the smoothing of spectra Is discussed In many texts (see for example 

 Bloomfleld, 1976) and will not be discussed in detail here. The spec- 

 trum presented in Figure 4-3 was in fact generated using the method of 

 Davis (1974) which utilizes prewhitening as well as low-pass filtering 

 of the spectral estimates. 



Another method of smoothing this rough spectrum is through the use 

 of regression techniques. Calculating a continuous mathematical func- 

 tion to describe the distribution of amplitudes would produce a smooth 

 representation of the spectrum while, depending upon the complexity of 

 the function used to fit the data, greatly reducing the number of param- 

 eters retained in the model. This least-squares representation will be 

 used in this study. 



By describing the spectrum with a simple, continuous mathematical 

 function, one can more easily take advantage of the many symmetry prop- 

 erties of the Fourier transform described by Bracewell (1978). For 

 example, Rayleigh's Theorem (or Parseval's Formula for discrete series) 

 states that the integral of the power spectrum equals the integral of 

 the squared modulus of the function, or 



r lf(x)I^ dx - r lF(s)l^ ds 



This is equivalent to stating that the total energy in one domain is 

 exactly equal to the total energy in the other. If one were Interested 

 in the total energy in a particular band of frequency (in order to study 

 the bottom Interaction of sound of a particular wavelength, for exam- 

 ple), the high and low frequencies of the band pass would become the 

 limits of integration of the power spectrum. With the spectrum repre- 



24 



