26 SAMPLED-DATA CONTROL SYSTEMS 
Much more will be said on the subject in later chapters dealing with the 
z transformation. 
2.6 Comparison with Carrier-modulation System 
In previous sections the process of sampling has been treated as a 
form of modulation in which the switching or sampling function p(f) is a 
periodic pulse train. The unusual feature of sampled signals of the type 
described is that the signal spectrum can be recovered by a linear filter. 
For instance, in the case of signals having finite spectra, a perfect low- 
|Flju)| 
-—w 0 @ —> We 
(b) 
Fia. 2.9. (a) Spectrum of signal. (b) Carrier-modulated signal. 
pass filter whose bandwidth extends from zero to half the sampling 
frequency can recover the signal perfectly, provided that the sampling 
rate is sufficiently high. It is of interest to compare this type of sampling 
with the more usual form employed in a-c feedback control systems. 
In the usual carrier system, the signal modulates a sinusoidal carrier 
whose expression is 
p(t)= E, cos wet (2.28) 
The expression for this carrier-modulated signal is thus 
f*(t) = Bf COS wet (2.29) 
Expressing the cosine in its exponential form, f*(t) becomes 
F(t) = fafQhe* + gfe] BE. (2.30) 
Obtaining the Fourier transform of f*(¢) by application of the shifting 
theorem, F'*(jw) is 
F* (jw) = [$F (go + joc) + $F (jw — jw.) |H. (2.31) 
