THE SAMPLING PROCESS 745) 
where wo is the sampling frequency and is equal to 27/T. Taking the 
Laplace transform of each term in the series and making use of the 
shifting theorem, the Laplace transform of f*(¢) becomes 
+ 0 
F*(s) = . » iG ae aed) (2.26) 
nrn=—o 
where F'(s) is the Laplace transform of the continuous function f(é) from 
which the sequence was obtained. 
It is of interest to note the equivalence between the expressions for 
F*(s) as given in (2.26) and (2.23): 
+0 
+ 0 
rd, Feet niw) = )) sore (2.27) 
n=0 
n=—-—-o 
This equivalence was discovered more than a century ago by Poisson 
and is essentially equivalent to the Poisson summation rule.4? To 
illustrate the form in which Laplace transforms of impulse-sampled 
sequences appear using this form of transform, an example will be used. 
EXAMPLE 
The function f(t) will be taken as the unit step u(¢) as in the pre- 
ceding example. The Laplace transform of the unit step is 1/s, so 
that the Laplace transform of the impulse sequence obtained from 
sampling a unit step is 
+o 
SQ) a BUBB 
eS aac, » 8 + njwo 
n=—o 
Unlike the previous example, this infinite summation cannot be readily 
expressed in closed form in terms of rational polynomials in s. 
The fact that the infinite summation in this example cannot be ex- 
pressed in closed form limits the usefulness of the form of F*(s) as given 
in (2.26). On the other hand, it shows much more clearly some of the 
periodic properties of the Laplace transform of a pulse sequence. F*(s) 
is seen to be periodic in jwo, a fact which is useful in establishing many 
of the theorems governing the manipulation of pulse transforms. In 
many cases, only a few of the terms of (2.26) produce significant effects 
in a linear system, and by taking into account only the first few terms of 
the series, applications to the synthesis of sampled-data systems can 
readily be made. It is true, nevertheless, that in most sampled-data- 
system problems, the form of F*(s) given in (2.23) finds more application. 
