I find it incredibly funny that this appears a couple of days after I finished my master's thesis. I realized I made a bit of a blunder a week before handing it in: I thought the Hilbert transform gets rid of the aliasing when a real signal's bandwidth reaches below the frequency origin and into negative frequencies, which it doesn't. The bandwidth "folding" is still there, but the negative frequency components are gone (which I did know). Since I got rid of the negative frequencies because of the symmetry of real signals about the frequency origin, there wasn't any point in doing the Hilbert transform. Thankfully I checked all the preprocessing steps I carried out in the thesis, and weeded out this unnecessary step.

In my defense, please do bear in mind that I have not had a thorough education in signal processing, physics degrees don't usually have courses like this. I know Hilbert spaces much more well than I do the Hilbert transform.