Processing of acoustical signals via a wavelet-based analysis

In the present paper, details are given on the implementation of a wavelet-based analysis tailored to the processing of acoustical signals. The family of the suitable wavelets (`Reimann wavelets') are obtained in the time domain from a Fourier transform, extracted in Ref.~\cite{r1} after invoking theoretical principles and time-frequency localisation constraints. A scheme is set forth to determine the optimal values of the parameters of this type of wavelet on the basis of the goodness of the reproduction of a $30$-s audio file containing harmonic signals corresponding to six successive $A$ notes of the chromatic musical scale, from $A_2$ to $A_7$. The quality of the reproduction over about six and a half octaves is investigated. Finally, details are given on the incorporation of the re-assignment method in the analysis framework, as the means a) to determine the important contributions of the wavelet transforms and b) to suppress noise present in the signal.