Robust Envelope Constrained Filtering

In Chapters 4 and 5, we studied numerical methods for finding a filter whose response to a specified signal fits into a given envelope. Assuming that the set of feasible filters does not contain the origin, i.e. no trivial solution, and since we seek a feasible filter with the smallest possible norm, it follows that the optimum filter always lies on the boundary of the feasible set. This means the response of the optimum filter to the prescribed input touches the output envelope at some points. Consequently, perturbations of the prescribed input or filter implementation errors could result in a filter output that violates the output constraints. The design of filters that are robust to such perturbations is very important in practice. Of course, one expects that there is some penalty for achieving robustness, for example, the noise gain of the optimal filter would increase in the robust case.