TM NO. 377 

 This may te vritten: 



4v,t.)-g'[<|5jTj-l,iI,c^^^,)^ 



(A-39) 



This equation provides a siniple relationship of the auto-covariance of the 

 driving motion u(t) with that of the instrument response function ^|)(t ) . 



Since Tp = K^ (l)~l and, from (A-'12)j eqvials (Tj.)-!, the limit may be 

 ■written : 



^^^ ^tr) 'ik 4^wtT), (A.40) 



Of coiorse, this limiting condition (Tj."*0) is not very realistic, since, for 

 any instrument, Tp is finite. 



Examining the relation (A-39) further — the formal equation for <*> (t) may 

 be written, from (lll»24)j ^ 



4l>CT). h^:t^C^-M^c^-^u^. 



).T,^ 



(A-2.1) 



Lee (i960) shows that the second derivative of ^^f^^(f)^^ given as: 



The second derivative of the auto-covariance function is another auto-covari- 

 ance function of the derivative of ^t) except for the negative sign. Thtis, 

 in (A-39) the correction term , /z 



is a positive bias in relation to the estimate of (PuCl)^ Tbe probable magnitudes 

 of the derivative term with respect to the auto-covariance in equation (A=39) can 

 now be examined. Assume that the driving function ^ is represented simply as : 



U/= ^(/sjJl-t J (A-43) 



where -/Z is the frequency of the oscillation. The auto-covariance function is 

 d>i^(Y) f which is approximated by: 



4^^*^^ ^r s«^-^*^"^-^^^^'''Jc/r, (A-4if) 



-'/J. 



A-8 



