212 ‘X’hc Fourier Traiisforin The same dcductioiis ri~nbe wriade for the imaginary part of thp spectrurn, which brings US to the coiiclusion that a r d f h c t i o n of time with cm imagiuary spectrim1 iiiust bc an odd hmction We have now sliowri the following rywnietry relationships for real signals: r,(t) real, e\wi sn(t)real, odd S(.7jj real, evw X ( j w ‘ ) imagiiiary,ocid c -i c i (955) (9.56) Tliese general principles ;we ~ ~ c a l by e d o u r observations ot the tranrforms of certairi signals xample 9.4 The rectangle fiinction is real a i d eveii ‘The m i n e is true for i t s ;pectrum (9.5 The rpal am1 odd fixnctioii of lirric ~ X 9.5 liiis a11 imaginary and odd spectmrn (9.58) t Signals 5,4 S y ~ ~ry~ofeImaginary The same deductions for time signals with real valws can also be made for irnagiiiary time signals Tlic equivalent rclatiorisliips t o (!J.S), (9.56) aPc ,E, { t )iinaginary evrri x < > ( firrmginary, ) octd c-) X ( , j u ) imaginary, + + X(j.i) real otld, Pwn (‘3.59) (9.60) The imaginary fuiictiori s(!) ancl its real iniagirrsry part Ini{%r(t)}should not bc confriscd For prircly imaginary functioiis x ( t ) = ,lh{x;(tj} Symmetry of Complex Signals ‘Tlic previous results for rwl arid for purely imaginary signals can now be uriittd t o give the symmetry for general complex signals Every signal c m be split into its even and odd parts, nxicl its real anti imtginary pitrts, aiitl ~