Solution to HW#3

1a)

b)

c)
i)

ii)

2a)

1. Re[s] > -1, \ we CAN use F.V.T. to find x(¥ )=0
2. Re[s] > Re[± j] > 0, \ we CANNOT use F.V.T. to find x(¥ )!!
3. Re[s] > 0, but for sX(s) Re[s] > -3!! Therefore we CAN use the F.V.T to find x(¥ )=2/3
4. Re[s] > 2, therefore we CANNOT use the FVT to find x(¥ )!!

1. The FT exists and we CAN make the substitution because ROC contains jw -axis!!
   
2. The FT exists but we CANNOT make the substitution because ROC does not contains jw -axis!! X(f)¹ X(s)|_{s=j2p f}
3. The FT exists but we CANNOT make the substitution because ROC does not contains jw -axis!! X(f)¹ X(s)|_{s=j2p f}
4. The FT does NOT exist!!

1a) i) T=1.0, x₁(t) = Au(t) - (A+B)u(t - ½) + Bu(t-1)

ii) T=1.0, x₁(t) = Asin(2p t)[u(t) - u(t - ½)] = Asin(2p t)u(t) - Asin(2p t - ½ + ½)u(t - ½) = Asin(2p t)u(t) + Asin(2p t - ½)u(t - ½)]

iii) T = ½, x₁(t) = Asin(2p t)[u(t) - u(t - ½)], same as above, with T = ½!!

2. i) T =4,

ii) T =4,

iii) T =4,

i)ii)iii)

2a)