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(3.7)
Changing the order and variables of integration, we obtain

(3.8)
where
Thus, the equivalent impulse response for the cascade is given by the convolution
of the cascaded impulse responses. The same conclusion can be reached by
inverse-transforming the product of the s-domain transfer functions. The time
domain expression shows more clearly that cascading results in a new form for
the impulse response. So if the output of the system is sampled to obtain
(3.9)
it is not possible to separate the three time functions that are convolved to produce
it.
By contrast, convolving an impulse-sampled function u(t) with a continuoustime
signal as shown in Figure 3.6 results in repetitions of the continuous-time
function, each of which is displaced to the location of an impulse in the train.
Unlike the earlier situation, the resultant time function is not entirely new,

FIGURE 3.5.
Cascade of two analog systems.

FIGURE 3.6
Analog system with sampled input.

CHAPTER 3 Modeling of Digital Control Systems

and there is hope of separating the functions that produced it. For a linear timeinvariant

(LTI) system with impulse-sampled input, the output is given by

(3.10)
Changing the order of summation and integration gives
(
(3.11)
Sampling the output yields the convolution summation
(3.12)
As discussed earlier, the convolution summation has the z-transform
(3.13)
or in s-domain notation
(3.14)
If a single block is an equivalent transfer function for the cascade of
Figure 3.5, then its components cannot be separated after sampling. However,
if the cascade is separated by samplers, then each block has a sampled output
and input as well as a z-domain transfer function. For n blocks not separated by
samplers, we use the notation

(3.15)
as opposed to n blocks separated by samplers where

(3.16)