Neha Patil (Editor)

Complex Mexican hat wavelet

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

In applied mathematics, the complex Mexican hat wavelet is a low-oscillation, complex-valued, wavelet for the continuous wavelet transform. This wavelet is formulated in terms of its Fourier transform as the Hilbert analytic signal of the conventional Mexican hat wavelet:

Ψ ^ ( ω ) = { 2 2 3 π 1 / 4 ω 2 e 1 2 ω 2 ω 0 0 ω 0.

Temporally, this wavelet can be expressed in terms of the error function, as:

Ψ ( t ) = 2 3 π 1 4 ( π ( 1 t 2 ) e 1 2 t 2 ( 2 i t + π erf [ i 2 t ] ( 1 t 2 ) e 1 2 t 2 ) ) .

This wavelet has O ( | t | 3 ) asymptotic temporal decay in | Ψ ( t ) | , dominated by the discontinuity of the second derivative of Ψ ^ ( ω ) at ω = 0 .

This wavelet was proposed in 2002 by Addison et al. for applications requiring high temporal precision time-frequency analysis.

References

Complex Mexican hat wavelet Wikipedia