scalograms

A scalogram is a time-scale energy representation of a signal obtained from the continuous wavelet transform. It displays how the signal’s energy is distributed across time and scale, typically by taking the squared magnitude of the wavelet coefficients. The scale parameter is related to frequency, so smaller scales capture higher-frequency content and larger scales capture lower-frequency content, depending on the chosen mother wavelet.

To compute a scalogram, one selects a mother wavelet such as Morlet or Mexican hat and computes the continuous wavelet transform W(a,b) = ∫ x(t) psi*( (t-b)/a ) dt. The scalogram is then formed as S(a,b) = |W(a,b)|^2, representing energy density across time b and scale a. Plots often use time on the horizontal axis and scale (or an associated frequency) on the vertical axis, with color or brightness indicating energy. Some variants display S(a,b) = |W(a,b)| rather than the squared magnitude.

Scalograms are used to analyze nonstationary signals whose spectral content evolves over time, including applications in seismology, speech and music analysis, EEG/MEG studies, and mechanical fault diagnostics. Compared with spectrograms, scalograms can offer improved localization of transient features due to the multi-resolution nature of wavelets, though results depend on the chosen wavelet and scale grid.

Limitations include sensitivity to the choice of mother wavelet, scale discretization, and edge effects near signal boundaries. They remain a widely used tool in time-frequency analysis for exploring how signal energy evolves across time and scale. Related concepts include the wavelet transform, the Morlet wavelet, and time-frequency representations.