14 SAMPLED-DATA CONTROL SYSTEMS 
where the various C;’s are the Fourier coefficients of the exponential 
series. Substituting (2.2) back in (2.1), there results a representation 
of the sampling process which is useful in giving an insight into its 
characteristics and implications. The expression so obtained is 
+ 0 
OS elie (2.3) 
k=—o 
It is of interest to observe the Fourier transform of the sampled 
function f*(¢) and to contrast it with the Fourier transform of the con- 
(a) 

(b) Time 
(c) 

Fic. 2.3. Finite pulse sampling operation. 
tinuous function f(¢) from which the sequence was obtained. Recalling 
the shifting theorem which states that 
S[e“fO] = FGw — d) (2.4) 
where F(jw) is the Fourier transform of f(é), and applying this relation- 
ship to the expression (2.3), there results the infinite summation 
i} 
F*(ju) = dor |; (. 2 ae (2.5) 
where F*(jw) is the Fourier transform of the sampled sequence f*(t). 
This expression is very important in determining the effects of sampling 
on the information content of the original signal f(t). F*(jw) is seen to 
