231 9.10 Time-Bandwidt h Product rlgaiii so cwrnplicatcd, however, that w e would likc to find an easier way We can read from (9.113) thai the Fourier transform of our auto-coirelatioir function is drwidy kiiown froin the Fourier t raiisforrii of tlrr si-fiinctioii The auto-correlation function p, ( r) then iollows by inverse trarisforinaliorr 15 F { p I l(I)} -= Ix(,,w)12 = 47r2r (9.114) i3 I p,,.(z) (9.115) = 2nsi To check tthe results we colisider the value at t = Here, tkie value ol the stito-eofrr!l;tt,ion yz.r ( ) is t>liesiinie its the energy E, of the signal z From (9.115) we obtairi with clcterrnined in (9.101) ,(0) = E , = 27r the saiiw value that we bad ct A t various tiriies we have seen that there is a close connection belwcAen the dura,tion of a signal and the appearance oi i(,s 511 rixii In Figlire 9.18 we made good iise of this proper 1y to tl iirie how long we needed to nieabure in the the-doniain, to get a clear rtsult e freyiieiicy-dorriai~i.This reciprocal roimection bc%uwn the length of tthe signal in time :mnd the width ofI,lic spe>ctriirnit;clearly iiot, limittd to thc rcctanglc functiori T1: ilarity throreiii (9.G9) that a sfretchtd t i m e signal corresponds to a comp spectrum (arid vice a) for any time signal We iiow wish to investigate this rtmricctioii rnore closely W e are cspc~ially interested in t h e r~lationshipbetween the period of a signal in time and the witltli of the corresporiciing spectriini The width of the spectrnm is kiiovrin as (lit ~ u ~ 1jilt we must make it clem m7hat; exactly siioulrl be understood by the tc\rms period and wiclth 111Figure 9.7 it is cvident what is meant by thc duralion of a rectangle function The corrcsponclirig sp are all, l-iowever strekhed out, iiifinitely Despite this, the spectrum of a rectangle fimvt iori does seein to become widn in a certain way, when llie impulse becomes shorter There are vxious possiblities to defiiie the length of a bigrial and iCs bantlwitlth for general signal We will consider three of them, and ~ l l bee l that they all lead l o the same funrlamcntal priiiciple ~