APPENDIX A 



RECENT TRENDS IN COMPUTER PROGRAMMING FOR SPECTRAL ANALYSIS 



An appreciable savings in computation time can be achieved by using 

 an arithmetic algorithm developed by Yates CI937) and expounded later for 

 use in Fourier analysis (Good, 1958). The algorithm was modified recently 

 to adapt it to the binary arithmetic used in modern digital computers 

 (Cooley and Tukey, 1955). A computer program using the Cooley-Tukey 

 method produces a series of complex Fourier coefficients, and additional 

 programming isolates the amplitudes and phase relationships of the co- 

 efficients. Thus it is ideally adapted to the analysis of repeating func- 

 tions such as the astronomic tides or water waves in the laboratory. The 

 Blackman-Tukey procedure, on the other hand, involves the auto-correlation 

 function whose cosine transform gives the spectrum as a function of lag. 

 This procedure was designed to detect low amplitude signals in a back- 

 ground of random noise, and is therefore useful for analyzing time series 

 such as those from complex ocean waves. 



The Cooley-Tukey procedure assumes that the record consists of repeat- 

 ing functions; and, since the data time series is of finite length, the same 

 care must be taken in its use as with the use of sine and cosine transforms 

 in the previous programming methods. The amplitudes of the Fourier harmonic 

 terms do not constitute a power spectrum. However, under certain conditions, 

 the absolute values (magnitudes) of the complex Fourier coefficients may be 

 squared to give an approximation of the power spectrum. 



The Cooley-Tukey procedure requires that the number of data points in 

 the time series be some arbitrary power of a small interger. For our pur- 

 poses it is convenient to use 2^^ which gives 4096 data points. Since the 

 last half of the coefficients are complex conjugates of the first half, 

 2048 data points would result from 4096 data points. It should be noted 

 that each point obtained from half of the Fourier transform, when squared, 

 represents only half of the energy density for its particular frequency 

 band . 



Waves generated in the laboratory are of single fundamental frequency 

 with very little spread of the spectral peak. The energy density estimate 

 can be calculated with no further processing. The computation of 2048 

 spectral points i s 'conven lent for laboratory data where high' resol ut ion 

 is necessary to isolate the very narrow frequency band of maximum energy 

 dens ity. 



Ocean waves, on the other hand, present a spectrum with random variables 

 having a standard deviation that is usually large with respect to the band 

 width of a single estimate. Each calculated coefficient in the Cooley-Tukey 

 procedure has two degrees of freedom and is nearly independent of its neigh- 

 bors. Since the fundamental frequency band is usually wide for ocean waves. 



37 



