TSM by Resampling STFTs Across Time
In view of Chapter 8, a natural implementation of TSM based on the STFT is as follows:
- Perform a short-time Fourier transform (STFT) using hop size
. Denote the STFT at frame
denote the result of TSM processing by
- To perform TSM by the factor
, advance the ``frame
during resynthesis instead of the usual
For example, if ( slow-down), the first STFT frame is processed normally, so that . However, the second output frame corresponds to a time , half way between the first two frames. This output frame may be created by interpolating (across time) the STFT magnitude magnitude spectra of the first. For example, using simple linear interpolation gives
where the phase is chosen to preserve continuity and/or the amplitude envelope from frame to frame under the overlap-add (more on this below). Generalizing to arbitrary TSM factors , we obtain
where , and is advanced by each frame-step.
In general, TSM methods based on STFT modification are classified as ``vocoder'' type methods (§G.5). Thus, the TSM implementation outlined above may be termed a weighted overlap-add (WOLA) phase-vocoder method.
TSM and S+N+T