SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 259 
10.3 Calculation of the Sampled Power Spectrum 
Since the sampled power spectrum defined in (10.25) is required in the 
analysis of sampled random signals, it is desirable to be able to calculate 
this function easily from given data. For example, one may be given the 
power spectrum of the continuous signal x(¢) and be required to find the 
sampled power spectrum. A simple formula is possible in this case if 
Sz2(s), the given power spectrum, is a rational function of s. Suppose 
the known power spectrum is 
Soo(s) = [~ _ ®sa(n)e" dr 
which may be written 
0 a 
Seo(s) = fo, Bealr)ew dr + f° Baalr)e—™ de 
Since (7) is an even function, the change of variable ¢ = —7 in the first 
integral leads to 
Seo(s) = [ Sec(o)e do + f° Beo(nyen" dr 
= G(—s) + G(s) (10.31) 
Where all the poles of G(s) le in the left half plane and all the poles of 
G(—s) lie in the right half plane. 
From (10.24) and (10.25) the sampled power spectrum is given by 
Lyes y &,,(nT)e 
xe Tr ox 
n=—2 
0 0 
=F | » b..(nT)2e" + » Bex(n Tz — #0) 
n=— n=0 
by substitution of k = —n in the first sum 
Ser(Z) = = Bs ®,.(kT)2® + >, ®,,(nT)z" — #.(0)| (10.32) 
k=0 n=0 
The formula for the sampled power spectrum depends upon the ability to 
express each of the three terms in (10.32) in terms of the function G(s) 
given in (10.31). A comparison of these two equations shows immedi- 
ately that 
Y bx(nT)e* = ZIG(s)] 
n=0 
which may be evaluated by complex convolution (4.10) as 
1 fete dd 
Z[G(s)] => a | ; G(A) T—ete» (10.33) 
c—jo 
