Continuous-time envelope constrained filter design via orthonormal filters

The envelope constrained (EC) filtering problem is concerned with designing a filter which minimises the gain to input noise while its response to a given signal fits into a prescribed envelope. This problem had been formulated as a constrained optimisation problem in Hilbert space. By restricting these filters to the span of a finite orthonormal set, the EC filtering problem can be posed as a finite-dimensional optimisation problem with a continuum of constraints. The constrained problem is approximated by an unconstrained problem which is then solved by descent direction based algorithms. It is shown that these algorithms converge globally, and one in particular has a quadratic rate of convergence. Numerical examples using the orthonormal Laguerre series approximation are studied.