260 SAMPLED-DATA CONTROL SYSTEMS 
It is also noted that 
ino] 
Pex(kT')2* = Z[G(—s)] 
k=0 
Ie Wy Ae dn 
~ aj la GN To ere ee 
1 c+je 
and ®,,(0) = =— / G(s) ds (10.35) 
277 c—jo 
The substitution of (10.33), (10.34), and (10.35) in (10.32) gives 
OO Nar aye (1 — e?7) dd 
The combination of three terms in the one integral (10.36) requires that a 
common path of integration be found for all the terms. This require- 
ment will be met if the function G(s) [which is half the partial-fraction 
expansion of S,,(s) as expressed in (10.31)] has no poles on or to the right 
of the imaginary axis. The desired common path is then the imaginary 
axis itself. The evaluation of (10.36) may be effected by equating the 
desired contour integral to the closed integral around the left half plane, 
in which case the poles of the integrand are the poles of G(A). A difficulty 
arises if G(A) tends to zero as \ tends to infinity only as1/\. In this case 
the integral around the semicircle completing the closed contour does not 
vanish unless a slight shift of g(t) @ sec to the left is made, which intro- 
duces a factor e® in the integrand which forces the integral over the semi- 
circle to be zero. ‘This artifice is necessary because of the way the z 
transform is defined and leads to no real difficulty. 
EXAMPLE 
As an example of the application of the technique developed above, 
suppose it is desired to find the sampled power spectrum of the random 
signal whose spectral density is 


a 2w4 = 1 1 
Szz(S) ‘Le wy? Grd 2 ss, wy a 8 Dats (10.37) 
= G(s) + G(—s) 
Substituting (10.37) in (10.36), | 
ie ap pee 
Selatan / 1. ie eluate 
T2029 J —jo or +ALL — e Te ][1 — et Te] 
oa ips SCaNiecsk Salelenis Bese 
~ TL — e tere, — etre] 
eT ao! alien Cae (10.38) 
TA = eP2-1)\ (1 — e- Tz) 
