Because the Gaussian process of the third-order cumulant spectrum is zero, it is often used to analyze the non Gaussian random process. Moreover, the third-order cumulant spectrum can detect the quadratic phase coupling, so it is often used to decouple the complex mechanical vibration signals. However, the third-order cumulant spectrum is a two-dimensional function. Due to the large amount of calculation, people usually use the one-dimensional diagonal slice spectrum of the third-order cumulant to analyze the vibration signal, which greatly reduces the amount of calculation.

The one-dimensional diagonal slice of the third-order cumulant of a non Gaussian stationary stochastic process x (T) is defined as:

Where: τ – delay; E [·] – statistical mean.

The one-dimensional Fourier transform of the one-dimensional diagonal slice of the third-order cumulant is called the one-dimensional diagonal slice spectrum or 1.5-dimensional spectrum of the third-order cumulant

Where: ω – angular frequency.