[jd20]

Not sure why it's not possible, though I could've had a defective unit. Specifically, I had a Fiio D3 ($20) which had bad background noise, even when not playing. I always assumed it was a power supply or shielding issue. The audio output from it actually sounded decent, just I could always hear the background noise. Don't think it was a grounding issue either, because it was the only DAC I ever had that consistently did that, no matter what system I plugged it into.

I've played around with the DAC built-in to the Raspberry Pi 3 as well, and that one was more of a distorted output problem, similar to what you describe with the $2 DAC.

Other DAC's I've tried (from around $75 and up) I've never heard noise / shielding issues, so again rapidly diminishing returns once you get past the super budget options, in my experience.

[zzzcpan]

If you hear high frequency noise it's just means the amplifier has too much gain for your headphones and doesn't automatically mute when there is no signal (many do). But all amplifiers have noise. If you use the line output though, it doesn't go through the amplifier and of course doesn't have its noise.