Envelope Estimation by Tangentially Constrained Spline

Estimating envelope of a signal has various applications including empirical mode decomposition (EMD) in which the cubic $C^{2}$ -spline based envelope estimation is generally used. While such functional approach can easily control smoothness of an estimated envelope, the so-called undershoot problem often occurs that violates the basic requirement of envelope. In this paper, a tangentially constrained spline with tangential points optimization is proposed for avoiding the undershoot problem while maintaining smoothness. It is defined as a quartic $C^{2}$ -spline function constrained with first derivatives at tangential points that effectively avoids undershoot. The tangential points optimization method is proposed in combination with this spline to attain optimal smoothness of the estimated envelope.