Envelope-constrained \(H_2\) FIR filter design (Q1969535)

The paper considers a finite impulse response (FIR) filtering problem with both frequency and time domain requirements. The FIR filter is designed such that to achieve a minimal \(H_2\) norm for the filtering error transfer function, while the filter output with a given input to the signal system is enelope-constrained (EC), i.e. contained or bounded by a prescribed envelope. The resulting \(H_2\) ECFIR filtering problem is formulated as a finite-dimensional optimization problem subject to several linear matrix inequality (LMI) constraints. More precisely, the optimization problem is expressed by introducing a penalty factor to force the output of the filter away from the envelope boundaries. The cost is the relaxation of the \(H_2\) norm of the system by a percentage that still falls within an acceptable level, formulated as an additional LMI constraint. Thus the proposed algorithm is able to minimize the \(H_2\) norm of the filtering error transfer function, while imposing the envelope constraint. Furthermore, at the cost of relaxing the \(H_2\) optimal norm, the proposed \(H_2\) ECFIR filtering algorithms are proved to be robust in providing a margin between the filter output and the envelope boundaries. A numerical example illustrates and supports the new approach.