Question

Determine Fourier Transform of f(t) = u(t - 2) + 8(t - 6)

Answer 1

Solution :- given function f(t) = uct-2) +$(t-6) Step 1 :- here fourier transform of act) is calculated fact) = uct) -> 1 fourier Transform of uct) :- fict) s Lt coswot uct) woro uct) is not fourier transfor- mable directly [as Note:- Formula for cos wat uct) Fourier Transforms ft {coscoot ucts} = T) [Scw+Wo) +s(w-wo).] tyw 2 woh new 2 Apply fourier transform for D on both sides F7{ fict)} = Lt woo } FT{ coswot uct) II (swtwol +S(W-W.] Ficw) - Lt woro t jw wo tw2 는 TL (S(w+0) +(w-o) ] + jw f(w) = [ ascws] + to 02004 1 f(w) = TT SCW) +Jw

Step2 -- farães transform of unit impulse function fact) = SCH) as sct) = 1 at tzo to as founcy transform of act) is -jwt F{x}} = facte dt ft{ sctsy 9 schy ejut at ] (Note: actisct-tol -actol set-to) . -jut -jut Sct.e =e Stt-0) -jwco) ce .867 stjejut SU) SU) at F7&scri] = IFQCW) = 1 Step 3:- given fet) = uct-2) + Sct-6) (Note: Shifting property in fourier transform ejwto FT act-to) •XCW) where XCW) is the fourer transform of acy

دے FOL) = fiet- fict) - UCE) flct-2) = uct-2) -jw (2) Fouriu transform of fact-2) FT{ f1c6-2)} = F (W) here filw) is founer transform of fact) - uct) FT{uct-2)} 11 ] » fach = 867) fact+6) - S67-6) ējwcz) (TTSCW) + - -> Fourier transform of fact-6) -jw (6) FT {fact-6)} =é F (W) here f2CW) is the founu transform of fact)=sch) ēju (6) [1] AT { SCE-633 =2jwcs) fcn= ult-2) +867-6) Apply fourcs transform on both sides fr{-fc1} = ft{uct-2) 4 + FT{$(-6) 48 @ in eJqw (Tsco) + jw ] tej6w Substitute Answer FCW)