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The nondispersive per-sample channel model for the optical fiber channel isconsidered. Under certain smoothness assumptions, the problem of finding theminimum amount of noise energy that can render two different input pointsindistinguishable is formulated. This minimum noise energy is then taken as ameasure of distance between the points in the input alphabet. Using themachinery of optimal control theory, necessary conditions that describe theminimum-energy noise trajectories are stated as a system of nonlineardifferential equations. It is shown how to find the distance between two inputpoints by solving this system of differential equations. The problem ofdesigning signal constellations with the largest minimum distance subject to apeak power constraint is formulated as a clique-finding problem. As an example,a 16-point constellation is designed and compared with conventional quadratureamplitude modulation. A computationally efficient approximation for theproposed distance measure is provided. It is shown how to use thisapproximation to design large constellations with large minimum distances.Based on the control-theoretic viewpoint of this paper, a new decoding schemefor such nonlinear channels is proposed.

A Signal-Space Distance Measure for Nondispersive Optical Fiber