Hilbert transform of a signal to measure skewness and asymmetry of a sinusoidal wave

Question

Thank you for taking the time to read this. I was hoping to get some assistance in understanding how these equations function:

$$As=\frac{\langle H(\eta)^3\rangle}{\langle\eta^2\rangle^{3/2}},\qquad Sk=\frac{\langle\eta^3\rangle}{\langle\eta^2\rangle^{3/2}}$$

The angle brackets are time average, H is Hilbert transform and eta is water level fluctuation. As and Sk are wave asymmetry and skewness.

I would just like to understand the why adding the Hilbert transform results in determining the asymmetry of the signal.

Answer 1

For positive frequencies, the Fourier transform of the Hilbert transform is equal to $-i$ times the Fourier transform (see this Wiki entry). Skewness and asymmetry are defined as, respectively, the real and imaginary part of the third-order correlator of the Fourier transformed time series, as explained by Steve Elgar in Relationships involving third moments and bispectra of a harmonic process, so taking the Hilbert transform carries out the desired switch from real to imaginary part.