A Hierarchical Constant Q Transform for Partial Tracking in Musical signals
This paper addresses a method for signal dependent timefrequency tiling of musical signals with respect to onsets and offsets of partials. The method is based on multi-level constant Q transforms where the calculation of bins in the higher levels of the transforms depend on the input signal content. The transform utilizes the signal energy in the subbands to determine whether the higher Q bins in the next level, that correspond roughly to the same frequency band in that level, will be calculated or not. At each higher level, the frequency resolution is increased by doubling the number of bins only for which there is significant energy in the previous level. The Q is adjusted accordingly for each level and is held constant within a level. Processing starts with a low Q that provides good time resolution and proceeds with higher levels until the desired maximum frequency resolution is achieved. The advantages of this method are twofold: First, the time resolution depends on the spacing of the frequency components in the input signal, potentially leading to reduced time smearing, and second, although signal dependent, the conditional calculation of higher Q levels of the transform has a direct consequence of reducing the number of operations in calculating the final spectrum for regular harmonic monophonic sounds. Partial tracking is performed using conventional peak picking and a birth-death strategy of frequency tracks. Testing is being carried out by resynthesizing the input sound from the extracted parameters using a sum of sinusoids with cubic interpolation for phase unwrapping between frames.