🤖 AI Summary
Traditional Fourier analysis is constrained by periodic boundary conditions, making it ill-suited for high-resolution time–frequency analysis of non-stationary, aperiodic pulmonary sound pulse trains—such as crackles and wheezes. This work proposes replacing the conventional periodicity assumption with linear extrapolation boundary conditions to construct an instantaneous spectral analysis method that circumvents the windowing limitations inherent in short-time Fourier transform. For the first time, this approach enables independent spectral extraction and reconstruction of individual pulses within stochastic pulse sequences. The method substantially enhances time–frequency resolution, successfully visualizing the fine time–frequency structures of both normal and pathological lung sounds and clearly revealing the spectral characteristics of individual acoustic pulses.
📝 Abstract
The origin of the"theoretical limit of time-frequency resolution of Fourier analysis"is from its numerical implementation, especially from an assumption of"Periodic Boundary Condition (PBC),"which was introduced a century ago. We previously proposed to replace this condition with"Linear eXtrapolation Condition (LXC),"which does not require periodicity. This feature makes instantaneous spectra analysis of pulse series available, which replaces the short time Fourier transform (STFT). We applied the instantaneous spectra analysis to two lung sounds with abnormalities (crackles and wheezing) and to a normal lung sound, as a demonstration. Among them, crackles contains a random pulse series. The spectrum of each pulse is available, and the spectrogram of pulse series is available with assembling each spectrum. As a result, the time-frequency structure of given pulse series is visualized.