Fourier Transform

Im wondering are Laplace transform and fourier transform actually the same, just that the exponential term of fourier is pure imaginary term? Can i always do fourier transform like i did laplace transform?

Besides, i need help on 2 questions below regarding the fourier transform...

1. f1(t) = u(t+1)*sin(pi*t)

2. Given F2(w) = 10 / [(2+jw)(5+jw)], find f(2t-1)

For question 2, i did find out the f2(t) =(10/3)(exp^-2t - exp^-5t)u(t)
however when i sub in t = 2t-1, i dont know how to deal with u(2t-1),

Thanks in advance. Regards.

for my question 1 i found that answer should be

F(w) = w*exp(jw) / (w^2 + 1)

does anyone mind to verify ?

Im wondering are Laplace transform and fourier transform actually the same, just that the exponential term of fourier is pure imaginary term?

Click to expand...

Yeah, pretty much! the Laplace transform contains a decaying exponential whose argument is of the form s = \(\sigma\) + j ω

When the real part (sigma) is zero, you have the fourier transform. Even further, the fourier transform of a function only exists when the imaginary axis is in the region of convergence of the laplace transform of that function. Otherwise, the fourier transform does not exist. Pretty interesting relationship.

For question 1,

your answer looks pretty reasonable to me. Except, did you account for the u(t+1) factor?


For question 2,

however when i sub in t = 2t-1, i dont know how to deal with u(2t-1)

Click to expand...

what do you mean? What are you trying to do with it? Doesn't the question just call for you to leave your answer just like that?

Hopefully that helps!

thx to guitarguy12387

however my answer for question 1 i did something so fatally wrong so the answer should be wrong as well

and for question 2, i found out there is a property called time scaling,
[ F {f(at)}= 1/|a| * F(w/a)], but i have problem applying it....

let say for same question given F(w) = 10 / [(2+jw)(5+jw)], find f(-3t)

here is what i did using the property
F{f(-3t)}= (1/3) * ( 10 / [(2+j (w/-3) )(5+j (w/-3) )]
= 10 / (6-jw) )(15-jw)

But the answer i found without using property should be
F{f(-3t)} = 30 / (6-jw) )(15-jw)

Does anyone mind to point out where i did wrong??

Hey,

For question 1, I'd probably just apply the definition of the FT and just solve the integral (using euler's formula on the sin(pi*t)). You can't use the time shifting property because the sin isn't shifted. There may be an easier way to do it, but i can't think of it off hand... maybe someone else can help out with that

Q2, you forgot the 1/|a| factor in the time scaling property. If you account for that, you'll get a 30 on top.