§ 15-2 The unit-impulse function
We define the unit impulse as a function of time which is zero
when its argument,generally (t-t0),is less than zero; which is also
zero when argument is greater than zero; which is infinite when
its argument is zero; and which has unit area.
??
???
??
??
? ????? ??00 1)(1)(
00)(
dttordtta n d
tt
??
?
0t
t
1
)( 0tt??
0
t
1
)(t?
0
??
???
????
???
? ????? ?
?
0
0
1)(1)(
0)(
00
00
t
t
dtttordtttan d
tttt
??
?
The strength of the impulse
5)(5 ?t?
5.2)3(5.2 ???? t?
t
5
)(t?
0
5.2?
3
The mathematical form ? ???? ? )0()()( fdtttf ?
? ???? ?? )()()( 00 tfdttttf ?
,3 6 8.0)2(2/ ??? te t ?
.707.0)()4/5s i n ( ?? tt ???
If the unit impulse is multiplied by a function of time,then
the strength of the impulse must be the value of that function at
the time for which the impulse argument is zero,In other
words,and the strength of the impulse
(a) A rectangular pulse of unit area which approaches a unit
impulse as,0??
(b) A modified(准 ) unit-step function.
(c) The derivative of the modified unit step.
The unit impulse may be regarded as the time derivative
of the unit step function.
dt
tdut )()( ?? ? ? ?? t tdtttuor 0 )0()()( ?
??21 ?21
?
1
)(a
t
)(tf
??21 ?21
1
)(b
t
)(tf
??21 ?21
?
1
)(c
t
dttdf /)(