38 15 Causality and the Hilbert Transform H(&J) = P(.jw) + jQ(.jw) oi a causal system a) IMerrniiie the corresponding iiripiilse response Ir (t j 0-c H ( g w ) €(O) = - SOW: b} Firid the imaginary function Q ( pwith ) (15.18) c) What symmetry p ( t ) 0-e r"(:p~), q ( t ) - Q(2.l) and Q ( j w ) show'! Exercise 15.8 An audio signal with thc spectrum S ( ~ Jis) to be trai1sniittc.d In ordex to bave bandwidth, the signal spec-trurn i s for \ U / > ~1~ = 27~.x kHz a) Thr signal s ( t ) is nrioclulsted by cos(wot) .m cos o'ot Giw the spectrum of s h f ( t ) b) How much baiidwidth does Snf(jw') use c.ornparect to S(SUJ)? c j To rc'inovc the disndvantagr foimd in part b), the signal will be trarisrruttced according t,o tlic following representation This method, known as single sideband modulation (SSB) takes advantage of the rcduudaney prcwni; in the spectrim of it red signal I>raw the speetrurn of SEn&w)