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3.3 The transfer function of the ZOH
To obtain the transfer function of the ZOH, we replace the number or discrete
impulse shown in Figure 3.3 by an impulse δ(t). The transfer function can then
be obtained by Laplace transformation of the impulse response. As shown in the
figure, the impulse response is a unit pulse of width T. A pulse can be represented
as a positive step at time zero followed by a negative step at time T. Using the
Laplace transform of a unit step and the time delay theorem for Laplace
transforms,

(3.2)
where 1(t) denotes a unit step.
Thus, the transfer function of the ZOH is

= (3.3)

Next, we consider the frequency response of the ZOH:

= (3.4)

CHAPTER 3 Modeling of Digital Control Systems



FIGURE 3.4
Magnitude of the frequency response of the zero-order hold with T=1 s.
We rewrite the frequency response in the form
=

= =
We now have

(3.5)
In the frequency range of interest where the sinc function is positive, the angle of
frequency response of the ZOH hold is seen to decrease linearly with frequency,
whereas the magnitude is proportional to the sinc function. As shown in Figure 3.4,
the magnitude is oscillatory with its peak magnitude equal to the sampling period

and occurring at the zero frequency.

3.4 Effect of the sampler on the transfer function

of a cascade
In a discrete-time system including several analog subsystems in cascade and several
samplers, the location of the sampler plays an important role in determining
3.4 Effect of the sampler on the transfer function of a cascade
the overall transfer function. Assuming that interconnection does not change the
mathematical models of the subsystems, the Laplace transform of the output of
the system of Figure 3.5 is given by

(3.6)
Inverse Laplace transforming gives the time response