24 SAMPLED-DATA CONTROL SYSTEMS 
For practical problems, it would be awkward if the Laplace transform 
were an infinite summation which could not be expressed in closed form. 
Fortunately, this is not the case for those time functions f(t) whose 
Laplace transform can be expressed as a ratio of polynomials ins. This 
will be proved in later chapters but will be demonstrated with a simple 
example here. 
EXAMPLE 
The function f(t) is the unit step function u(t). This means that 
all values of f(n7’) are equal to unity for positive n and the Laplace 
transform F'*(s) given in (2.23) is simply 
+00 
Pi) = » e77Ts 
n=0 
The infinite summation is recognized as a geometric progression in 
e-Ts whose sum is 
1 
* = 
F (s) = quater: 
It is seen in this example that F*(s) could be readily expressed in 
closed form in terms of e-7*. This simplification was in consequence 
of the assumption of impulse sampling, with the result that the Laplace 
transform of each term of the sequence was simply the value f(n7’) and 
a delay factor e~"?*. Had the switching function p(t) been of different 
form, the Laplace transform of the typical term of the sequence would 
have contained contributions of the switching pulses themselves and the 
resultant expressions would have been much more complex. Another 
point which is brought out in this simple example is the mathematical 
desirability of having all equal sampling intervals. Had they been 
unequal, the infinite series would not have been a simple geometric 
progression with a constant ratio and its summation could not have been 
expressed in closed form. Fortunately, the assumptions which had to be 
made are also representative of the practical situation. 
An alternate form of Laplace-transform representation of the impulse 
sequence can be obtained by using the form for f*(¢) given in (2.12): 
FQ = fe)sr() (2.24) 
Since 67(t) is a periodic function, it is expressed in terms of a Fourier 
series, all of whose coefficients are equal to 1/7’, as shown in (2.14) and 
(2.15). Thus (2.20) can be written as 
+ 
$0) =), Mee (2.25) 
n=—0 
