discrete approximation to (8„ 14), Q(p) is expanded as a periodic even function 

 about p equal to zero. Thus Q received a weight of one, Q, through Q , 

 receive double weight, and Q^ received a weight of one. 



The values of L are the discrete estimates of the Fourier coefficients of 

 the even expansion of Q. 



Due to the fact that the L's are only estimates of the spectrum since the 

 series of readings is finite, they have to be filtered to recover a smoothed 

 estimate of the spectrum in terms of the U's. 



There is^, of course, another way to estimate the spectrum. The original 

 series of points could be expanded in a Fourier series. Sine and cosine co- 

 efficients a^^ and b^^, for periods of nAt/1, nAt/2, nAt/3, etc. would then be 



2 2 2 



computed. The quantity c^^ = a + b is then a very unstable estimate of 



the energy at that particular frequency. A proper running weighted average 

 of the values of c^ would then recover the spectrum as determined by the 



Tukey method. The number of degrees of freedom (f) is a measure of the num- 



2 

 ber of values of c weighted in the average and of the shape of the weighting 



process. The U's have a Chi Square distribution with f degrees of freedom. 



The values of U have the dimensions of (length) , and U-, as given 

 above is an estimate of the contribution to the total variance made by frequen- 

 cies in the range from 2iT(h --1)/Atm to 2iT(h +"2)/Atm. 



The theoretical equations in this two- variable problem are given by 



+ 

 For h 3 and h = m the values of U must be halved since one of the 



frequencies defined above is not applicable. 



66 



