formula and its application in dual branch egc performance analysis a novel identity for the gaussian q function of the sum of two non negative variables is introduced as an extension of craig s q function formula it is shown that this new identity provides a unified framework in the performance analysis of dual branch equal gain combining egc diversity similar to that of craig s q function formula for maximum ratio combining diversity then the performance of dual correlated branch egc in nakagami m fading is studied using this novel approach for coherent modulations with error probabilities in terms of the gaussian q function it is shown that the average error probability of the dual correlated branch egc in nakagami m fading is the same as the average error probability of a modeled single channel receiver in the same fading environment using this fact simple exact and approximate closed form expressions for this average error probability are derived the analytical results are verified and illustrated by numerical results and computer simulations

explicit constructions of optimal access mds codes with nearly optimal sub packetization an n k l maximum distance separable mds array code of length n dimension k n r and sub packetization l is formed of l times n matrices over a finite field f with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword repair of a failed node recovery of one erased column can be performed by accessing a set of d le n 1 surviving helper nodes the code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity for optimal access mds codes with d n 1 the sub packetization l satisfies the bound l ge r k 1 r in our previous work ieee trans inf theory vol 63 no 4 2017 for any n and r we presented an explicit construction of optimal access mds codes with sub packetization l r n 1 in this paper we take up the question of reducing the sub packetization value l to make it to approach the lower bound we construct an explicit family of optimal access codes with l r lceil n r rceil which differs from the optimal value by at most a factor of r 2 these codes can be constructed over any finite field f as long as f ge r lceil n r rceil and afford low complexity encoding and decoding procedures we also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints lrc codes which we call group repair with optimal access in this variation we assume that the set of n sm nodes is partitioned into m repair groups of size s and require that the amount of accessed data for repair is the smallest possible whenever the d s k 1 helper nodes include all the other s 1 nodes from the same group as the failed node for this problem we construct a family of codes with the group optimal access property these codes can be constructed over any field f of size f ge n and also afford low complexity encoding and decoding procedures

revisit to the scaling theory of transient phenomena generalization to correlated noise and singular perturbation expansion up to infinite order abstract a scaling theory is formulated in nonlinear langevin equations with colored noise near the instability point from which the center of the initial distribution is deviated slightly an intuitive derivation is given as well as perturbational calculations up to the infinite order of the noise moments of all orders are calculated in the approximation in which only the most dominant terms are retained in the scaling limit hence the scaling distribution function is also obtained the onset time t 0 is derived as a functional of the correlation function of the noise and it is shown to increase when the noise correlation length increases

search for the symmetry of the fokker planck equation abstract it is shown that the fokker planck equations with linear drift x and nonlinear drift x x i e the rayleigh process have exact symmetries on their own solution manifolds the sl 2 r weyl algebra and the sl 2 r i respectively the symmetries of these two types of processes for 1 2 dimensional cases are explicitly found out too

on the distribution of the sum of nakagami m random variables in this paper we investigate the classical problem of finding the probability density function pdf of the sum of nakagami m random variables exact infinite series representations are derived for the sum of three and four identically and independently distributed i i d nakagami m random variables and subsequently it is extended to the sum of general number of random variables a useful pattern emerged as a result of these extensions and a new fourier transform pair that involves parabolic cylinder function and gauss hypergeometric function is obtained bounds on the error resulting from truncation of the infinite series of pdfs are also presented these pdfs are used to analyze the performance of dual triple and quadruple branch predetection equal gain combining egc receivers over nakagami m fading environment furthermore a hypergeometric relation is derived as a result of those derivations subsequently the analysis is extended to the case of l branch egc receiver performance as well selected simulation plots are provided to compare the results with the approximate results available in the literature and to illustrate the validity of the formulation

on the sum of kappa stochastic variates and applications to equal gain combining in this paper we study the statistics of the sum of not necessarily identically distributed kappa that is k random variables rv s assuming half integer values for the shaping parameters novel closed form expressions for the probability density function pdf of the sum of independent k rvs are obtained while for arbitrary values of the shaping parameters a corresponding pdf expression is derived in terms of fast converging infinite series furthermore an infinite series representation for the pdf of the sum of two arbitrarily correlated k rvs is derived the proposed analysis is employed to the performance analysis of equal gain combining egc receivers operating over composite fading shadowing channels modeled by the k distribution more specifically the outage and the average bit error probabilities as well as the average channel capacity of egc receivers operating over such composite environment are studied considering different channel fading shadowing conditions and correlation effects various numerical performance evaluation results are presented these results are complemented by equivalent computer simulated ones that validate the accuracy of the proposed analysis

energy detection of unknown signals in fading and diversity reception a comprehensive performance analysis of the energy detector over fading channels with single antenna reception or with antenna diversity reception is developed for the no diversity case and for the maximal ratio combining mrc diversity case with either nakagami m or rician fading expressions for the probability of detection are derived by using the moment generating function mgf method and probability density function pdf method the former which avoids some difficulties of the latter uses a contour integral representation of the marcum q function for the equal gain combining egc diversity case with nakagami m fading expressions for the probability of detection are derived for the cases l 2 3 4 and l 4 where l is the number of diversity branches for the selection combining sc diversity with nakagami m fading expressions for the probability of detection are derived for the cases l 2 and l 2 a discussion on the comparison between mgf and pdf methods is presented we also derive several series truncation error bounds that allow series termination with a finite number of terms for a given figure of accuracy these results help quantify and understand the achievable improvement in the energy detector s performance with diversity reception numerical and simulation results are also provided

