Fourier Transform: Solve Homework Equations for fd

[Lengalicious]

Homework Statement

See Attachment

Homework Equations

The Attempt at a Solution

Ok so in a previous question I worked out f_d = e^-ipd*2*sinc(pa)/√(2∏), also worked out its Fourier transform if that helps.

Now I really am stuck on the question, any guidance would be appreciated, I don't understand how the function f_d is summed over from -N to N, like I say any help to just send me in the right direction would be great.

 

[MisterX]

The problem suggests that the function f_d should be a function of x, but I don't see an x in your expression.

 

[Lengalicious]

The problem suggests that the function f_d should be a function of x, but I don't see an x in your expression.

Well the original function was f(x) which was a function of x that I then Fourier transformed with limits of x =-a to x=a, this replaced the x's with a's. f_d(x) was then a slight variation of the original f(x) function, ultimately I ended up with said function where f_d(x) = f(x - d) which transforms to e^(-ipd)*f(p) where f(p) was the Fourier transform of f(x).

 

[Lengalicious]

I made an issue with the question, not sure how to edit. Anyway f_d(x) = 1/a when |x-d|< a. What I said in the 1st post is actually the Fourier transform of this.

 

[paranormal]

Dear student,

The Fourier transform is a mathematical operation that allows us to convert a function from its original domain (usually time or space) to its frequency domain. It is a powerful tool in signal processing and has applications in many fields of science and engineering.

In this problem, it seems that you have already calculated the Fourier transform of fd. However, the question now asks you to solve for fd itself. This can be done by using the inverse Fourier transform, which allows us to convert a function from its frequency domain back to its original domain.

To solve for fd, you will need to use the inverse Fourier transform equation, which is given by:

fd = ∫F(fd) eipd dω/2π

Where F(fd) is the Fourier transform of fd and ω is the frequency variable. You can use the expression you have already calculated for F(fd) to solve for fd. You will also need to use the limits of integration (-N to N) to sum over all the frequencies in the Fourier transform.

I hope this helps to guide you in the right direction. Good luck with your homework! Remember to always double check your calculations and make sure you understand the concepts behind them. Happy solving!