Optimization of Signal Sets for Partial-Response Channels—Part I: Numerical Techniques

Given a linear, time-invariant, discrete-time channel, the problem of constructing N input signals of finite length K that maximize minimum l₂ distance between pairs of outputs is considered. Two constraints on the input signals are considered: A power constraint on each of the N inputs (hard constraint) and an average power constraint over the entire set of inputs (soft constraint). The hard constraint problem is equivalent to packing N points in an ellipsoid in min(C, N-1) dimensions to maximize the minimum Euclidean distance between pairs of points. Gradient-based numerical algorithms and a constructive technique based on dense lattices are used to find locally optimal solutions to the preceding signal design problems. Numerical results, consisting of minimum distance vs. input length for different information rates, are given for the soft constraint problem. The channels considered are the identity channel, the 1-D channel, and the 1 - D2channel. Signal constellations found via gradient search are superior to the multidimensional lattice constructions when the number of points per dimension is small (i.e., when the information rate is 1 bit/T or less, 1/T being the symbol rate). The average spectra of optimized signal sets is examined. It is shown that transmitted energy is concentrated into frequency bands where the channel attenuation is relatively small. The measure of this frequency band increases with information rate. It is observed that the average spectrum of a signal set is primarily determined by the shape, or boundary of the signal constellation, assuming the points are uniformly distributed throughout this region. Two numerical examples are shown for which the average spectrum of an optimized signal set resembles the water pouring spectrum that achieves Shannon capacity, assuming additive white Gaussian noise.