I ran into this a while a go, and found a fix, but it has happened again, and the fix isn't so easy this time.

A few months ago TDS Telcom installed a Dynamic T1 at a customer site, 1.5 to the internet and 6 pots lines off a Cisco router/channel bank.
Brand new IP 500 (Avaya IP Office) customer dials 9, and the number, system times out and gives the trunk....and there is a hum on the line. Test at demarc, no hum.... Change ground, lift ground, try different power, let entire rack run off UPS, swap control unit, swap analog trunk cards, ALL to no avail...... Just seconds before going "postal" i pulled the 110V power on the Cisco unit and let that run off of just battery, WOW, hum gone. Plug power back in, hum returns.
Call TDS, they tell me not possible...but tech returns, and tests it by lifting the power, no hum. Re-plugs in 110V power, hum back.

Tech swaps out the power supply, no hum anymore. The tech did call me a few weeks later and stated that he used that power supply at a different customer site that was using an older Toshiba system, and there was NO hint of hum. Must have just been a combination of my system and THAT power supply.

Fast forward to last week. Same thing, different customer. IP 500 with pots lines. Customer reports hum on lines. Test the same as above, by lifting the 110V power and trying test calls....no hum. Call in a ticket to TDS, who showed up this am. They tried grounding the battery backup (wasn't previously grounded) tried swapping the power supply, But nothing is fixing this one.....

Would a "hum filter" fix this? Can anyone recommend one?

Also, a week or 2 after we found the faulty power supply, we got a call from another TDS branch that was having the same problem, the techs had seen our notes, and cured problem by swapping the power supply....So, this might be happening more frequently that we know....