Question

4. Consider the signal co(t) = et, 0<t<1 , elsewhere Determine the Fourier transform of each of the signals shown in Figure 2. You should be able to do this by explicitly evaluating only the transform of co(t) and then using properties of the Fourier transform. X(t) X2(t) Xolt) Xp(t) -Xol-t) X3(t) Xolt +1) X4(t) Xolt) txo(t) My Lane 1 0

Answer 1

Solution:

ajw Consider the signal, Kolu) set asli Lo else where Using Fourier transform pair erat ulte t for Re{a} so and time shifting property se [+-to) = eJoto x (jw) we have x(4) = xo It) cert[ult) - ult-1)] retult) - et ult-1) aert ult) re-le-Lt-1) u (t-1) Xow). I - THJW -o la) X, Lt) = molt) + salt) using time seversal properly belt) = xl-jw) and ineqe we have X, (jw) - Xoljw) + Xo L-jw) - - - Titjo = l-e-lities) Z t-elltju) + l-eli- Hjw 14jw 7 2-2ecosw-wesinca ( It w2 x,Lt) = xo(t)- Xol-t) Using time revesisal poopesty se (t) = x l-J6), linearity posperity axlu) + by s axljwt by (w) x, ju) = X (Uu) - Yo[-Ju) = l-e Litje) i-e-(i-jw) . . tjw w Scanned with CamScanner +2W +2e-sinw +2we-leosw' item .

and X3 (t)= volt) + Xo 1t+) lesing linearity, time shifting pooperty equation ③ we have, Xs(jw) = Xo (jw) te iw to (jw) El-e-21+jw) Give-11+jw) 7. leuw It jw 1tju Elteswe'(lteires) Htw (d) Xult = txolt) Using o the the differentiation in propenly talt) jd (jw) frequency domain and in equ ③ we haves Xu Low) a (L-en beton cinzele-jw jwerte -ju (14.jw) 2 z Scanned with CamScanner