Fourier Transform Homework: Determine F(k) & Plot Result

[rogeralms]

Homework Statement

Determine the Fourier Transform of the function shown. Plot the result using excel, MathCad, or Matlab. See attachment for figure of triangle above x-axis from -X₀/2 tp X₀/2 with a max height of 1 at x=0.

Homework Equations

The answer is F(k) = X₀/2 [sin(kX₀/4) / (kX₀/4) ]² =X₀ / 2 sinc²(kX₀/4)

The Attempt at a Solution

f(x) = mx + k = Δy/x + 1 = 2/X₀ X + 1, x≤0
=-2/X₀ X + , x≥0

F(x)= ∫_-X0/2 ⁰ (2/X₀ X + 1) e^ikx dx + ∫₀^X₀/2(-2/X₀*X+1)e^ikxdx


I am very confused by the change of variables taking place in the different problems.

The PDF is the assignment as it was given. We worked problem 2 in class. I am confused by problem 1. Could someone give me a few pointers as to how to proceed, where I might have gone wrong, etc. This is due on Wed. I have not been idle but still am not quite there for a solution. Additional work is included in the Word files.
Thank you.

 

[TSny]

Note that f(x) is an even function of x. Also,

e^ikx = coskx + i sinkx

which is a sum of an even and an odd function.

So, f(x)e^ikx can be written as a sum of an even function and an odd function.

What happens when you integrate an odd function over an interval that is symmetrical about the origin?

Also, when you integrate the even part, you can just integrate from 0 to x_o/2 and multiply the result by 2.