Question

I need help modeling this function in order to do Fourier series

Answer 1

My first approach was to define x(t) in unit step functions .
My second approach was to take the derivative of x(t) so that it becomes a combination of unit impulses .

Answer 2

and then by doing this I can apply the time shifting property of : e^-j w t

Answer 3

Since the signal appears to be even That would mean That I would only have to solve for the coefficients a_0 and a_n , while b_n = 0 .

Answer 4

You're just trying to find the fourier series?

Answer 5

yes

Answer 6

I think I am over thinking it ,haha . I think there is an easier approach .

Answer 7

So, yes. You can take \(b_n = 0\).

Also, what is the period of this function?

Answer 8

calculating....

Answer 9

period is 2 pi

Answer 10

Then here's the best approach:
\[
a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) cos(nx) \ dx
\]
You're familiar with that formula?

Answer 11

yes

Answer 12

just not sure how to model f(x)

Answer 13

Well think more along the lines of changing integration bounds... f is either 1 or 0 right? What does that mean

Answer 14

We can split up the integral into separate ones like this:
\[
a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos{nx} \ dx = \frac{2}{\pi} \int_{0}^{\pi} f(x)\cos{nx} \ dx \\ = \frac{2}{\pi} \int_{0}^{\frac{\pi}{2}} f(x)\cos{nx} \ dx + \frac{2}{\pi} \int_{\frac{\pi}{2}}^{\pi} f(x)\cos{nx} \ dx
\]

Answer 15

you cant model its as sin(t)U(t)

Answer 16

if that was what you were refering to @inkyvoyd

Answer 17

buuuuuut I might be wrong and , if so , then this problem is a piece of cake

Answer 18

Does what I did above make sense?

Answer 19

yes it does make sense

Answer 20

Okay, so now all we have to do is evaluate those two integrals at the bottom

Answer 21

I understand the mechanics of the fourier series .

Answer 22

So plug in the values for \(f(x)\) along those intervals and evaluate the integral

Answer 23

Yes , okay . So what is f(x) ??

Answer 24

From the graph, \(f(x)\) = 1 over the first interval and -1 over the second.

Answer 25

ohh , okay , Ok thanks . That is all I wanted to know . Much appreciated .

Answer 26

Okay, no problem