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：傅里叶分析中“时频分辨率理论极限”的根源在于其数值实现，尤其是源于一个世纪前引入的“周期边界条件（PBC）”假设。我们曾提出用“线性外推条件（LXC）”替代该条件，此方法无需周期性假设。这一特性使得脉冲序列的瞬时频谱分析成为可能，从而取代了短时傅里叶变换（STFT）。我们将瞬时频谱分析应用于两种异常肺音（湿啰音与哮鸣音）及一种正常肺音作为演示。其中，湿啰音包含随机脉冲序列。通过该方法可获取每个脉冲的频谱，并可通过整合各频谱绘制脉冲序列的声谱图。最终，给定脉冲序列的时频结构得以可视化呈现。