34 SAMPLED-DATA CONTROL SYSTEMS 
For instance, if a very simple form of data extrapolation in which only 
the first term, r(n7'), is taken, the extrapolator is referred to as a zero-order 
hold since the polynomial generated by this system is of zero order. 
Similarly, if the first two terms are implemented, the classification is 
first-order since the polynomial which will be extrapolated is of first order. 
It is recognized that the zero-order data hold is also known as a data clamp 
and that it operates on the assumption that the value of the function 
in a given sampling interval is equal to the function at the beginning of 
that interval. This extrapolation is perfect only for functions which are 
constants. Practical systems rarely employ data extrapolators which 
are beyond first order, both for reasons of economy and because if too 
many back differences are taken, an excessive settling time results and 
noise effects are increased. These points will be discussed later in more 
detail. 
3.3 The Zero-order Data Hold 
As indicated in the previous section, the zero-order data hold includes 
only the first term of the series as expressed in (3.12). This form of 
data hold is important from a practical point of view because of its 
simplicity and the fact that it is readily implemented. A standard 
electronic clamp circuit will set its output at a level equal to or propor- 
tional to the magnitude of an input pulse and then reset itself when a 
new pulse is applied. Such circuits maintain a constant output between 
pulses and thus implement the zero-order-hold relationship. Similarly, a 
digital register will hold a number until a reset pulse is applied and a new 
number set up. In all cases, the output of the device essentially assumes 
that the continuous function within a sampling interval is constant and 
equal to the value of the function at the preceding sampling instant. 
The form of the reconstructed function at the output of a zero-order 
hold is shown in Fig. 3.4. The extrapolation in each case is a constant 
and is refreshed at each sampling instant. Because of the appearance 
of the reconstructed function, the data hold is sometimes referred to as a 
“staircase” or “‘boxcar’’ data system. 
In order to include the effects of a zero-order data hold in a dynamical 
system, it is necessary that a mathematical description of its effect be 
obtained and, if possible, a transfer function derived. It is recalled that 
the transfer function of a linear system is the Laplace transform of the 
impulsive response of the system. In the case of a zero-order data hold, 
it is assumed that the short pulse which is applied to its input is approxi- 
mated by an impulse of an area equal to the magnitude of the pulse. 
If an impulse of unit area is applied to the data hold, its response should 
be a unit-magnitude continuous function which is maintained until 
