Apart from the prospective applications of spectral envelopes mentioned in chapter 8, which would all be possible with the results of the project as is, other interesting ideas and enhancements would need more basic research. Some of these are described in the following.
Further reasearch is possible in the optimization of the representation of spectral envelopes as a sampled curve in the frequency-amplitude plane. The number of points neccessary to precisely represent a spectral envelope depends on the frequency range to cover, and on the complexity of the envelope. It could be automatically checked if the number of points is high enough by an analysis of the spectral envelope to be represented, and the frequency grid could be adapted to the requirements. Even with the current spectral representation, nothing constrains us to store the same number of points at each frame. Thus, for frames containing silence, only one point would be stored to indicate that a spectral envelope is present but on the level of silence. This would take up almost no space, while frames with complicated spectral envelopes could be stored more accurately.
An interesting perspective, which has not been explored in this project, is the application of the theory of wavelets to spectral envelopes. Without going into details, wavelets are a way to analyse a signal on multiple time-frequency resolutions simultaneously. While Fourier analysis uses sinusoids with an infinite time support, wavelet analysis uses different possible sets of basis functions with limited time support. See [Cha95] for an introduction to wavelets, [Mal97] for a detailed description, and [Hub97] for a insightful account of the history of wavelets.
Wavelets could be applied to the representations of spectral envelopes in two ways. First, they could be used for data compression in the framework of sampled (spectral) representation of spectral envelopes.
Second, after an idea of Rémi Gribonval, a completely wavelet-based representation might be possible, which could render manipulation very easy: Figure 9.1 shows an example spectral envelope. After the analysis of the spectral envelope, interpreted as a signal, with wavelets at different scales [MZ92], the inflection points (where the curving of the spectral envelope changes direction) are available, equally at different scales, as shown in figure 9.2. This information is sufficient to reconstruct the spectral envelope with good precision.
It can be seen that each peak in the spectral envelope in figure 9.1 is identified by two inflection lines in the analysis in figure 9.2.10.1 It is now easy to imagine moving these lines to shift formant peaks, or changing the spacing to modify their bandwidth.
It was said in section 4.1 that the representation should offer a parameterization of a spectral envelope that is flexible and easy to manipulate. However, throughout the whole project it was not entirely clear what it means that a parameterization is easy to manipulate. For singing voice and speech applications it is quite clear, but for musical applications I could only make a guess and offer the greatest possible flexibility. With more applications of the spectral envelope tools developed in this project, and when more people will be using them, it will become clearer if there is such a thing as the best parameter set for manipulation.