Preprocessing for the Automated Transcription of Polyphonic Music: Linking Wavelet Theory and Auditory Filtering

In this paper a fast method for the calculation of a linear time-frequency distribution based on the gammatone lter auditory model is introduced as a preprocessing step for the automated transcription of music and auditory source separation. Examples show that this method has a promising potential for the analysis of music pieces with a limited spectral overlap of the diierent signal components. In the past few years wavelet transforms have become an important tool for signal processing. See (Rioul and Vetterli, 1991) for an overview. An important property of both the continuous wavelet transform (CWT) and the short-time Fourier transform (STFT) is their linearity, which makes them more suitable for the analysis of multicomponent signals than quadratic time-frequency distributions (TFD) suuering from cross-term artefacts. The CWT is given by a dd; a > 0: (1) For admissibility of a time function g(t) as a mother wavelet, it must satisfy Z 1 ?1 jg(t)j 2 dt < 1; and G(0) = 0; (2) where G(f) is the Fourier transform of g(t). Functions satisfying these conditions look like short waves, which has been the reason for naming them wavelets. The main diierence between the STFT and the CWT lies in the fact, that for the STFT the analysis window remains unaltered, whereas the CWT window changes its scale due to the scaling factor a. The quasi-logarithmic organization of musical scales and of the frequency resolution in the human cochlea makes the CWT a more appropriate TFD of acoustic signals than the STFT. Since eq.1 cannot be evaluated everywhere, it has to be modiied by picking certain xed values for a and b yielding a discrete approximation of the CWT. Furthermore, the wavelets used throughout our work are close to being analytic (progressive), that is they satisfy 8f < 0 : G(f) 0. Thus, given the complex-valued lter outputs the instantaneous frequencies can be estimated from the phases and the signal envelopes can be estimated from the moduli, if the signal components have a negligible overlap. Due to the high computational burden of the quasi-continuous CWT, innnite impulse response (IIR) realizations like our gammatone approach are desirable. As stated by Patterson (1992) the gammatone lter can be a good approximation of the ltering in the human cochlea if the parameters are properly adjusted. See g.1 for an example of a gammatone impulse response. A quasi-analytic version of the gammatone lter is of the …