Instantaneous Spectra Analysis of Pulse Series - Application to Lung Sounds with Abnormalities

TL;DR

The paper addresses the theoretical time–frequency resolution limit imposed by conventional periodic boundary conditions in Fourier analysis and proposes Linear eXtrapolation Condition (LXC) as a nonperiodic alternative that enables instantaneous spectra analysis. It develops LXC-Fourier analysis, which extends instantaneous frequency with AM–FM decomposition, and uses a locally linearized approach with LPC-based estimation to obtain unique time-varying parameters $f_m(t)$ and $\lambda_m(t)$ for each mode. The instantaneous spectrum is defined in discrete form as $F_{ m disc}(f,t_k)$, allowing mode-specific spectra and power interpretations, with a variant $F_{\pm}$ capturing growth/decay. Applied to lung sounds, the method analyzes crackles, wheezes, and normal sounds, revealing high-frequency central components for abnormalities and a broad, pulse-driven structure in crackles, with higher effective time–frequency resolution than conventional STFT-based methods. The results underscore the need for wider bandwidth recordings and suggest broad applicability of LXC-Fourier analysis to nonperiodic, pulse-like signals in other fields.

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.

Figures (6)

• Figure 1: Waveform inside hatched area is given time series for analysis. (a) Conventional Periodic Boundary Condition (PBC), which repeats given waveform infinitely walker, and (b) proposed Linear eXtrapolation Condition (LXC), which linearly extrapolates given waveform cspa2025.
• Figure 2: Two spectrograms for single waveform cspa2025b.
• Figure 3: (a) Waveform, and 12 samples for analysis. (b) Obtained spectra with LXC- and PBC-Fourier analysis git.
• Figure 4: Spectrogram of LXC-Fourier analysis for FM time series cspa2025. Frequency-width$\times$time-width is always 1 Hz$\cdot$s for PBC-Fourier analysis.
• Figure 5: Obtained results for three lung sounds (crackles, wheezing, normal). (L) Central frequencies, and (R) corresponding instantaneous spectra.