Question

2 a) Use direct integral and find s() (in real form) if O-10 [ reetf+4)/2+reet(f-4)2 Use direct integral and find the Fourier transform or x(t)-4 [u(t +4)-ut-6n

Answer 1

(a) X(t) = 10[rect (f + 4)/2 + rect (f - 4)/2] x(t) = integral_-infinity^infinity x(t) e^k 2 pi ft df = integral_5^-3 e^j 2 pi ft df + integral_3^5 e^j 2 pi ft df = e^j 2 pi ft]_-5^-3/j 2 pi t + e^j 2 pi ft]_3^5/j 2 pi t = e^-j 2 pi 3t - e^-j 2 pi 5t + e^j 2 pi 5t - e^j 2 pi 3t/j 2 pi t = sin 10 pi t - sin 6 pi t/pi t (b) X(t) = integral_-infinity^infinity x(t) e^-j 2 pi ft dt = integral_-4^6 4 e^-j 2 pi ft dt = 4e^-j 2 pi ft]_-4^6/-j 2 pi f = 4[e^j 2 pi f times 4 - e^-j 2 pi f times 6/j 2 pi f