34 Sgnats and Systems Chap.
kh()
Figure 1.35 Continuous Figure 1.36 Sca.ed im
ome untt ampulse. pulse.
Note that Ss1) 1s a short pulse, of duration A and with unit area for any value of A.
As A ’ 0, S,(r) becomes narrower and higher, maintaining its unit area. Its limiting form,
S(1) = (1.74)
can then be thought of as an idealization of the short pulse &s(1) as the duration Abecomes
insignåcant. Since 8(i) has, in effect, no duration but unit area, we adop1 the graphical
notation for it shown in Fgure I.35. where the arow at t = 0inhcates that the area of the
pulse is concentrated at = 0 and the height of the arow and the "1" next to he arrow
are used to represent the area of the impukse. More generally. a scaled smpulse kõe) will
have an area k, and thås,
kS(r) dr- ku(i).
Ascaled impulse with arca k 1s shown in Figure 1.36, where the height of the urrow used
to depict the scalcd impuise is chosen to be prop1rtional to the area of the impulse.
As with d1serete time, we can providea simple graphical interpretation of the running
integral of cq (1.71), this is shown in Figurc ).37. Since the area of the continuou_-time
unil impulse ð(r) is conccatrated at t = 0, we seethat the runoing integral is 0 for t<o
and I for t >0. Ako, we note thal the relationskip in eq., (1.71) between the continuous
time unt stcp and impulse can be rewritten in a different form, analogous to the discrete
time fom in cq. (1.67), by changing the vuriable of integration fromT too =- :
uit) = S7) dr - S(1 - o\-da),
or cquivalently,
u(/) = |l.75)
The graphical interpretation of this form of the clationship berween utri and Sit) is
given in Figure 1.38. Since in this case the area of B( -o s concentrated at the point
o = l, we again see that the inlegral in eq. (1.75) is 0 for r <0 and I for 4> 0. This type
ot graphical interpretation of the behavior of the unit impulse under integration will be
extremely useiul in Chapler 2.
Sec. 1.4 The Unit mpulse and Unt Step Functions 35
Interval of untegration Interval of integralion
80-o)
B()
(a)
Interval of integration Interval of integralion
8t-o1
x()8(r) = x(0)8(). (176)
By the same argument, we have an analogous expressim for an impulse concentated at
an arbirary puint, say, ty. That is,
Signals and Systems Chap.
36
S,(t
x()
S,(t)
XID)
The product x()8,(t).
Figure 1.39
functions; (bË en
(a) graphs af both
the nonzero portiTn
of
larged view of
'their product
(b)
somewhat in
in this section has been be
of the unit
impuse this signal that will
Although our discussion somc important intuition about should be viewed as an
provide us with the unit impulse any real physi
furmal, it does book. As we have stated, detail in Section 2.5, instantancously
throughoul he discuss in more sespond
useful illustrate and thus does not
idealization. As we asSoCiated with it and to sch a sys
uration is applieddurativn
systerm has some inertia suficicntly shont pulse's or by
cal Consequently, if a pulse of noticeably infucnced by he primary characteristic
to inputs.system rcsponse will not be Instead, the
pulsei.e., ts area. For
tem, the the pulse, for that matter. of the
shape of integrated effect be of much
the
the details of will matter is the net, others, the pulse will have tomatter Nev
that
of the pulse respond much more quickly
han no longer
shape or ics duration "short enough."
systems that before the details of the pulsealways find a pulse that is is short enough
shorter durativn pbysical system, we canthis concept-the pulse thatto this idealized
ertheless, for any then is an ídealization of thc responsc of a systemprocess of deve]
The unit
impulse Chapter 2. analysis, and in the
system. As we
will sec in into the ideal1zed
for any In slgnat l adairional insight
pulse
pu plays u crucial rolethis ole, we will
understanding sungularity
oping and cullectivcly refarcdnamso as
of general
Signal.3 (which are cften the alternat1ve Distnbutnon
rdated functions unahematics under subject, see
other
impulsc and studied in the freld of detailcd discussivns of this 1965). Gen
The unit thonNughly ore MeGraw-Hil Book Coopany,text, Founer
distrnbutions Fo advanced
fnctions) have becn theury of Zeman1an (New
York
or che tureUnverssty Press.
1979).Carnbndge
l958)
ifunctns and the Analyss, by A. H. Halsicd Press,
York. tork: mathematical thoory
Transform Hcskins (New Lighthill (New related in spirit to heunderle this 1pic a4
Theoy and by R.FFunnions, by M. J
eral1sed Functions. closely
25 1s ntroductxon to conccpts that
Generultzed n Scctva
Analysis and singu:anty fuNcions
pruvides an
infomal
discussion of
Our texts and thus
dcscribcd n these
malhcmalies
