Time-Frequency Spectra with the S-transform

Most interesting data series consist of signal and noise series that are usually non-stationary, i.e. the properties fluctuate with time.Fourier analysis (FT) of the whole time series provides the spectrum of the whole time series, but is not capable of showing the time variation of the spectrum.The S-transform, also known as the Stockwell Transform (ST), uses a scalable, translating Gaussian window to determine the local spectrum at every point on the time series.The local spectrum supplements the local temporal information in the time series, and aids in the detection of onsets and cessations of events.For a N point time series, the output of the FT is a N point complex spectrum; the output of ST is a N x N, 2D time-frequency matrix, giving a N point spectrum at every point on the time series.Since the original time series contains only N points, the additional N² - N points is a measure of the non-independent, redundant information computed for the ST.The redundancy is useful in presenting the similarities in the neighbouring local spectra and contributes to the visual continuity and smoothness to the 2-D time-frequency spectrum. The computation and storage of N² - N additional points is a major drag on the usage of ST. Several approaches to reducing the computational burden are presented.Local spectra aids in analysis of 1D data. In images, the 2D local spectra aids in definition of texture and image segmentation.For 3 and 4 colour images, Trinion ST and Quaternion ST have been defined.In use since 1997, ST has found applications in numerous disciplines, including medical data series and images; power quality disturbance; atmospheric physics; exploration geophysics etc.The ST has been implemented in ARM and Raspberry Pi processors by several researchers.

This presentation is introductory, for the interested practitioner.Mathematical content will be at the absolute minimum.

What this presentation is about and why it matters

This talk introduces the S-transform (also called the Stockwell Transform), a time–frequency analysis tool that produces a full time–frequency map for a sampled signal. Unlike a single global Fourier spectrum, the S-transform produces a local spectrum at every time sample by viewing the signal through a frequency-scaled Gaussian window. That makes it easy to see when frequencies start, stop, or change — which is crucial in real-world engineering problems such as fault detection in power systems, seismic arrival-time analysis, radar/communications, and audio/bio-signal analysis.

Why it matters for practicing engineers: many signals are non-stationary (their frequency content changes over time). Relying only on a global Fourier transform can be misleading for these signals. The S-transform is a practical, invertible method that gives amplitude and absolute phase at each time–frequency point, helping with diagnostics, feature detection, and visualization during exploratory analysis.

Who will benefit the most from this presentation

• Signal processing engineers and scientists who analyze non-stationary signals (seismology, biomedical, audio, power monitoring, radar).
• Students and practitioners learning time–frequency methods who want a conceptual, low-math introduction.
• Engineers looking for interpretable visual diagnostics (onset detection, chirp tracking, dispersion) and practical trade-offs between resolution and computation.
• Developers who plan to implement or optimize S-transform variants (DOST, Hartley S-transform) on embedded platforms like ARM or Raspberry Pi.

What you need to know

This talk is deliberately introductory, but a few core ideas will help you get more from it:

• Sampling basics: A sampled record of N points yields N independent pieces of information. Frequency answers are periodic in the discrete domain and are limited by the Nyquist frequency (highest resolvable) and the Rayleigh frequency (lowest resolvable over the record length).
• Gabor uncertainty: Time and frequency localization trade off with each other. The talk frames this using the Gabor statement that the product of time and frequency uncertainty cannot be made arbitrarily small. In symbols: $\Delta t\cdot\Delta f\gtrsim 1$. This underlies why windows must be chosen carefully.
• Windowed local spectra: The S-transform computes a local spectrum at every time sample by multiplying the signal with a Gaussian window whose width depends on frequency. The normalized Gaussian can be written as $\frac{1}{\sigma\sqrt{2\pi}}e^{-t^{2}/(2\sigma^{2})}$ where $\sigma$ scales with frequency.
• Comparison to related methods: STFT (Short-Time Fourier Transform) uses a fixed window length; wavelet transforms use scaled wavelets (different normalization conventions exist); DOST (Discrete Orthonormal Stockwell Transform) remaps the time–frequency plane for efficiency. The S-transform sits between STFT and wavelets conceptually: it uses sinusoidal kernels modulated by frequency-dependent Gaussians.
• Practical trade-offs: The classical S-transform produces an N×N complex matrix (N voices × N time points), which is rich but heavy: storage grows as O(N^2) and naive computation can be costly. The talk surveys practical enhancements (DOST, Hartley S-transform, asymmetric windows, quaternion/trinion extensions for color images) that address accuracy or resource use.

Glossary

• S-transform / Stockwell Transform: A time–frequency transform that uses a frequency-scaled Gaussian window to compute a local Fourier spectrum at each time point.
• STFT (Short-Time Fourier Transform): Local Fourier spectra computed using a fixed-length window sliding along time.
• Wavelet Transform: A transform using scaled and shifted prototype wavelets; good for multi-resolution analysis of frequency content.
• DOST (Discrete Orthonormal Stockwell Transform): An efficient, discrete basis for approximating the S-transform with reduced computation and storage.
• Gaussian window: The bell-shaped weighting function used in the S-transform; its width is frequency-dependent in the S-transform.
• Gabor uncertainty principle: The trade-off rule that limits simultaneous precision in time and frequency.
• Nyquist frequency: The highest frequency that can be represented for a given sampling rate (half the sampling frequency).
• Rayleigh frequency: The lowest resolvable frequency given the finite record length (one cycle over the full record).
• Instantaneous frequency: Local frequency value as a function of time (what chirps and dispersive waves exhibit).
• Hartley transform / Hartley S-transform: A real-valued transform alternative that can reduce storage and compute by avoiding complex arithmetic.

Why you should watch

This presentation is a clear, practitioner-friendly tour of the S-transform and its place among time–frequency tools. The speaker balances intuition, simple math, and concrete examples — including seismogram analysis and bird-song audio — that demonstrate why a time–frequency first step is valuable for unfamiliar signals. If you want a low-math explanation of how the S-transform works, why redundancy in the time–frequency plane is useful, and what practical variants exist to tame computation, this talk delivers good guidance and pointers to further reading and MATLAB code.

Enjoy the talk: it’s a concise, well-illustrated introduction that connects theory with hands-on concerns and applications you can reuse in engineering practice.

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