It's not a "faulty fan" sound, rather a skrilling high pitch sound that goes up and down every now and then. Sounds to me more like a capacitor that is making that noise, or another electronic component chip or transistor. I would simply now blindly assume it could be the power supply of your PC !
To test this try the below (advanced tech, use caution) :
1.) Remove your PSU completely with all the cables. On your PSU main plug-to-mainboard, contact the green pin and any black pin. Then connect the power supply cable. The FAN of the power supply should now spin up. See if the noise is there.
2.) On cold boot, briefly attempt to stop the FAN's of your machine. Make sure you don't damage any FAN fins and only to keep them "off" for 5 seconds max, so that you can try and identify the location of the noize.
3.) Also check on your mainboard for bulged (round) capacitors. Usually all the caps on teh mainboard should be flat ontop. If they are round or seem to have brown/black liquid coming out then they need to be replaced. Check your mainboard thoroughly...