We analyze a mobile wireless link comprising M transmitter and N receiver antennas operating in a Rayleigh flatfading environment. The propagation coefficients between pairs 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 1 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 2M transmitted signal matrix is equal to the product of two statistically independent matrices: a T 2 T isotropically distributed unitary matrix times a certain T 2M 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. Index Terms---Multielement antenna arrays, space--time modulation, wireless communications. I.

In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in three-dimensional sensor networks. As in other large-scale 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 3-dimensional 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 hop-diameter 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; hop-diameter; minimum/maximum degrees; transmitting/sensing ranges; analytical methods; energy consumption; topology control. I.

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 piecewise-constant 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...

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.

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.

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.

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.

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 time-consu3J9 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 lti-sensor 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.

Abstract. We prove upper and lower bounds for the complementary incomplete gamma function Γ(a, z) with complex parameters a and z. Our bounds are refined within the circular hyperboloid of one sheet {(a, z) : |z |> c|a − 1|} with a real and z complex. Our results show that within the hyperboloid, |Γ(a, z) | is of order |z | a−1 e − Re(z) , and extends an upper estimate of Natalini and Palumbo to complex values of z.

On the occasion of the conference we mention examples of Stieltjes' work on asymptotics of special functions. The remaining part of the paper gives a selection of asymptotic methods for integrals, in particular on uniform approximations. We discuss several "standard" problems and examples, in which known special functions (error functions, Airy functions, Bessel functions, etc.) are needed to construct uniform approximations. Finally, we discuss the recent interest and new insights in the Stokes phenomenon. An extensive bibliography on uniform asymptotic methods for integrals is given, together with references to recent papers on the Stokes phenomenon for integrals and related topics.