CHANNEL SHAPING TO MAXIMIZE MINIMUM DISTANCE

Suppose that N inputs to a linear, time-invariant channel are designed to maximize the minimum L2 distance between channel outputs. It is as sumed that all inputs are zero outside the finite time window [-T, T] 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 L2 norm of H(f) is bounde d. This leads to an ellipse packing problem in which N - 1 axis length s, which define an ellipse in R(N - 1), and N points inside the ellips e are to be chosen to maximize the minimum Euclidean distance between points, subject to the constraint that the sum of the squared axis len gths is constant. An optimality condition is derived, and it is conjec tured that the optimal ellipse in which the N points must lie is an n- dimensional sphere, where n less-than-or-equal-to N. An approximate vo lume calculation suggests that n increases as O(log N). As T --> infin ity, this implies that an optimal channel response is ideal bandlimite d with bandwidth 2R' Hz, where R' = (log(e) N)/(2T) is the information rate.