applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemat ...

, the complex Mexican hat wavelet is a low-oscillation,

complex-valued In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...

for the

continuous wavelet transform Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

. This wavelet is formulated in terms of its

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

as the Hilbert

analytic signal In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components. The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hil ...

of the conventional

Mexican hat wavelet In mathematics and numerical analysis, the Ricker wavelet :\psi(t) = \frac \left(1 - \left(\frac\right)^2 \right) e^ is the negative normalized second derivative of a Gaussian function, i.e., up to scale and normalization, the second Hermite fu ...

: :

\hat(\omega) = \begin
  2\sqrt\pi^\omega^2 e^ & \omega\geq0 \\
  0 & \omega\leq 0.
\end

Temporally, this wavelet can be expressed in terms of the

error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non- elementa ...

, as: :

left(1 - t^2\right)e^\right)\right).

This wavelet has

O\left(, t, ^\right)

asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

temporal decay in

, \Psi(t),

, dominated by the discontinuity of the second

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

\hat(\omega)

\omega = 0

. This wavelet was proposed in 2002 by Addison ''et al.''P. S. Addison, ''et al.'', ''The Journal of Sound and Vibration'', 2002
for applications requiring high temporal precision time-frequency analysis.

References

{{reflist Continuous wavelets