y^(nAt) = y(nAt) 



y^CnAt) = y[(n + j) At] (23) 



y.(AAt) = y{[n*+ (i - 1) J] At} , J < N 



Differences between successive spectra computed by this approach can be 

 attributed to the nonoverlapped sections of record; hence, frequency con- 

 stituents of a short section of record can be inferred. The approach produced 

 inconsistent results and was abandoned. 



Another difficulty identified in preliminary FFT analysis was the vari- 

 ability of spectral phases, (^^ in equation (13), between adjacent 

 frequencies which appeared in the amplitude spectrum to be part of a single 

 spectral peak. Phase relationships in the spectrum certainly cannot be 

 studied unless an unambiguous phase value can be assigned to each concentra- 

 tion of energy in the detailed spectrum. Tests with sinusoidal waves were 

 analyzed to more clearly identify what was happening to phases. Tests with a 

 single sinusoidal component indicated a gradual variation of phase up to the 

 spectral peak, a jump in phase of n radians at the peak, and a gradual 

 variation on the other side of the peak. 



The same behavior can be demonstrated analytically from equation (17) 

 for a single sinusoidal wave. 6, (j), and m in equation (17) are fixed 

 for a given sinusoidal time series. Hence, a and b can change only in 

 response to changes in tan iT(m - m + 6)/N as m varies. However, tan ir(m - m 

 + 6)/N changes gradually with m over the entire range 1 $ m 5 N. As m 

 approaches m from below, tan ir(m - m + 6)/N is positive but approaches zero. 

 If < 6 < 1/2, tan •ir(m - m + 6)/N becomes less than zero when m = m + 1, 

 although it is still very small in magnitude. If -1/2 < 6 < 0, the change of 

 sign in the tangent function occurs when m = m. Thus, both a^^^ and h^^ in 

 equation (17) change sign when m = m or m = m+l, depending on the value 

 of 6. When ^ in equation (13) is computed over an interval of 2ii , the 

 change in sign of both a and b near the spectral peak appears as an 

 abrupt phase shift of ir. 



More insight on spectral phase can be obtained by substituting equation 

 (17) into equation (13) to get 



<},^ = TTfi - 4, (24) 



which is correct only for values of m very close to m. Since -1/2 < 6 < 

 1/2, the phase estimates returned by the FFT analysis at the spectral peak can 

 differ from the correct value <t> by as much as -n/l. This behavior is illus- 

 trated in Figure 10 for an artificial signal composed of three sinusoidal 

 waves. It is clear that the FFT phase estimate is determined by equation 

 (24), and it can differ greatly from the actual phase of the component waves. 



31 



