Wiley signals and systems e book TLFe BO 197

8. Con~olut~ion
arid Impiilsc Respoase

182

One of the two step responses miist also be differentiatetl hcfort conivolut ion
takes placc. Alterriatively, the convolution of the step response can be carried ont.
and their the rrsult can be dilTerentiater1. Equation (8.67) can easily be expanded
to cover N cascaded systems. Tire coniplete step respoiisr is ohtaincd by t-fiffcrenlisting the component step I
or the complete prodixt of tonvohitioii
( N - 1) h i e s .

by

O ~ V Oion
~ ~ t

In Section 8.4.3 WY biicfly dealt with the ctdcul&ion of tlw couvohition integral.
This iwthod always works, biit it is cuinbersorne for funclions that are rlrfiiied by
mull iple cases, a5 tEic convolution integial takes a tliffereiit form tor each case, and
w e b must bc calculated indivithially
In this section we will desvxibe a 5implc mctliod that is well suited to signals
with corisl ant scvtions, rzs it is not necessary to cwduate tlir conivolution integral for
tlrcse sections. With sortie practicP it i s possible to find the convolution product
with this method, called convolut/on hy insperf7on. ‘ r h ~
reader should try to
acquire this skill, because it bririgs an intuitive Luldeistandirig of the c oiivolutiori
operation wliich is esseiitial fo1 practical work.
To tiemonstratc the process. we will tonsidn tlre two rectanglP c;ignaIs from
Figure 8.27 and fiist evaluate thc convolittion integral as in Section 8.4.3.
~ x ~ r ~ i 8.6

ple

_
I
I

\ t

*2

4

1

Figure 8.27: Examplc 8.6 of‘ coiivdution by inspe~ctioii

The two signals from Figure 8.27 are defined by the func‘tioris

1 o5t0 otherwisr

2
arid y(t)

=

o
0 otherwise

(8.68)

‘To calculate i,he conk olut ion integral, however. both sigiids rmst bc fulictkms of