21 The Fourier Transform The c m s of ~ this similarity i s that both the tinit and trcquency pararn real valurs 'Vhc significant diffeimces between the t ransfu~urma i d its inverse are thc intcgrat ion variables e ireqiirncy), the sign in front of tlie rxponeiitinl ftmction (-/+) arid lie I 1/2n before the intrgral Lilw thr iriveise Laplace transform, the inverse Fourier t r ansforin can rcprcsent a suyeriiripositiori of cigcnfinictioiis r J w tof an LTl-systelrl, but in this c a i e only in~tlampcdostillat ions are $1errriitt d Irk fac*t, (9.62) call br interpn\tcd as an invcrse 1,aplncc trarlsfornl (5.34) for which the irn;igiiiary axis s = J W has been chosen as the path of integrntion This iritrrprctatiori is valid if thc conditions fox X(s) = X ( J W 3w ) ~ (9.9) are fulfilld; the region of convcrgeilce [or X ( s ) encloscs the iniagiriary axis arid X ( ~ Jcan ) hc differentiated any nurnbcr of times To vt?rify that (9.62) actua11y yidds the corresponding furiction oi iiriic for a possibly discontimious spectrum S(p), we put tlle defirrition of X(p)irito (9.62) in accordance with (9.1), and exchange the sequence o T integrations Thcrc i s an exprrwSon (ill square braxkets) within the outcr integral that we can iiiterpiei (see Section 9.4.3) as a clelta impulse shifted in the time-dorniiin In contrast to Section 9.4.3, the roles of ui arid t have been swapped Willi tlir selectivr propcrtv, we cwd up with r ( f ) Tlius we have s e w that thc inttgral in (9.62) leads to a function of time from which (with (9.1)) X ( p ) c5tli be cal(~ulatec1 b(t = 7) (9.63) s(i) This der ivatiori requires that both improper integrals can be swapped, which is tlie case for normal convergence of intc Just as foi the Ilaplece transfoim, tlie irnent of a time signal to its Fourier transform is irnarnbiguoiIs, if discontiriuities are foresccii and clealt with (Chnpter 4.6.2) Deviations at individual points of cliscontiniiities cio not change tlie value of the intcgrals in (9.63), ;md whcii solving p r a c t h l prohlerns, this degree of unarnbiguoiisriess i s sufficient rie sfor As ~ ~ f lasl lthe symiiietry properties that m7e have already dealt with, the Fouricr transfoiml has many propelties that need t o be known to take advantage of the