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.

翻译：傅里叶分析“时频分辨率理论极限”的根源在于其数值实现，特别是源于一个世纪前引入的“周期性边界条件”假设。我们先前提出用“线性外推条件”替代此条件，该条件不要求周期性。这一特性使得脉冲序列的瞬时频谱分析成为可能，从而替代了短时傅里叶变换。作为演示，我们将瞬时频谱分析应用于两种异常肺音（爆裂音和哮鸣音）及一种正常肺音。其中，爆裂音包含随机脉冲序列。每个脉冲的频谱均可获得，通过组合各频谱即可得到脉冲序列的谱图。最终，给定脉冲序列的时频结构得以可视化呈现。