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Capacity of a Mobile MultipleAntenna Communication Link in Rayleigh Flat Fading
"... We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flatfading environment. The propagation coefficients between every pair of transmitter and receiver antennas are statistically independent and unknown; they remain constant for a coherence int ..."
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Cited by 398 (21 self)
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We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flatfading environment. The propagation coefficients between every pair of transmitter and receiver antennas are statistically independent and unknown; they remain constant for a coherence interval of T symbol periods, after which they change to new independent values which they maintain for another T symbol periods, and so on. Computing the link capacity, associated with channel coding over multiple fading intervals, requires an optimization over the joint density of T M complex transmitted signals. We prove that there is no point in making the number of transmitter antennas greater than the length of the coherence interval: the capacity for M> Tis equal to the capacity for M = T. Capacity is achieved when the T M transmitted signal matrix is equal to the product of two statistically independent matrices: a T T isotropically distributed unitary matrix times a certain T M random matrix that is diagonal, real, and nonnegative. This result enables us to determine capacity for many interesting cases. We conclude that, for a fixed number of antennas, as the length of the coherence interval increases, the capacity approaches the capacity obtained as if the receiver knew the propagation coefficients.
Extremal properties of threedimensional sensor networks with applications
 IEEE Transactions on Mobile Computing
"... In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in threedimensional sensor networks. As in other largescale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a cri ..."
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Cited by 23 (3 self)
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In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in threedimensional sensor networks. As in other largescale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a critical threshold, corresponding to the minimum amount of the communication effort or power expenditure by individual nodes, above (resp. below) which the property exists with high (resp. a low) probability. For sensor networks, properties of interest include simple and multiple degrees of connectivity/coverage. First, we investigate the network topology according to the region of deployment, the number of deployed sensors and their transmitting/sensing ranges. More specifically, we consider the following problems: Assume that n nodes, each capable of sensing events within a radius of r, are randomly and uniformly distributed in a 3dimensional region R of volume V, how large must the sensing range rSense be to ensure a given degree of coverage of the region to monitor? For a given transmission range rTrans, what is the minimum (resp. maximum) degree of the network? What is then the typical hopdiameter of the underlying network? Next, we show how these results affect algorithmic aspects of the network by designing specific distributed protocols for sensor networks. Keywords Sensor networks, ad hoc networks; coverage, connectivity; hopdiameter; minimum/maximum degrees; transmitting/sensing ranges; analytical methods; energy consumption; topology control. I.
The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asy ..."
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Cited by 18 (1 self)
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The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.
Asymptotic Estimates of Elementary Probability Distributions
 Studies in Applied Mathematics
, 1996
"... Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions. ..."
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Cited by 15 (6 self)
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Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions.
Coding Requirements for MultipleAntenna Channels with Unknown Rayleigh Fading
"... Multiple transmitter and receiver antennas can boost the reliability and capacity of wireless fading channels. When the channel is unknown to the transmitter and receiver, it is known that the fading coherence time in a piecewiseconstant fading channel ultimately limits the channel capacity. In thi ..."
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Cited by 12 (1 self)
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Multiple transmitter and receiver antennas can boost the reliability and capacity of wireless fading channels. When the channel is unknown to the transmitter and receiver, it is known that the fading coherence time in a piecewiseconstant fading channel ultimately limits the channel capacity. In this work, we examine coding requirements for the unknown fading channel by computing the random coding error exponent. We show that the fading coherence time also plays a fundamental role in the error exponent, by proving that the error exponent is not increased by having more transmitter antennas than the number of samples in the coherence time. The signal structure that maximizes the exponent is computed and is shown to be very similar to the signal structure that achieves capacity. We calculate the minimum coding block length requirements as a function of probability of error for various fading coherence times. We conclude that coding over a certain number of independent fades is always nee...
Computing Budget Allocation for Efficient Ranking and Selection of Variances with Application to Target Tracking Algorithms
"... This paper addresses the problem of ranking and selection for stochastic processes,su h as target tracking algorithms, where variance is the performance metric. Comparison of di#erent tracking algorithms or parameter sets within one algorithm relies on timeconsu3J9 andcompu9yj3J3J9y demanding simu ..."
