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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
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW      443
functions as t,he number of discontinuities and the magnitude of each impulse
funaion will be equal to the magnitude of the corresponding discontinuity.
     2) Multiplication of f(t) by e-ot. The Laplace transform ofthe function e-ar  f (t)
is given by
~[e-ar f (t)] =  f.    f(t)e-'yre-sr dt = f(s + cv)            (5.17)
Thus, multiplying the function  f (t) by e-at has the effect of replacing the Laplace
variable s by s + a. Here, ct may be real or complex.
   3) Change of time scale. Suppose the time t is changed to t/a, then
,[f(-.)]= f, f( )e-nd,
Let ti = tlcy and si - as. Then,
(5.18)
,[f (-.)] = J- f (ti)e-si"d(ati) = a f (si) = ar f (as)          (5.19)
As an example, consider f (t) ~ e-t so that f (t/4) = e-0.25r_ We have ct -. 4 and
s\ - 4s. Then,
/[ f (t)] = f (s) = ~l
[f(4)]
=a f (as) =
4
4s +
(5.20)
(5,21)
4) DVj'erentiation. The Laplace transform of the derivative of a funct,ion is given
,[dd f (t)] = s f (s) - f (0)
To prove this theorem, we proceed as follows:
Then,
so that
       e-sr
F fct)e-stdt=f(t)-.
f (s) =  f CO)
(5_22)
t = oo
              a, df (t) e-sr  .
                           - - dt             (5.23)
          dr  -s
t-0     -
+l/  d f (t) ]
 s  dt ]
,[dfd( )] = s f (s) - f (0)
(5.24)
(5.25)
~r
         tl; .
          : <::. .
    - .;r.:
        : \u
  ;.l{
 lim  : [A(l - e~s'o)] = A
ro ~ O         : (tOS)
8(t - ti) : O     t 7! ti
          - oo     t - tI
   [:.8(t - ti)dt = 1
PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
   In a sinular fashion, we can obtain the Laplace transforms of higher order
derrvatives of f (t). For example,
,[ddf  )] =s2 f(s)-sf(0)- f(0)
where  f (0) is the value of d f (t)ldt at t = 0.
   -5) Final value theorem. This theorem gives
 f (co) = tlun f (t) = sluno [s f (s)]
                                             s -.
To prove this theorem, take the limit as s approaches zero in Eq. (5,25).
(5.26)
(5.27)
sl., f, '   d:( )]e-s,dt = sl_,[s f (s)] - f (0)              (5.28)
Because e-s. : 1 as s -* 0, we have
Hence,
[,- [d~.( )]d, = f (oo) - f (0) = l.o [s f (s)] - f (0)        (5.29)
                                                                          s -+
　
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