Channel shaping to maximize minimum distance

Abstract

Suppose that N inputs to a linear, time-invariant channel are designed to maximize the minimum L , distance between channel outputs. It is assumed that all inputs are zero outside the finite time window [-T , TI and are constrained in energy. The jointly optimal inputs and channel frequency response H(f) for which the minimum distance is maximized is studied, subject to the constraint that the L , norm of H(f) is bounded. This leads to an ellipse packing problem in which N-1 axis lengths, which define an ellipse in RN-', and N points inside the ellipse are to be chosen to maximize the minimum Euclidean distance between points, subject to the constraint that the sum of the squared axis lengths is constant. An optimality condition is derived, and it is conjectured that the optimal ellipse in which the N points must lie is an n-dimensional sphere, where n I N. An approximate volume calculation suggests that n increases as O(log N). As T + x , this implies that an optimal channel response is ideal bandlimited with bandwidth 2R' Hz, where R' = (log, N) / (Z T) is the information rate.

An algorithm for maximising covered area

“…A large body of literature on this topic has concentrated on packing spheres of identical size in Euclidean space (Kershner 1939;Krieger and Schaffner 1971;Honig 1993;Convay and Sloane 1998;Graham and Lubachevsky 1998;Zeger 1997a,b, 2002;Aoki and Gaborit 1999;Zong 1999;Hsiang 2001;Bajic and Woods 2003;Szpiro 2003). The packing methods involve constructing high dimensional lattices, which serve as centres for spheres.…”