Help with evaluating Fourier transform of triangle wave

Question

I am told that I need to evaluate and sketch the Fourier transform of a triangle wave, shown below.

And I am given that the Fourier transform formula is given generally by

$$F(u) = \int^{\infty}_{-\infty} f(x)e^{-iux}dx$$

where $e^{ix} = \cos(x) + i\sin(x)$ and $i = \sqrt-1$.

However, I am flummoxed as to how I am supposed to proceed with this problem and would appreciate any hints/suggestions. I guess I'm primarily confused because I don't understand how to proceed when I'm not given a formula for $f(x)$ that I can plug into my formula for $F(u)$. Do I need to find a formula for $f(x)$ within each period of the function?

Answer 1

Consider cutting your graph into 6 pieces:

• $[0, 1/2]$
• $[1/2, 3/2]$
• $[3/2, 5/2]$
• $[5/2, 7/2]$
• $[7/2, 9/2]$
• $[9/2, 5]$

On each of the intervals, $f(x)$ is just a line. Can you find the equations and integrate?

Since this triangular wave is also periodic, can you use this to reduce the number of intervals needed?