# Fourier Series and Energy Density

1. Oct 23, 2012

### chief10

dealing with absolute functions that are limited always throws me off so lets consider this

f(x)=|x| for -∏ ≤ x < ∏
f(t)= f(t+2∏)

it's not too bad however finding the energy density is throwing me off a little..
the questions tend to be generally phrased as below:

Find the energy density of f(t).

-thanks guys and girls

Last edited: Oct 23, 2012
2. Oct 23, 2012

### tiny-tim

hi chief10!

(try using the Quick Symbols box next to the Reply box )

(i'm not sure what you're asking , but …)

f(t) = f(t + 2π) means that the function has period 2π,

so it will have a fourier series, of coss and sins

3. Oct 23, 2012

### chief10

hey there thanks for the hello :)

i'll make the question more clear

4. Oct 23, 2012

### HallsofIvy

Please do! For one thing, "energy" is a physics concept, not mathematics so you will have to say what you mean by the "energy density".

5. Oct 23, 2012

### chief10

what i'm talking about is energy density of a periodic function ~ f(t)

you know, Parseval and all that - it corresponds to Fourier in mathematics.

Last edited: Oct 23, 2012
6. Oct 23, 2012

### PhilDSP

Hi chief10,

You seem to be looking for the "power spectrum of a signal" that is used often by electrical engineers. It's called other names such as spectral density, power spectral density and energy spectral density. The idea is that the power varies according to what frequency components exist in the signal in addition to their amplitude. The power for each frequency component varies according to the square of the frequency.

$$power \quad = \quad \int_{- \infty}^{\infty} \! |f(t) |^2 \, \mathrm{d} t$$

Taking the Fourier transform (the conjugate of F{f(t)} is needed because the wave equation is complex)

$$power \quad = \quad \int_{- \infty}^{\infty} \! |F(\omega) |^2 \, \mathrm{d} \omega \quad = \quad F(\omega)F^*(\omega)$$

The meaning of F() on the left is specific to the frequency under the integral whereas on the right it means the summation of every frequency that exists in the signal. Though this is an over-simplified explanation. See a derivation of Parseval's theorem for the details (http://en.wikipedia.org/wiki/Parseval's_theorem for example)

Last edited: Oct 23, 2012
7. Oct 23, 2012

### chief10

so probably a good idea to find the Fourier Series for f(t) first?

8. Oct 23, 2012

### PhilDSP

Yes, the result of doing the Fourier transform gives you the Fourier series.

9. Oct 23, 2012

### chief10

hmm i'm having trouble computing this series

the absolute is making it difficult, any ideas?

10. Oct 23, 2012

### PhilDSP

11. Oct 23, 2012

### chief10

I solved the series. I'm hoping I did it correctly.
This is what I got.

(∏/2) - 2*Σ [((-1)n/n)*sin(nx)]

I'll see what I can do with the above to find Energy Density of f(t)