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Cited by 8 (1 self)
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This paper addresses the problem of ranking and selection for stochastic processes,su h as target tracking algorithms, where variance is the performance metric. Comparison of di#erent tracking algorithms or parameter sets within one algorithm relies on timeconsu3J9 andcompu9yj3J3J9y demanding simu lations. We present a method to minimize simuy3F90 time, yet to achieve a desirable confidence of the obtainedresune byapplying ordinal optimization and compu07y buu0 allocation ideas and techniqu80 while taking intoaccou t statistical properties of the variance. The developed method is applied to a general tracking problem of N s sensors tracking T sing a sequ# tial mu ltisensor datafuayD tracking algorithm. The optimization consists of finding the order of processing sensor information thatresuDF in the smallest variance of the position error. Resur. that we obtained with high confidence levels and inredu99 simu lation times confirm the findings from ou previou research (where we considered onlytwo sensors) that processing the best available sensor the last performs the best, on average. The presented method can be applied to anyranking and selection problem where variance is the performance metric.
Uniform Asymptotics for the Incomplete Gamma Functions Starting From Negative Values of the Parameters
"... We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the i ..."
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Cited by 5 (1 self)
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We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for fl(a; z) and \Gamma(a; z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section.
Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula
 Advances in Applied Probability
, 2007
"... This paper presents new Gaussian approximations for the cumulative distribution function P(Aλ ≤ s) of a Poisson random variable Aλ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasiGaussian form, which permits evaluation in terms of the normal distrib ..."
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Cited by 4 (1 self)
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This paper presents new Gaussian approximations for the cumulative distribution function P(Aλ ≤ s) of a Poisson random variable Aλ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasiGaussian form, which permits evaluation in terms of the normal distribution function Φ. The quasiGaussian form contains an implicitly defined function y, which is closely related to the Lambert W function. A detailed analysis of y leads to a powerful asymptotic expansion and sharp bounds on P(Aλ ≤ s). The results for P(Aλ ≤ s) differ from most classical results related to the central limit theorem in that the leading term Φ(β), with β = (s − λ)/ λ, is replaced by Φ(α), where α is a simple function of s that converges to β as s → ∞. Changing β into α turns out to increase precision for small and moderately large values of s. The results for P(Aλ ≤ s) lead to similar results related to the Erlang B formula. The asymptotic expansion for Erlang’s B is shown to give rise to accurate approximations; the obtained bounds seem to be the sharpest in the literature thus far.
Interference in Cellular Networks: Sum of Lognormals Modeling
, 2007
"... ii We examine the existing methods for evaluating the distribution of the sum of lognormal random variables, focusing on closedform results. We find that there are no results in literature that are both simple and accurate. We then derive a new closedform expression for the lower tail of the distr ..."
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Cited by 3 (2 self)
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ii We examine the existing methods for evaluating the distribution of the sum of lognormal random variables, focusing on closedform results. We find that there are no results in literature that are both simple and accurate. We then derive a new closedform expression for the lower tail of the distribution, and use it to construct a new method using a powerlognormal distribution. We apply both basic momentmatching and our new method the problem of the total interference power in a cellular system. For both methods, we derive equations that find the interference distribution essentially in closed form, using minimal numerical integration. We apply both methods to the uplink and downlink in systems with and without power control, for various cellular layouts, channel models and user activity probability. We compare distributions obtained by MonteCarlo simulation directly with those obtained by our method, and find very good matches in many useful cases. iii Acknowledgements
Asymptotics of Zeros of Incomplete Gamma Functions
 Annals of Numerical Mathematics
, 1994
"... We consider the complex zeros with respect to z of the incomplete gamma functions fl(a; z) and \Gamma(a; z), with a real and positive. In particular we are interested in the case that a is large. The zeros are obtained from approximations that are computed by using uniform asymptotic expansions of ..."
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Cited by 3 (2 self)
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We consider the complex zeros with respect to z of the incomplete gamma functions fl(a; z) and \Gamma(a; z), with a real and positive. In particular we are interested in the case that a is large. The zeros are obtained from approximations that are computed by using uniform asymptotic expansions of the incomplete gamma functions. The complex zeros of the complementary error function are used as a first approximations. Applications are discussed for the zeros of the partial sums s n (z) = P n j=0 z j =j! of exp(z). 1991 Mathematics Subject Classification: 33B15, 33B20, 41A60, 65U05. Keywords & Phrases: incomplete gamma functions, zeros of incomplete gamma functions, error function, uniform asymptotic expansion. 1.