BEHAVIOR OF SYSTEMS BETWEEN SAMPLING INSTANTS 219 
they be unacceptable. Various techniques have been developed which 
will permit partial or selective evaluation of the time function between 
sampling instants. The least accurate of these methods uses the infinite- 
summation form of the Laplace transform of the sampled signal combined 
with the continuous plant transfer function. The resultant expression is 
inverted approximately by taking only the first few significant terms of 
the infinite summation. This method is very difficult to apply to feed- 
back systems and has only limited value. 
A more satisfactory method involves the use of multiple-rate sampling 
at the output of the system. Ey obtaining exactly the values of the 
extra samples which lie between the unit rate samples of the variable a 
measure of the ripple is obtained. The method gives only limited though 
accurate information describing the ripple, but because of its simplicity, 
it can be readily applied and interpreted. 
The partial-fraction technique is an exact method which gives the 
Laplace transform of the continuous output in any chosen interval. 
Because the particular sampling interval can be chosen at will, this 
method can be applied directly to feedback control systems whose peak 
overshoot is being sought. Furthermore, because the continuous-element 
transfer function is decomposed into partial fractions, the components 
which contribute most to the ripple output can be readily identified. 
Corrective compensation can then be applied, though even with this 
method it is not readily accomplished. 
A closely related method is the one in which advanced or modified 
z transforms are used. In this approach, an exact expression for the 
ripple is obtained by using methods which are entirely analogous to those 
for ordinary z transforms. A disadvantage of the method as compared 
with the partial-fraction-expansion method is that it is not readily evident 
which parts of the plant transfer function contribute most significantly to 
the ripple. On the other hand, the one-to-one relationship to the ordi- 
nary z-transform method makes the advanced z-transform technique 
attractive. 
A phenomenon which is observed in sampled-data systems is that of 
“hidden oscillations.”” These can occur as a result of the presence of 
poles in the continuous elements, which happen to produce oscillations 
whose periods are exactly equal to or multiples of the sampling period. 
From a practical viewpoint, these can be readily anticipated and avoided 
by the designer. In conclusion, the behavior of sampled-data systems 
between sampling instants can be studied by one or more methods, each 
of which has advantages and disadvantages. They should be applied 
with a view of obtaining the desired information with the least effort. 
