[Discussion] CQT, VQT, energy preservation, and frequency band alignment

I wanted to make a separate issue to consolidate the discussion around a number of related points that have popped up in the implementation of VQT #1018, inverse CQT #165, and so on. @lostanlen and I have discussed these things in various places (often offline), but it would be helpful to have a more permanent record.

Frequency band definitions

In the CQT implementation, we follow the SK2010 definition of the frequency bands, which are essentially left-aligned: [f, a * f] (for some constant a). This works well enough, but it makes some calculations awkward, eg in the VQT. Later on in the CQT toolbox, the definition shifted to a centered representation, eg [f / sqrt(a), f * sqrt(a)] (again for some constant a), so that the frequency of each filter is centered within the band.

Some questions:

• Should we add support for centered frequency bands? I don’t think it would change much, except possibly some boundary effects at the top end of octaves.
• Should we make centering the default? Should we even bother with left-aligned bands anymore?

Energy preservation

Our CQT implementation has historically had a bunch of headaches around normalization and energy preservation. Of course we can’t exactly preserve energy with a lossy frequency representation, but our current cqt/icqt round trip gets pretty close. However, other feature representations (notably Mel) are not so forgiving, and it would be nice if we could strive for consistency here.

[bmcfee]

Yeah, that’s loosely speaking what I had in mind. A couple of caveats:

1. We can’t just take the max frequency anymore, because the max frequency could have a narrower bandwidth than the one below it. Instead, we can do something like this inside wavelet_length:

    f_cutoff = max(freqs * (1 + 0.5 * window_bandwidth(window) / Q) + 0.5 * gamma)
    
    if f_cutoff > sr / 2.0:
        raise ParameterError(f"Filter at frequency={???:.2f} would exceed Nyquist={sr/2}")

(I guess we’re cool with a per-frequency gamma?)

1. We actually need this f_cutoff in vqt to enable early downsampling. Actually, it might make sense to migrate the Nyquist check out of the filter constructor and into the vqt function itself. This would effectively mean that wavelet_lengths should return both the filter lengths and the cutoff frequency. A little awkward, but probably a better compromise overall.

[lostanlen]

I think @lostanlen’s post above had a typo in the alpha_high calculation, where it should subtract freq[-2] not freq[-1].

Yes, sorry, i hadn’t tested it. freq[-2] of course