Wiley signals and systems e book TLFe BO 251

The exact wine ol D3U%depends on the forin of (lie sigiial ~ ( t )The mininiurn is found for a Gauss irnpiilse [5l.Becniixc of forxnttl analogies wit,h qiiantiirii IIICiiics, this eqiiation i s i11s0 called the unrer As ;I Gausi impulre has it particularly g width proctiirt U f33 if is t.iriplo)yed whcncwr it IS irnpoutaril to pack i f s m i d i riicrgy as possible ilia11 frcqucncy b a d ovei a sm:tll amount of time This is, for examplt., a, demand of a digital traiisrnisdion syst eni In short-lime spectral analysis good tirnc aiid frequency resoliition i s uiittl at thc same t i i n c b , arid Gaii windows clre widely eniplowcl From t h various definitions of duration and hatlitlwidtli and the results obtained we can cliaw ~ O I W i m p t ant c.oiirliisions 1)uration aiid bandwidth of a signal are rr.ciprocal It is tliercforr not possible to fiiitl a signal that has any deqired short duration and at lhe same t h e ally dcsired small bandwicltli Shortwnig the diiration of t kit) higiial always inrreasos (lie haiitiwidth, aiid vice vcwa This statement i s very irnyortitrit for signa.1 transmission a i d spectritl aiialy ( w e Ext~inplr9.11) It is forrnally related L o the uncertainty relation from quaiiturri mecliztnics Exercise 9.1 Calculate the Fourier transforms of thc tollowing signals with the Fouriei integral as long B S it coriwrgcs For coriipa~isoi, givc also the Laplacc transforms with the regions of convergence a) z(t) = z(t) e-lWof c) n(t) = &(-4t) d) Z(t) = s ( - t ) ) r ( t )== c-/.JlJl Note: b) Piuperties of the Laplarr traiisform arc in Cliapkr c) IMes for cnlculations with the delta irnpulsr are in Chaptrr 8.3.4