Loss of Time-Domain Localization in STFT When Window Function Has Infinite Width

Short-Time Fourier Transform (STFT) is a fundamental time-frequency analysis method used to study signal variations across both time and frequency domains. The core algorithmic approach involves segmenting the signal into short time intervals using a sliding window function, followed by applying Fourier Transform to each segment to generate a joint time-frequency representation. In Python implementations, this is typically achieved using libraries like scipy.signal.stft with parameters controlling window type and length.

The window function plays a critical role in STFT by determining the weight distribution for each signal segment. When the window width becomes infinitely large (equivalent to applying a rectangular window spanning the entire signal duration), STFT degenerates into conventional Fourier Transform. From a computational perspective, this scenario corresponds to setting the window length equal to the signal length in code implementations, where functions like numpy.fft.fft would produce identical results. However, Fourier Transform only provides global frequency components, completely losing time-domain localization capabilities and failing to track how frequency components evolve over time.

Therefore, STFT's effectiveness hinges on using finite-width window functions. Narrower windows enhance time resolution at the cost of frequency resolution (shorter FFT bins), while wider windows improve frequency resolution but reduce time localization. In practical implementations using MATLAB's spectrogram function or Python's librosa.stft, engineers must carefully select window parameters (like Hanning window length) based on signal characteristics to balance the time-frequency resolution trade-off, typically through iterative parameter tuning and visualization of resulting spectrograms.