Complex Mexican hat wavelet - Alchetron, the free social encyclopedia

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

(Text) CC BY-SA