Electronic – How to filter hum from transformer for audio rectifier

Ah you didn't read the reviews for this amp - it seems that it does cause loud pops at power-up and one user has implemented a switch that disables the speakers but this could be done with a delaying relay circuit fairly simply.

As for the "hum" - if it is "hum" and not degradation of the signal or high frequency whistles (etc.) this is likely caused by your power supply having too much ripple on it. The amp is a class D type and these are notoriously poor at coping with power supply ripple. Power supply ripple will come from a standard transformer/bridge/smoothing capacitor and is inevitable in these configurations - you can try putting a 10,000 uF cap across the supply rails close to the power amp.

Why are class D amps poor at rejecting power ripple - the output transistors are basically used as switches and they alternate rapidly (above audio frequencies) and this means any power supply ripple is superimposed onto the speaker wires. A conventional amp can easily reject ripple because it operates linearly and will only push ripple to the output when driven at very high levels (because the transistors saturate when the ripple causes the power voltage to be cyclically at a minimum).

It could be wiring but it's easy just to put a big capacitor across the 12V power rails first then take it from there if that doesn't solve the problem to an acceptable level.

Co-ax and twisted pairs are usually good at removing common mode noise.

Ground loops in coax installations can be a problem as they cause unbalanced noise. This becomes more of a problem if the co-ax shield carries any stray or intended 50/60Hz current.

Make sure signal co-ax screen is not used or involved in current paths that include conductors other than internal core. Running the Co-ax parallel with unbalanced high current 50/60Hz conductors could ealisy cause stray currents to flow in the screen if it is grounded at both ends. Any current on the screen that is not on the core will show up as noise.

Also not that with co-ax capacitive pickup should be minimal if the sensor is shielded. Electromagnetic pickup should be cancelled out except at the terminal points.

You should make sure the grounding at the aquisition end is as good as possible and the co-ax screen is isolated from ground at the sensor end. Grounding the sensor housing locally is usually still a good idea (unless it is small and can be left to float with at co-ax screen potential) but having the co-ax shield grounded to two unequal grounds is always a bad idea.