Analysis with the FFT Algorithm

For about 30 years, routine analysis of wave data has been dominated by the fast Fourier transform (FFT) algorithm. The algorithm is incredibly fast, easy to use, and highly appropriate within the context of linear directional mixtures of waves. If the water level elevation time history is denoted by $[\eta (n \Delta t); n = 0, 1,2, \ldots, N-1]$, the FFT representation or Fourier series is

 

$$\eta (n \Delta t) = \Delta f \sum^{N-1}_{m=0} A (m\Delta f) e^{i2\pi mn/N}$$ (1)

where

 

$$A (m\Delta f) = \Delta t \sum^{N-1}_{n=0} \eta (n \Delta t) e^{-2\pi mn/N}$$ (2)

 

$$\Delta f = \frac1{N\Delta t}$$ (3)

Time is given as $t = n \Delta t$ and frequency is specified as $f = m \Delta f$.

Since

 

$$e^{\pm i 2 \pi mn/N} = \cos \left( \frac{2\pi mn}{N} \right) \pm i \sin \left( \frac{2\pi mn}{N} \right)$$ (4)

both $\eta (n \Delta t)$ and $A(m \Delta f)$ are formally periodic in time, with period $N\Delta t$, and in frequency, with period $N \Delta f$. If stationarity is imposed on this periodicity, to extend the formulas formally outside the data interval of $(0,N\Delta t)$, the computational structure for spectral estimation becomes remarkably simple and convenient.

However, there is a dark side to this. For if the representation is both periodic and stationary, then the Fourier coefficient A_m = U_m - i V_m is uncorrelated with any other coefficient A_m' with 0 < m, m' < N/2. Furthermore, U_m and V_m are uncorrelated with each other [3]. This is true whether the sea surface is Gaussian or not. Thus, the FFT conventions have imposed a certain structure on the analytical results which is an artifact of the analysis, and not a necessary feature of the actual wave hydrodynamics.

One of the features of nonlinear unidirectional waves of permanent form [7,8] is that second, third, and higher harmonics of the first harmonic of the wave profile travel at the same celerity as the first harmonic. This maintains the wave profile in a skewed, sharp crest and flat trough shape as it travels. It also implies that higher harmonics are correlated in time with the first harmonic. The Fourier coefficients at these harmonics are not uncorrelated.

It appears obvious that directional mixtures of severely nonlinear waves should also have nonzero intercorrelation between harmonics. But the FFT representation has destroyed this! For spectral estimation, this artifact is relatively unimportant. However, if one is seeking to study nonlinear effects, it becomes highly significant.