SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 263 
of the real frequency, 
1 sinh wT 2(0 > cosiol) 
Swe) = T cosh oil’ — cos wT w? (10.46) 

It is possible to verify by integration of (10.46) over all frequencies that 
the mean-square value of the output is unity, which is the same as the 
mean-square value of the input. This particular calculation is really 

--—--—— S,,(jw) input 
\ ——-— S,;,Vw) output of sampler 
\ On Sy,y(jw) output of clamp / 

ee 
na 
4 
37 
ie ae 
Fia. 10.5. Plots of spectra showing the effects of sampling and clamping. 
unnecessary, however, because by inspection of Fig. 10.3 with the gain 
factor removed and y = T itis obvious that the mean-square value of the 
output of a zero-order hold is always equal to the mean-square value of 
the input signal. For very small 7, the spectrum of the output closely 
resembles that of the input and, in the limit as 7’ — 0, one obtains 
1 sinh w,T 2(1 — cos wT’) 

pas Sw(je) = pm T cosh wiT — cos wT" w?” 
pbs Qa 
“orto 
= Szz(jw) (10.47) 
Plots of S,2(jw), T?.S*,(jw), and S,,(jw) for the example under consideration 
are given in Fig. 10.5 for the particular case of w:1 = 1, 7 = 1. The ratio 
