The exact wine ol D3U% depends on the forin of lie sigiial ~ t.. This is, for examplt., a, demand of a digital traiisrnisdion syst eni.. 1uration aiid bandwidth of a signal are rr.cipr
Trang 1The 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 IIIC-
width proctiirt U 3 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
iiics, this eqiiation i s i11s0 called the unrer
A s ; Gausi impulre has it particularly g
good tirnc aiid frequency resoliition i s
windows clre widely eniplowcl
uiittl at thc same t i i n c b , arid Gaii
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~inplr 9.11) 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 - l W o f
c) n(t) = & ( - 4 t )
d) Z(t) = s ( - t )
.) r ( t ) == c-/.JlJl
Note: b) Piuperties of the Laplarr traiisform arc i n Cliapkr 4
cnlculations with the delta irnpulsr are in Chaptrr 8.3.4
c) IMes for