Correspondence: 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 assumed 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 bounded. This leads to an ellipse packing problem in which N - 1 axis lengths, which define an ellipse in R1-1, 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 V points must lie is an NDimensional sphere, where n N. An approximate volume calculation suggests that n increases as O(log N). As T, this implies that an optimal channel response is ideal bandlimited with bandwidth 2R' Hz, where R' = (loge N)/(2T) is the information rate.