2
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Can the real-valued wavelet reconstruct the signal by a single integral? $\endgroup$
    – Wang Yun
    Jul 1 at 7:45
  • $\begingroup$ @WangYun Unsure, seems possible on real-valued inputs; I could take a look in another post. $\endgroup$ Jul 1 at 14:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.