theoretical equations in the one-dimensional case are given by 



(8.13) • Q(t') = lim ^ I -r^t) 1 (t + t')dt 



ts>oo 



and 



CD 



( 



(8.14) [A(|JL)]^=-i 



IT 



Q(t') cos fxt' dt' 



-oo 



The equations due to Tukey [1949] are given by equations (8. 15) to (8. 17) 

 where Ni , No, N^j ..... N^ are given values equally spaced usually in time. 



(8. 15) Q(p) = -^^ Y N(^) N(k + p) 



^-1 p = 0,lj...<,5m. 



Q;'' = Qq. Qp* = 2Qp. (p = 1 to m - 1), and Q^ = Q^ 



'o 



<^-^^> 1 "^ « ^ph 



JLv =~ S Q cos 



ii m "^p m 



P=u h = 0, 1, o . . o s m 



Let L_-j^ = L._|_j^ , and Lm-l = ^m+l 



(8.17) Uj^ = 0.23 1^_^ + 0.54 1^ + O.ZSI^^],, 



h=0, 1, .,.., m, 



Define U^* =• U^/Z, Uj, = Uh (h = 1 to 19), U^ = U^/2, 



In the above equations (8. 15) is the discrete approximation to equation 

 (8.13). Also in equation (8.13), Q(t') equals Q(-t') and to obtain the 



65 



