and the s i cp fuiict ion ~ t \lit\ will deal wit,h the evaluation of tlic Foiirier tmniornis iii thc coursc of this chapter.. Up t o iiow, TW ltax7e encounimeti it in the time-tlorri
Trang 1and the s i cp fuiict ion ~ ( t )
\lit\ will deal wit,h the evaluation of tlic Foiirier tmniornis iii thc coursc of this chapter
The Fourier tr;tnsform of the s k p fimction contains a delta impitlse 6(d) Up t o
iiow, TW ltax7e encounimeti it in the time-tlorriain, biit not in the ~requenc.y-dorrii~in
As Laplace trimsforms in the region of corivcrgeiice are analyt ic fimctions there
is no room there Erw gmrrafiscd functions Ro-,wver, with Fourier traxisionns that arc fiuietions of a real parameter w (jiist as with fimci ioris of tirtci), thc extension
to cover gewra1isc.d functions does make smse T h e popularity of t hc Fouricr
transform conies from the posqiblity of assigning a spcctnin7 with the help of the dcltit irripiilsc to riiariy iniporkd prattica.21 functions for which 1 hew is no Laplacc transiorm Sonir eltmcntctry examples are con nts ( ~ g r ( t ) = 1) arid t,he trigovtornctric frmctions siriwi,t and cob wet
it can bc scen immediately that llie FoinieI transform of a function of tiinc is the
sitirit a5 i t s Lixplace transtoriii on the imagina axis s = j w of the coniplex plane
That is of coiiise only triie if the r,aplacc transiorm cxists as well T h e coimcct,iou bctween Laplace and Fourier t ransl'orrns is: if the region of coriwrgerie<' of E { - r ( f ) }
contains tkic imaginary axis s = 71*', tlml
j
s=.7w
(9.9)
Wit12 (9.9) the corivention sliould r i m 7 i~iake sense, t h a l t , h ~ Forrrier trarisforrn is defined on thc imaginary axis ancl not 0x1 the real axis