Channel shaping to maximize minimum distance

The problem of jointly selecting optimal transmitted signals and the channel frequency response, assuming the channel is linear and time-invariant, is considered where the minimum L₂ distance between channel outputs is the optimization criterion. Results indicate that, for a fixed information rate, as the length of the inputs goes to infinity the optimal channel frequency response is a constant wherever it is positive. A crude but simple volume estimate suggests that the optimal channel bandwidth is equal to twice the information rate. A second problem considered is the joint optimization of channel inputs and a transmitter filter, given a fixed channel response. This problem remains unsolved in general, although the present discussion indicates that the optimal filter transfer function should simply be constant over the bandwidth where the asymptotic transmitted spectrum is nonzero