Improve phase vocoder

[jurihock]

1. Phase Vocoder Done Right
2. Pitch-shifting algorithm design and applications in music
3. Efficient approximations for the arctangent function
4. Phase-locked vocoder

[jurihock]

Just as an alternative idea to the phase vocoder...

DFT magnitude based phase estimation:

• Single Pass Spectrogram Inversion (SPSI) + lonce/SPSI_Python
• Phase Gradient Heap Integration (PGHI) + rrlyman/phase-reconstruction

However:

• SPSI doesn't work well with SDFT and STFT at smaller hops like 2048/32 (as tested in voyx).
• PGHI appears to be 6-8 times slower than SPSI.

[jurihock]

The arctan approximation [3] is still faster than std::arg (about 50ms difference in case of the default voice sample), but of course less accurate e.g. compared to the python implementation. Since accuracy is more important to me at this point, the arctan approximation will be not yet implemented. Done in #40.

Regarding [1], my current observation is that the sliding vocoder generally produces less artifacts especially if pitching instrumental recordings. So it makes more sense to discover the sliding DFT first instead of obfuscating the vocoder...

[jurihock]

Idea:

Use $log(a \cdot e^{j \phi}) = log(a) + j \phi$ instead of explicit std::abs and std::arg calls, since both log-amplitude and phase are needed anyway.

[jurihock]

The pitch shifting result is comparable to the signalsmith:

make
./out/main hybrid-phase --trim --freq=2 input.wav output.wav

where input.wav is the original dno-solo example converted to 16-bit mono wav.

Actually the default signalsmith configuration uses 6144 (3072 without zero padding) DFT bins, which is not power of two. Disabling multipleTimeObservations and zeroPadding makes no noticeable difference.

The comparable stftPitchShift configuration is -w 8k -v 4.