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

Khuong, Ho Van; Kong, Hyung Yun

doi:10.1109/lcomm.2006.1603370

Cited by 71 publications

(9 citation statements)

References 3 publications

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“…If all {k n } are equal, d 2 X;X À Á is a central Chi-square distributed random variable and the PDF has been derived in [10,; if all {k n } are distinct, the PDF has been derived in [10, Eq. 14-5-26]; if some of {k n } are identical and the rest are all distinct, then the PDF is derived in [11]. However, to our knowledge, the general case of {k n } or the general PDF of d 2 ðX;XÞ remains open, which is considered in the next section 3.2 PDF of the Modified Euclidean Distance d 2 X;X À Á Suppose among the N eigenvalues {k n } in (3), there are K non-zero eigenvalues, i.e., {k 1 …k K }.…”

Section: Conditional Pepmentioning

confidence: 99%

“…However, as we stated Table 1 Numbers of (R-1)-factor products required to determine A 1;1 for M = R=3-7 and s = 1 using (8) It should also be noted that the PDF of d 2 X;X À Á , or the sum of independent exponential distributed random variables, is a useful tool to drive different types of probabilities for communications systems. For example, it can be used to derive the error probability for diversity systems such as the PEP for S-T codes as shown in this paper, the outage probability for repetition codes as shown in [11], and the BER of RAKE receiver as shown in [10]. Since the close-form PDF of d 2 X;X À Á in the general case remains an open problem, the authors attempt to apply the CF method to solve the problem.…”

Section: Exact Close-form Pepmentioning

confidence: 99%

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An Exact Close-form PEP and a new PEP for Space-Time Codes in Rayleigh Fading Channels

Zhi

Cheung

Yuk

2009

Int J Wireless Inf Networks

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A close-form expression for the exact Pair-wise Error Probability (PEP) of Space-Time (S-T) codes in Rayleigh fading channel is derived using the general and close-form solution for the probability-density function (PDF) of a sum of independent exponential distributed random variables. The expression requires evaluating the coefficients for partial fraction expansion, so an easy analytical way is proposed for doing this. The exact PEP is subsequently used to develop a simple PEP using the upper bound. Both PEPs are used in the Union bound for error rate evaluation. Numerical calculations and Monte Carlo computer simulation are used to study the accuracies of these Union bounds for error rate evaluation of a rotationbased diagonal S-T code (D code) in Rayleigh fading channels. Four other PEPs based on different bounds, i.e., the Chernoff bound, the asymptotic bound, the tight asymptotic bound, and the Eigen-Geometric-Mean (EGM) bound, are also studied for comparison. Results show that our derived close-form PEP is an exact PEP and our proposed PEP is a very tight bound to the exact PEP.

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Section: Conditional Pepmentioning

confidence: 99%

Section: Exact Close-form Pepmentioning

confidence: 99%

An Exact Close-form PEP and a new PEP for Space-Time Codes in Rayleigh Fading Channels

Zhi

Cheung

Yuk

2009

Int J Wireless Inf Networks

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“…Amari and Misra [2] propose a simplification of Scheuer [1]'s formula using Laplace transforms and multi-function generalization of the Lebnitz rule for higher order derivatives of products of two functions. For a particular case with constraints on the values of the k i s, Van Khuong and Kong [3] provide the probability distribution function by inverting its Fourier transform. Using the Wilk's integral representation of the distribution of the product of independent beta random variables, Favaro and Walker [4] provide an alternative formula for FðÁÞ.…”

Section: Introductionmentioning

confidence: 99%

A linear algebraic approach for the computation of sums of Erlang random variables

Legros

Jouini

2015

Applied Mathematical Modelling

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a b s t r a c tWe propose a matrix analysis approach to analytically provide the cumulative distribution function of the sum of independent Erlang random variables. This reduces to the characterization of the exponential of the involved generator matrix. We propose a particular basis of vectors in which we write the generator matrix. We find, in the new basis, a Jordan-Chevalley decomposition allowing to simplify the calculation of the exponential of the generator matrix. This is a simpler alternative approach to the existing ones in the literature.

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“…Parzen in [42] formulated the estimation of PDFs using kernel approaches. Later on, new approaches emerged to overcome the specification of diverse applications such as the sum of Gamma densities in Risk Theory [43] or the sum of exponential random variables in wireless communication [44]. Other researchers have focused on the choice of the kernel and smoothing functions [45,46].…”

Section: Introductionmentioning

confidence: 99%

Blind estimation of statistical properties of non-stationary random variables

Mansour

Mesleh

Aggoune

2014

EURASIP J. Adv. Signal Process.

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To identify or equalize wireless transmission channels, or alternatively to evaluate the performance of many wireless communication algorithms, coefficients or statistical properties of the used transmission channels are often assumed to be known or can be estimated at the receiver end. For most of the proposed algorithms, the knowledge of transmission channel statistical properties is essential to detect signals and retrieve data. To the best of our knowledge, most proposed approaches assume that transmission channels are static and can be modeled by stationary random variables (uniform, Gaussian, exponential, Weilbul, Rayleigh, etc.). In the majority of sensor networks or cellular systems applications, transmitters and/or receivers are in motion. Therefore, the validity of static transmission channels and the underlying assumptions may not be valid. In this case, coefficients and statistical properties change and therefore the stationary model falls short of making an accurate representation. In order to estimate the statistical properties (represented by the high-order statistics and probability density function, PDF) of dynamic channels, we firstly assume that the dynamic channels can be modeled by short-term stationary but long-term non-stationary random variable (RV), i.e., the RVs are stationary within unknown successive periods but they may suddenly change their statistical properties between two successive periods. Therefore, this manuscript proposes an algorithm to detect the transition phases of non-stationary random variables and introduces an indicator based on high-order statistics for non-stationary transmission which can be used to alter channel properties and initiate the estimation process. Additionally, PDF estimators based on kernel functions are also developed. The first part of the manuscript provides a brief introduction for unbiased estimators of the second and fourth-order cumulants. Then, the non-stationary indicators are formulated. Finally, simulation results are presented and conclusions are derived.

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

Cited by 71 publications

References 3 publications

An Exact Close-form PEP and a new PEP for Space-Time Codes in Rayleigh Fading Channels

An Exact Close-form PEP and a new PEP for Space-Time Codes in Rayleigh Fading Channels

A linear algebraic approach for the computation of sums of Erlang random variables

Blind estimation of statistical properties of non-stationary random variables

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