Timefrequency

Time-frequency analysis is a branch of signal processing that represents a signal in both time and frequency, capturing how spectral content evolves over time. It is particularly useful for non-stationary signals whose frequencies change, such as speech, music, biomedical data, and sonar.

Common time-frequency representations include the short-time Fourier transform (STFT) and the spectrogram, which use a sliding window to compute local Fourier transforms. The window size controls trade-offs between time and frequency resolution. The wavelet transform provides multi-resolution analysis, offering good time localization for high-frequency content and good frequency localization for low-frequency content. Beyond linear methods, quadratic representations such as the Wigner–Ville distribution and Cohen’s class offer higher resolution but can introduce cross-terms, which can complicate interpretation.

Key properties and limitations center on the uncertainty principle: a fixed window (as in STFT) imposes a constant time–frequency trade-off, while wavelets adapt resolution across scales. The Wigner–Ville distribution achieves high resolution but generates interference terms between components, requiring smoothing or other techniques to manage artifacts.

Applications of time-frequency analysis span many fields. In speech and music analysis, it enables feature extraction and transcription. In engineering and biomedical domains, it aids in diagnostic and monitoring tasks using EEG, EMG, or heart-rate signals. In radar, sonar, and seismology, time-frequency methods help detect transient events and characterize signal content.

Historically, time-frequency analysis emerged from Gabor’s work on windowed Fourier transforms and the Wigner distribution, evolving into a diverse set of tools for modern signal processing.