Understanding Fourier Transforms for Translated Functions

[bodensee9]

Homework Statement

I am supposed to find the Fourier Transform of the following:

suppose t(subu) is a translation of the function f by u, so that f*t(subu) = f(t-u).
suppose also that 1 means denotes a characteristic function so that the characteristic function has the value 1 from -T to T and 0 everywhere else.

1. f=t(subu)*1.

Would I evaluate the Fourier Transform with f = 1 from (-T+u) to T. My reasoning would be that since 1 means that f has value from -T to T, but then f is translated by u, so that you would need to add u to -T for f to have a value?

2. suppose that H is the heaviside function that takes the value of 1 from 0 to inf.

f = t(subu)*1*H.

would I evaluate the Fourier Transform from u to T? My reasoning is that because of H, f can only have a value from 0 to inf, but that is shifted by u, so you need to evaluate it from u to inf.

3. f = 1*t(subu)*H

would I get the same answer as 2?

Thanks very much.

Homework Equations

The Attempt at a Solution

 

[lippyka]

1. Yes, you would evaluate the Fourier Transform with f = 1 from (-T+u) to T. This is because the translation of the function by u means that the function's value is shifted to the right by u. Therefore, in order for the function to have a value from -T to T, you need to add u to -T. Therefore, the Fourier Transform should be evaluated with f = 1 from (-T+u) to T.2. Yes, you would evaluate the Fourier Transform from u to T. This is because the heaviside function H means that the function can only have a value from 0 to inf. However, since the function is translated by u, you need to shift the range of the function to the right by u. Therefore, the Fourier Transform should be evaluated from u to T.3. Yes, you would get the same answer as 2. This is because the order of the operations does not matter, so the order of the multiplication does not matter. Therefore, the Fourier Transform should still be evaluated from u to T.