galilei invariant gauge symmetries in fokker planck dynamics with logarithmic diffusion and drift terms a tensorial approach to galilean invariance is utilized together with lie symmetries of differential equations in order to derive equations of fokker planck type containing a logarithmic diffusion tensor and drift term the formalism is based on the projection from an extended by one space like dimension minkowski manifold to the usual newtonian spacetime so that non relativistic models are described by manifestly covariant lagrangians in this paper we obtain the fokker planck equations from the euler lagrange equations with the extended manifold by using a specific choice of the gauge condition we work in 1 1 spacetime and carry out the analysis for both abelian and non abelian symmetries

infinite series representations of the bivariate rayleigh and nakagami m distributions despite the usefulness of the bivariate correlated nakagami m cumulative distribution function cdf in communications systems analysis an infinite series for the computation of this function is not known in this letter an infinite series representation of the bivariate cdf is derived bounds on the error resulting from truncation of the infinite series are also derived

on lowest density mds codes let f sub q denote the finite field gf q and let h be a positive integer mds maximum distance separable codes over the symbol alphabet f sub q sup b are considered that are linear over f sub q and have sparse low density parity check and generator matrices over f sub q that are systematic over f sub q sup b lower bounds are presented on the number of nonzero elements in any systematic parity check or generator matrix of an f sub q linear mds code over f sub q sup b along with upper bounds on the length of any mds code that attains those lower bounds a construction is presented that achieves those bounds for certain redundancy values the building block of the construction is a set of sparse nonsingular matrices over f sub q whose pairwise differences are also nonsingular bounds and constructions are presented also for the case where the systematic condition on the parity check and generator matrices is relaxed to be over f sub q rather than over f sub q sup b

analysis of equal gain diversity on nakagami fading channels an infinite series for the complementary probability distribution function cdf of the signal to noise ratio snr at the output of l branch equal gain eg diversity combiners in nakagami 1960 fading channels is derived the bit error rate for a matched filter receiver is analyzed for the l branch eg combiner and different fading parameters both coherent phase shift keying cpsk and differential coherent phase shift keying dcpsk are considered the effects of gain unbalance between branches on the probability distribution of the snr and on the bit error rates are investigated bit error rate results are also obtained for coherent and noncoherent reception of frequency shift keying fsk the effects of gain unbalances on fsk modulations are also investigated bit error rates for eg combining on rayleigh fading channels are obtained for l 2 these results are presented as a special case of the more generalized nakagami fading model

two classes of binary mds array codes with asymptotically optimal repair for any single column an m times k r binary maximum distance separable mds array code contains k information columns and r parity columns with each entry being a bit where any k out of k r columns can recover the k information columns when there is a failed column it is critical to minimize the repair bandwidth that is the total number of bits downloaded from d out of k r 1 surviving columns in repairing the failed column in this article we first propose two explicit constructions of binary mds array codes that have asymptotically optimal repair bandwidth for any information column where r geq 2 and d k r 1 for the first construction and r geq 4 is an even number and d k frac r 2 1 for the second construction by applying a generic transformation for the proposed two classes of binary mds array codes we then obtain two classes of new binary mds array codes that also have optimal repair bandwidth for any parity column and asymptotically optimal repair bandwidth for any information column

equal gain diversity receiver performance in wireless channels performance analysis of equal gain combining egc diversity systems is notoriously difficult only more so given that the closed form probability density function pdf of the egc output is only available for dual diversity combining in rayleigh fading a powerful frequency domain approach is therefore developed in which the average error rate integral is transformed into the frequency domain using parseval s theorem such a transformation eliminates the need for computing or approximating the egc output pdf which is unknown but instead requires the knowledge of the corresponding characteristic function which is readily available the frequency domain method also circumvents the need to perform multiple fold convolution integral operations usually encountered in the calculation of the pdf of the sum of the received signal amplitudes we then derive integral expressions for the average symbol error rate of an arbitrary two dimensional signaling scheme with egc reception in rayleigh rician nakagami m 1960 and nakagami q fading channels for practically important cases of second and third order diversity systems in nakagami fading both coherent and noncoherent detection methods for binary signaling are analyzed using the appell hypergeometric function a number of closed form solutions are derived in which the results put forward by zhang see ibid vol 45 p 270 73 1997 are shown to be special cases

further results on the beaulieu series a frequent problem in digital communications is the computation of the probability density function pdf and cumulative distribution function cdf given the characteristic function chf of a random variable rv this problem arises in signal detection equalizer performance equal gain diversity combining intersymbol interference and elsewhere often it is impossible to analytically invert the chf to get the pdf and cdf in closed form beaulieu 1990 1991 has derived an infinite series for the cdf of a sum of rvs that has been widely used we rederive his series using the gil pelaez 1951 inversion formula and the poisson sum formula this derivation has several advantages including both the bridging of the well known sampling theorem with beaulieu s series and yielding a simple expression for calculating the truncation error term it is also shown that the pdf and cdf can be computed directly using a discrete fourier transform

an infinite series for the computation of the complementary probability distribution function of a sum of independent random variables 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

phased burst error correcting array codes various aspects of single phased burst error correcting array codes are explored these codes are composed of two dimensional arrays with row and column parities with a diagonally cyclic readout order they are capable of correcting a single burst error along one diagonal optimal codeword sizes are found to have dimensions n sub 1 n sub 2 such that n sub 2 is the smallest prime number larger than n sub 1 these codes are capable of reaching the singleton bound a new type of error approximate errors is defined in q ary applications these errors cause data to be slightly corrupted and therefore still close to the true data level phased burst array codes can be tailored to correct these codes with even higher rates than before