What is the transmitter? What is the receiver?
In a wireless microphone system, the transmitter is the handheld mic, or the bodypack (worn by the talker) that connects to a lapel mic or a headworn mic. The transmitter always has a battery of some type. The receiver is the "box" that connects into the sound system. A receiver typically has two external antennas; in some systems, the antennas are hidden inside the box. The receiver requires AC power from a wall socket, though some receivers can be battery powered.
In a wireless in-ear monitor system (Shure PSM products), the transmitter is the "box" that connects to the mixing board. The transmitter requires AC power from a wall socket. The receiver is the bodypack worn by the performer and provides the signal to earphones. The receiver always has a battery of some type.