the statistical stability is low. Therefore, more statistical stability 

 is achieved with lower resolution and more degrees of freedom for each 

 estimate of energy density if the individual estimates are merged. This 

 is accomplished by programming the computer to combine the calculated 

 values of energy density into groups of, say, eight each, thus forming a 

 new series of 256 terms where each value represents the energy in a much 

 wider frequency band. Since the energy density of a power spectrum uses 

 mean-square amplitudes, the grouping of the Fourier components is the sum 

 of the squares of their members. The phase of each complex Fourier com- 

 ponent does not enter into the power spectrum and need not be considered 

 in the grouping process. 



The phases of the Fourier coefficients have an arbitrary zero reference 

 depending on the beginning of the time series. However, when two series 

 have the same beginning time and length, the phase differences between the 

 two are real and each phase difference is equivalent to that between two 

 records obtained from cross-spectral analysis using the Blackman-Tukey 

 approach . 



The computed complex Fourier coefficients of a single time series 

 provide a useful tool for laboratory studies of wave profile characteris- 

 tics since they contain information for calculating the phase relationship 

 between the fundamental frequency and each of its harmonics. For exarflple, 

 the wave profile from a train of laboratory-generated waves that are exactly 

 repeating, is completely defined by the amplitudes and relative phase angles 

 of its harmonics. 



38 



