Wavelet analysis uses variable-length windows for better time resolution at high frequencies

Wavelet analysis divides the time-frequency plane into nonuniform regions, unlike the STFT’s uniform grid. Short wavelets analyze high frequencies with good time resolution but poor frequency resolution. Long wavelets analyze low frequencies with good frequency resolution but poor time resolution. This multiresolution property makes wavelets well-suited for analyzing transients in music: a cymbal crash is detected by short, fast wavelets while a bass note is resolved by long, slow wavelets. The Morlet wavelet (used in audio analysis) has a Gaussian envelope similar to a grain; its duration scales inversely with frequency, always containing a constant number of cycles. The fast wavelet transform (FWT) computes this efficiently in O(n log n).

Examples

A 10 kHz transient is detected by a 0.1ms wavelet. A 50 Hz bass note is resolved by a 20ms wavelet. Both are analyzed simultaneously in the wavelet transform.

Assessment

How does the wavelet transform’s time-frequency grid differ from the STFT’s? In what musical analysis scenario is the wavelet transform superior to a fixed-window STFT?