I noticed that there are two types of wavelet functions, i.e. the real-valued, such as the Mexican hat wavelet, and the complex-valued, such as the Morlet wavelet. How was the complex-valued wavelet function proposed? What are their advantages?
Complex, or analytic wavelets enable:
- Instantaneous frequency, amplitude, and phase extraction - detailed post.
- Robust feature extraction for classification, stable against time-warping deformations (and, if coupled with time averaging, robust to time shifts), and averaged information recovery via higher-order transforms -- paper, lecture
- Exact analyticity enables superior time localization - example on hyperbolic chirp
Although lacking negative frequencies by definition, they aren't limited to real inputs: complex inputs are handled with the anti-analytic complement that lacks positive frequencies.