APPENDIX C 



ASYMPTOTIC CHI-SQUARED PROPERTIES OF THE FFT SPECTRAL LINES 

 FOR NON-GAUSSIAN, M-DEPENDENT WAVE TRAINS 



1 . Basic Definitions and Assumptions . 



Let n • , j - 0,±1 ,±2 ,±3, . . . be water level elevations about mean 

 water level. It will be assumed that {ri • } is a stationary second-order 

 stochastic process which is not necessarily Gaussian and that the random 

 sequence (n - } has the properties that, uniformly in n, 



E [r]^] = (C-1) 



E [n^] < M < 0° . (C-2) 



Let Yj^ = ria+n ^°^ " = 0, 1 , 2, 3, . . . ,N-1 . That is {Y^} is a finite 



sequence of the water level elevations starting at ri and terminating 



with n ., 1 . The time interval between water level elevation values is 

 a+N- 1 



denoted by At. The finite Fourier transform coefficients are defined as: 



N-1 



(N) _ ,, V Y e"^^""™/^ 



n 



n=0 



= yf^^ - i v^^^ , (c-3) 



where i = /^ . The superscript N is attached to Am to indicate 

 that the FFT coefficients are computed on the basis of a sequence of 

 length N. 



It will be assumed that the sequence n • is m-dependent. That is 



{ri, , n. 1 , ■ • . , lu } and {ri , n .,..., r\ } are statistically 

 b b+1 b+s c-r' c-r+1' ' c ^ 



independent sets of random variables if b - c > m (Rosen, 1967). 



The probability density for r\ . will be assumed to satisfy either 



conditions (a) or (b) under item 5 given in the following. The FFT 



coefficients will be said to be degenerate if p = and hence 



^ ^m 



U^ = V„ E 0. 



93 



