A ground loop in your AV system caused by antenna connection or TV cable is very common if you have your computer connected to the same system. This type of ground loop problem can be solved by using suitable isolation between your AV system and the antenna cable. The simplest way to get rid of the hum is to disconnect the antenna cable from the AV system. If you still want to watch cabe TV or listen to your radio an keep the system hum free then you have to install isolators to all those antenna cable connections your system has.
Best solution to antenna/cable caused ground loop is to add a 1:1 transformer in the antenna signal, floating the VCR with respect to the cable tv ground. This solved the hum, no need for a messing with the audio signal, and the tv image quality did not suffer (most tv's get an overdose of signal from the cable tv anyway).
You can simply make such a 1:1 transformer:
Take two strands of insulated wire, length: 10-20 cm.
Twist the two wires, about 1-2 twists per cm is enough.
Next you need a small transformer core suitable for UHF frequencies. I used a donut shaped ferrite core. Try looking in an old TV or radio. Wrap the twisted wire a few turns around the core.
One wire is the primary winding, the other the secondary (so it helps to color code the wires).
Make the leads between the coaxial antenna plugs and the core as short as possible.
You're making a transformer, i.e. if you have a red and a blue wire: connect the blue ends to one antenna plug, and the red wire to the other plug - not two colors on 1 plug.
For TV transmission high frequency signals such a simple transformer (using a suitable core) is almost perfect. Do not try to use other types of transformers because your circuit would not work then and they would cause annoying interference to other TVs connected to same antenna network (and in worst case you get cable TV people hunting for you because you messed with their cable network). I have seen...