Hilbert-Hermitian wavelet

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The Hilbert-Hermitian wavelet was designed by Jon Harrop in 2004 [the wavelets are a special case of the Morse wavelets, introduced by Daubechies and Paul (1988, Inverse Problems, vol 4, p 661-80)] for the reliable time-frequency analysis of signals that contain rapidly amplitude- or frequency-modulated components. This wavelet is formulated in Fourier space as the Hilbert analytic function of the Hermitian wavelet:

\hat{\Psi}_\sigma (\omega) = \begin{cases} c_\sigma \omega^\sigma e^{- \frac{1}{2} \sigma \omega^{2}} & \omega\geq0 \\ 0 & \omega\leq0 \end{cases}

where:

c_\sigma = \sqrt{2} \left( \sigma^{- (\sigma + \frac{1}{2})} \Gamma(\sigma + \frac{1}{2}) \right)

Whereas the parameter of the Hermitian wavelet is an integer, the parameter σ of this wavelet is generalised to any real number σ > 0. The wavelet is strictly admissible for these parameters.

As the Fourier transform \hat{\Psi}_\sigma (\omega) of this wavelet is zero for negative frequencies, time-frequency representations computed using this wavelet suffer less from the phase-dependent artefacts that can plague analyses performed using the Morlet wavelet with high temporal localisation. As this wavelet is based upon the Hermitian wavelet, this wavelet benefits from near-minimal time-frequency uncertainty.

The temporal variance Δ2 of this wavelet is minimised when \sigma = \frac{1}{4} (2 + \sqrt 2).

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