ZLA Controller reported clearing a pilot for an approach who then went below minimum altitude for the area. Reporter stated the quality of the radio frequency may have caused them to miss an incorrect clearance readback and added that the radio sites in the sector have had problems for many years; making it difficult to understand communications.

Narrative

I cleared Aircraft X for the VOR-A Approach into HII. I cleared the aircraft to cross EED AOA FL059 and then cleared for the approach. In my opinion it sounded like the pilot read back the correct restriction at EED although the frequencies are terrible at that sector and might have caused it to sound like a correct read back. The MSAW alert did NOT go off and I happen to notice the aircraft was below the MEA and at FL057. I issued a low altitude alert and told the aircraft to climb to FL059. Once I observed the aircraft at a safe altitude (FL060); I recleared the aircraft for the VOR-A approach. The pilot did seem to be in a training flight which might have also caused this. We really need to get better radio sites at sector 6. This has been an issue for several years and multiple reports have been filed with no resolve. Area D has eventually given up on filing reports for the radios since we have been told nothing can be done. If there was a bad read back then better radios would have certainly prevented this from happening. There are times when mains/standbys/BUECS (Back Up Emergency Communication System) do not work and the only way to reach a pilot is by aircraft relay. Aircraft relays are to the point where they have to be used almost every session at this sector. As a regular occurrence; when we are able to reach pilots it is hard to understand them.