Hall thrusters are known to exhibit a large variety of instabilities. Their physical mechanisms have been identified at low (kHz) and intermediate (MHz) frequencies, even though they are still not fully understood. Furthermore, electromagnetic radiations generated by Hall thrusters, named “self-emission” of the thruster, have been measured from kHz to MHz as expected from the identified instabilities, but also at higher frequencies. The origin of the high frequency (GHz) self-emission remains for now unknown. Assessing this self-emission, that is important for understanding the physics of Hall thrusters as well as for electromagnetic compatibility issues with the spacecraft, is challenging. Another aspect that makes the understanding of the physics of Hall thrusters complex comes from the eventual coupling between instabilities, which has been recently suggested and observed. The aim of this paper is to explore the possibility of characterizing simultaneously instabilities in Hall thrusters on a broadband frequency range (from kHz to GHz) in situ, meaning in a conventionally used vacuum chamber where Hall thrusters are usually operated. We show in this paper that, despite the reverberant nature of the vacuum metallic chamber, useful information is extracted at low and intermediate frequencies and even at high frequency from the measurements done with an antenna in this environment.