Question

1)

2cost,tl s 1 2 lt > 2 (a) The Fourier transform of the function: f(t)=(cost, 1 < is 0, 3 cos(w+5) (b) The inverse Fourier transform of the function F (w)22 is

do both a & b subparts thank u

Answer 1

a) The Fourier fransform of 2 cost, Itl s 1 0, tl > 2 is given by cost e jotdt +coste-jodt In the first integral cos t is an even function and hence frorm e-jwt the cosine part will survive, and therefore -22 cos t cos(wt) dt cos t [cos(ot) - i sinwt)]dt 0 Now observe that cos t cos(out)--[cos(1 + ω) t + cos( 1-ω)t] cost sin(wt)--[sin( 1 + ω)t + sin(-1 + ω)t]

l hus F(a))-2 | [cos(1 + ω)t + cos(1-a))t]dt sin(1+ω)t 1 [sin(1 + ω) t _ cos(1 + ω)t sin(1+ω) sin(1-ω)t sin(1-u)t cos(1-ω)t 2 1) sin(1-ω) 1[sin 2(1+w) sin 2(1-ω) sin(1+ω) sin(1-ω) 1 [-cos 2(1 + ω) _ cos 2(1-ω) cos(1 + a)) cos(1-ω) (1-ω)

sin(1+ω) sin(1-ω) 1[sin 2(1+w) sin 2(1-ω) sin(1+ω) sin(1-ω) 1 [ cos 2(1+w) cos 2(1-a)) cos(1+ω) 1+ω cos(1-ω) 3 cos( ω+ (b) The inverse Fourier transform of the function F (a 1 3 cos 3V2π6(t) sin(a)