temporalspectral

Temporalspectral is a concept and set of techniques for representing and analyzing signals in a joint time–frequency domain. It refers to the characterization of a signal’s evolution over time together with its evolving spectral content, enabling examination of nonstationary signals whose frequency components change over time.

In practice, temporalspectral representations map a real-time signal x(t) into a two-dimensional function X(t,f) of time t and frequency f. The most common example is the spectrogram, obtained as the magnitude squared of the short-time Fourier transform: S(t,f) = |∫ x(τ) w(τ−t) e^{-j2π f τ} dτ|^2, where w is a window function. Another example is the Wigner–Ville distribution, which provides high time–frequency concentration but introduces cross-terms for multi-component signals.

Key properties include energy conservation and the resolution trade-off implied by the uncertainty principle: increasing time localization reduces frequency localization and vice versa. The STFT uses a fixed window, while multi-resolution approaches such as wavelet transforms provide variable resolution across scales. Cohen’s class offers a family of time–frequency distributions by kernel shaping to suppress cross-terms.

Applications span speech and music analysis, biomedical signal processing (for example EEG and EMG), radar and sonar, and nonstationary signal characterization in engineering and science.

Limitations include cross-term interference in certain representations and sensitivity to noise. Ongoing research focuses on higher-resolution methods, adaptive and sparse representations, and machine learning approaches to interpret temporalspectral data.