3.1. A Third-Order Volterra Series Model

The Volterra series is a functional extension of linear (first-order) systems to bilinear (second-order), trilinear (third-order), and higher-order systems.36 These extensions require the multi-dimensional FRF, H( f, g) and H( f, g, h) respectively, to describe the bilinear and trilinear systems in place of the simple one dimensional FRF H( f) for a linear system.

In the time-domain, the general third-order Volterra equivalent system can be described as36 ^, 43:

(24)

where x( t) is a Gaussian input; y _i( t) ( i = 1, 2, 3) are the first-, second- and third-order response components respectively; h ₁( ?), h ₂( ? ₁, ? ₂) and h ₃( ? ₁, ? ₂, ? ₃) are the linear, bilinear and trilinear kernels of the impulse response functions respectively. Each kernel and the corresponding FRF form a Fourier transform pair, where for an nth-order model:

(25)

For Gaussian input, the odd-order response components will be mutually correlated, but uncorrelated with even-order system outputs. The even-order system outputs will also be mutually correlated. For non-Gaussian input data, all output components are usually correlated.

3.2. Application to Spectral Analysis of Wave Force

The total wave modal force Q acting on a platform leg can be decomposed into that described by Morison equation up to SWL, F, plus...