Time-Frequency Signal Analysis

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Abstractions about representing signals jointly in time and frequency — transform families (discrete-time Fourier transform, S transform, Wigner distribution function), techniques for suppressing artifacts or localizing content (modified Wigner distribution function, Hilbert spectral analysis), and elementary analyzing functions such as the modified Morlet wavelet.

10 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Discrete-time Fourier transform — A periodic continuous-frequency function obtained by summing a discrete-time sequence against complex exponentials, representing the sequence by its spectral amplitudes.
• Hilbert spectral analysis — A time–frequency analysis that forms analytic signals, often from empirical mode components, and derives instantaneous amplitude and frequency from Hilbert phase.
• Modified Morlet wavelet — A localized oscillatory analyzing function formed by multiplying a cosine carrier by a hyperbolic-secant envelope and normalizing it for wavelet use.
• Modified Wigner distribution function — A member of a modified quadratic time–frequency representation family designed to suppress the interference cross-terms of the Wigner distribution while retaining useful concentration.
• Rectangular function — A piecewise-constant pulse equal to one on a centered finite interval and zero outside, with a convention-dependent value at the endpoints.
• S transform — A time–frequency transform combining a frequency-dependent Gaussian window with Fourier phase reference to localize nonstationary signal content.
• Time–frequency analysis — Represent a nonstationary signal jointly over time and frequency so changing spectral content, transients, and localization tradeoffs remain visible instead of being collapsed into one global spectrum.
• Time–frequency representation — A representation that distributes a signal’s amplitude, energy or power jointly over time and frequency.
• Wigner distribution function — Represent a signal or quantum state bilinearly over a conjugate-variable plane, preserving informative marginals and covariance while accepting negative values or interference cross-terms.
• WSSUS Model — The WSSUS channel model combines time-lag-stationary second-order behavior with uncorrelated delay components, enabling a delay–Doppler scattering description.