How does Audacity’s “Isolate Center” functionality work?

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I’m going to go into a bit of detail about my current understanding, in case there’s a misstep along the way.

For a while now, I’ve known how to manually remove the center of a stereo audio file in Audacity. What you do is, split the stereo track into two mono tracks, and then invert one of them. This has the effect of removing everything that was panned center, since it was in both sides, but is now being cancelled out against itself.

So that’s cool. And my intuition told me at first that I should be able to then take that isolated file, invert it, and play it against the original recording (in mono), in order to cancel out everything *but* center.

I quickly learned that this doesn’t seem to work, and I’ll explain my understanding of why (again – call me out if I’m wrong here):

We start with two tracks, left and right. Left has the left-panned audio (which we’ll call `L`) as well as the center-panned audio (`C`), and then right obviously has `R` and `C`. So these are our two tracks:

`LC CR`

Then we invert, say, the right one.

`LC cr`

Canceling those out gives us `Lr`, which effectively sounds like the original song in mono, minus the center track. But it’s `Lr`, not `LR`, so we can’t actually use it to cancel out both sides of the stereo mix and leave only `C`. It *sounds* to our ears like `LR`, but since one side is always inverted, we can’t actually isolate the center. We’ll end up with `LLC` or `CRR`.

So anyway, imagine my surprise when I find that Audacity has an “Isolate Center” feature that achieves this exact effect. How is it doing it? Is my understanding of why it shouldn’t be possible incorrect? Or is Audacity doing something much more sophisticated, like maybe something AI-based that actually tries to “hear” what the two sides have in common?

(P.S. Yes, I also noticed that Audacity can remove the center from the same menu, rather than having to do my manual process. TIL)

In: Technology
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