Optimization of signal sets for partial-response channels

Summary form only given, as follows. Given a linear, time-invariant, discrete-time channel with impulse response h[k], the problem of constructing N input signals of finite length T 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(T,N - 1) dimensions to maximize the minimum Euclidean distance between pairs of points. Gradient-based numerical techniques are used to find locally optimal solutions to the preceding signal design problems with both hard and soft constraints. In the case of hard constraints, feasible descent directions are found by solving linear programs. For the soft constraint problem, a penalty function that approximates the minimium-distance cost function is maximized. The penalty function is similar in form to the error criterion used by Foschini, Gitlin, and Weinstein, where two-dimensional signal sets are optimized for a nondispersive channel with additive white Gaussian noise. Numerical results, consisting of minimum distance versus 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 - D² channel. The computer-generated signal constellations are superior to multidimensional constructions based on dense lattices when the number of points per dimension is small. The numerical results are compared with upper and lower bounds on ε-capacity, which for a given linear time-invariant channel and information rate gives the maximum achievable minimum distance, or coding gain, as the input length tends to infinity.