Analysis/Synthesis Using Time-Varying Windows and Chirped Atoms
A common assumption that is often made regarding audio signals is that they are short-term stationary. In other words, it is typically assumed that the statistical properties of audio signals change slowly enough that they can be considered nearly constant over a short interval. However, using a fixed analysis window (which is typical in practice) we have no way to change the analysis parameters over time in order to track the slowly evolving properties of the audio signal. For example, while a long window may be appropriate for analyzing tonal phenomena it will smear subsequent note onsets. Furthermore, the audio signal may not be completely stationary over the duration of the analysis window. This is often true of sounds containing glissando, vibrato, and other transient phenomena. In this paper we build upon previous work targeted at non-stationary analysis/synthesis. In particular, we discuss how to simultaneously adapt the window length and the chirp rate of the analysis frame in order to maximally concentrate the spectral energy. This is done by a) finding the analysis window that leads to the minimum entropy spectrum; and, b) estimating the chirp rate using the distribution derivative method. We also discuss a fast method of analysis/synthesis using the fan-chirp transform and overlap-add. Finally, we analyze several real and synthetic signals and show a qualitative improvement in the spectral energy concentration.