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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?

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Complex, or analytic wavelets enable:

  1. Instantaneous frequency, amplitude, and phase extraction - detailed post.
  2. 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
  3. 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.

In contrast, real wavelets are advantageous for detecting fast transients and analyzing fractals (e.g. perturbation, noise analysis) - ch6 of Wavelet Tour, and ch4 for analytic.

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  • $\begingroup$ Can the real-valued wavelet reconstruct the signal by a single integral? $\endgroup$
    – Wang Yun
    Commented Jul 1, 2021 at 7:45
  • $\begingroup$ @WangYun Unsure, seems possible on real-valued inputs; I could take a look in another post. $\endgroup$ Commented Jul 1, 2021 at 14:20

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