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Unformatted text preview: a Function by an Impulse Let us now consider what happens when we multiply the u nit impulse 6(t) by
a function ¢ (t) t hat is known t o b e continuous a t t = O. Since t he impulse exists
only a t t = 0, a nd t he value of ¢ (t) a t t = 0 is ¢(O), we o btain ¢(t)8(t) = ¢(0)6(t) (1.23a) Similarly, if ¢ (t) is multiplied by a n impulse 6(t - T) (impulse located a t t = T ),
t hen ¢ (t)6(t - T) = ¢ (T)6(t - T)
provided ¢ (t) is continuous a t t = T . (1.24a) provided ¢ (t) is continuous a t t = O. T his result means t hat t he area under t he
product o f a function with an impulse 8(t) is equal to the value o f t hat function a t
t he instant where the unit impulse is located. T his p roperty is very i mportant and
useful, a nd is known as t he s ampling o r s ifting p roperty of the unit impulse.
From Eq. (1.23b) i t follows t hat I: t- 8(t) d t = ¢(O) (b) (a) I: (1.23b) ¢ (t)6(t - T) d t = ¢ (T) (1.24b) Equation (1.24b) is j ust a nother form of sampling or sifting property. In the case
of Eq. (1.24b), the impulse 8(t - T) is l ocated a t t = T . Therefore, the area under
¢ (t)6(t - T) is ¢ (T), t he value of ¢ (t) a t t he instant where t he impulse is located
( at t = T ). I n these derivations we have assumed t hat t he function is continuous
a t t he instant where the impulse is located.
U nit Impulse a s a Generalized Function T he definition of the unit impulse function given in Eq. (1.21) is n ot m athe·
matically rigorous, which leads t o serious difficulties. First, the impulse function
does not define a unique function: for example, i t c an be shown t hat 6(t) + 6(t)
also satisfies Eq. (1.21).1 Moreover, 6(t) is not even a t rue function in the ordinary
sense. An ordinary function is specified by its values for all time t. T he impulse
function is zero everywhere except a t t = 0, a nd a t t his only interesting p art of its
range i t is undefined. These difficulties are resolved by defining the impulse as a
generalized function r ather t han a n ordinary function. A g eneralized f unction is
defined by its effect on other functions instead of by its value a t every instant of
In this approach the impulse function is defined by the sampling property [Eq.
(1.24)J. We say nothing about what t he impulse function is or what i t looks like.
Instead, t he impulse function is defined in terms of its effect on a t est function
¢ (t). We define a unit impulse as a function for which the area under its product
with a function ¢ (t) is equal t o t he value of the function ¢ (t) a t t he instant where
t he impulse is located. I t is assumed t hat ¢ (t) is continuous a t t he location of the
impulse. Therefore, either Eq. (1.24a) or (1.24b) can serve as a definition of the
impulse function in this approach. Recall t hat t he sampling property [Eq. (1.24)J
is t he consequence of the classical (Dirac) definition of impulse in Eq. (1.21). In
contrast, the sampling property [Eq. (1.24)J defines the impulse function in the
generalized function approach.
We now present an interesting application of the...
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