15 Causality and the Hilbert Tiansform 376 s(t) !H Tt I I 2(t) = e(t) i Figure 15.3: IIiIbert-traiwforrrl in the the-domain and freqrrertcy-domain The complex signal of time ~1 ( t ) is called the nizalytrcal sigrml corresponding to thc w a l signal x.(t).'The analytical signal allows a simple description cif impart ant, communications arid sigml processing systcrns lilw inodrilathn, sanipling of band-pass signals, filter banks antl others The following example shows il typical applicaticm Example 15.2 Figwe 15.4 shows art arrangement ftx transmission of a real signal ~ ( ) It,s spectrimi X ( , p )has i+ band-pass characterist ic with limit frequeiicies w I;ind "'2 In general, X ( j u ) is complex Init 111 Figure 15.4 only a real v-ainlucd spcct,rurri is shown, far the sake of simplicity X ( j w ) has conjugatc symmetry (9~49),as z { t ) is real, antl hccause of this symmetry, tlie lcft sideband is a reflection of lhe riglit it;nd contains nu further information It is advantageous to tramniit uiily one of the two sidebancts so that only half the baritlwitlth is needed The rnissirig sidrband can be reconstruc ivrr a5 the syxnrnctry is known It is also useful to reducp the barid signal is going to be sampled (set C1-titptt.r 11.3.2) ljroni Table 11.2 we t a n that a conqdex band-pass signal with a unilateral s p e c t r i m (shown in Figurell.14) requires only half tlie sanipling frequency that a rcal band-pass signal nevds Tt now remains to be sbowri how a complex band-pass signal with a imilatcral spectrum is for med from a real band-pass signal This is where the Hilbert transform comes in The signal I ( t ) (15 45) has thc rcal and imaginary p't't)s Re{si(t)} = z ( t ) , Xrn{a~(t))= ' k { x ( t ) } ( 15.46) and the desired right-sided spwtruin XI ( p ) It can bc critically sarripled with - W I , rcgardless of the spccific values of w l antl wa 'Yhe resultant signal xz(l) has a periodic spectrnm that does not overlap Tlie original signal ~ ( tc )m w, = w.2