Wiley signals and systems e book TLFe BO 241

9 The F’ourier Transform 226 For w = 0, tlie expoiitkntial term 0x1 the left side of llie Fouric>rintegral disappears ail tl (9.95) For the fiirictions of time S ( t j and q(t) we will also perinii corriplex fimctions We can rmd from the SyJklnictry property (9.61) a i d Examplc 9.7 that for a signal g ( t ) arid the correspondiug complex corijiigate signal g*(1) t h a t the relationship betwt.cn tlie spcywa G ( p ) arid G * ( p j is (9.96) Froni liere we ran obtain the geiirral form of Parsed’s theorem for coiiiplex funciions of t h i e (9.97) For g ( t ) = S(t); we obtain the easy t o rcniember relationship (9.98) x I To iiilerpret this formula wr define Lhe energy of a tirrw signal 15: Energy of a time signal The linergy E f of n siqnnl f ( t ) is gzven by T h e reasori that this integral is called the energy of a sigiial betonic~sclear when WP imagine j ( t ) , for example,, to be a decreasing voltage acioss an ohmic rrsistaircnc The ciiergv converted to heat in the resistance is proportional to the iiitegral over tlir sqnarc of the voltage IJhirig the rriagnit iidc sqtiart.ti permits the use ol coniplex xigiids Parsevnl’s theorem says that tlie energy of a t i m e signal can be calculated not only in the tiriic~-tlomai~i, hut also in the frequency-tiori~aiu,by iritc.grating the rncignitude squared specdrum Only the magnitude is irivolwtl; thr phase clearly has no effect on the eriergv of the signal