Cubic constraint approximation based adaptive algorithms for envelope-constrained filtering

The design of envelope-constrained filters is formulated as a constrained optimization problem. In this paper the constraint approximation is realized through a cubic natural spline, which results in an unconstrained optimization problem for envelope-constrained filter design. The solution of this unconstrained problem is suitable for real-time update. It is shown that, compared with the previously used quadratic natural spline, the cubic natural spline leads to the establishment of the adaptive algorithms with much more desirable performance. In particular, the step size of the cubic constraint approximation based adaptive algorithms can be chosen in a more flexible manner so as to achieve faster convergence. Numerical examples illustrate the main results.