[Paper Review] Instantaneous Spectra Analysis of Pulse Series - Application to Lung Sounds with Abnormalities

Fumihiko Ishiyama|arXiv (Cornell University)|Feb 3, 2026

Phonocardiography and Auscultation Techniques0 citations

TL;DR

The paper introduces Linear eXtrapolation Condition (LXC) for Fourier analysis to obtain instantaneous spectra, applies it to lung sounds (crackles, wheezing, normal), and visualizes time-frequency structure of pulse-type signals.

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.

Motivation & Objective

• Motivate replacing periodic boundary condition (PBC) Fourier analysis with LXC-Fourier analysis to overcome time-frequency resolution limits.
• Develop a locally linearized AM-FM series framework to decompose signals into evolving frequency and amplitude terms.
• Apply instantaneous spectra analysis to lung sounds with abnormalities to demonstrate visualization of pulse-like structures.

Proposed method

• Model S(t) as a sum of M complex-exponential terms S(t)=sum_m exp(H_m(t)) with H_m'(t)=2π i f_m(t)+λ_m(t).
• Estimate local derivatives H_m'(t_k) via a non-standard linear predictive coding (LPC) method that respects the Linear eXtrapolation Condition (LXC).
• Compute complex amplitudes c_m(t_k) by minimizing the reconstruction error across a short window around t_k.
• Obtain instantaneous spectrum F_disc(f,t_k)=sum_m | c_m(t_k)λ_m(t_k) / (λ_m(t_k)+2π i(|f_m(t_k)|-f)) | and optionally F_±(f,t_k) to capture growth/decay.
• Apply a frequency filter and other filters (amplitude, spectral width, power) to highlight relevant components.
• Demonstrate that LXC-Fourier analysis includes conventional PBC-Fourier as a special case and offers higher time-frequency resolution.

Figure 1: Waveform inside hatched area is given time series for analysis. (a) Conventional Periodic Boundary Condition (PBC), which repeats given waveform infinitely [ 3 ] , and (b) proposed Linear eXtrapolation Condition (LXC), which linearly extrapolates given waveform [ 1 ] .

Experimental results

Research questions

• RQ1 Can LXC-Fourier analysis provide instantaneous spectra for non-periodic pulse series without requiring periodic extension?
• RQ2 How does the instantaneous spectrum reveal time-frequency structures in lung sounds with abnormalities like crackles and wheezing compared to normal sounds?
• RQ3 What are the practical considerations and filters needed to extract meaningful spectra from real-world lung sound data?
• RQ4 Does the method reveal growth/decay structures in pulse sequences not visible with conventional Fourier analysis?

Key findings

• LXC-Fourier analysis theoretically includes conventional PBC-Fourier analysis as a special case.
• The method yields instantaneous spectra for each AM-FM component, enabling pulse-by-pulse spectrum visualization.
• In lung sounds, crackles show central frequencies scattered 300–700 Hz with broad instantaneous spectra due to pulse decay.
• Wheezing shows central frequencies around 300 Hz with spectra concentrated near that frequency.
• Normal lung sounds exhibit no significant content above 100 Hz after frequency filtering.
• The approach reveals time-frequency structures (e.g., growth/decay patterns) not visible with traditional PBC-Fourier analysis.

Figure 2: Two spectrograms for single waveform [ 11 ] .