Optimal signal processing with constraints

The design of filters is considered for signal processing problems where the shape of the filter output is constrained to lie between upper and lower pulse shape boundaries, and the output noise power is minimized. A number of practical signal processing problems are shown to be meaningfully formulated using this pulse shaping approach. The pulse shaping problem is formulated as a constrained convex optimization problem and the Kuhn-Tucker Theorem provides necessary and sufficient conditions for an optimal solution. Using Duality Theorem the constrained primal problem is transformed to an unconstrained concave nondifferentiable dual problem and a convergent algorithm based on the subsifferentiability property of concave functions is developed for its solution. Several important properties of the optimal pulse shaping filter are derived and discussed. These include the observation that the optimal filter has the structure of a matched filter in cascade with another filter. Finally, an adaptive implementation of the dual algorithm us presented and its convergence properties are discussed in both deterministic and stochastic environments. The adaptive pulse shaping filter is shown to be analogous in a pulse shaping context to the adaptive least squares filter developed by Widrow.