Question

NB: In this Webwork problem, take sinc(t) = sin(t)/t (in contrast, in Signal Processing literature, sinc(t) = sin(mt)/at). Find the Fourier transform Xı(w), X2(w), and X3(W) of the signals xi(t), x2(t), and x3(t), using the Fourier transform pair X(t) = u(t + 1) – ult – 1) + X(W) = 2 sinc(w). Then select the Fourier transform property you used for each signal, from the corresponding drop-down menu. In your answers, enter “w” for omega. a) x1(t) = -3u(t + 2) + 7u(t) – 4uſt – 2) X1(w) = b) x2(t) = cos(24t)[u(t + 1) – uſt – 1)] X2(W) = c) xz(t) = 5[u(t + }) – uſt – })] X3(w) = Answers to the drop-downs will only be marked correct if their corresponding FT's are correct.

Answer 1

Givenir act) & ult+1) - ult-1) cws = 2 (a) a Ct = -3 u Ct+2) & tult) – 4 uct-2) T act st 2

we know alth = ultt) - ult) 4 x (+12) 7 x (tt) 4 & ult12) - 72(t+) 2, (+1 = 42CH2)-+* (t +1) XI(W) = 1 *C2w) – + x(w)ej

XI(W) 3. Bx (22) » 7 X (w) elvo iw X1W) = 16 sine (26) - 14 e sincew) | 2 (dt) FI IX (old) | This property pypen is used in this problem 1 x (H) t e jwx (w)

© 22 (+) = cosaat Puct H) -ult-1)] het acti= u(t+1) - ult-1) cosant = ejert éjant, elzate con evantact Fots Z(w-26) FIT 2 (WF 20) ے (+) 2 د (ه) لا ult+1)-ult-1) a s 2 sinc(w) t 2 sinc Cw-2x) jaat e 2 t) ajzatz(+) cosant 2012 2 sine (w+21) 2 Sine Cw-20) + sine (wt2n] cosant 2 (t) t sine (w-21) + sine (wt 20) cosant [uCtrl-u(t-1)] sinc (w-27) t sine (w+20) cosant [uct +1)-u Ct-n] o XC6-20+x6w+20)

© 234) = 5 [uch+*) - ult-ko] i 13 (1) -15 15 we know ЕТ 2sine (w) 52(54) -> Scaling operation. - 115 15 xCat) t I & (wsl) Fourier transfor -mproperty 52 (54) - 5 x (05) 151 5 x 15t) t & (w/5) 13(+) _FT, X (w/5) az (+) 2 sine (co/5)