38 SAMPLED-DATA CONTROL SYSTEMS 
jected to a pulse train, so that preceding the pulse applied at zero time 
there had been another pulse applied 7 sec earlier. This prior pulse 
generated an impulsive response which contributed a negative slope in 
the interval under consideration equal to the previous sample divided 
by the sampling interval 7. In this manner, the slope of the extrapolated 
signal has the correct slope as re- 
quired by (38.17). 
The impulsive response shown in 
Fig. 3.9 can be decomposed into a 
set of component step and ramp func- 
tions, as shown in Fig. 3.10. It is 
readily verified that the sum of these 
components produces the correct 
form for g,(t). The Laplace trans- 
form of the impulsive response is the 
sum of the Laplace transforms of the 

Fic. 3.9. Impulsive response of a first- 
order data hold. components as follows: 
= 1 xls = 2 —Ts __ Za —Ts 1 —2Ts a —2T's 
Gis) = + TO ee 73 ¢ + ee = T° (3.18) 
Combining terms, 
— p—Ta\2 
Gi(s) = Td + Ts) = (8.19) 
This transfer function may be used in the analysis of systems incorporat- 
ing this type of hold system. 
It is of interest to obtain the frequency response of the first-order hold 
by replacing s by jw. Doing so and simplifying the resulting expression, 
this response is given by 
G(jo) = PJD oT ie ae 
wT /2 

2 
) /-wT + tan“! wT (38.20) 
This response is sketched in Fig. 3.11, where it is seen that the frequency 
passband from zero to the first zero-transmission frequency is 2/T, 
which is the same as that of the zero-order hold. The major difference 
between the two frequency-response characteristics is that the first-order 
hold accentuates some of the frequencies in the passband excessively. 
In addition, the transmission of the higher-frequency components is 
greater and is reflected in the time domain by higher-ripple components 
under certain circumstances. On the other hand, the first-order hold 
is capable of reproducing a ramp function perfectly, as contrasted to the 
zero-order hold, which can reproduce only a step function or constant 
signal perfectly after a transient period. 
