inverse CQT

(cc @ebattenberg @dpwe)

Seems like we ought to be able to do this, if we preserve phase in the cqt.

The main barrier comes from variable_hop, which uses oversampling and aggregation to handle large hop-sizes with small filters. This works fine in magnitude, but I don’t see a clean way to make it work out in the complex domain.

Perhaps we can add a dtype parameter to cqt: if the user requests a complex-type output, we preserve phase, and throw an exception on variable-hop aggregation. Otherwise, we let it through in the current state.

Are there other barriers to icqt that I’m not seeing at the moment?

[bmcfee]

I’ve updated the notebook to work on the latest #417 PR, which fixes a number of attenuation continuity errors in addition to normalizing filters by length.

The results now look and sound good enough (IMO) to actively work toward implementation in librosa, once #417 is merged.

Here’s an example of cqt(y) (left) and cqt(icqt(cqt(y))) (right):

More plots and examples are included in the notebook. There are still a couple of issues to work out (eg the unscaled case), and I think there might be a small amount of delay introduced by the icqt.

[bmcfee]

What about adding Holighaus’ CQT? It’s invertible.

I don’t think we should have multiple CQT implementations.

As I mentioned above, we have a working implementation of inverse cqt. I’m holding off on making a PR because it requires phase information, which is currently not produced by default.

If we provide icqt functionality, I want it to work “out of the box”. This will be possible in 0.5.0 when the deprecated default parameter settings are replaced, and cqt is complex-valued by default.