STFTs

The Short-Time Fourier Transform (STFT) is a signal processing technique used to determine the spectral content of a signal over time. It is a variation of the classical Fourier Transform, which analyzes the entire signal at once. The STFT addresses the limitation of the Fourier Transform in that it cannot represent non-stationary signals, whose frequency content changes over time.

To perform an STFT, the original signal is divided into short, overlapping segments. A window function is then applied to each segment to reduce spectral leakage. The Fourier Transform is then computed for each windowed segment. This process results in a sequence of Fourier Transforms, each representing the frequency content of the signal within its corresponding time window. The output of the STFT is often visualized as a spectrogram, which plots frequency on one axis, time on another, and the magnitude of the frequency components as intensity or color.

The choice of window length and overlap is crucial in STFT analysis. A shorter window provides better time resolution but poorer frequency resolution, while a longer window offers improved frequency resolution at the expense of time resolution. This trade-off is a fundamental aspect of the Heisenberg uncertainty principle as applied to signal processing. The STFT is widely used in various fields, including audio analysis, speech recognition, radar signal processing, and seismic data analysis, where understanding the temporal evolution of spectral characteristics is essential.