I’m considering building a directional FM antenna for an FM station that has almost no signal in most of the house. Otherwise I must put the receiver near the window and put the antenna in a specific position. (related thread)

At the same time I am also tempted to build a omni-directional FM antenna that would also suit DAB frequencies (if possible). Since I would prefer to only run one coax cable throughout the house I guess I need to know if the directional and omni-directional antennas can simply be spliced together to share the same coax.

Is that feasible?