Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation

G. Fichera [1] and other authors have investigated partial differential equations of the form [Eq. 1.1] in which the matrix (aij(x)) is required to be semidefinite. Equations of this type occur in the theory of random processes. A numerical analysis of some equations of this type has been by Cannon and Hill [9]. In this paper we consider a particular boundary value problem [Eq. 1.2] where we require [Eq. 1.3] and [Eq. 1.4]. A problem of this sort was discussed analytically by W. Fleming [2], but he did not obtain an explicit solution for T(x,0). The solution T(x,y) is related to a randomly-accelerated particle whose position ξ(t) satisfies the stochastic differential equation [Eq. 1.5] where w(t) is white Gaussian noise. If the initial position and velocity are ξ(0) = x and ξ'(0) = y, where |x| < 1, then T(x,y) is the expected value of the first time at which the position ξ(t) equals ±1. We obtain an analytic solution for T(x,y) in terms of hypergeometric functions and confluent hypergeometric functions. We use this analytic solution to test the validity of numerical methods which are applicable to general elliptic-parabolic equations (1.1). We show that, even though the truncation error for the difference equations does not tend to zero, nevertheless the difference methods give convergence of the difference methods. Each difference method requires the solution of a large number of simultaneous linear difference equations. We give iterative methods for solving these equations, and we prove that the iterations converge

An entropy maximization problem related to optical communication

Motivated by a problem in optical communication, we consider the general problem of maximizing the entropy of a stationary random process that is subject to an average transition cost constraint. Using a recent result of Justenson and Hoholdt, we present an exact solution to the problem and suggest a class of finite state encoders that give a good approximation to the exact solution

A Study of Optimal Abstract Jamming Strategies vs. Noncoherent MFSK

We introduce an abstract model for studying MFSK jammers. We conclude that Houston's partial-band tone jammers are optimal among all energy-restricted jamming threats vs. orthodox MFSK, but that if the communicator uses random amplitude modulation as a countermeasure, a gain of 3dB vs. optimal jamming (which is no longer tone jamming) is achievable

New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities

With the Delsarte-MacWilliams inequalities as a starting point, an upper bound is obtained on the rate of a binary code as a function of its minimum distance. This upper bound is asymptotically less than Levenshtein's bound, and so also Elias's

Performance of binary block codes at low signal-to-noise ratios

The performance of general binary block codes on an unquantized additive white Gaussian noise (AWGN) channel at low signal-to-noise ratios is considered. Expressions are derived for both the block error and the bit error probabilities near the point where the bit signal-to-noise ratio is zero. These expressions depend on the global geometric structure of the code, although the minimum distance still seems to play a crucial role. Examples of codes such as orthogonal codes, biorthogonal codes, the (24,12) extended Golay code, and the (15,6) expurgated BCH code are discussed. The asymptotic coding gain at low signal-to-noise ratios is also studied

A Study of Optimal Abstract Jamming Strategies vs. Noncoherent MFSK

We introduce an abstract model for studying MFSK jammers. We conclude that Houston's partial-band tone jammers are optimal among all energy-restricted jamming threats vs. orthodox MFSK, but that if the communicator uses random amplitude modulation as a countermeasure, a gain of 3dB vs. optimal jamming (which is no longer tone jamming) is achievable