Signal representations are essential tools and paradigms for sound analysis and synthesis, audio processing, coding and music information retrieval. Popular examples are the STFT (Short-Time Fourier Transform) and the WT (Wavelet Transform), which lead to time-frequency and time-scale representations, respectively.

However, the definitions of the transforms and the underlying localization characteristics are often dictated by mathematical simplifications. For example, in the STFT, both time and frequency resolutions are constant or, in the simplest case of wavelet expansions, the frequency resolution is limited to one octave.

In order to incorporate physical or perceptual characteristics in the representation, essential to the valuable interpretation of the representative elements -- windowed sinusoids for the STFT and wavelets for the WT -- one can resort to domain mappings offered by time and frequency warping operators. In most cases, in order to preserve the perfect reconstruction properties of the transform, one desires to map any of the domains in a one-to-one fashion, so that information is preserved.

In linear transforms, where the analysis process is accomplished by orthogonal signal projections (scalar products) over the representative elements, remapping the signal is equivalent to inverse mapping the representative elements. This results in a modification of the localization properties of the representative elements. This is exemplified in the figure, where the uniform frequency domain cosine window elements of STFT are mapped into windows having 1/3 octave frequency resolution.

While warping provides an effective method for designing the localization characteristics of the representation, unfortunately, it also affects the organization of the transform. For example, warping in frequency introduces dispersion in time, so that the time organization is disrupted in which the various frequency components are represented on a frequency dependent time axis.

In order to circumvent the problem of dispersion, we devised redressing methods consisting in further warping in the transform domain, described in recent papers found in the References.

The benefits of the method, when applied to a warped STFT achieving single tone resolution in a 12-tone scale bandwidth allocation, are shown in the figures. In the first figure (left), the non-uniform spectrogram of a short excerpt of music piece is computed using purely warped STFT is shown. In the next figure (right), the redressed spectrogram, with same frequency resolution, is shown. As one can see, in the redressed spectrogram one is able to follow the score of the piece.

References

@inproceedings{Eva17,
	author = {{E}vangelista, G.},
	title = {{R}edressing {W}arped {W}avelets and {O}ther {S}imilar {W}arped {T}ime-{S}omething {R}epresentations},
	booktitle = {{P}roc. of Digital Audio Effects Conference ({D}{A}{F}{x}'17)},
	year = {2017},
	pages = {260--267},
	address = {{E}dinburgh, UK}
}

@inproceedings{MejEva16,
  author = {Thomas Mejstrik and Gianpaolo Evangelista},
  title = {Estimates of the Reconstruction Error in Partially Redressed Warped Frames Expansions},
  booktitle = {Proc. of Digital Audio Effect Conf. (DAFx'16)},
  year = {2016},
  pages = {9--16}
}

@inproceedings{Eva14a,
  author = {Gianpaolo Evangelista},
  title = {Approximations for Online Computation of Redressed Frequency Warped Vocoders},
  booktitle = {Proc. of Digital Audio Effect Conf. (DAFx'14)},
  year = {2014},
  pages = {85--91}
}

@article{EvaDorMat13,
  author = {G. Evangelista and M. Dörfler and E. Matusiak},
  title = {Arbitrary Phase Vocoders by means of Warping},
  journal = {Music / Technology},
  year = {2013},
  volume = {VII},
  pages = {91--118}
}

@inproceedings{Eva13,
  author = {Gianpaolo Evangelista},
  title = {Warped Frames: dispersive vs. non-dispersive sampling},
  booktitle = {Proceedings of the 10th Sound and Music Computing Conference},
  publisher = {Logos Verlag Berlin},
  year = {2013},
  pages = {553--560}
}

@inproceedings{EvaDoeMat12,
  author = {Gianpaolo Evangelista and Monika Dörfler and Ewa Matusiak},
  title = {Phase Vocoders With Arbitrary Frequency Band Selection},
  booktitle = {Proceedings of the 9th Sound and Music Computing Conference},
  year = {2012},
  pages = {442--449}
}

@article{Eva01a,
  author = {G. Evangelista},
  title = {Dyadic Warped Wavelets},
  journal = {Advances in Imaging and Electron Physics},
  publisher = {Academic Press},
  year = {2001},
  volume = {117},
  pages = {73--171}
}

@article{EvaCav98b,
  author = {G. Evangelista and S. Cavaliere},
  title = {Discrete Frequency Warped Wavelets: Theory and Applications},
  journal = {IEEE Trans. on Signal Processing},
  year = {1998},
  volume = {46},
  number = {4},
  pages = {874--885},
  note = {special issue on Theory and Applications of Filter Banks and Wavelets}
}

@article{EvaCav98a,
  author = {G. Evangelista and S. Cavaliere},
  title = {Frequency Warped Filter Banks and Wavelet Transform: A Discrete-Time Approach Via Laguerre Expansions},
  journal = {IEEE Trans. on Signal Processing},
  year = {1998},
  volume = {46},
  number = {10},
  pages = {2638--2650}
}