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~[ f (t)] = f (s) = l- f (t)e-sr dt (5.1)
where s is a complex variable, equal to cr + ja . Quite often, s is also called the
Laplace variable. We assume that f (t) satisfies all the conditions for the existence
of its Laplace transform.
The inverse of a Laplace transform is defined as
f (t) = ~-J[ f (s)] = /="j f. f(s)es' d, (5.2)
Some of the important theorems on Laplace transform are summarized in the
following.
1) Translatedt'unction. The Laplace transform ofthe translated function f (t- a)
(Fig. 5.1), where f (t - a) : O for O < t < ct can be obtained as follows.
0 a
Fig. 5.1 Translated function.
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF'REVIEW 441
A
We have
[o t
0
a) Pulse function b) Impulse funchon
Let -c ~ t - a. Then,
Fig.5.2 Pulse and impulse functions.
L[ f (r)] = f (s) = f.'n f (r)e-s' dC
(5.3)
f (s) = r f (t - a,)e-S('-LY)d, = e'' r f (t - a,)e-srdt = ea."L[ f (t _ a)]
(5.4)
Therefore,
L[ f (t - cr)] = e-aS f (s)
(5.5)
This theorem states that the translation ofthe time function f (t) by a corresponds
to a multiplication of its transform by e-as.
Using this theorem, one can obtain the Laplace transform of a pulse function as
follows. A pulse function (Fig. 5.2a) is given by
f (t) = A
-O
0 < t < to
t < O,to < t
(5.6)
The given pulse function can be expressed as a sum of two-step functions, each
of height A, one positive step function beginning at t -. 0 and the other negative
step function at t - to. Thus,
f (t) : Al(t) - Al(t - to)
(5.7)
where /(t) and l(t - to) are the unit step functions originating at t : 0 and t -. to,
:~
l -J
:B
:t~
i<
.G
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,: .. ,;
~~
'/. ,
jll J
:L.
'. .
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.:
442 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
respectively. The Laplace cransform of a unit step function is given by
Then,
L[ f (t)]
= F lct)e-s,dt = F le-s'dt = ~
(5.8)
L[ f (t)] = L[Al(t)] - L[Al(t - to)] = C(l - e-sro) (5.9)
The impulse function is a special limiting case of a pulse function. Consider the
pulse function given by
f (t) = ,hmo
(A)
O< t < to
-0 t<0 to<t
(5.10)
(5.11)
The height of the impulse is A/to and its duration is to. Therefore, the area under
the impulse is equal to A. As the duration to approaches zef0, the height of the
pulse approaches infinity giving us an impulse (Fig. 5.2b). Note that, even though
the height of the impulse tends to infinity, the area remains finite.
The Laplace transform of an impulse function is given by
L [ f (t)] = rl.0 A (1 - e-sr") =
(5.12)
An impulse function of infinite magnitude and zero duration is a mathematical
fiction. However, if the magnitude of a pulse input is very large and its duration is
very small, then we can approximate it as an impulse input. An impulse function
whose area is equal to unity is called a unit-impulse function or Dirac delta func-
tion. The Laplace transform of a unit-impulse function is unity. A unit-impulse
function occurring at t = ti is usually denoted by 8(t - ti), which has the following
properties:
(5.13)
(5.14)
(5.15)
The concept of the unit-impulse is very useful in differentiating discontinuous
functions. For example,
8(t) = ddt l(t)
(5.16)
where 8(t) and l(t) are the unit-impulse and unit-step functions, respectively, and
both occur at the origin. Thus, integrating a unit-impulse function, we get a unit-
step function* The concept ofimpulse function helps us represent functions involv-
ing multiple discontinuities. Such a representation will have that many impulse
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