Wiley signals and systems e book TLFe BO 312

aling v-ith some simple disrrtlte siguals; x w
Itnow thrir continuous-tinit. coiirti carparts. 'L'heso sigrials are the inipnlsc,
expoiieiitial series.

~

1
The discrete unit imp-tllse is defined by

(L2.3)
Figure 12.2 shows thib series. The discrele unit impulse Iiah the helertiw propel t y
"U

I
(12.4)

that, r i z ~1 ) easily
~
checked

k
Figure 12.2:Discrc4t. u r r i t impulse

Comparing (,hiswith the relwtivc property of f , h continuoiih-time
~
delta impulse
shows a close relatioiiship between the two:
(12.5)
Integration with resp
to tlie continuous variable z in (12.5) correspoiids 1 o
smiiniatioii with respect to tJie di ete inrlcpcirdeiit variwhle K ; in (12.4). TVe

will sec that replacing integration in the clefinit,ion of a continuous sigrlal with
summat ion often leads to the equivalent discrete signal.
Thcre is a m o l e \ significant, diffcrence bet8weerit,lw disrrete nnit impulse arid ~ I I P
contiinioiis deka impulse: the unit, irnpulse 6jk] is not a distribution. For k = 0 it
has the finite valiie 6[0]= I . which makes calculations. simple. In contrast to thcl
continuous tlelta irripiilse 6 ( t ) , ilie unit impulse
a forriitila. and does not wquirc the use of the SP