9 ’I‘he Foririer Trarisforrn - 220 duration, nncl we wish to determine for how long it needs to be olrxc~vdfor u s t o able to distitiguish t,he two signals from each other P’igitre 9.18 shmw the two sigiinlci whose superiiiipositioii we will be ohserving froin time - ‘ / I o T,’2 Figure :I 18: Siriusoidal sigrids 91 ( t ) ,yA(t) and hc window of observation S ( t ) W e already know that the spectruiri of a siiiusoidal signal corrsists of two delta irnpulbcs o i r the frequerrcy axis The location of the irnpiilses corresponds to l,hr frequency of the signal As both frequencies arc nlmost the sailits, the delta impulses will lie close to eat41 othcr: g ( t ) = 9l(t) + gz(t) = c o s d l t + coswzt I (9.76) The firiitr observation time can be exprrssrd inathernatically as mn~~ltipliral,ioIi hy a rectarrgle function in the tiinc-domain The fimction tlescribcs tlrc finite window of observation through which WP sw thc sinusoidal function (9.77) ?Ye a,lready know the spectrum of the window: a si-function The obselved signal i s t (9.78) A4 = f ( t ).fJ(i) = f i t ) b r ( t ) + gz(t)l ? as shown in Figure 9.19 Tlie spectrurn i~(*tually ol~servetlt m be rrpr the multiplication rule, as the convolution of the spectrum G ( j w ) wliicli contains t he itlral delti-1irnpulsc>with the spectrtini of the window of observation Because of tlie convuliition wilh foul shifted delta impulses, the spect r i m of the measured sigriat contains f o i terms ~ that each have the smic form as the spectrum of’ tlie window of observation and are located a t the signal flequency Y(jW) = - 2n F(:jw)* G(:J’hi) (9.79)