A method for manufacturing a semiconductor die, comprising the steps of: providing a MEMS device having a structural body, provided with a cavity, and a membrane structure suspended over the cavity; coupling the structural body to a filtering module via direct bonding or fusion bonding so that a first portion of the filtering module extends over the cavity and a second portion of the filtering module extends seamlessly as a prolongation of the structural body; and etching selective portions of the filtering module in an area corresponding to the first portion, to form filtering openings fluidically coupled to the cavity. The semiconductor die is, for example, a microphone.