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.



Hilbert transform of a signal to measure skewness and asymmetry of a sinusoidal wave
For positive frequencies, the Fourier transform of the Hilbert transform is equal to $-i$ times the Fourier transform [(see this Wiki entry)](https://en.wikipedia.org/wiki/Hilbert_transform#Relationship_with_the_Fourier_transform). 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,](https://www.whoi.edu/science/AOPE/dept/Publications/007.pdf) so taking the Hilbert transform carries out the desired switch from real to imaginary part.


