Lawrence A. Shepp

A Mathematical Theory of Network Interference and Its Applications

In this paper, we introduce a mathematical framework for the characterization of network interference in wireless systems. We consider a network in which the interferers are scattered according to a spatial Poisson process and are operating asynchronously in a wireless environment subject to path loss, shadowing, and multipath fading. We start by determining the statistical distribution of the aggregate network interference. We then investigate four applications of the proposed model: 1) interference in cognitive radio networks; 2) interference in wireless packet networks; 3) spectrum of the aggregate radio-frequency emission of wireless networks; and 4) coexistence between ultrawideband and narrowband systems. Our framework accounts for all the essential physical parameters that affect network interference, such as the wireless propagation effects, the transmission technology, the spatial density of interferers, and the transmitted power of the interferers.

On the performance of wide-bandwidth signal acquisition in dense multipath channels

This paper investigates important properties of acquisition receivers that employ commonly used serial-search strategies. In particular, we focus on the properties of the mean acquisition time (MAT) for wide bandwidth signals in dense multipath channels. We show that a lower bound of the MAT over all possible search strategies is the solution to an integer programming problem with a convex objective function. We also give an upper bound expression for the MAT over all possible search strategies. We demonstrate that the MAT of the fixed-step serial search (FSSS) does not depend on the timing delay of the first resolvable path, thereby simplifying the evaluation of the MAT of the FSSS. The results in this paper can be applied to design and analysis of fast acquisition systems in various wideband scenarios.

Neural substrates, experimental evidences and functional hypothesis of acupuncture mechanisms.

Objectives –  Athough acupuncture therapy has demonstrated itself to be effective in several clinical areas, the underlying mechanisms of acupuncture in general and the analgesic effect in particular are, however, still not clearly delineated. We, therefore, have studied acupuncture analgesic effect through fMRI and proposed a hypothesis, based on the obtained result, which will enlighten the central role of the brain in acupuncture therapy.

On the best finite set of linear observables for discriminating two Gaussian signals

Consider the problem of discriminating two Gaussian signals by using only a finite number of linear observables. How to choose the set of n observables to minimize the error probability P_{e} , is a difficult problem. Because H , the Hellinger integral, and H^{2} form an upper and a lower bound for P_{e} , we minimize H instead. We find that the set of observables that minimizes H is a set of coefficients of the simultaneously orthogonal expansions of the two signals. The same set of observables maximizes the Hajek J -divergence as well.

The complex zeros of random polynomials

Mark Kac gave an explicit formula for the expectation of the number, vn (a), of zeros of a random polynomial, n-I Pn(z) = E ?tj, j=O in any measurablc subset Q of the reals. Here, ... ?In-I are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function gn for which E vn(L) = Jgn(x) dx. Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset Q of the complex plane C. Namely, we show that E vn (Ki) = J hn(x, y) dxdy + J gn(x) dx, where hn is an explicit intensity function. We also study the asymptotics of hn showing that for large n its mass lies close to, and is uniformly distributed around, the unit circle.

Optimal Stopping Rules and Maximal Inequalities for Bessel Processes

We consider, for Bessel processes

On the SNR penalty of MPSK with hybrid selection/maximal ratio combining over i.i.d. Rayleigh fading channels

X \in {\operatorname{Bes}}^\alpha (x)

A New Look at Pricing of the "Russian Option"

with arbitrary order (dimension)

ON THE NUMBER OF LEAVES OF A EUCLIDEAN MINIMAL SPANNING TREE

\alpha \in {\bf R}

Sets of Uniqueness and Additivity in Integer Lattices

, the problem of the optimal stopping (1.4) for which the gain is determined by the value of the maximum of the process X and the cost which is proportional to the duration of the observation time. We give a description of the optimal stopping rule structure (Theorem 1) and the price (Theorem 2). These results are used for the proof of maximal inequalities of the type \[ {\bf E}\mathop {\max }\limits_{r \leq \tau} X_r \leq \gamma (\alpha )\sqrt {{\bf E}\tau }, \] where