Module dsp::cwt

Continuous Wavelet Transform (Morlet wavelet)

The CQT (cqt.h) picks logarithmically-spaced frequency bins; the CWT instead picks a set of scales directly (the caller chooses them, typically log-spaced for the same musically/perceptually-relevant reason the CQT uses log-spaced bins) and correlates a scaled/shifted copy of a mother wavelet against the signal at every sample position, producing a full time-scale "scalogram" rather than one value per frame. The Morlet wavelet — a complex sinusoid windowed by a Gaussian, psi(t) = pi^-0.25 * exp(-t^2/2) * exp(iw0t) — is the standard choice for time-frequency analysis (its Gaussian envelope gives it the best achievable time-frequency localization for a sinusoid-based wavelet).

A scale s corresponds to a center frequency of approximately f = w0fs / (2pi*s) Hz (fs = the signal's sample rate) — smaller scales resolve higher frequencies with finer time resolution, larger scales resolve lower frequencies with coarser time resolution (the same time/frequency trade-off the CQT's per-bin window lengths express, just parameterized directly by scale here instead of by bin index).

Direct (not FFT-accelerated) correlation, kernel support truncated to +/-4 standard deviations of the Gaussian envelope (negligible energy beyond that) — O(numScales * signalLen * scale) in the worst case; fine for offline/analysis use.