Fourier Series and Energy Density

[chief10]

dealing with absolute functions that are limited always throws me off so let's 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

 

[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

 

[chief10]

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

hey there thanks for the hello :)

i'll make the question more clear

 

[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".

 

[chief10]

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".

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.

 

[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)

 

[chief10]

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{x(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)

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)

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

 

[PhilDSP]

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

 

[chief10]

hmm I'm having trouble computing this series

the absolute is making it difficult, any ideas?

 

[PhilDSP]

Not sure what you mean. Instead of the absolute value of the Fourier series you multiply the (complex) Fourier series by its complex conjugate. (You want to use the polar form)

http://en.wikipedia.org/wiki/Complex_conjugate

 

[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)